The GLIMMIX Procedure SAS/STAT User’s Guide (Book Excerpt)

The GLIMMIX Procedure SAS/STAT User’s Guide (Book Excerpt)
®
SAS/STAT 9.2 User’s Guide
The GLIMMIX Procedure
(Book Excerpt)
®
SAS Documentation
This document is an individual chapter from SAS/STAT® 9.2 User’s Guide.
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Chapter 38
The GLIMMIX Procedure
Contents
Overview: GLIMMIX Procedure . . . . . . . . . . . . . . . . . . . . . . . .
Basic Features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Notation for the Generalized Linear Mixed Model . . . . . . . . . . .
The Basic Model . . . . . . . . . . . . . . . . . . . . . . . .
G-Side and R-Side Random Effects and Covariance Structures
Relationship with Generalized Linear Models . . . . . . . . .
PROC GLIMMIX Contrasted with Other SAS Procedures . . . . . . .
Getting Started: GLIMMIX Procedure . . . . . . . . . . . . . . . . . . . . .
Logistic Regressions with Random Intercepts . . . . . . . . . . . . . .
Syntax: GLIMMIX Procedure . . . . . . . . . . . . . . . . . . . . . . . . .
PROC GLIMMIX Statement . . . . . . . . . . . . . . . . . . . . . . .
BY Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
CLASS Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . .
CONTRAST Statement . . . . . . . . . . . . . . . . . . . . . . . . .
COVTEST Statement . . . . . . . . . . . . . . . . . . . . . . . . . .
EFFECT Statement (Experimental) . . . . . . . . . . . . . . . . . . .
ESTIMATE Statement . . . . . . . . . . . . . . . . . . . . . . . . . .
FREQ Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ID Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
LSMEANS Statement . . . . . . . . . . . . . . . . . . . . . . . . . .
LSMESTIMATE Statement . . . . . . . . . . . . . . . . . . . . . . .
MODEL Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Response Variable Options . . . . . . . . . . . . . . . . . . .
Model Options . . . . . . . . . . . . . . . . . . . . . . . . .
NLOPTIONS Statement . . . . . . . . . . . . . . . . . . . . . . . . .
OUTPUT Statement . . . . . . . . . . . . . . . . . . . . . . . . . . .
PARMS Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . .
RANDOM Statement . . . . . . . . . . . . . . . . . . . . . . . . . .
WEIGHT Statement . . . . . . . . . . . . . . . . . . . . . . . . . . .
Programming Statements . . . . . . . . . . . . . . . . . . . . . . . .
User-Defined Link or Variance Function . . . . . . . . . . . . . . . .
Implied Variance Functions . . . . . . . . . . . . . . . . . . .
Automatic Variables . . . . . . . . . . . . . . . . . . . . . .
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Details: GLIMMIX Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Generalized Linear Models Theory . . . . . . . . . . . . . . . . . . . . . .
Maximum Likelihood . . . . . . . . . . . . . . . . . . . . . . . . .
Scale and Dispersion Parameters . . . . . . . . . . . . . . . . . . .
Quasi-likelihood for Independent Data . . . . . . . . . . . . . . . .
Effects of Adding Overdispersion . . . . . . . . . . . . . . . . . .
Generalized Linear Mixed Models Theory . . . . . . . . . . . . . . . . . .
Model or Integral Approximation . . . . . . . . . . . . . . . . . .
Pseudo-likelihood Estimation Based on Linearization . . . . . . . .
Maximum Likelihood Estimation Based on Laplace Approximation
Maximum Likelihood Estimation Based on Adaptive Quadrature . .
Aspects Common to Adaptive Quadrature and Laplace Approximation
Notes on Bias of Estimators . . . . . . . . . . . . . . . . . . . . .
GLM Mode or GLMM Mode . . . . . . . . . . . . . . . . . . . . . . . . .
Statistical Inference for Covariance Parameters . . . . . . . . . . . . . . . .
The Likelihood Ratio Test . . . . . . . . . . . . . . . . . . . . . .
One- and Two-Sided Testing, Mixture Distributions . . . . . . . . .
Handling the Degenerate Distribution . . . . . . . . . . . . . . . .
Wald Versus Likelihood Ratio Tests . . . . . . . . . . . . . . . . .
Confidence Bounds Based on Likelihoods . . . . . . . . . . . . . .
Satterthwaite Degrees of Freedom Approximation . . . . . . . . . . . . . .
Empirical Covariance (“Sandwich”) Estimators . . . . . . . . . . . . . . . .
Residual-Based Estimators . . . . . . . . . . . . . . . . . . . . . .
Design-Adjusted MBN Estimator . . . . . . . . . . . . . . . . . .
Exploring and Comparing Covariance Matrices . . . . . . . . . . . . . . . .
Processing by Subjects . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Radial Smoothing Based on Mixed Models . . . . . . . . . . . . . . . . . .
From Penalized Splines to Mixed Models . . . . . . . . . . . . . .
Knot Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Odds and Odds Ratio Estimation . . . . . . . . . . . . . . . . . . . . . . .
The Odds Ratio Estimates Table . . . . . . . . . . . . . . . . . . .
Odds or Odds Ratio . . . . . . . . . . . . . . . . . . . . . . . . . .
Odds Ratios in Multinomial Models . . . . . . . . . . . . . . . . .
Parameterization of Generalized Linear Mixed Models . . . . . . . . . . . .
Intercept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Interaction Effects . . . . . . . . . . . . . . . . . . . . . . . . . .
Nested Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Implications of the Non-Full-Rank Parameterization . . . . . . . .
Missing Level Combinations . . . . . . . . . . . . . . . . . . . . .
Notes on the EFFECT Statement . . . . . . . . . . . . . . . . . . .
Positional and Nonpositional Syntax for Contrast Coefficients . . .
Response-Level Ordering and Referencing . . . . . . . . . . . . . . . . . .
Comparing the GLIMMIX and MIXED Procedures . . . . . . . . . . . . .
Singly or Doubly Iterative Fitting . . . . . . . . . . . . . . . . . . . . . . .
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The GLIMMIX Procedure F 2077
Default Estimation Techniques . . . . . . . . . . . . . . . . . . . . . . . .
Default Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Model Information . . . . . . . . . . . . . . . . . . . . . . . . . .
Class Level Information . . . . . . . . . . . . . . . . . . . . . . .
Number of Observations . . . . . . . . . . . . . . . . . . . . . . .
Response Profile . . . . . . . . . . . . . . . . . . . . . . . . . . .
Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Optimization Information . . . . . . . . . . . . . . . . . . . . . . .
Iteration History . . . . . . . . . . . . . . . . . . . . . . . . . . .
Convergence Status . . . . . . . . . . . . . . . . . . . . . . . . . .
Fit Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Covariance Parameter Estimates . . . . . . . . . . . . . . . . . . .
Type III Tests of Fixed Effects . . . . . . . . . . . . . . . . . . . .
Notes on Output Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . .
ODS Table Names . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ODS Graphics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Diagnostic Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Graphics for LS-Mean Comparisons . . . . . . . . . . . . . . . . .
ODS Graph Names . . . . . . . . . . . . . . . . . . . . . . . . . .
Examples: GLIMMIX Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . .
Example 38.1: Binomial Counts in Randomized Blocks . . . . . . . . . . .
Example 38.2: Mating Experiment with Crossed Random Effects . . . . . .
Example 38.3: Smoothing Disease Rates; Standardized Mortality Ratios . .
Example 38.4: Quasi-likelihood Estimation for Proportions with Unknown
Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Example 38.5: Joint Modeling of Binary and Count Data . . . . . . . . . .
Example 38.6: Radial Smoothing of Repeated Measures Data . . . . . . . .
Example 38.7: Isotonic Contrasts for Ordered Alternatives . . . . . . . . .
Example 38.8: Adjusted Covariance Matrices of Fixed Effects . . . . . . .
Example 38.9: Testing Equality of Covariance and Correlation Matrices . .
Example 38.10: Multiple Trends Correspond to Multiple Extrema in Profile
Likelihoods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Example 38.11: Maximum Likelihood in Proportional Odds Model with Random Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Example 38.12: Fitting a Marginal (GEE-Type) Model . . . . . . . . . . .
Example 38.13: Response Surface Comparisons with Multiplicity Adjustments
Example 38.14: Generalized Poisson Mixed Model for Overdispersed Count
Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Example 38.15: Comparing Multiple B-Splines . . . . . . . . . . . . . . .
Example 38.16: Diallel Experiment with Multimember Random Effects . .
Example 38.17: Linear Inference Based on Summary Data . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Overview: GLIMMIX Procedure
The GLIMMIX procedure fits statistical models to data with correlations or nonconstant variability and where the response is not necessarily normally distributed. These models are known as
generalized linear mixed models (GLMM).
GLMMs, like linear mixed models, assume normal (Gaussian) random effects. Conditional on these
random effects, data can have any distribution in the exponential family. The exponential family
comprises many of the elementary discrete and continuous distributions. The binary, binomial,
Poisson, and negative binomial distributions, for example, are discrete members of this family. The
normal, beta, gamma, and chi-square distributions are representatives of the continuous distributions
in this family. In the absence of random effects, the GLIMMIX procedure fits generalized linear
models (fit by the GENMOD procedure).
GLMMs are useful for the following applications:
estimating trends in disease rates
modeling CD4 counts in a clinical trial over time
modeling the proportion of infected plants on experimental units in a design with randomly
selected treatments or randomly selected blocks
predicting the probability of high ozone levels in counties
modeling skewed data over time
analyzing customer preference
joint modeling of multivariate outcomes
Such data often display correlations among some or all observations as well as nonnormality. The
correlations can arise from repeated observation of the same sampling units, shared random effects
in an experimental design, spatial (temporal) proximity, multivariate observations, and so on.
The GLIMMIX procedure does not fit hierarchical models with nonnormal random effects. With the
GLIMMIX procedure you select the distribution of the response variable conditional on normally
distributed random effects.
For more information about the differences between the GLIMMIX procedure and SAS procedures
that specialize in certain subsets of the GLMM models, see the section “PROC GLIMMIX Contrasted with Other SAS Procedures” on page 2083.
Basic Features F 2079
Basic Features
The GLIMMIX procedure enables you to specify a generalized linear mixed model and to perform
confirmatory inference in such models. The syntax is similar to that of the MIXED procedure and
includes CLASS, MODEL, and RANDOM statements. For instructions on how to specify PROC
MIXED REPEATED effects with PROC GLIMMIX, see the section “Comparing the GLIMMIX
and MIXED Procedures” on page 2266. The following are some of the basic features of PROC
GLIMMIX.
SUBJECT= and GROUP= options, which enable blocking of variance matrices and parameter
heterogeneity
choice of linearization approach or integral approximation by quadrature or Laplace method
for mixed models with nonlinear random effects or nonnormal distribution
choice of linearization about expected values or expansion about current solutions of best
linear unbiased predictors
flexible covariance structures for random and residual random effects, including variance
components, unstructured, autoregressive, and spatial structures
CONTRAST, ESTIMATE, LSMEANS, and LSMESTIMATE statements, which produce hypothesis tests and estimable linear combinations of effects
NLOPTIONS statement, which enables you to exercise control over the numerical optimization. You can choose techniques, update methods, line search algorithms, convergence criteria, and more. Or, you can choose the default optimization strategies selected for the particular
class of model you are fitting.
computed variables with SAS programming statements inside of PROC GLIMMIX (except
for variables listed in the CLASS statement). These computed variables can appear in the
MODEL, RANDOM, WEIGHT, or FREQ statement.
grouped data analysis
user-specified link and variance functions
choice of model-based variance-covariance estimators for the fixed effects or empirical (sandwich) estimators to make analysis robust against misspecification of the covariance structure
and to adjust for small-sample bias
joint modeling for multivariate data. For example, you can model binary and normal responses from a subject jointly and use random effects to relate (fuse) the two outcomes.
multinomial models for ordinal and nominal outcomes
univariate and multivariate low-rank mixed model smoothing
2080 F Chapter 38: The GLIMMIX Procedure
Assumptions
The primary assumptions underlying the analyses performed by PROC GLIMMIX are as follows:
If the model contains random effects, the distribution of the data conditional on the random
effects is known. This distribution is either a member of the exponential family of distributions or one of the supplementary distributions provided by the GLIMMIX procedure. In
models without random effects, the unconditional (marginal) distribution is assumed to be
known for maximum likelihood estimation, or the first two moments are known in the case
of quasi-likelihood estimation.
The conditional expected value of the data takes the form of a linear mixed model after a
monotonic transformation is applied.
The problem of fitting the GLMM can be cast as a singly or doubly iterative optimization
problem. The objective function for the optimization is a function of either the actual log
likelihood, an approximation to the log likelihood, or the log likelihood of an approximated
model.
For a model containing random effects, the GLIMMIX procedure, by default, estimates the parameters by applying pseudo-likelihood techniques as in Wolfinger and O’Connell (1993) and Breslow
and Clayton (1993). In a model without random effects (GLM models), PROC GLIMMIX estimates the parameters by maximum likelihood, restricted maximum likelihood, or quasi-likelihood.
See the section “Singly or Doubly Iterative Fitting” on page 2269 about when the GLIMMIX procedure applies noniterative, singly and doubly iterative algorithms, and the section “Default Estimation Techniques” on page 2271 about the default estimation methods. You can also fit generalized
linear mixed models by maximum likelihood where the marginal distribution is numerically approximated by the Laplace method (METHOD=LAPLACE) or by adaptive Gaussian quadrature
(METHOD=QUAD).
Once the parameters have been estimated, you can perform statistical inferences for the fixed effects
and covariance parameters of the model. Tests of hypotheses for the fixed effects are based on Waldtype tests and the estimated variance-covariance matrix. The COVTEST statement enables you to
perform inferences about covariance parameters based on likelihood ratio tests.
PROC GLIMMIX uses the Output Delivery System (ODS) for displaying and controlling the output
from SAS procedures. ODS enables you to convert any of the output from PROC GLIMMIX into a
SAS data set. See the section “ODS Table Names” on page 2278 for more information.
The GLIMMIX procedure now uses ODS Graphics to create graphs as part of its output. For general
information about ODS Graphics, see Chapter 21, “Statistical Graphics Using ODS.” For specific
information about the statistical graphics available with the GLIMMIX procedure, see the PLOTS
options in the PROC GLIMMIX and LSMEANS statements.
Notation for the Generalized Linear Mixed Model F 2081
Notation for the Generalized Linear Mixed Model
This section introduces the mathematical notation used throughout the chapter to describe the generalized linear mixed model (GLMM). See the section “Details: GLIMMIX Procedure” on page 2210
for a description of the fitting algorithms and the mathematical-statistical details.
The Basic Model
Suppose Y represents the .n 1/ vector of observed data and is a .r 1/ vector of random effects.
Models fit by the GLIMMIX procedure assume that
EŒYj D g
1
.Xˇ C Z/
where g./ is a differentiable monotonic link function and g 1 ./ is its inverse. The matrix X is an
.n p/ matrix of rank k, and Z is an .n r/ design matrix for the random effects. The random
effects are assumed to be normally distributed with mean 0 and variance matrix G.
The GLMM contains a linear mixed model inside the inverse link function. This model component
is referred to as the linear predictor,
D Xˇ C Z
The variance of the observations, conditional on the random effects, is
VarŒYj D A1=2 RA1=2
The matrix A is a diagonal matrix and contains the variance functions of the model. The variance
function expresses the variance of a response as a function of the mean. The GLIMMIX procedure determines the variance function from the DIST= option in the MODEL statement or from
the user-supplied variance function (see the section “Implied Variance Functions” on page 2206).
The matrix R is a variance matrix specified by the user through the RANDOM statement. If the
conditional distribution of the data contains an additional scale parameter, it is either part of the
variance functions or part of the R matrix. For example, the gamma distribution with mean has
the variance function a./ D 2 and VarŒY j D 2 . If your model calls for G-side random
effects only (see the next section), the procedure models R D I, where I is the identity matrix.
Table 38.15 identifies the distributions for which 1.
G-Side and R-Side Random Effects and Covariance Structures
The GLIMMIX procedure distinguishes two types of random effects. Depending on whether the
parameters of the covariance structure for random components in your model are contained in G or
in R, the procedure distinguishes between “G-side” and “R-side” random effects. The associated
covariance structures of G and R are similarly termed the G-side and R-side covariance structure,
respectively. R-side effects are also called “residual” effects. Simply put, if a random effect is an
element of , it is a G-side effect and you are modeling the G-side covariance structure; otherwise,
you are modeling the R-side covariance structure of the model. Models without G-side effects are
2082 F Chapter 38: The GLIMMIX Procedure
also known as marginal (or population-averaged) models. Models fit with the GLIMMIX procedure
can have none, one, or more of each type of effect.
Note that an R-side effect in the GLIMMIX procedure is equivalent to a REPEATED effect in the
MIXED procedure. The R-side covariance structure in the GLIMMIX procedure is the covariance
structure that you would formulate with the REPEATED statement in the MIXED procedure. In
the GLIMMIX procedure all random effects and their covariance structures are specified through
the RANDOM statement. See the section “Comparing the GLIMMIX and MIXED Procedures” on
page 2266 for a comparison of the GLIMMIX and MIXED procedures.
The columns of X are constructed from effects listed on the right side in the MODEL statement.
Columns of Z and the variance matrices G and R are constructed from the RANDOM statement.
The R matrix is by default the scaled identity matrix, R D I. The scale parameter is set to
one if the distribution does not have a scale parameter, such as in the case of the binary, binomial,
Poisson, and exponential distribution (see Table 38.15). To specify a different R matrix, use the
RANDOM statement with the _RESIDUAL_ keyword or the RESIDUAL option. For example,
to specify that the Time effect for each patient is an R-side effect with a first-order autoregressive
covariance structure, use the RESIDUAL option:
random time / type=ar(1) subject=patient residual;
To add a multiplicative overdispersion parameter, use the _RESIDUAL_ keyword:
random _residual_;
You specify the link function g./ with the LINK= option in the MODEL statement or with programming statements. You specify the variance function that controls the matrix A with the DIST=
option in the MODEL statement or with programming statements.
Unknown quantities subject to estimation are the fixed-effects parameter vector ˇ and the covariance parameter vector that comprises all unknowns in G and R. The random effects are not
parameters of the model in the sense that they are not estimated. The vector is a vector of random
variables. The solutions for are predictors of these random variables.
Relationship with Generalized Linear Models
Generalized linear models (Nelder and Wedderburn 1972; McCullagh and Nelder 1989) are a special case of GLMMs. If D 0 and R D I, the GLMM reduces to either a generalized linear
model (GLM) or a GLM with overdispersion. For example, if Y is a vector of Poisson variables
so that A is a diagonal matrix containing EŒY D on the diagonal, then the model is a Poisson
regression model for D 1 and overdispersed relative to a Poisson distribution for > 1. Because
the Poisson distribution does not have an extra scale parameter, you can model overdispersion by
adding the following statement to your GLIMMIX program:
random _residual_;
If the only random effect is an overdispersion effect, PROC GLIMMIX fits the model by (restricted)
maximum likelihood and not by one of the methods specific to GLMMs.
PROC GLIMMIX Contrasted with Other SAS Procedures F 2083
PROC GLIMMIX Contrasted with Other SAS Procedures
The GLIMMIX procedure generalizes the MIXED and GENMOD procedures in two important
ways. First, the response can have a nonnormal distribution. The MIXED procedure assumes that
the response is normally (Gaussian) distributed. Second, the GLIMMIX procedure incorporates random effects in the model and so allows for subject-specific (conditional) and population-averaged
(marginal) inference. The GENMOD procedure allows only for marginal inference.
The GLIMMIX and MIXED procedure are closely related; see the syntax and feature comparison
in the section “Comparing the GLIMMIX and MIXED Procedures” on page 2266. The remainder
of this section compares the GLIMMIX procedure with the GENMOD, NLMIXED, LOGISTIC,
and CATMOD procedures.
The GENMOD procedure fits generalized linear models for independent data by maximum likelihood. It can also handle correlated data through the marginal GEE approach of Liang and Zeger
(1986) and Zeger and Liang (1986). The GEE implementation in the GENMOD procedure is a
marginal method that does not incorporate random effects. The GEE estimation in the GENMOD
procedure relies on R-side covariances only, and the unknown parameters in R are estimated by
the method of moments. The GLIMMIX procedure allows G-side random effects and R-side covariances. PROC GLIMMIX can fit marginal (GEE-type) models, but the covariance parameters
are not estimated by the method of moments. The parameters are estimated by likelihood-based
techniques. When the GLIMMIX and GENMOD procedures fit a generalized linear model where
the distribution contains a scale parameter, such as the normal, gamma, inverse gaussian, or negative binomial distribution, the scale parameter is reported in the “Parameter Estimates” table. For
some distributions, the parameterization of this parameter differs. See the section “Scale and Dispersion Parameters” on page 2213 for details about how the GLIMMIX procedure parameterizes
the log-likelihood functions and information about how the reported quantities differ between the
two procedures.
Many of the fit statistics and tests in the GENMOD procedure are based on the likelihood. In a
GLMM it is not always possible to derive the log likelihood of the data. Even if the log likelihood
is tractable, it might be computationally infeasible. In some cases, the objective function must be
constructed based on a substitute model. In other cases, only the first two moments of the marginal
distribution can be approximated. Consequently, obtaining likelihood-based tests and statistics is
difficult for many generalized linear mixed models. The GLIMMIX procedure relies heavily on
linearization and Taylor-series techniques to construct Wald-type test statistics and confidence intervals. Likelihood ratio tests and confidence intervals for covariance parameters are available in
the GLIMMIX procedure through the COVTEST statement.
The NLMIXED procedure fits nonlinear mixed models where the conditional mean function is
a general nonlinear function. The class of generalized linear mixed models is a special case of
the nonlinear mixed models; hence some of the models you can fit with PROC NLMIXED can
also be fit with the GLIMMIX procedure. The NLMIXED procedure relies by default on approximating the marginal log likelihood through adaptive Gaussian quadrature. In the GLIMMIX
procedure, maximum likelihood estimation by adaptive Gaussian quadrature is available with the
METHOD=QUAD option in the PROC GLIMMIX statement. The default estimation methods
thus differ between the NLMIXED and GLIMMIX procedures, because adaptive quadrature is
possible for only a subset of the models available with the GLIMMIX procedure. If you choose
2084 F Chapter 38: The GLIMMIX Procedure
METHOD=LAPLACE or METHOD=QUAD(QPOINTS=1) in the PROC GLIMMIX statement for
a generalized linear mixed model, the GLIMMIX procedure performs maximum likelihood estimation based on a Laplace approximation of the marginal log likelihood. This is equivalent to the
QPOINTS=1 option in the NLMIXED procedure.
The LOGISTIC and CATMOD procedures also fit generalized linear models; PROC LOGISTIC
accommodates the independence case only. Binary, binomial, multinomial models for ordered data,
and generalized logit models that can be fit with PROC LOGISTIC can also be fit with the GLIMMIX procedure. The diagnostic tools and capabilities specific to such data implemented in the
LOGISTIC procedure go beyond the capabilities of the GLIMMIX procedure.
Getting Started: GLIMMIX Procedure
Logistic Regressions with Random Intercepts
Researchers investigated the performance of two medical procedures in a multicenter study. They
randomly selected 15 centers for inclusion. One of the study goals was to compare the occurrence
of side effects for the procedures. In each center nA patients were randomly selected and assigned
to procedure “A,” and nB patients were randomly assigned to procedure “B.” The following DATA
step creates the data set for the analysis:
data multicenter;
input center group$ n sideeffect;
datalines;
1 A 32 14
1 B 33 18
2 A 30
4
2 B 28
8
3 A 23 14
3 B 24
9
4 A 22
7
4 B 22 10
5 A 20
6
5 B 21 12
6 A 19
1
6 B 20
3
7 A 17
2
7 B 17
6
8 A 16
7
8 B 15
9
9 A 13
1
9 B 14
5
10 A 13
3
10 B 13
1
11 A 11
1
11 B 12
2
Logistic Regressions with Random Intercepts F 2085
12
12
13
13
14
14
15
15
;
A
B
A
B
A
B
A
B
10
9
9
9
8
8
7
8
1
0
2
6
1
1
1
0
The variable group identifies the two procedures, n is the number of patients who received a given
procedure in a particular center, and sideeffect is the number of patients who reported side effects.
If YiA and YiB denote the number of patients in center i who report side effects for procedures A
and B, respectively, then—for a given center—these are independent binomial random variables.
To model the probability of side effects for the two drugs, iA and iB , you need to account for the
fixed group effect and the random selection of centers. One possibility is to assume a model that
relates group and center effects linearly to the logit of the probabilities:
iA
D ˇ0 C ˇA C i
log
1 iA
iB
log
D ˇ0 C ˇB C i
1 iB
In this model, ˇA ˇB measures the difference in the logits of experiencing side effects, and the
i are independent random variables due to the random selection of centers. If you think of ˇ0 as
the overall intercept in the model, then the i are random intercept adjustments. Observations from
the same center receive the same adjustment, and these vary randomly from center to center with
variance VarŒi  D c2 .
Because iA is the conditional mean of the sample proportion, EŒYiA =niA ji  D iA , you can
model the sample proportions as binomial ratios in a generalized linear mixed model. The following
statements request this analysis under the assumption of normally distributed center effects with
equal variance and a logit link function:
proc glimmix data=multicenter;
class center group;
model sideeffect/n = group / solution;
random intercept / subject=center;
run;
The PROC GLIMMIX statement invokes the procedure. The CLASS statement instructs the procedure to treat the variables center and group as classification variables. The MODEL statement
specifies the response variable as a sample proportion by using the events/trials syntax. In terms
of the previous formulas, sideeffect/n corresponds to YiA =niA for observations from group A and
to YiB =niB for observations from group B. The SOLUTION option in the MODEL statement requests a listing of the solutions for the fixed-effects parameter estimates. Note that because of the
events/trials syntax, the GLIMMIX procedure defaults to the binomial distribution, and that distribution’s default link is the logit link. The RANDOM statement specifies that the linear predictor
contains an intercept term that randomly varies at the level of the center effect. In other words, a
random intercept is drawn separately and independently for each center in the study.
2086 F Chapter 38: The GLIMMIX Procedure
The results of this analysis are shown in Figure 38.1–Figure 38.9.
The “Model Information Table” in Figure 38.1 summarizes important information about the model
you fit and about aspects of the estimation technique.
Figure 38.1 Model Information
The GLIMMIX Procedure
Model Information
Data Set
Response Variable (Events)
Response Variable (Trials)
Response Distribution
Link Function
Variance Function
Variance Matrix Blocked By
Estimation Technique
Degrees of Freedom Method
WORK.MULTICENTER
sideeffect
n
Binomial
Logit
Default
center
Residual PL
Containment
PROC GLIMMIX recognizes the variables sideeffect and n as the numerator and denominator in the
events/trials syntax, respectively. The distribution—conditional on the random center effects—is
binomial. The marginal variance matrix is block-diagonal, and observations from the same center
form the blocks. The default estimation technique in generalized linear mixed models is residual
pseudo-likelihood with a subject-specific expansion (METHOD=RSPL).
The “Class Level Information” table lists the levels of the variables specified in the CLASS statement and the ordering of the levels. The “Number of Observations” table displays the number of
observations read and used in the analysis (Figure 38.2).
Figure 38.2 Class Level Information and Number of Observations
Class Level Information
Class
center
group
Levels
15
2
Number
Number
Number
Number
of
of
of
of
Values
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
A B
Observations Read
Observations Used
Events
Trials
30
30
155
503
There are two variables listed in the CLASS statement. The center variable has fifteen levels, and
the group variable has two levels. Because the response is specified through the events/trial syntax,
the “Number of Observations” table also contains the total number of events and trials used in the
analysis.
The “Dimensions” table lists the size of relevant matrices (Figure 38.3).
Logistic Regressions with Random Intercepts F 2087
Figure 38.3 Dimensions
Dimensions
G-side Cov. Parameters
Columns in X
Columns in Z per Subject
Subjects (Blocks in V)
Max Obs per Subject
1
3
1
15
2
There are three columns in the X matrix, corresponding to an intercept and the two levels of the
group variable. For each subject (center), the Z matrix contains only an intercept column.
The “Optimization Information” table provides information about the methods and size of the optimization problem (Figure 38.4).
Figure 38.4 Optimization Information
Optimization Information
Optimization Technique
Parameters in Optimization
Lower Boundaries
Upper Boundaries
Fixed Effects
Starting From
Dual Quasi-Newton
1
1
0
Profiled
Data
The default optimization technique for generalized linear mixed models with binomial data is the
quasi-Newton method. Because a residual likelihood technique is used to compute the objective
function, only the covariance parameters participate in the optimization. A lower boundary constraint is placed on the variance component for the random center effect. The solution for this
variance cannot be less than zero.
The “Iteration History” table displays information about the progress of the optimization process.
After the initial optimization, the GLIMMIX procedure performed 15 updates before the convergence criterion was met (Figure 38.5). At convergence, the largest absolute value of the gradient
was near zero. This indicates that the process stopped at an extremum of the objective function.
2088 F Chapter 38: The GLIMMIX Procedure
Figure 38.5 Iteration History and Convergence Status
Iteration History
Iteration
Restarts
Subiterations
Objective
Function
Change
Max
Gradient
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
5
3
2
1
1
1
1
1
1
1
1
1
1
1
1
0
79.688580269
81.294622554
81.438701534
81.444083567
81.444265216
81.444277364
81.444266322
81.44427636
81.444267235
81.44427553
81.44426799
81.444274844
81.444268614
81.444274277
81.444269129
81.444273808
0.11807224
0.02558021
0.00166079
0.00006263
0.00000421
0.00000383
0.00000348
0.00000316
0.00000287
0.00000261
0.00000237
0.00000216
0.00000196
0.00000178
0.00000162
0.00000000
7.851E-7
8.209E-7
4.061E-8
2.292E-8
0.000025
0.000023
0.000021
0.000019
0.000017
0.000016
0.000014
0.000013
0.000012
0.000011
9.772E-6
9.102E-6
Convergence criterion (PCONV=1.11022E-8) satisfied.
The “Fit Statistics” table lists information about the fitted model (Figure 38.6).
Figure 38.6 Fit Statistics
Fit Statistics
-2 Res Log Pseudo-Likelihood
Generalized Chi-Square
Gener. Chi-Square / DF
81.44
30.69
1.10
Twice the negative of the residual log likelihood in the final pseudo-model equaled 81.44. The ratio
of the generalized chi-square statistic and its degrees of freedom is close to 1. This is a measure of
the residual variability in the marginal distribution of the data.
The “Covariance Parameter Estimates” table displays estimates and asymptotic estimated standard
errors for all covariance parameters (Figure 38.7).
Figure 38.7 Covariance Parameter Estimates
Covariance Parameter Estimates
Cov Parm
Subject
Intercept
center
Estimate
Standard
Error
0.6176
0.3181
Logistic Regressions with Random Intercepts F 2089
The variance of the random center intercepts on the logit scale is estimated as b
2c D 0:6176.
The “Parameter Estimates” table displays the solutions for the fixed effects in the model
(Figure 38.8).
Figure 38.8 Parameter Estimates
Solutions for Fixed Effects
Effect
group
Intercept
group
group
A
B
Estimate
Standard
Error
DF
t Value
Pr > |t|
-0.8071
-0.4896
0
0.2514
0.2034
.
14
14
.
-3.21
-2.41
.
0.0063
0.0305
.
Because of the fixed-effects parameterization used in the GLIMMIX procedure, the “Intercept”
effect is an estimate of ˇ0 C ˇB , and the “A” group effect is an estimate of ˇA ˇB , the log odds
ratio. The associated estimated probabilities of side effects in the two groups are
1
D 0:2147
1 C expf0:8071 C 0:4896g
1
b
B D
D 0:3085
1 C expf0:8071g
b
A D
There is a significant difference between the two groups (p=0.0305).
The “Type III Tests of Fixed Effect” table displays significance tests for the fixed effects in the
model (Figure 38.9).
Figure 38.9 Type III Tests of Fixed Effects
Type III Tests of Fixed Effects
Effect
group
Num
DF
Den
DF
F Value
Pr > F
1
14
5.79
0.0305
Because the group effect has only two levels, the p-value for the effect is the same as in the “Parameter Estimates” table, and the “F Value” is the square of the “t Value” shown there.
2090 F Chapter 38: The GLIMMIX Procedure
You can produce the estimates of the average logits in the two groups and their predictions on the
scale of the data with the LSMEANS statement in PROC GLIMMIX:
ods select lsmeans;
proc glimmix data=multicenter;
class center group;
model sideeffect/n = group / solution;
random intercept / subject=center;
lsmeans group / cl ilink;
run;
The LSMEANS statement requests the least squares means of the group effect on the logit scale.
The CL option requests their confidence limits. The ILINK option adds estimates, standard errors,
and confidence limits on the mean (probability) scale (Figure 38.10).
Figure 38.10 Least Squares Means
The GLIMMIX Procedure
group Least Squares Means
group
A
B
Estimate
Standard
Error
DF
t Value
Pr > |t|
Alpha
Lower
Upper
-1.2966
-0.8071
0.2601
0.2514
14
14
-4.99
-3.21
0.0002
0.0063
0.05
0.05
-1.8544
-1.3462
-0.7388
-0.2679
group Least Squares Means
group
A
B
Mean
Standard
Error
Mean
Lower
Mean
Upper
Mean
0.2147
0.3085
0.04385
0.05363
0.1354
0.2065
0.3233
0.4334
The “Estimate” column displays the least squares mean estimate on the logit scale, and the “Mean”
column represents its mapping onto the probability scale. The “Lower” and “Upper” columns
are 95% confidence limits for the logits in the two groups. The “Lower Mean” and “Upper Mean”
columns are the corresponding confidence limits for the probabilities of side effects. These limits are
obtained by inversely linking the confidence bounds on the linear scale, and thus are not symmetric
about the estimate of the probabilities.
Syntax: GLIMMIX Procedure F 2091
Syntax: GLIMMIX Procedure
You can specify the following statements in the GLIMMIX procedure:
PROC GLIMMIX < options > ;
BY variables ;
CLASS variables ;
CONTRAST ’label’ contrast-specification < , contrast-specification > < , . . . > < / options > ;
COVTEST < ’label’ > < test-specification > < / options > ;
EFFECT effect-specification ;
ESTIMATE ’label’ contrast-specification < (divisor =n) >
< , ’label’ contrast-specification < (divisor =n) > > < , . . . > < / options > ;
FREQ variable ;
ID variables ;
LSMEANS fixed-effects < / options > ;
LSMESTIMATE fixed-effect < ’label’ > values < divisor =n >
< , < ’label’ > values < divisor =n > > < , . . . > < / options > ;
MODEL response< (response-options) > = < fixed-effects > < / model-options > ;
MODEL events/trials = < fixed-effects > < / model-options > ;
NLOPTIONS < options > ;
OUTPUT < OUT=SAS-data-set >
< keyword< (keyword-options) > < =name > >. . .
< keyword< (keyword-options) > < =name > > < / options > ;
PARMS (value-list) . . . < / options > ;
RANDOM random-effects < / options > ;
WEIGHT variable ;
Programming statements ;
The CLASS, CONTRAST, COVTEST, EFFECT, ESTIMATE, LSMEANS, LSMESTIMATE, and
RANDOM statements and the programming statements can appear multiple times. The PROC
GLIMMIX and MODEL statements are required, and the MODEL statement must appear after the
CLASS statement if a CLASS statement is included. The EFFECT statements must appear before
the MODEL statement.
2092 F Chapter 38: The GLIMMIX Procedure
PROC GLIMMIX Statement
PROC GLIMMIX < options > ;
The PROC GLIMMIX statement invokes the procedure. Table 38.1 summarizes some important
options in the PROC GLIMMIX statement by function. These and other options in the PROC
GLIMMIX statement are then described fully in alphabetical order.
Table 38.1
PROC GLIMMIX Statement Options
Option
Description
Basic Options
DATA=
METHOD=
NOFIT
NOPROFILE
NOREML
ORDER=
OUTDESIGN
specifies the input data set
determines estimation method
does not fit the model
includes scale parameter in optimization
determines computation of scale parameters in GLM models
determines the sort order of CLASS variables
writes X and/or Z matrices to a SAS data set
Displayed Output
ASYCORR
HESSIAN
ITDETAILS
NOBSDETAIL
NOCLPRINT
ODDSRATIO
PLOTS
SUBGRADIENT
displays the asymptotic correlation matrix of the covariance parameter estimates
displays the asymptotic covariance matrix of the covariance parameter estimates
displays the gradient of the objective function with respect to the
parameter estimates
displays the Hessian matrix
adds estimates and gradients to the “Iteration History”
shows data exclusions
suppresses “Class Level Information” completely or in part
requests odds ratios
produces ODS statistical graphics
writes subject-specific gradients to a SAS data set
Optimization Options
MAXOPT=
specifies the number of optimizations
Computational Options
EMPIRICAL
INFOCRIT
INITGLM
INTITER=
NOBOUND
SCORING=
computes empirical (“sandwich”) estimators
affects the computation of information criteria
uses fixed-effects starting values via generalized linear model
sets the number of initial GLM steps
unbounds the covariance parameter estimates
applies Fisher scoring where applicable
ASYCOV
GRADIENT
PROC GLIMMIX Statement F 2093
Table 38.1
continued
Option
Singularity Tolerances
ABSPCONV=
Description
SINGCHOL=
SINGRES=
SINGULAR=
determines the absolute parameter estimate convergence criterion
for PL
specifies significant digits in computing objective function
specifies the relative parameter estimate convergence criterion for
PL
tunes singularity for Cholesky decompositions
tunes singularity for the residual variance
tunes general singularity criterion
Debugging Output
LIST
lists model program and variables
FDIGITS=
PCONV=
You can specify the following options in the PROC GLIMMIX statement.
ABSPCONV=r
specifies an absolute parameter estimate convergence criterion for doubly iterative estimation methods. For such methods, the GLIMMIX procedure by default examines the relative
change in parameter estimates between optimizations (see PCONV=). The purpose of the
ABSPCONV= criterion is to stop the process when the absolute change in parameter estimates is less than the tolerance criterion r. The criterion is based on fixed effects and
covariance parameters.
Note that this convergence criterion does not affect the convergence criteria applied within
any individual optimization. In order to change the convergence behavior within an optimization, you can change the ABSCONV=, ABSFCONV=, ABSGCONV=, ABSXCONV=,
FCONV=, or GCONV= option in the NLOPTIONS statement.
ASYCORR
produces the asymptotic correlation matrix of the covariance parameter estimates. It is
computed from the corresponding asymptotic covariance matrix (see the description of the
ASYCOV option, which follows).
ASYCOV
requests that the asymptotic covariance matrix of the covariance parameter estimates be displayed. By default, this matrix is the observed inverse Fisher information matrix, which
equals mH 1 , where H is the Hessian (second derivative) matrix of the objective function.
The factor m equals 1 in a GLM and equals 2 in a GLMM.
When you use the SCORING= option and PROC GLIMMIX converges without stopping the
scoring algorithm, the procedure uses the expected Hessian matrix to compute the covariance
matrix instead of the observed Hessian. Regardless of whether a scoring algorithm is used
or the number of scoring iterations has been exceeded, you can request that the asymptotic
covariance matrix be based on the expected Hessian with the EXPHESSIAN option in the
PROC GLIMMIX statement. If a residual scale parameter is profiled from the likelihood
equation, the asymptotic covariance matrix is adjusted for the presence of this parameter;
2094 F Chapter 38: The GLIMMIX Procedure
details of this adjustment process are found in Wolfinger, Tobias, and Sall (1994) and in the
section “Estimated Precision of Estimates” on page 2219.
CHOLESKY
CHOL
requests that the mixed model equations be constructed and solved by using the Cholesky root
of the G matrix. This option applies only to estimation methods that involve mixed model
equations. The Cholesky root algorithm has greater numerical stability but also requires
more computing resources. When the estimated G matrix is not positive definite during a
particular function evaluation, PROC GLIMMIX switches to the Cholesky algorithm for that
b becomes positive definite again. When
evaluation and returns to the regular algorithm if G
the CHOLESKY option is in effect, the procedure applies the algorithm all the time.
DATA=SAS-data-set
names the SAS data set to be used by PROC GLIMMIX. The default is the most recently
created data set.
EMPIRICAL< =CLASSICAL | HC0 >
EMPIRICAL< =DF | HC1 >
EMPIRICAL< =MBN< (mbn-options) > >
EMPIRICAL< =ROOT | HC2 >
EMPIRICAL< =FIRORES | HC3 >
EMPIRICAL< =FIROEEQ< (r ) > >
requests that the covariance matrix of the parameter estimates be computed as one of the
asymptotically consistent estimators, known as sandwich or empirical estimators. The name
stems from the layering of the estimator. An empirically based estimate of the inverse variance of the parameter estimates (the “meat”) is wrapped by the model-based variance estimate
(the “bread”).
Empirical estimators are useful for obtaining inferences that are not sensitive to the choice of
the covariance model. In nonmixed models, they can help, for example, to allay the effects
of variance heterogeneity on the tests of fixed effects. Empirical estimators can coarsely
be grouped into likelihood-based and residual-based estimators. The distinction arises from
the components used to construct the “meat” and “bread” of the estimator. If you specify
the EMPIRICAL option without further qualifiers, the GLIMMIX procedure computes the
classical sandwich estimator in the appropriate category.
Likelihood-Based Estimator
Let H.˛/ denote the second derivative matrix of the log likelihood for some parameter vector
˛, and let gi .˛/ denote the gradient of the log likelihood with respect
P to ˛ for the i th of m
independent sampling units. The gradient for the entire data is m
i D1 gi .˛/. A sandwich
estimator for the covariance matrix of b
˛ can then be constructed as (White 1982)
!
m
X
1
0
H.b
˛/
gi .b
˛/gi .b
˛/ H.b
˛/ 1
i D1
PROC GLIMMIX Statement F 2095
If you fit a mixed model by maximum likelihood with Laplace or quadrature approximation (METHOD=LAPLACE, METHOD=QUAD), the GLIMMIX procedure constructs this
likelihood-based estimator when you choose EMPIRICAL=CLASSICAL. If you choose EMPIRICAL=MBN, the likelihood-based sandwich estimator is further adjusted (see the section “Design-Adjusted MBN Estimator” on page 2242 for details). Because Laplace and
quadrature estimation in GLIMMIX includes the fixed-effects parameters and the covariance parameters in the optimization, this empirical estimator adjusts the covariance matrix of both types of parameters. The following empirical estimators are not available with
METHOD=LAPLACE or with METHOD=QUAD: EMPIRICAL=DF, EMPIRICAL=ROOT,
EMPIRICAL=FIRORES, and EMPIRICAL=FIROEEQ.
Residual-Based Estimators
For a general model, let Y denote the response with mean and variance †, and let D be
the matrix of first derivatives of with respect to the fixed effects ˇ. The classical sandwich
estimator (Huber 1967; White 1980) is
!
m
X
0
1
0
1
b
b
b ei e †
b b
b

D†
Di 
i
i
i
i
1 D/
, ei D yi
i D1
where  D .D0 †
units.
b
i , and m denotes the number of independent sampling
Since the expected value of ei e0i does not equal †i , the classical sandwich estimator is biased,
particularly if m is small. The estimator tends to underestimate the variance of b̌. The EMPIRICAL=DF, ROOT, FIRORES, FIROEEQ, and MBN estimators are bias-corrected sandwich estimators. The DF estimator applies a simple sample size adjustment. The ROOT,
FIRORES, and FIROEEQ estimators are based on Taylor series approximations applied to
residuals and estimating equations. For uncorrelated data, the EMPIRICAL=FIRORES estimator can be motivated as a jackknife estimator.
In the case of a linear regression model, the various estimators reduce to the
heteroscedasticity-consistent covariance matrix estimators (HCMM) of White (1980) and
MacKinnon and White (1985). The classical estimator, HC0, was found to perform poorly
in small samples. Based on simulations in regression models, MacKinnon and White (1985)
and Long and Ervin (2000) strongly recommend the HC3 estimator. The sandwich estimators
computed by the GLIMMIX procedure can be viewed as an extension of the HC0—HC3
estimators of MacKinnon and White (1985) to accommodate nonnormal data and correlated
observations.
The MBN estimator, introduced as a residual-based estimator by Morel (1989) and Morel,
Bokossa, and Neerchal (2003), applies an additive adjustment to the residual crossproduct.
It is controlled by three suboptions. The valid mbn-options are as follows: a sample size
adjustment is applied when the DF suboption is in effect. The NODF suboption suppresses
this component of the adjustment. The lower bound of the design effect parameter 0 r 1 can be specified with the R= option. The magnitude of Morel’s ı parameter is partly
determined with the D= option (d 1).
2096 F Chapter 38: The GLIMMIX Procedure
For details about the general expression for the residual-based estimators and their relationship, see the section “Empirical Covariance (“Sandwich”) Estimators” on page 2241. The
MBN estimator and its parameters are explained for residual- and likelihood-based estimators in the section “Design-Adjusted MBN Estimator” on page 2242.
The EMPIRICAL=DF estimator applies a simple, multiplicative correction factor to the classical estimator (Hinkley 1977). This correction factor is
m=.m k/
m>k
cD
1
otherwise
where k is the rank of X, and m equals the sum of all frequencies when PROC GLIMMIX is
in GLM mode and equals the number of subjects in GLMM mode. For example, the following
statements fit an overdispersed GLM:
proc glimmix empirical;
model y = x;
random _residual_;
run;
PROC GLIMMIX is in GLM mode, and the individual observations are the independent
sampling units from which the sandwich estimator is constructed. If you use a SUBJECT=
effect in the RANDOM statement, however, the procedure fits the model in GLMM mode
and the subjects represent the sampling units in the construction of the sandwich estimator.
In other words, the following statements fit a GEE-type model with independence working
covariance structure and subjects (clusters) defined by the levels of ID:
proc glimmix empirical;
class id;
model y = x;
random _residual_ / subject=id type=vc;
run;
See the section “GLM Mode or GLMM Mode” on page 2231 for information about how the
GLIMMIX procedure determines the estimation mode.
The EMPIRICAL=ROOT estimator is based on the residual approximation in Kauermann
and Carroll (2001), and the EMPIRICAL=FIRORES estimator is based on the approximation
in Mancl and DeRouen (2001). The Kauermann and Carroll estimator requires the inverse
square root of a nonsymmetric matrix. This square root matrix is obtained from the singular
value decomposition in PROC GLIMMIX, and thus this sandwich estimator is computationally more demanding than others. In the linear regression case, the Mancl-DeRouen estimator
can be motivated as a jackknife estimator, based on the “leave-one-out” estimates of b̌; see
MacKinnon and White (1985) for details.
The EMPIRICAL=FIROEEQ estimator is based on approximating an unbiased estimating
equation (Fay and Graubard 2001). It is computationally less demanding than the estimator
of Kauermann and Carroll (2001) and, in certain balanced cases, gives identical results. The
optional number 0 r < 1 is chosen to provide an upper bound on the correction factor. The
default value for r is 0:75.
When you specify the EMPIRICAL option with a residual-based estimator, PROC GLIMMIX
adjusts all standard errors and test statistics involving the fixed-effects parameters.
PROC GLIMMIX Statement F 2097
Sampling Units
Computation of an empirical variance estimator requires that the data can be processed by
independent sampling units. This is always the case in GLMs. In this case, m, the number
of independent units, equals the sum of the frequencies used in the analysis (see “Number
of Observations” table). In GLMMs, empirical estimators can be computed only if the data
comprise more than one subject as per the “Dimensions” table. See the section “Processing
by Subjects” on page 2245 for information about how the GLIMMIX procedure determines
whether the data can be processed by subjects. If a GLMM comprises only a single subject
for a particular BY group, the model-based variance estimator is used instead of the empirical
estimator, and a message is written to the log.
EXPHESSIAN
requests that the expected Hessian matrix be used in computing the covariance matrix of the
nonprofiled parameters. By default, the GLIMMIX procedure uses the observed Hessian matrix in computing the asymptotic covariance matrix of covariance parameters in mixed models
and the covariance matrix of fixed effects in models without random effects. The EXPHESSIAN option is ignored if the (conditional) distribution is not a member of the exponential
family or is unknown. It is also ignored in models for nominal data.
FDIGITS=r
specifies the number of accurate digits in evaluations of the objective function. Fractional
values are allowed. The default value is r D log10 , where is the machine precision.
The value of r is used to compute the interval size for the computation of finite-difference
approximations of the derivatives of the objective function. It is also used in computing the
default value of the FCONV= option in the NLOPTIONS statement.
GRADIENT
displays the gradient of the objective function with respect to the parameter estimates in the
“Covariance Parameter Estimates” table and/or the “Parameter Estimates” table.
HESSIAN
HESS
H
displays the Hessian matrix of the optimization.
INFOCRIT=NONE | PQ | Q
IC=NONE | PQ | Q
determines the computation of information criteria in the “Fit Statistics” table. The GLIMMIX procedure computes various information criteria that typically apply a penalty to the
(possibly restricted) log likelihood, log pseudo-likelihood, or log quasi-likelihood that depends on the number of parameters and/or the sample size. If IC=NONE, these criteria are
suppressed in the “Fit Statistics” table. This is the default for models based on pseudolikelihoods.
2098 F Chapter 38: The GLIMMIX Procedure
The AIC, AICC, BIC, CAIC, and HQIC fit statistics are various information criteria. AIC
and AICC represent Akaike’s information criteria (Akaike 1974) and a small sample bias corrected version thereof (for AICC, see Hurvich and Tsai 1989; Burnham and Anderson 1998).
BIC represents Schwarz’s Bayesian criterion (Schwarz 1978). Table 38.2 gives formulas for
the criteria.
Table 38.2
Criteria
AIC
Information Criteria
Formula
Reference
2` C 2d
Akaike (1974)
AICC
2` C 2d n =.n
HQIC
2` C 2d log log n
Hannan and Quinn (1979)
2` C d log n
Schwarz (1978)
2` C d.log n C 1/
Bozdogan (1987)
BIC
CAIC
d
1/
Hurvich and Tsai (1989)
Burnham and Anderson (1998)
Here, ` denotes the maximum value of the (possibly restricted) log likelihood, log pseudolikelihood, or log quasi-likelihood, d is the dimension of the model, and n, n reflect the size
of the data.
The IC=PQ option requests that the penalties include the number of fixed-effects parameters,
when estimation in models with random effects is based on a residual (restricted) likelihood.
For METHOD=MSPL, METHOD=MMPL, METHOD=LAPLACE, and METHOD=QUAD,
IC=Q and IC=PQ produce the same results. IC=Q is the default for linear mixed models
with normal errors, and the resulting information criteria are identical to the IC option in the
MIXED procedure.
The quantities d , n, and n depend on the model and IC= option in the following way:
GLM:
IC=Q and IC=PQ options have no effect on the computation.
GLMM, IC=Q:
d equals the number of parameters in the optimization whose solutions do not fall on the boundary or are otherwise constrained. The
scale parameter is included, if it is part of the optimization. If you
use the PARMS statement to place a hold on a scale parameter, that
parameter does not count toward d .
n equals the sum of the frequencies (f ) for maximum likelihood and
quasi-likelihood estimation and f rank.X/ for restricted maximum
likelihood estimation.
n equals n, unless n < d C 2, in which case n D d C 2.
d equals the number of effective covariance parameters—that is, covariance parameters whose solution does not fall on the boundary. For
estimation of an unrestricted objective function (METHOD=MMPL,
METHOD=MSPL, METHOD=LAPLACE, METHOD=QUAD), this
value is incremented by rank.X/.
PROC GLIMMIX Statement F 2099
n equals the effective number of subjects as displayed in the “Dimensions” table, unless this value equals 1, in which case n equals the
number of levels of the first G-side RANDOM effect specified. If the
number of effective subjects equals 1 and there are no G-side random
effects, n is determined as
f rank.X/ METHOD D RMPL; METHOD D RSPL
nD
f
otherwise
where f is the sum of frequencies used.
n equals f or f
rank.X/ (for METHOD=RMPL and
METHOD=RSPL), unless this value is less than d C 2, in which
case n D d C 2.
GLMM, IC=PQ: For METHOD=MSPL, METHOD=MMPL, METHOD=LAPLACE,
and METHOD=QUAD, the results are the same as for IC=Q. For
METHOD=RSPL and METHOD=RMPL, d equals the number of effective covariance parameters plus rank.X/, and n D n equals f rank.X/.
The formulas for the information criteria thus agree with Verbeke and
Molenberghs (2000, Table 6.7, p. 74) and Vonesh and Chinchilli (1997, p.
263).
INITGLM
requests that the estimates from a generalized linear model fit (a model without random effects) be used as the starting values for the generalized linear mixed model. This option is the
default for METHOD=LAPLACE and METHOD=QUAD.
INITITER=number
specifies the maximum number of iterations used when a generalized linear model is fit initially to derive starting values for the fixed effects; see the INITGLM option. By default, the
initial fit involves at most four iteratively reweighted least squares updates. You can change
the upper limit of initial iterations with number. If the model does not contain random effects,
this option has no effect.
ITDETAILS
adds parameter estimates and gradients to the “Iteration History” table.
LIST
requests that the model program and variable lists be displayed. This is a debugging feature
and is not normally needed. When you use programming statements to define your statistical
model, this option enables you to examine the complete set of statements submitted for processing. See the section “Programming Statements” for more details about how to use SAS
statements with the GLIMMIX procedure.
MAXLMMUPDATE=number
MAXOPT=number
specifies the maximum number of optimizations for doubly iterative estimation methods
based on linearizations. After each optimization, a new pseudo-model is constructed through
a Taylor series expansion. This step is known as the linear mixed model update. The
2100 F Chapter 38: The GLIMMIX Procedure
MAXLMMUPDATE option limits the number of updates and thereby limits the number of
optimizations. If this option is not specified, number is set equal to the value specified in the
MAXITER= option in the NLOPTIONS statement. If no MAXITER= value is given, number
defaults to 20.
METHOD=RSPL
METHOD=MSPL
METHOD=RMPL
METHOD=MMPL
METHOD=LAPLACE
METHOD=QUAD< (quad-options) >
specifies the estimation method in a generalized linear mixed model (GLMM). The default is
METHOD=RSPL.
Pseudo-Likelihood
Estimation methods ending in “PL” are pseudo-likelihood techniques. The first letter of the
METHOD= identifier determines whether estimation is based on a residual likelihood (“R”)
or a maximum likelihood (“M”). The second letter identifies the expansion locus for the
underlying approximation. Pseudo-likelihood methods for generalized linear mixed models
can be cast in terms of Taylor series expansions (linearizations) of the GLMM. The expansion
locus of the expansion is either the vector of random effects solutions (“S”) or the mean of
the random effects (“M”). The expansions are also referred to as the “S”ubject-specific and
“M”arginal expansions. The abbreviation “PL” identifies the method as a pseudo-likelihood
technique.
Residual methods account for the fixed effects in the construction of the objective function,
which reduces the bias in covariance parameter estimates. Estimation methods involving Taylor series create pseudo-data for each optimization. Those data are transformed to have zero
mean in a residual method. While the covariance parameter estimates in a residual method
are the maximum likelihood estimates for the transformed problem, the fixed-effects estimates are (estimated) generalized least squares estimates. In a likelihood method that is not
residual based, both the covariance parameters and the fixed-effects estimates are maximum
likelihood estimates, but the former are known to have greater bias. In some problems, residual likelihood estimates of covariance parameters are unbiased.
For more information about linearization methods for generalized linear mixed models, see
the section “Pseudo-likelihood Estimation Based on Linearization” on page 2218.
Maximum Likelihood with Laplace Approximation
If you choose METHOD=LAPLACE with a generalized linear mixed model, PROC GLIMMIX approximates the marginal likelihood by using Laplace’s method. Twice the negative of
the resulting log-likelihood approximation is the objective function that the procedure minimizes to determine parameter estimates. Laplace estimates typically exhibit better asymptotic
PROC GLIMMIX Statement F 2101
behavior and less small-sample bias than pseudo-likelihood estimators. On the other hand, the
class of models for which a Laplace approximation of the marginal log likelihood is available
is much smaller compared to the class of models to which PL estimation can be applied.
To determine whether Laplace estimation can be applied in your model, consider the marginal
distribution of the data in a mixed model
Z
p.y/ D p.yj/ p./ d Z
D exp flogfp.yj/g C logfp./gg d Z
D exp fnf .y; /g d The function f .y; / plays an important role in the Laplace approximation: it is a function of the joint distribution of the data and the random effects (see the section “Maximum
Likelihood Estimation Based on Laplace Approximation” on page 2222). In order to construct a Laplace approximation, PROC GLIMMIX requires a conditional log-likelihood
logfp.yj/g as well as the distribution of the G-side random effects. The random effects
are always assumed to be normal with zero mean and covariance structure determined by
the RANDOM statement. The conditional distribution is determined by the DIST= option
of the MODEL statement or the default associated with a particular response type. Because a valid conditional distribution is required, R-side random effects are not permitted
for METHOD=LAPLACE in the GLIMMIX procedure. In other words, the GLIMMIX procedure requires for METHOD=LAPLACE conditional independence without R-side overdispersion or covariance structure.
Because the marginal likelihood of the data is approximated numerically, certain features
of the marginal distribution are not available—for example, you cannot display a marginal
variance-covariance matrix. Also, the procedure includes both the fixed-effects parameters
and the covariance parameters in the optimization for Laplace estimation. Consequently, this
setting imposes some restrictions with respect to available options for Laplace estimation.
Table 38.3 lists the options that are assumed for METHOD=LAPLACE, and Table 38.4 lists
the options that are not compatible with this estimation method.
The section “Maximum Likelihood Estimation Based on Laplace Approximation” contains
details about Laplace estimation in PROC GLIMMIX.
Maximum Likelihood with Adaptive Quadrature
If you choose METHOD=QUAD in a generalized linear mixed model, the GLIMMIX procedure approximates the marginal log likelihood with an adaptive Gauss-Hermite quadrature. Compared to METHOD=LAPLACE, the models for which parameters can be estimated
by quadrature are further restricted. In addition to the conditional independence assumption and the absence of R-side covariance parameters, it is required that models suitable for
METHOD=QUAD can be processed by subjects. (See the section “Processing by Subjects”
on page 2245 about how the GLIMMIX procedure determines whether the data can be processed by subjects.) This in turn requires that all RANDOM statements have SUBJECT=
2102 F Chapter 38: The GLIMMIX Procedure
effects and in the case of multiple SUBJECT= effects that these form a containment hierarchy.
In a containment hierarchy each effect is contained by another effect, and the effect contained
by all is considered “the” effect for subject processing. For example, the SUBJECT= effects
in the following statements form a containment hierarchy:
proc glimmix;
class A B block;
model y = A B A*B;
random intercept / subject=block;
random intercept / subject=A*block;
run;
The block effect is contained in the A*block interaction and the data are processed by block.
The SUBJECT= effects in the following statements do not form a containment hierarchy:
proc glimmix;
class A B block;
model y = A B A*B;
random intercept / subject=block;
random block
/ subject=A;
run;
The section “Maximum Likelihood Estimation Based on Adaptive Quadrature” on page 2225
contains important details about the computations involved with quadrature approximations.
The section “Aspects Common to Adaptive Quadrature and Laplace Approximation” on
page 2228 contains information about issues that apply to Laplace and adaptive quadrature,
such as the computation of the prediction variance matrix and the determination of starting
values.
You can specify the following quad-options for METHOD=QUAD in parentheses:
EBDETAILS
reports details about the empirical Bayes suboptimization process should this suboptimization fail.
EBSSFRAC=r
specifies the step-shortening fraction to be used while computing empirical Bayes estimates of the random effects. The default value is r D 0:8, and it is required that
r > 0.
EBSSTOL=r
specifies the objective function tolerance for determining the cessation of step shortening while computing empirical Bayes estimates of the random effects, r 0. The
default value is r D 1E 8.
EBSTEPS=n
specifies the maximum number of Newton steps for computing empirical Bayes estimates of random effects, n 0. The default value is n D 50.
EBSUBSTEPS=n
specifies the maximum number of step shortenings for computing empirical Bayes estimates of random effects. The default value is n D 20, and it is required that n 0.
PROC GLIMMIX Statement F 2103
EBTOL=r
specifies the convergence tolerance for empirical Bayes estimation, r 0. The default
value is r D 1E4, where is the machine precision. This default value equals
approximately 1E 12 on most machines.
INITPL=number
requests that adaptive quadrature commence after performing up to number pseudolikelihood updates. The initial pseudo-likelihood (PL) steps (METHOD=MSPL) can
be useful to provide good starting values for the quadrature algorithm. If you choose
number large enough so that the initial PL estimation converges, the process is equivalent to starting a quadrature from the PL estimates of the fixed-effects and covariance
parameters. Because this also makes available the PL random-effects solutions, the
adaptive step of the quadrature that determines the number of quadrature points can
take this information into account.
Note that you can combine the INITPL option with the NOINITGLM option in the
PROC GLIMMIX statement to define a precise path for starting value construction
to the GLIMMIX procedure. For example, the following statement generates starting
values in these steps:
proc glimmix method=quad(initpl=5);
1.
2.
3.
4.
A GLM without random effects is fit initially to obtain as starting values for the
fixed effects. The INITITER= option in the PROC GLIMMIX statement controls
the number of iterations in this step.
Starting values for the covariance parameters are then obtained by MIVQUE0 estimation (Goodnight 1978b), using the fixed-effects parameter estimates from step
1.
With these values up to five pseudo-likelihood updates are computed.
The PL estimates for fixed-effects, covariance parameters, and the solutions for
the random effects are then used to determine the number of quadrature points and
used as the starting values for the quadrature.
The first step (GLM fixed-effects estimates) is omitted, if you modify the previous
statement as follows:
proc glimmix method=quad(initpl=5) noinitglm;
The NOINITGLM option is the default of the pseudo-likelihood methods you select
with the METHOD= option.
QCHECK
performs an adaptive recalculation of the objective function ( 2 log likelihood) at the
solution. The increment of the quadrature points, starting from the number of points
used in the optimization, follows the same rules as the determination of the quadrature
point sequence at the starting values (see the QFAC= and QMAX= suboptions). For
example, the following statement estimates the parameters based on a quadrature with
seven nodes in each dimension:
proc glimmix method=quad(qpoints=7 qcheck);
2104 F Chapter 38: The GLIMMIX Procedure
Because the default search sequence is 1; 3; 5; 7; 9; 11; 21; 31, the QCHECK option
computes the 2 log likelihood at the converged solution for 9; 11; 21; and 31 quadrature points and reports relative differences to the converged value and among successive
values. The ODS table produced by this option is named “QuadCheck.”
C AUTION : This option is useful to diagnose the sensitivity of the likelihood approximation at the solution. It does not diagnose the stability of the solution under changes
in the number of quadrature points. For example, if increasing the number of points
from 7 to 9 does not alter the objective function, this does not imply that a quadrature with 9 points would arrive at the same parameter estimates as a quadrature with 7
points.
QFAC=r
determines the step size for the quadrature point sequence. If the GLIMMIX procedure determines the quadrature nodes adaptively, the log likelihoods are computed for
nodes in a predetermined sequence. If Nmi n and Nmax denote the values from the
QMIN= and QMAX= suboptions, respectively, the sequence for values less than 11 is
constructed in increments of 2 starting at Nmi n . Values greater than 11 are incremented
in steps of r. The default value is r D 10. The default sequence, without specifying
the QMIN=, QMAX=, or QFAC= option, is thus 1; 3; 5; 7; 9; 11; 21; 31. By contrast,
the following statement evaluates the sequence 8; 10; 30; 50:
proc glimmix method=quad(qmin=8,qmax=51,qfac=20);
QMAX=n
specifies an upper bound for the number of quadrature points. The default is n D 31.
QMIN=n
specifies a lower bound for the number of quadrature points. The default is n D 1 and
the value must be less than the QMAX= value.
QPOINTS=n
determines the number of quadrature points in each dimension of the integral. Note
that if there are r random effects for each subject, the GLIMMIX procedure evaluates
nr conditional log likelihoods for each observation to compute one value of the objective function. Increasing the number of quadrature nodes can substantially increase
the computational burden. If you choose QPOINTS=1, the quadrature approximation
reduces to the Laplace approximation. If you do not specify the number of quadrature
points, it is determined adaptively by increasing the number of nodes at the starting
values. See the section “Aspects Common to Adaptive Quadrature and Laplace Approximation” on page 2228 for details.
QTOL=r
specifies a relative tolerance criterion for the successive evaluation of log likelihoods
for different numbers of quadrature points. When the GLIMMIX procedure determines
the number of quadrature points adaptively, the number of nodes are increased until the
QMAX=n limit is reached or until two successive evaluations of the log likelihood have
a relative change of less than r. In the latter case, the lesser number of quadrature nodes
is used for the optimization.
PROC GLIMMIX Statement F 2105
The EBSSFRAC, EBSSTOL, EBSTEPS, EBSUBSTEPS, and EBTOL suboptions affect the
suboptimization that leads to the empirical Bayes estimates of the random effects. Under
normal circumstances, there is no reason to change from the default values. When the suboptimizations fail, the optimization process can come to a halt. If the EBDETAILS option is
in effect, you might be able to determine why the suboptimization fails and then adjust these
values accordingly.
The QMIN, QMAX, QTOL, and QFAC suboptions determine the quadrature point search
sequence for the adaptive component of estimation.
As for METHOD=LAPLACE, certain features of the marginal distribution are not available
because the marginal likelihood of the data is approximated numerically. For example, you
cannot display a marginal variance-covariance matrix. Also, the procedure includes both the
fixed-effects and covariance parameters in the optimization for quadrature estimation. Consequently, this setting imposes some restrictions with respect to available options. Table 38.3
lists the options that are assumed for METHOD=QUAD and METHOD=LAPLACE, and
Table 38.4 lists the options that are not compatible with these estimation methods.
Table 38.3
Defaults for METHOD=LAPLACE and METHOD=QUAD
Statement
Option
PROC GLIMMIX
PROC GLIMMIX
MODEL
NOPROFILE
INITGLM
NOCENTER
Table 38.4
Options Incompatible with METHOD=LAPLACE and METHOD=QUAD
Statement
Option
PROC GLIMMIX
PROC GLIMMIX
PROC GLIMMIX
PROC GLIMMIX
MODEL
MODEL
MODEL
RANDOM
RANDOM _RESIDUAL_
RANDOM
RANDOM
RANDOM
RANDOM
RANDOM
EXPHESSIAN
SCOREMOD
SCORING
PROFILE
DDFM=KENWARDROGER
DDFM=SATTERTHWAITE
STDCOEF
RESIDUAL
All R-side random effects
V
VC
VCI
VCORR
VI
In addition to the options displayed in Table 38.4, the NOBOUND option in the PROC
GLIMMIX and the NOBOUND option in the PARMS statements are not available with
METHOD=QUAD. Unbounding the covariance parameter estimates is possible with
METHOD=LAPLACE, however.
2106 F Chapter 38: The GLIMMIX Procedure
No Random Effects Present
If the model does not contain G-side random effects or contains only a single overdispersion component, then the model belongs to the family of (overdispersed) generalized linear
models if the distribution is known or the quasi-likelihood models for independent data if the
distribution is not known. The GLIMMIX procedure then estimates model parameters by the
following techniques:
normally distributed data: residual maximum likelihood
nonnormal data: maximum likelihood
data with unknown distribution: quasi-likelihood
The METHOD= specification then has only an effect with respect to the divisor used in estimating the overdispersion component. With a residual method, the divisor is f k, where f
denotes the sum of the frequencies and k is the rank of X. Otherwise, the divisor is f .
NAMELEN=number
specifies the length to which long effect names are shortened. The default and minimum value
is 20.
NOBOUND
requests the removal of boundary constraints on covariance and scale parameters in mixed
models. For example, variance components have a default lower boundary constraint of 0,
and the NOBOUND option allows their estimates to be negative.
The NOBOUND option cannot be used for adaptive quadrature estimation with
METHOD=QUAD. The scaling of the quadrature abscissas requires an inverse Cholesky
root that is possibly not well defined when the G matrix of the mixed model is negative
definite or indefinite. The Laplace approximation (METHOD=LAPLACE) is not subject to
this limitation.
NOBSDETAIL
adds detailed information to the “Number of Observations” table to reflect how many observations were excluded from the analysis and for which reason.
NOCLPRINT< =number >
suppresses the display of the “Class Level Information” table, if you do not specify number.
If you specify number, only levels with totals that are less than number are listed in the table.
NOFIT
suppresses fitting of the model. When the NOFIT option is in effect, PROC GLIMMIX
produces the “Model Information,” “Class Level Information,” “Number of Observations,”
and “Dimensions” tables. These can be helpful to gauge the computational effort required to
fit the model. For example, the “Dimensions” table informs you as to whether the GLIMMIX
procedure processes the data by subjects, which is typically more computationally efficient
than processing the data as a single subject. See the section “Processing by Subjects” for
more information.
PROC GLIMMIX Statement F 2107
If you request a radial smooth with knot selection by k-d tree methods, PROC GLIMMIX also
computes the knot locations of the smoother. You can then examine the knots without fitting
the model. This enables you to try out different knot construction methods and bucket sizes.
See the KNOTMETHOD=KDTREE option (and its suboptions) of the RANDOM statement.
If you combine the NOFIT option with the OUTDESIGN option, you can write the X and/or
Z matrix of your model to a SAS data set without fitting the model.
NOINITGLM
requests that the starting values for the fixed effects not be obtained by first fitting a generalized linear model. This option is the default for the pseudo-likelihood estimation methods and
for the linear mixed model. For the pseudo-likelihood methods, starting values can be implicitly defined based on an initial pseudo-data set derived from the data and the link function.
For linear mixed models, starting values for the fixed effects are not necessary. The NOINITGLM option is useful in conjunction with the INITPL= suboption of METHOD=QUAD in
order to perform initial pseudo-likelihood steps prior to an adaptive quadrature.
NOITPRINT
suppresses the display of the “Iteration History” table.
NOPROFILE
includes the scale parameter into the optimization for models that have such a parameter
(see Table 38.15). By default, the GLIMMIX procedure profiles scale parameters from the
optimization in mixed models. In generalized linear models, scale parameters are not profiled.
NOREML
determines the denominator for the computation of the scale parameter in a GLM for normal
data and for overdispersion parameters. By default, the GLIMMIX procedure computes the
scale parameter for the normal distribution as
b
D
n
X
fi .yi
i D1
f
b
y i /2
k
where k is the rank of X, fi is the frequency associated with the i th observation, and f D
P
fi . Similarly, the overdispersion parameter in an overdispersed GLM is estimated by the
ratio of the Pearson statistic and .f k/. If the NOREML option is in effect, the denominators
are replaced by f , the sum of the frequencies. In a GLM for normal data, this yields the
maximum likelihood estimate of the error variance. For this case, the NOREML option is a
convenient way to change from REML to ML estimation.
In GLMM models fit by pseudo-likelihood methods, the NOREML option changes the estimation method to the nonresidual form. See the METHOD= option for the distinction between residual and nonresidual estimation methods.
ODDSRATIO
OR
requests that odds ratios be added to the output when applicable. Odds ratios and their confidence limits are reported only for models with logit, cumulative logit, or generalized logit
link. Specifying the ODDSRATIO option in the PROC GLIMMIX statement has the same
effect as specifying the ODDSRATIO option in the MODEL statement and in all LSMEANS
2108 F Chapter 38: The GLIMMIX Procedure
statements. Note that the ODDSRATIO option in the MODEL statement has several suboptions that enable you to construct customized odds ratios. These suboptions are available
only through the MODEL statement. For details about the interpretation and computation of
odds and odds ratios with the GLIMMIX procedure, see the section “Odds and Odds Ratio
Estimation” on page 2254.
ORDER=DATA
ORDER=FORMATTED
ORDER=FREQ
ORDER=INTERNAL
specifies the sorting order for the levels of CLASS variables. This ordering determines which
parameters in the model correspond to each level in the data, so the ORDER= option can be
useful when you use CONTRAST or ESTIMATE statements.
When the default ORDER=FORMATTED is in effect for numeric variables for which you
have supplied no explicit format, the levels are ordered by their internal values. To order
numeric class levels with no explicit format by their BEST12. formatted values, you can
specify this format explicitly for the CLASS variables.
The following table shows how PROC GLIMMIX interprets values of the ORDER= option.
Value of ORDER=
Levels Sorted By
DATA
order of appearance in the input data set
FORMATTED
external formatted value, except for numeric variables
with no explicit format, which are sorted by their unformatted (internal) value
FREQ
descending frequency count; levels with the most observations come first in the order
INTERNAL
unformatted value
For FORMATTED and INTERNAL values, the sort order is machine dependent.
When the response variable appears in a CLASS statement, the ORDER= option in the PROC
GLIMMIX statement applies to its sort order. When you specify a response-option in the
MODEL statement, this overrides the ORDER= option in the PROC GLIMMIX statement.
For example, in the following statements the sort order of the wheeze variable is determined
by the formatted value (default):
proc glimmix order=data;
class city;
model wheeze = city age / dist=binary s;
run;
The ORDER= option in the PROC GLIMMIX statement has no effect on the sort order of the
wheeze variable because it does not appear in the CLASS statement. However, in the following statements the sort order of the wheeze variable is determined by the order of appearance
in the input data set because the response variable appears in the CLASS statement:
PROC GLIMMIX Statement F 2109
proc glimmix order=data;
class city wheeze;
model wheeze = city age / dist=binary s;
run;
For more information about sort order, see the chapter on the SORT procedure in the Base SAS
Procedures Guide and the discussion of BY-group processing in SAS Language Reference:
Concepts.
OUTDESIGN< (options) >< =SAS-data-set >
creates a data set that contains the contents of the X and Z matrix. If the data are processed
by subjects as shown in the “Dimensions” table, then the Z matrix saved to the data set corresponds to a single subject. By default, the GLIMMIX procedure includes in the OUTDESIGN
data set the X and Z matrix (if present) and the variables in the input data set. You can specify
the following options in parentheses to control the contents of the OUTDESIGN data set:
NAMES
produces tables associating columns in the OUTDESIGN data set with fixed-effects
parameter estimates and random-effects solutions.
NOMISS
excludes from the OUTDESIGN data set observations that were not used in the analysis.
NOVAR
excludes from the OUTDESIGN data set variables from the input data set. Variables listed in the BY and ID statements and variables needed for identification of
SUBJECT= effects are always included in the OUTDESIGN data set.
X< =prefix >
saves the contents of the X matrix. The optional prefix is used to name the columns.
The default naming prefix is “_X”.
Z< =prefix >
saves the contents of the Z matrix. The optional prefix is used to name the columns.
The default naming prefix is “_Z”.
The order of the observations in the OUTDESIGN data set is the same as the order of the
input data set. If you do not specify a data set with the OUTDESIGN option, the procedure
uses the DATAn convention to name the data set.
PCONV=r
specifies the parameter estimate convergence criterion for doubly iterative estimation methods. The GLIMMIX procedure applies this criterion to fixed-effects estimates and covariance
.u/
parameter estimates. Suppose bi denotes the estimate of the i th parameter at the uth optimization. The procedure terminates the doubly iterative process if the largest value
.u/
2
b.u
jbi
.u/
jbi j
i
C
1/
j
.u 1/
jbi
j
2110 F Chapter 38: The GLIMMIX Procedure
is less than r. To check an absolute convergence criteria as well, you can set the
ABSPCONV= option in the PROC GLIMMIX statement. The default value for r is 1E8
times the machine epsilon, a product that equals about 1E 8 on most machines.
Note that this convergence criterion does not affect the convergence criteria applied within any
individual optimization. In order to change the convergence behavior within an optimization,
you can use the ABSCONV=, ABSFCONV=, ABSGCONV=, ABSXCONV=, FCONV=, or
GCONV= option in the NLOPTIONS statement.
PLOTS < (global-plot-options) > < = plot-request < (options) > >
PLOTS < (global-plot-options) > < = (plot-request < (options) > < ... plot-request < (options) > >) >
requests that the GLIMMIX procedure produce statistical graphics via the Output Delivery
System, provided that the ODS GRAPHICS statement has been specified. For general information about ODS Graphics, see Chapter 21, “Statistical Graphics Using ODS.” For examples of the basic statistical graphics produced by the GLIMMIX procedure and aspects of
their computation and interpretation, see the section “ODS Graphics” on page 2280 in this
chapter. You can also request statistical graphics for least squares means through the PLOTS
option in the LSMEANS statement, which gives you more control over the display compared
to the PLOTS option in the PROC GLIMMIX statement.
Global Plot Options
The global-plot-options apply to all relevant plots generated by the GLIMMIX procedure.
The global-plot-options supported by the GLIMMIX procedure are as follows:
OBSNO
uses the data set observation number to identify observations in tooltips, provided that
the observation number can be determined. Otherwise, the number displayed in tooltips
is the index of the observation as it is used in the analysis within the BY group.
UNPACKPANEL
UNPACK
breaks a graphic that is otherwise paneled into individual component plots.
Specific Plot Options
The following listing describes the specific plots and their options.
ALL
requests that all plots appropriate for the analysis be produced. In models with G-side
random effects, residual plots are based on conditional residuals (by using the BLUPs
of random effects) on the linear (linked) scale. Plots of least squares means differences
are produced for LSMEANS statements without options that would contradict such a
display.
PROC GLIMMIX Statement F 2111
ANOMPLOT
ANOM
requests an analysis of means display in which least squares means are compared
against an average least squares mean (Ott 1967; Nelson 1982, 1991, 1993). See the
DIFF= option in the LSMEANS statement for the computation of this average. Least
squares mean ANOM plots are produced only for those fixed effects that are listed in
LSMEANS statements that have options that do not contradict the display. For example, if you request ANOM plots with the PLOTS= option in the PROC GLIMMIX
statement, the following LSMEANS statements produce analysis of mean plots for effects A and C:
lsmeans A / diff=anom;
lsmeans B / diff;
lsmeans C;
The DIFF option in the second LSMEANS statement implies all pairwise differences.
When differences against the average LS-mean are adjusted for multiplicity with the
ADJUST=NELSON option in the LSMEANS statement, the ANOMPLOT display is
adjusted accordingly.
BOXPLOT < (boxplot-options) >
requests box plots for the effects in your model that consist of classification effects only.
Note that these effects can involve more than one classification variable (interaction and
nested effects), but cannot contain any continuous variables. By default, the BOXPLOT
request produces box plots of (conditional) residuals for the qualifying effects in the
MODEL and RANDOM statements. See the discussion of the boxplot-options in a
later section for information about how to tune your box plot request.
CONTROLPLOT
CONTROL
requests a display in which least squares means are visually compared against a reference level. LS-mean control plots are produced only for those fixed effects that are
listed in LSMEANS statements that have options that do not contradict with the display. For example, the following statements produce control plots for effects A and C if
you specify PLOTS=CONTROL in the PROC GLIMMIX statement:
lsmeans A / diff=control(’1’);
lsmeans B / diff;
lsmeans C;
The DIFF option in the second LSMEANS statement implies all pairwise differences.
When differences against a control level are adjusted for multiplicity with the
ADJUST= option in the LSMEANS statement, the control plot display is adjusted accordingly.
DIFFPLOT< (diffplot-options) >
DIFFOGRAM < (diffplot-options) >
DIFF< (diffplot-options) >
requests a display of all pairwise least squares mean differences and their significance.
2112 F Chapter 38: The GLIMMIX Procedure
When constructed from arithmetic means, the display is also known as a “mean-mean
scatter plot” (Hsu 1996; Hsu and Peruggia 1994). For each comparison a line segment,
centered at the LS-means in the pair, is drawn. The length of the segment corresponds
to the projected width of a confidence interval for the least squares mean difference.
Segments that fail to cross the 45-degree reference line correspond to significant least
squares mean differences.
If you specify the ADJUST= option in the LSMEANS statement, the lengths of the line
segments are adjusted for multiplicity.
LS-mean difference plots are produced only for those fixed effects listed in LSMEANS
statements that have options that do not conflict with the display. For example, the
following statements request differences against a control level for the A effect, all
pairwise differences for the B effect, and the least squares means for the C effect:
lsmeans A / diff=control(’1’);
lsmeans B / diff;
lsmeans C;
The DIFF= type in the first statement contradicts a display of all pairwise differences.
Difference plots are produced only for the B and C effects if you specify PLOTS=DIFF
in the PROC GLIMMIX statement.
You can specify the following diffplot-options. The ABS and NOABS options determine the positioning of the line segments in the plot. When the ABS option is in effect
(this is the default) all line segments are shown on the same side of the reference line.
The NOABS option separates comparisons according to the sign of the difference. The
CENTER option marks the center point for each comparison. This point corresponds
to the intersection of two least squares means. The NOLINES option suppresses the
display of the line segments that represent the confidence bounds for the differences
of the least squares means. The NOLINES option implies the CENTER option. The
default is to draw line segments in the upper portion of the plot area without marking
the center point.
MEANPLOT< (meanplot-options) >
requests a display of the least squares means of effects specified in LSMEANS statements. The following meanplot-options affect the display. Upper and lower confidence
limits are plotted when the CL option is used. When the CLBAND option is in effect,
confidence limits are shown as bands and the means are connected. By default, least
squares means are not joined by lines. You can achieve that effect with the JOIN or
CONNECT option. Least squares means are displayed in the same order in which they
appear in the “Least Squares Means” table. You can change that order for plotting purposes with the ASCENDING and DESCENDING options. The ILINK option requests
that results be displayed on the inverse linked (the data) scale.
Note that there is also a MEANPLOT suboption of the PLOTS= option in the
LSMEANS statement. In addition to the meanplot-options just described, you can also
specify classification effects that give you more control over the display of interaction
means through the PLOTBY= and SLICEBY= options. To display interaction means,
you typically want to use the MEANPLOT option in the LSMEANS statement. For
PROC GLIMMIX Statement F 2113
example, the next statement requests a plot in which the levels of A are placed on the
horizontal axis and the means that belong to the same level of B are joined by lines:
lsmeans A*B / plot=meanplot(sliceby=b join);
NONE
requests that no plots be produced.
ODDSRATIO < (oddsratioplot-options) >
requests a display of odds ratios and their confidence limits when the link function
permits the computation of odds ratios (see the ODDSRATIO option in the MODEL
statement). Possible suboptions of the ODDSRATIO plot request are described below
under the heading “Odds Ratio Plot Options.”
RESIDUALPANEL< (residualplot-options) >
requests a paneled display constructed from raw residuals. The panel consists of a
plot of the residuals against the linear predictor or predicted mean, a histogram with
normal density overlaid, a Q-Q plot, and a box plot of the residuals. The residualplotoptions enable you to specify which type of residual is being graphed. These are further
discussed below under the heading “Residual Plot Options.”
STUDENTPANEL< (residualplot-options) >
requests a paneled display constructed from studentized residuals. The same panel
organization is applied as for the RESIDUALPANEL plot type.
PEARSONPANEL< (residualplot-options) >
requests a paneled display constructed from Pearson residuals. The same panel organization is applied as for the RESIDUALPANEL plot type.
Residual Plot Options
The residualplot-options apply to the RESIDUALPANEL, STUDENTPANEL, and
PEARSONPANEL displays. The primary function of these options is to control which
type of a residual to display. The four types correspond to keyword-options as for output statistics in the OUTPUT statement. The residualplot-options take on the following
values:
BLUP
CONDITIONAL
uses the predictors of the random effects in computing the residual.
ILINK
NONLINEAR
computes the residual on the inverse linked scale (the data scale).
NOBLUP
MARGINAL
does not use the predictors of the random effects in computing the residual.
2114 F Chapter 38: The GLIMMIX Procedure
NOILINK
LINEAR
computes the residual on the linked scale.
UNPACK
produces separate plots from the elements of the panel.
You can list a plot request one or more times with different options. For example, the
following statements request a panel of marginal raw residuals, individual plots generated from a panel of the conditional raw residuals, and a panel of marginal studentized
residuals:
ods graphics on;
proc glimmix plots=(ResidualPanel(marginal)
ResidualPanel(unpack conditional)
StudentPanel(marginal));
The default is to compute conditional residuals on the linear scale if the model
contains G-side random effects (BLUP NOILINK). Not all combinations of the
BLUP/NOBLUP and ILINK/NOILINK suboptions are possible for all residual types
and models. For details, see the description of output statistics for the OUTPUT statement. Pearson residuals are always displayed against the linear predictor; all other
residuals are graphed versus the linear predictor if the NOILINK suboption is in effect (default), and against the corresponding prediction on the mean scale if the ILINK
option is in effect. See Table 38.11 for a definition of the residual quantities and exclusions.
Box Plot Options
The boxplot-options determine whether box plots are produced for residuals or for
residuals and observed values, and for which model effects the box plots are constructed. The available boxplot-options are as follows:
BLOCK
BLOCKLEGEND
displays levels of up to four classification variables of the box plot effect by using
block legends instead of axis tick values.
BLUP
CONDITIONAL
constructs box plots from conditional residuals—that is, residuals that use the
estimated BLUPs of random effects.
FIXED
produces box plots for all fixed effects (MODEL statement) consisting entirely
of classification variables.
PROC GLIMMIX Statement F 2115
GROUP
produces box plots for all GROUP= effects in RANDOM statements consisting
entirely of classification variables.
ILINK
NONLINEAR
computes the residual on the scale of the data (the inverse linked scale).
NOBLUP
MARGINAL
constructs box plots from marginal residuals.
NOILINK
LINEAR
computes the residual on the linked scale.
NPANELPOS=number
specifies the number of box positions on the graphic and provides the capability
to break a box plot into multiple graphics. If number is negative, no balancing of
the number of boxes takes place and number is the maximum number of boxes
per graphic. If number is positive, the number of boxes per graphic is balanced.
For example, suppose that variable A has 125 levels. The following statements
request that the number of boxes per plot results be balanced and result in six
plots with 18 boxes each and one plot with 17 boxes:
ods graphics on;
proc glimmix plots=boxplot(npanelpos=20);
class A;
model y = A;
run;
If number is zero (this is the default), all levels of the effect are displayed in a
single plot.
OBSERVED
adds box plots of the observed data for the selected effects.
PEARSON
constructs box plots from Pearson residuals rather than from the default residuals.
PSEUDO
adds box plots of the pseudo-data for the selected effects. This option is available
only for the pseudo-likelihood estimation methods that construct pseudo-data.
RANDOM
produces box plots for all effects in RANDOM statements that consist entirely of
classification variables. This does not include effects specified in the GROUP=
or SUBJECT= option of the RANDOM statements.
2116 F Chapter 38: The GLIMMIX Procedure
RAW
constructs box plots from raw residuals (observed minus predicted).
STUDENT
constructs box plots from studentized residuals rather than from the default residuals.
SUBJECT
produces box plots for all SUBJECT= effects in RANDOM statements consisting
entirely of classification variables.
USEINDEX
uses as the horizontal axis label the index of the effect level, rather than the formatted value(s). For classification variables with many levels or model effects
that involve multiple classification variables, the formatted values identifying the
effect levels might take up too much space as axis tick values, leading to extensive thinning. The USEINDEX option replaces tick values constructed from
formatted values with the internal level number.
By default, box plots of residuals are constructed from the raw conditional residuals
(on the linked scale) in linear mixed models and from Pearson residuals in all other
models. Note that not all combinations of the BLUP/NOBLUP and ILINK/NOILINK
suboptions are possible for all residual types and models. For details, see the description of output statistics for the OUTPUT statement.
Odds Ratio Plot Options
The oddsratioplot-options determine the display of odds ratios and their confidence
limits. The computation of the odds ratios follows the ODDSRATIO option in the
MODEL statement. The available oddsratioplot-options are as follows:
LOGBASE= 2 | E | 10
log-scales the odds ratio axis.
NPANELPOS=n
provides the capability to break an odds ratio plot into multiple graphics having
at most jnj odds ratios per graphic. If n is positive, then the number of odds ratios
per graphic is balanced. If n is negative, then no balancing of the number of odds
ratios takes place. For example, suppose you want to display 21 odds ratios.
Then NPANELPOS=20 displays two plots, the first with 11 and the second with
10 odds ratios, and NPANELPOS= 20 displays 20 odds ratios in the first plot
and a single odds ratio in the second. If n D 0 (this is the default), then all odds
ratios are displayed in a single plot.
ORDER=ASCENDING | DESCENDING
displays the odds ratios in sorted order. By default the odds ratios are displayed
in the order in which they appear in the “Odds Ratio Estimates” table.
PROC GLIMMIX Statement F 2117
RANGE=(< min > < ,max >) | CLIP
specifies the range of odds ratios to display. If you specify RANGE=CLIP, then
the confidence intervals are clipped and the range contains the minimum and
maximum odds ratios. By default the range of view captures the extent of the
odds ratio confidence intervals.
STATS
adds the numeric values of the odds ratio and its confidence limits to the graphic.
PROFILE
requests that scale parameters be profiled from the optimization, if possible. This is the default
for generalized linear mixed models. In generalized linear models with normally distributed
data, you can use the PROFILE option to request profiling of the residual variance.
SCOREMOD
requests that the Hessian matrix in GLMMs be based on a modified scoring algorithm, provided that PROC GLIMMIX is in scoring mode when the Hessian is evaluated. The procedure
is in scoring mode during iteration, if the optimization technique requires second derivatives,
the SCORING=n option is specified, and the iteration count has not exceeded n. The procedure also computes the expected (scoring) Hessian matrix when you use the EXPHESSIAN
option in the PROC GLIMMIX statement.
The SCOREMOD option has no effect if the SCORING= or EXPHESSIAN option is not
specified. The nature of the SCOREMOD modification to the expected Hessian computation
is shown in Table 38.17, in the section “Pseudo-likelihood Estimation Based on Linearization” on page 2218. The modification can improve the convergence behavior of the GLMM
compared to standard Fisher scoring and can provide a better approximation of the variability of the covariance parameters. For more details, see the section “Estimated Precision of
Estimates” on page 2219.
SCORING=number
requests that Fisher scoring be used in association with the estimation method up to iteration number. By default, no scoring is applied. When you use the SCORING= option and
PROC GLIMMIX converges without stopping the scoring algorithm, the procedure uses the
expected Hessian matrix to compute approximate standard errors for the covariance parameters instead of the observed Hessian. If necessary, the standard errors of the covariance
parameters as well as the output from the ASYCOV and ASYCORR options are adjusted.
If scoring stopped prior to convergence and you want to use the expected Hessian matrix in
the computation of standard errors, use the EXPHESSIAN option in the PROC GLIMMIX
statement.
Scoring is not possible in models for nominal data. It is also not possible for GLMs with
unknown distribution or for those outside the exponential family. If you perform quasilikelihood estimation, the GLIMMIX procedure is always in scoring mode and the SCORING= option has no effect. See the section “Quasi-likelihood for Independent Data” for a
description of the types of models where GLIMMIX applies quasi-likelihood estimation.
The SCORING= option has no effect for optimization methods that do not involve second
derivatives. See the TECHNIQUE= option in the NLOPTIONS statement and the section
2118 F Chapter 38: The GLIMMIX Procedure
“Choosing an Optimization Algorithm” on page 405 in Chapter 18, “Shared Concepts and
Topics,” for details about first- and second-order algorithms.
SINGCHOL=number
tunes the singularity criterion in Cholesky decompositions. The default is 1E4 times the
machine epsilon; this product is approximately 1E 12 on most computers.
SINGRES=number
sets the tolerance for which the residual variance is considered to be zero. The default is 1E4
times the machine epsilon; this product is approximately 1E 12 on most computers.
SINGULAR=number
tunes the general singularity criterion applied by the GLIMMIX procedure in divisions and inversions. The default is 1E4 times the machine epsilon; this product is approximately 1E 12
on most computers.
STARTGLM
is an alias of the INITGLM option.
SUBGRADIENT< =SAS-data-set >
SUBGRAD< =SAS-data-set >
creates a data set with information about the gradient of the objective function. The contents
and organization of the SUBGRADIENT= data set depend on the type of model. The following paragraphs describe the SUBGRADIENT= data set for the two major estimation modes.
See the section “GLM Mode or GLMM Mode” on page 2231 for details about the estimation
modes of the GLIMMIX procedure.
GLMM Mode
If the GLIMMIX procedure operates in GLMM mode, the SUBGRADIENT= data set contains as many observations as there are usable subjects
in the analysis. The maximum number of usable subjects is displayed in
the “Dimensions” table. Gradient information is not written to the data set
for subjects who do not contribute valid observations to the analysis. Note
that the objective function in the “Iteration History” table is in terms of the
2 log (residual, pseudo-) likelihood. The gradients in the SUBGRADIENT= data set are gradients of that objective function.
The gradients are evaluated at the final solution of the estimation problem.
If the GLIMMIX procedure fails to converge, then the information in the
SUBGRADIENT= data set corresponds to the gradient evaluated at the
last iteration or optimization.
The number of gradients saved to the SUBGRADIENT= data set equals
the number of parameters in the optimization. For example, with
METHOD=LAPLACE or METHOD=QUAD the fixed-effects parameters
and the covariance parameters take part in the optimization. The order in
which the gradients appear in the data set equals the order in which the
gradients are displayed when the ITDETAILS option is in effect: gradients for fixed-effects parameters precede those for covariance parameters,
and gradients are not reported for singular columns in the X0 X matrix.
In models where the residual variance is profiled from the optimization,
BY Statement F 2119
a subject-specific gradient is not reported for the residual variance. To
decompose this gradient by subjects, add the NOPROFILE option in the
PROC GLIMMIX statement. When the subject-specific gradients in the
SUBGRADIENT= data set are summed, the totals equal the values reported by the GRADIENT option.
GLM Mode
When you fit a generalized linear model (GLM) or a GLM with overdispersion, the SUBGRADIENT= data set contains the observation-wise gradients of the negative log-likelihood function with respect to the parameter
estimates. Note that this corresponds to the objective function in GLMs
as displayed in the “Iteration History” table. However, the gradients displayed in the “Iteration History” for GLMs—when the ITDETAILS option
is in effect—are possibly those of the centered and scaled coefficients.
The gradients reported in the “Parameter Estimates” table and in the SUBGRADIENT= data set are gradients with respect to the uncentered and
unscaled coefficients.
The gradients are evaluated at the final estimates. If the model does not
converge, the gradients contain missing values. The gradients appear in
the SUBGRADIENT= data set in the same order as in the “Parameter
Estimates” table, with singular columns removed.
The variables from the input data set are added to the SUBGRADIENT=
data set in GLM mode. The data set is organized in the same way as
the input data set; observations that do not contribute to the analysis are
transferred to the SUBGRADIENT= data set, but gradients are calculated
only for observations that take part in the analysis. If you use an ID statement, then only the variables in the ID statement are transferred to the
SUBGRADIENT= data set.
BY Statement
BY variables ;
You can specify a BY statement with PROC GLIMMIX to obtain separate analyses for observations
in groups defined by the BY variables. When a BY statement appears, the procedure expects the
input data set to be sorted in order of the BY variables. The variables are one or more variables in
the input data set.
If your input data set is not sorted in ascending order, use one of the following alternatives:
Sort the data by using the SORT procedure with a similar BY statement.
Specify the BY statement options NOTSORTED or DESCENDING in the BY statement for
the GLIMMIX procedure. The NOTSORTED option does not mean that the data are unsorted
but rather that the data are arranged in groups (according to values of the BY variables) and
that these groups are not necessarily in alphabetical or increasing numeric order.
2120 F Chapter 38: The GLIMMIX Procedure
Create an index on the BY variables by using the DATASETS procedure (in Base SAS software).
Because sorting the data changes the order in which PROC GLIMMIX reads observations, the
sorting order for the levels of the CLASS variable might be affected if you have specified
ORDER=DATA in the PROC GLIMMIX statement. This, in turn, affects specifications in the
CONTRAST, ESTIMATE, or LSMESTIMATE statement.
For more information about the BY statement, see SAS Language Reference: Concepts. For more
information about the DATASETS procedure, see the Base SAS Procedures Guide.
CLASS Statement
CLASS variables ;
The CLASS statement names the classification variables to be used in the analysis. If the CLASS
statement is used, it must appear before the MODEL statement.
Classification variables can be either character or numeric. By default, class levels are determined
from the entire formatted values of the CLASS variables. Note that this represents a slight change
from previous releases in the way in which class levels are determined. Prior to SAS 9, class levels
were determined by using no more than the first 16 characters of the formatted values. If you want
to revert to this previous behavior you can use the TRUNCATE option in the CLASS statement. In
any case, you can use formats to group values into levels. Refer to the discussion of the FORMAT
procedure in the Base SAS Procedures Guide and the discussions of the FORMAT statement and
SAS formats in SAS Language Reference: Dictionary. You can adjust the order of CLASS variable
levels with the ORDER= option in the PROC GLIMMIX statement.
You can specify the following option in the CLASS statement after a slash (/).
TRUNCATE
specifies that class levels should be determined by using no more than the first 16 characters
of the formatted values of CLASS variables. When formatted values are longer than 16
characters, you can use this option in order to revert to the levels as determined in releases
previous to SAS 9.
CONTRAST Statement F 2121
CONTRAST Statement
CONTRAST ’label’ contrast-specification
< , contrast-specification > < , . . . >
< / options > ;
The CONTRAST statement provides a mechanism for obtaining custom hypothesis tests. It is patterned after the CONTRAST statement in PROC MIXED and enables you to select an appropriate
inference space (McLean, Sanders, and Stroup 1991). The GLIMMIX procedure gives you greater
flexibility in entering contrast coefficients for random effects, however, because it permits the usual
value-oriented positional syntax for entering contrast coefficients, as well as a level-oriented syntax
that simplifies entering coefficients for interaction terms and is designed to work with constructed
effects that are defined through the experimental EFFECT statement. The differences between the
traditional and new-style coefficient syntax are explained in detail in the section “Positional and
Nonpositional Syntax for Contrast Coefficients” on page 2262.
You can test the hypothesis L0 D 0, where L0 D ŒK0 M0  and 0 D Œˇ 0 0 , in several inference
spaces. The inference space corresponds to the choice of M. When M D 0, your inferences
apply to the entire population from which the random effects are sampled; this is known as the
broad inference space. When all elements of M are nonzero, your inferences apply only to the
observed levels of the random effects. This is known as the narrow inference space, and you can
also choose it by specifying all of the random effects as fixed. The GLM procedure uses the narrow
inference space. Finally, by zeroing portions of M corresponding to selected main effects and
interactions, you can choose intermediate inference spaces. The broad inference space is usually the
most appropriate; it is used when you do not specify random effects in the CONTRAST statement.
In the CONTRAST statement,
label
identifies the contrast in the table. A label is required for every contrast specified.
Labels can be up to 200 characters and must be enclosed in quotes.
contrast-specification identifies the fixed effects and random effects and their coefficients from
which the L matrix is formed. The syntax representation of a contrastspecification is
< fixed-effect values . . . > < | random-effect values . . . >
fixed-effect
identifies an effect that appears in the MODEL statement. The keyword INTERCEPT can be used as an effect when an intercept is fitted in the model. You do
not need to include all effects that are in the MODEL statement.
random-effect
identifies an effect that appears in the RANDOM statement. The first random
effect must follow a vertical bar (|); however, random effects do not have to be
specified.
values
are constants that are elements of the L matrix associated with the fixed and
random effects. There are two basic methods of specifying the entries of the L
matrix. The traditional representation—also known as the positional syntax—
relies on entering coefficients in the position they assume in the L matrix. For
example, in the following statements the elements of L associated with the b
main effect receive a 1 in the first position and a 1 in the second position:
2122 F Chapter 38: The GLIMMIX Procedure
class a b;
model y = a b a*b;
contrast ’B at A2’ b 1 -1
a*b 0
0
1 -1;
The elements associated with the interaction receive a 1 in the third position and
a 1 in the fourth position. In order to specify coefficients correctly for the interaction term, you need to know how the levels of a and b vary in the interaction,
which is governed by the order of the variables in the CLASS statement. The
nonpositional syntax is designed to make it easier to enter coefficients for interactions and is necessary to enter coefficients for effects constructed with the
experimental EFFECT statement. In square brackets you enter the coefficient
followed by the associated levels of the CLASS variables. If B has two and
A has three levels, the previous CONTRAST statement, by using nonpositional
syntax for the interaction term, becomes
contrast ’B at A2’ b 1 -1 a*b [1, 2 1] [-1, 2 2];
It assigns value 1 to the interaction where A is at level 2 and B is at level 1, and
it assigns 1 to the interaction where both classification variables are at level
2. The comma separating the entry for the L matrix from the level indicators
is optional. Further details about the nonpositional contrast syntax and its use
with constructed effects can be found in the section “Positional and Nonpositional Syntax for Contrast Coefficients” on page 2262. Nonpositional syntax is
available only for fixed-effects coefficients.
The rows of L0 are specified in order and are separated by commas. The rows of the K0 component
of L0 are specified on the left side of the vertical bars (|). These rows test the fixed effects and are,
therefore, checked for estimability. The rows of the M0 component of L0 are specified on the right
side of the vertical bars. They test the random effects, and no estimability checking is necessary.
If PROC GLIMMIX finds the fixed-effects portion of the specified contrast to be nonestimable (see
the SINGULAR= option), then it displays missing values for the test statistics.
If the elements of L are not specified for an effect that contains a specified effect, then the elements
of the unspecified effect are automatically “filled in” over the levels of the higher-order effect.
This feature is designed to preserve estimability for cases where there are complex higher-order
effects. The coefficients for the higher-order effect are determined by equitably distributing the
coefficients of the lower-level effect as in the construction of least squares means. In addition, if
the intercept is specified, it is distributed over all classification effects that are not contained by any
other specified effect. If an effect is not specified and does not contain any specified effects, then all
of its coefficients in L are set to 0. You can override this behavior by specifying coefficients for the
higher-order effect.
If too many values are specified for an effect, the extra ones are ignored; if too few are specified,
the remaining ones are set to 0. If no random effects are specified, the vertical bar can be omitted;
otherwise, it must be present. If a SUBJECT effect is used in the RANDOM statement, then the
coefficients specified for the effects in the RANDOM statement are equitably distributed across the
levels of the SUBJECT effect. You can use the E option to see exactly what L matrix is used.
PROC GLIMMIX handles missing level combinations of classification variables similarly to PROC
CONTRAST Statement F 2123
GLM and PROC MIXED. These procedures delete fixed-effects parameters corresponding to missing levels in order to preserve estimability. However, PROC MIXED and PROC GLIMMIX do
not delete missing level combinations for random-effects parameters, because linear combinations
of the random-effects parameters are always estimable. These conventions can affect the way you
specify your CONTRAST coefficients.
The CONTRAST statement computes the statistic
F D
b̌
b
0
L.L0 CL/
1 L0
b̌
b
r
where r D rank.L0 CL/, and approximates its distribution with an F distribution unless
DDFM=NONE. If you select DDFM=NONE as the degrees-of-freedom method in the MODEL
statement, and if you do not assign degrees of freedom to the contrast with the DF= option, then
PROC GLIMMIX computes the test statistic r F and approximates its distribution with a chisquare distribution. In the expression for F , C is an estimate of VarŒb̌; b
; see the section
“Estimated Precision of Estimates” on page 2219 and the section “Aspects Common to Adaptive
Quadrature and Laplace Approximation” on page 2228 for details about the computation of C in a
generalized linear mixed model.
The numerator degrees of freedom in the F approximation and the degrees of freedom in the chisquare approximation are equal to r. The denominator degrees of freedom are taken from the “Tests
of Fixed Effects” table and correspond to the final effect you list in the CONTRAST statement. You
can change the denominator degrees of freedom by using the DF= option.
You can specify the following options in the CONTRAST statement after a slash (/).
BYCATEGORY
BYCAT
requests that in models for nominal data (generalized logit models) the contrasts not be combined across response categories but reported separately for each category. For example,
assume that the response variable Style is multinomial with three (unordered) categories. The
following GLIMMIX statements fit a generalized logit model relating the preferred style of
instruction to school and educational program effects:
proc glimmix data=school;
class School Program;
model Style(order=data) = School Program / s ddfm=none
dist=multinomial link=glogit;
freq Count;
contrast ’School 1 vs. 2’ school 1 -1;
contrast ’School 1 vs. 2’ school 1 -1 / bycat;
run;
The first contrast compares school effects in all categories. This is a two-degrees-of-freedom
contrast because there are two nonredundant categories. The second CONTRAST statement
produces two single-degree-of-freedom contrasts, one for each nonreference Style category.
The BYCATEGORY option has no effect unless your model is a generalized (mixed) logit
model.
2124 F Chapter 38: The GLIMMIX Procedure
CHISQ
requests that chi-square tests be performed for all contrasts in addition to any F tests. A chisquare statistic equals its corresponding F statistic times the numerator degrees of freedom,
and these same degrees of freedom are used to compute the p-value for the chi-square test.
This p-value will always be less than that for the F test, because it effectively corresponds to
an F test with infinite denominator degrees of freedom.
DF=number
specifies the denominator degrees of freedom for the F test. For the degrees of freedom
methods DDFM=BETWITHIN, DDFM=CONTAIN, and DDFM=RESIDUAL, the default is
the denominator degrees of freedom taken from the “Tests of Fixed Effects” table and corresponds to the final effect you list in the CONTRAST statement. For DDFM=NONE, infinite
denominator degrees of freedom are assumed by default, and for DDFM=SATTERTHWAITE
and DDFM=KENWARDROGER, the denominator degrees of freedom are computed separately for each contrast.
E
requests that the L matrix coefficients for the contrast be displayed.
GROUP coeffs
sets up random-effect contrasts between different groups when a GROUP= variable appears in
the RANDOM statement. By default, CONTRAST statement coefficients on random effects
are distributed equally across groups. If you enter a multiple row contrast, you can also enter
multiple rows for the GROUP coefficients. If the number of GROUP coefficients is less
than the number of contrasts in the CONTRAST statement, the GLIMMIX procedure cycles
through the GROUP coefficients. For example, the following two statements are equivalent:
contrast ’Trt 1 vs 2 @ x=0.4’ trt 1 -1 0 | x 0.4,
trt 1 0 -1 | x 0.4,
trt 1 -1 0 | x 0.5,
trt 1 0 -1 | x 0.5 /
group 1 -1, 1 0 -1, 1 -1, 1 0 -1;
contrast ’Trt 1 vs 2 @ x=0.4’ trt 1 -1 0 | x
trt 1 0 -1 | x
trt 1 -1 0 | x
trt 1 0 -1 | x
group 1 -1, 1 0 -1;
0.4,
0.4,
0.5,
0.5 /
SINGULAR=number
tunes the estimability checking. If v is a vector, define ABS(v) to be the largest absolute value
of the elements of v. If ABS(K0 K0 T) is greater than c*number for any row of K0 in the
contrast, then K0 ˇ is declared nonestimable. Here, T is the Hermite form matrix .X0 X/ X0 X,
and c is ABS(K0 ), except when it equals 0, and then c is 1. The value for number must be
between 0 and 1; the default is 1E 4.
SUBJECT coeffs
sets up random-effect contrasts between different subjects when a SUBJECT= variable appears in the RANDOM statement. By default, CONTRAST statement coefficients on random
effects are distributed equally across subjects. Listing subject coefficients for multiple row
CONTRAST statements follows the same rules as for GROUP coefficients.
COVTEST Statement F 2125
COVTEST Statement
COVTEST < ’label’ > < test-specification > < / options > ;
The COVTEST statement provides a mechanism to obtain statistical inferences for the covariance
parameters. Significance tests are based on the ratio of (residual) likelihoods or pseudo-likelihoods.
Confidence limits and bounds are computed as Wald or likelihood ratio limits. You can specify
multiple COVTEST statements.
The likelihood ratio test is obtained by fitting the model subject to the constraints imposed by the
test-specification. The test statistic is formed as twice the difference of the (possibly restricted) log
(pseudo-) likelihoods of the full and the reduced models. Note that fitting the null model does not
necessarily require fewer computer resources compared to fitting the full model. The optimization
settings for refitting the model are the same as for the full model and can be controlled with the
NLOPTIONS statement.
Common questions in mixed modeling are whether variance components are zero, whether random
effects are independent, and whether rows (columns) can be added or removed from an unstructured
covariance matrix. When the parameters under the null hypothesis fall on the boundary of the
parameter space, the distribution of the likelihood ratio statistic can be a complicated mixture of
distributions. In certain situations it is known to be a relatively straightforward mixture of central
chi-square distributions. When the GLIMMIX procedure recognizes the model and hypothesis as a
case for which the mixture is readily available, the p-value of the likelihood ratio test is determined
accordingly as a linear combination of central chi-square probabilities. The Note column in the
“Likelihood Ratio Tests for Covariance Parameters” table along with the table’s footnotes informs
you about when mixture distributions are used in the calculation of p-values. You can find important
statistical and computational details about likelihood ratio testing of covariance parameters with the
GLIMMIX procedure in the section “Statistical Inference for Covariance Parameters” on page 2232.
In generalized linear mixed models that depend on pseudo-data, the GLIMMIX procedure fits the
null model for a test of covariance parameters to the final pseudo-data of the converged optimization.
Test Specification
The test-specification in the COVTEST statement draws on keywords that represent a particular null
hypothesis, lists or data sets of parameter values, or general contrast specifications. Valid keywords
are as follows:
GLM | INDEP
tests the model against a null model of complete independence. All
G-side covariance parameters are eliminated and the R-side covariance
structure is reduced to a diagonal structure.
DIAGG
tests for a diagonal G matrix by constraining off-diagonal elements in G
to zero. The R-side structure is not modified.
DIAGR | CINDEP
tests for conditional independence by reducing the R-side covariance
structure to diagonal form. The G-side structure is not modified.
2126 F Chapter 38: The GLIMMIX Procedure
HOMOGENEITY
tests homogeneity of covariance parameters across groups by imposing
equality constraints. For example, the following statements fit a oneway model with heterogeneous variances and test whether the model
could be reduced to a one-way analysis with the same variance across
groups:
proc glimmix;
class A;
model y = a;
random _residual_ / group=A;
covtest ’common variance’ homogeneity;
run;
See Example 38.9 for an application with groups and unstructured covariance matrices.
START | INITIAL
compares the final estimates to the starting values of the covariance parameter estimates. This option is useful, for example, if you supply
starting values in the PARMS statement and want to test whether the
optimization produced significantly better values. In GLMMs based on
pseudo-data, the likelihoods that use the starting and the final values are
based on the final pseudo-data.
ZEROG
tests whether the G matrix can be reduced to a zero matrix. This eliminates all G-side random effects from the model.
Only a single keyword is permitted in the COVTEST statement. To test more complicated hypotheses, you can formulate tests with the following specifications.
TESTDATA=data-set
TDATA=data-set
reads in covariance parameter values from a SAS data set. The data set should contain the
numerical variable Estimate or numerical variables named Covpi. The GLIMMIX procedure
associates the values for Covpi with the ith covariance parameter.
For data sets containing the numerical variable Estimate, the GLIMMIX procedure fixes the
ith covariance parameter value at the value of the ith observation in the data set. A missing
value indicates not to fix the particular parameter. PROC GLIMMIX performs one likelihood
ratio test for the TESTDATA= data set.
For data sets containing numerical variables named Covpi, the procedure performs one likelihood ratio test for each observation in the TESTDATA= data set. You do not have to specify
a Covpi variable for every covariance parameter. If the value for the variable is not missing,
PROC GLIMMIX fixes the associated covariance parameter in the null model. Consider the
following statements:
data TestDataSet;
input covp1 covp2 covp3;
datalines;
. 0 .
0 0 .
. 0 0
0 0 0
;
COVTEST Statement F 2127
proc glimmix method=mspl;
class subject x;
model y = x age x*age;
random intercept age / sub=subject type=un;
covtest testdata=TestDataSet;
run;
Because the G matrix is a .2 2/ unstructured matrix, the first observation of the TestDataSet
corresponds to zeroing the covariance between the random intercept and the random slope.
When the reduced model is fit, the variances of the intercept and slope are reestimated. The
second observation reduces the model to one with only a random slope in age. The third
reduces the model to a random intercept model. The last observation eliminates the G matrix
altogether.
Note that the tests associated with the first and last set of covariance parameters in TestDataSet can also be obtained by using keywords:
proc glimmix;
class subject x;
model y = x age x*age;
random intercept age / sub=subject type=un;
covtest DiagG;
covtest GLM;
run;
value-list
supplies a list of values at which to fix the covariance parameters. A missing value in the
list indicates that the covariance parameter is not fixed. If the list is shorter than the number of covariance parameters, missing values are assumed for all parameters not specified.
The COVTEST statements that test the random intercept and random slope in the previous
example are as follows:
proc glimmix;
class subject x;
model y = x age x*age;
random intercept age / sub=subject type=un;
covtest 0 0;
covtest . 0 0;
run;
GENERAL coefficients < ,coefficients > < ,. . . >
CONTRAST coefficients < ,coefficients > < ,. . . >
provides a general facility to test linear combinations of covariance parameters. You can
specify one or more sets of coefficients. The position of a coefficient in the list corresponds
to the position of the parameter in the “Covariance Parameter Estimates” table. The linear
combination of covariance parameters that is implied by each set of coefficients is tested
against zero. If the list of coefficients is shorter than the number of covariance parameters, a
zero coefficient is assumed for the remaining parameters.
For example, in a heterogeneous variance model with four groups, the following statements
test the simultaneous hypothesis H W 12 D 22 ; 32 D 42 :
2128 F Chapter 38: The GLIMMIX Procedure
proc glimmix;
class A;
model y = a;
random _residual_ / group=A;
covtest ’pair-wise homogeneity’
general 1 -1 0 0,
0 0 1 -1;
run;
In a repeated measures study with four observations per subject, the COVTEST statement in
the following example tests whether the four correlation parameters are identical:
proc glimmix;
class subject drug time;
model y = drug time drug*time;
random _residual_ / sub=subject type=unr;
covtest ’Homogeneous correlation’
general 0 0 0 0 1 -1
,
0 0 0 0 1 0 -1
,
0 0 0 0 1 0 0 -1
,
0 0 0 0 1 0 0 0 -1
,
0 0 0 0 1 0 0 0 0 -1;
run;
Notice that the variances (the first four covariance parameters) are allowed to vary. The null
model for this test is thus a heterogeneous compound symmetry model.
The degrees of freedom associated with these general linear hypotheses are determined as the
rank of the matrix LL0 , where L is the k q matrix of coefficients and q is the number of
covariance parameters. Notice that the coefficients in a row do not have to sum to zero. The
following statement tests H W 1 D 32 ; 3 D 0:
covtest general 1 -3, 0 0 1;
Covariance Test Options
You can specify the following options in the COVTEST statement after a slash (/).
CL< (suboptions) >
requests confidence limits or bounds for the covariance parameter estimates. These limits are
displayed as extra columns in the “Covariance Parameter Estimates” table.
The following suboptions determine the computation of confidence bounds and intervals.
See the section “Statistical Inference for Covariance Parameters” on page 2232 for details
about constructing likelihood ratio confidence limits for covariance parameters with PROC
GLIMMIX.
ALPHA=number
determines the confidence level for constructing confidence limits for the covariance
parameters. The value of number must be between 0 and 1, the default is 0.05, and the
confidence level is 1 number.
COVTEST Statement F 2129
LOWERBOUND
LOWER
requests lower confidence bounds.
TYPE=method
determines how the GLIMMIX procedure constructs confidence limits for covariance
parameters. The valid methods are PLR (or PROFILE), ELR (or ESTIMATED), and
WALD.
TYPE=PLR (TYPE=PROFILE) requests confidence bounds by inversion of the profile
(restricted) likelihood ratio (PLR). If is the parameter of interest, L denotes the likelihood (possibly restricted and possibly a pseudo-likelihood), and 2 is the vector of
the remaining (nuisance) parameters, then the profile likelihood is defined as
L.2 je
/ D sup L.e
; 2 /
2
for a given value e
of . If L.b
/ is the overall likelihood evaluated at the estimates b
,
the .1 ˛/ 100% confidence region for satisfies the inequality
n
o
2 L.b
/ L.2 je
/ 21;.1 ˛/
where 21;.1 ˛/ is the cutoff from a chi-square distribution with one degree of freedom and ˛ probability to its right. If a residual scale parameter is profiled from the
estimation, and is expressed in terms of a ratio with during estimation, then profile likelihood confidence limits are constructed for the ratio of the parameter with the
residual variance. A column showing the ratio estimates is added to the “Covariance
Parameter Estimates” table in this case. To obtain profile likelihood ratio limits for the
parameters, rather than their ratios, and for the residual variance, use the NOPROFILE
option in the PROC GLIMMIX statement. Also note that METHOD=LAPLACE or
METHOD=QUAD implies the NOPROFILE option.
The TYPE=ELR (TYPE=ESTIMATED) option constructs bounds from the estimated
likelihood (Pawitan 2001), where nuisance parameters are held fixed at the (restricted)
maximum (pseudo-) likelihood estimates of the model. Estimated likelihood intervals
are computationally less demanding than profile likelihood intervals, but they do not
take into account the variability of the nuisance parameters or the dependence among
the covariance parameters. See the section “Statistical Inference for Covariance Parameters” on page 2232 for a geometric interpretation and comparison of ELR versus
PLR confidence bounds. A .1 ˛/ 100% confidence region based on the estimated
likelihood is defined by the inequality
n
o
2 L.b
/ L.e
;b
2 / 21;.1 ˛/
where L.e
;b
2 / is the likelihood evaluated at e
and the component of b
that corresponds to 2 . Estimated likelihood ratio intervals tend to perform well when the correlations between the parameter of interest and the nuisance parameters is small. Their
coverage probabilities can fall short of the nominal coverage otherwise. You can display the correlation matrix of the covariance parameter estimates with the ASYCORR
option in the PROC GLIMMIX statement.
2130 F Chapter 38: The GLIMMIX Procedure
If you choose TYPE=PLR or TYPE=ELR, the GLIMMIX procedure reports the righttail probability of the associated single-degree-of-freedom likelihood ratio test along
with the confidence bounds. This helps you diagnose whether solutions to the inequality could be found. If the reported probability exceeds ˛, the associated bound does
not meet the inequality. This might occur, for example, when the parameter space
is bounded and the likelihood at the boundary values has not dropped by a sufficient
amount to satisfy the test inequality.
The TYPE=WALD method requests confidence limits based on the Wald-type statistic
Z D b
=ease.b
/, where ease is the estimated asymptotic standard error of the covariance parameter. For parameters that have a lower boundary constraint of zero, a
Satterthwaite approximation is used to construct limits of the form
b
2;1 ˛=2
b
2;˛=2
where D 2Z 2 , and the denominators are quantiles of the 2 distribution with degrees of freedom. Refer to Milliken and Johnson (1992) and Burdick and Graybill
(1992) for similar techniques. For all other parameters, Wald Z-scores and normal
quantiles are used to construct the limits. Such limits are also provided for variance
components if you specify the NOBOUND option in the PROC GLIMMIX statement
or the PARMS statement.
UPPERBOUND
UPPER
requests upper confidence bounds.
If you do not specify any suboptions, the default is to compute two-sided Wald confidence
intervals with confidence level 1 ˛ D 0:95.
CLASSICAL
requests that the p-value of the likelihood ratio test be computed by the classical method. If
b
is the realized value of the test statistic in the likelihood ratio test,
p D Pr 2 b
where is the degrees of freedom of the hypothesis.
DF=value-list
enables you to supply degrees of freedom 1 ; ; k for the computation of p-values from
chi-square mixtures. The mixture weights w1 ; ; wk are supplied with the WGHT= option.
If no weights are specified, an equal weight distribution is assumed. If b
is the realized value
of the test statistic in the likelihood ratio test, PROC GLIMMIX computes the p-value as
(Shapiro 1988)
pD
k
X
wi Pr 2i b
i D1
Note that 20 0 and that mixture weights are scaled to sum to one. If you specify more
weights than degrees of freedom in value-list, the rank of the hypothesis (DF column) is
substituted for the missing degrees of freedom.
COVTEST Statement F 2131
Specifying a single value for value-list without giving mixture weights is equivalent to
computing the p-value as
p D Pr 2 b
For example, the following statements compute the p-value based on a chi-square distribution
with one degree of freedom:
proc glimmix noprofile;
class A sub;
model score = A;
random _residual_ / type=ar(1) subject=sub;
covtest ’ELR low’ 30.62555 0.7133361 / df=1;
run;
The DF column of the COVTEST output will continue to read 2 regardless of the DF= specification, however, because the DF column reflects the rank of the hypothesis and equals the
number of constraints imposed on the full model.
ESTIMATES
EST
displays the estimates of the covariance parameters under the null hypothesis. Specifying the
ESTIMATES option in one COVTEST statement has the same effect as specifying the option
in every COVTEST statement.
MAXITER=number
limits the number of iterations when you are refitting the model under the null hypothesis to
number iterations. If the null model does not converge before the limit is reached, no p-values
are produced.
PARMS
displays the values of the covariance parameters under the null hypothesis. This option is useful if you supply multiple sets of parameter values with the TESTDATA= option. Specifying
the PARMS option in one COVTEST statement has the same effect as specifying the option
in every COVTEST statement.
RESTART
specifies that starting values for the covariance parameters for the null model are obtained by
the same mechanism as starting values for the full models. For example, if you do not specify
a PARMS statement, the RESTART option computes MIVQUE(0) estimates under the null
model (Goodnight 1978b). If you provide starting values with the PARMS statement, the
starting values for the null model are obtained by applying restrictions to the starting values
for the full model.
By default, PROC GLIMMIX obtains starting values by applying null model restrictions
to the converged estimates of the full model. Although this is computationally expedient,
the method does not always lead to good starting values for the null model, depending on
the nature of the model and hypothesis. In particular, when you receive a warning about
parameters not specified under H0 falling on the boundary, the RESTART option can be
useful.
2132 F Chapter 38: The GLIMMIX Procedure
TOLERANCE=r
Values within tolerance r 0 of the boundary of the parameter space are considered on the
boundary when PROC GLIMMIX examines estimates of nuisance parameters under H0 and
determines whether mixture weights and degrees of freedom can be obtained. In certain cases,
when parameters not specified under the null hypothesis are on boundaries, the asymptotic
distribution of the likelihood ratio statistic is not a mixture of chi-squares (see, for example,
case 8 in Self and Liang 1987). The default for r is 1E4 times the machine epsilon; this
product is approximately 1E 12 on most computers.
WALD
produces Wald Z tests for the covariance parameters based on the estimates and asymptotic
standard errors in the “Covariance Parameter Estimates” table.
WGHT=value-list
enables you to supply weights for the computation of p-values from chi-square mixtures. See
the DF= option for details. Mixture weights are scaled to sum to one.
EFFECT Statement (Experimental)
EFFECT effect-specification ;
The experimental EFFECT statement enables you to construct special collections of columns for X
or Z matrices in your model. These collections are referred to as constructed effects to distinguish
them from the usual model effects formed from continuous or classification variables.
For details about the syntax of the EFFECT statement and how columns of constructed effects are
computed, see the section “Constructed Effects and the EFFECT Statement (Experimental)” on
page 377 of Chapter 18, “Shared Concepts and Topics.” For specific details concerning the use
of the EFFECT statement with the GLIMMIX procedure, see the section “Notes on the EFFECT
Statement” on page 2260.
ESTIMATE Statement
ESTIMATE ’label’ contrast-specification < (divisor =n) >
< , ’label’ contrast-specification < (divisor =n) > > < , . . . >
< / options > ;
The ESTIMATE statement provides a mechanism for obtaining custom hypothesis tests. As in the
CONTRAST statement, the basic element of the ESTIMATE statement is the contrast-specification,
which consists of MODEL and G-side random effects and their coefficients. Specifically, a contrastspecification takes the form
< fixed-effect values . . . > < | random-effect values . . . >
ESTIMATE Statement F 2133
Based on the contrast-specifications in your ESTIMATE statement, PROC GLIMMIX constructs
the matrix L0 D ŒK0 M0 , as in the CONTRAST statement, where K is associated with the fixed
effects and M is associated with the G-side random effects. The GLIMMIX procedure supports
nonpositional syntax for the coefficients of fixed effects in the ESTIMATE statement. For details
see the section “Positional and Nonpositional Syntax for Contrast Coefficients” on page 2262.
PROC GLIMMIX then produces for each row l of L0 an approximate t test of the hypothesis
H W l D 0, where D Œˇ 0 0 0 . You can also obtain multiplicity-adjusted p-values and confidence
limits for multirow estimates with the ADJUST= option. The output from multiple ESTIMATE
statements is organized as follows. Results from unadjusted estimates are reported first in a single
table, followed by separate tables for each of the adjusted estimates. Results from all ESTIMATE
statements are combined in the “Estimates” ODS table.
Note that multirow estimates are permitted. Unlike the CONTRAST statement, you need to specify
a ’label’ for every row of the multirow estimate, because PROC GLIMMIX produces one test per
row.
PROC GLIMMIX selects the degrees of freedom to match those displayed in the “Type III Tests of
Fixed Effects” table for the final effect you list in the ESTIMATE statement. You can modify the
degrees of freedom by using the DF= option. If you select DDFM=NONE and do not modify the
degrees of freedom by using the DF= option, PROC GLIMMIX uses infinite degrees of freedom,
essentially computing approximate z tests. If PROC GLIMMIX finds the fixed-effects portion of
the specified estimate to be nonestimable, then it displays “Non-est” for the estimate entry.
ADJDFE=SOURCE
ADJDFE=ROW
specifies how denominator degrees of freedom are determined when p-values and confidence
limits are adjusted for multiple comparisons with the ADJUST= option. When you do not
specify the ADJDFE= option, or when you specify ADJDFE=SOURCE, the denominator degrees of freedom for multiplicity-adjusted results are the denominator degrees of freedom for
the final effect listed in the ESTIMATE statement from the “Type III Tests of Fixed Effects”
table.
The ADJDFE=ROW setting is useful if you want multiplicity adjustments to take into account
that denominator degrees of freedom are not constant across estimates. This can be the case,
for example, when the DDFM=SATTERTHWAITE or DDFM=KENWARDROGER degreesof-freedom method is in effect.
ADJUST=BON
ADJUST=SCHEFFE
ADJUST=SIDAK
ADJUST=SIMULATE< (simoptions) >
ADJUST=T
requests a multiple comparison adjustment for the p-values and confidence limits for the estimates. The adjusted quantities are produced in addition to the unadjusted quantities. Adjusted
confidence limits are produced if the CL or ALPHA= option is in effect. For a description of
the adjustments, see Chapter 39, “The GLM Procedure,” and Chapter 58, “The MULTTEST
Procedure,” of the SAS/STAT User’s Guide and the documentation for the ADJUST= option
in the LSMEANS statement. The ADJUST= option is ignored for generalized logit models.
2134 F Chapter 38: The GLIMMIX Procedure
If the STEPDOWN option is in effect, the p-values are further adjusted in a step-down fashion.
ALPHA=number
requests that a t-type confidence interval be constructed with confidence level 1 number.
The value of number must be between 0 and 1; the default is 0.05. If DDFM=NONE and
you do not specify degrees of freedom with the DF= option, PROC GLIMMIX uses infinite
degrees of freedom, essentially computing a z interval.
BYCATEGORY
BYCAT
requests that in models for nominal data (generalized logit models) estimates be reported
separately for each category. In contrast to the BYCATEGORY option in the CONTRAST
statement, an ESTIMATE statement in a generalized logit model does not distribute coefficients by response category, because ESTIMATE statements always correspond to single
rows of the L matrix.
For example, assume that the response variable Style is multinomial with three (unordered)
categories. The following GLIMMIX statements fit a generalized logit model relating the
preferred style of instruction to school and educational program effects:
proc glimmix data=school;
class School Program;
model Style(order=data) = School Program / s ddfm=none
dist=multinomial link=glogit;
freq Count;
estimate ’School 1 vs. 2’ school 1 -1 / bycat;
estimate ’School 1 vs. 2’ school 1 -1;
run;
The first ESTIMATE statement compares school effects separately for each nonredundant
category. The second ESTIMATE statement compares the school effects for the first nonreference category.
The BYCATEGORY option has no effect unless your model is a generalized (mixed) logit
model.
CL
requests that t-type confidence limits be constructed. If DDFM=NONE and you do not specify degrees of freedom with the DF= option, PROC GLIMMIX uses infinite degrees of freedom, essentially computing a z interval. The confidence level is 0.95 by default. These
intervals are adjusted for multiplicity when you specify the ADJUST= option.
DF=number
specifies the degrees of freedom for the t test and confidence limits. The default is the denominator degrees of freedom taken from the “Type III Tests of Fixed Effects” table and
corresponds to the final effect you list in the ESTIMATE statement.
DIVISOR=value-list
specifies a list of values by which to divide the coefficients so that fractional coefficients can
be entered as integer numerators. If you do not specify value-list, a default value of 1.0 is
assumed. Missing values in the value-list are converted to 1.0.
ESTIMATE Statement F 2135
If the number of elements in value-list exceeds the number of rows of the estimate, the extra
values are ignored. If the number of elements in value-list is less than the number of rows of
the estimate, the last value in value-list is copied forward.
If you specify a row-specific divisor as part of the specification of the estimate row, this value
multiplies the corresponding divisor implied by the value-list. For example, the following
statement divides the coefficients in the first row by 8, and the coefficients in the third and
fourth row by 3:
estimate ’One
’One
’One
’One
vs.
vs.
vs.
vs.
two’
three’
four’
five’
A
A
A
A
2 -2 (divisor=2),
1 0 -1
,
3 0 0 -3
,
1 0 0 0 -1 / divisor=4,.,3;
Coefficients in the second row are not altered.
E
requests that the L matrix coefficients be displayed.
EXP
requests exponentiation of the estimate. When you model data with the logit, cumulative
logit, or generalized logit link functions, and the estimate represents a log odds ratio or log
cumulative odds ratio, the EXP option produces an odds ratio. See “Odds and Odds Ratio
Estimation” on page 2254 for important details about the computation and interpretation of
odds and odds ratio results with the GLIMMIX procedure. If you specify the CL or ALPHA=
option, the (adjusted) confidence bounds are also exponentiated.
GROUP coeffs
sets up random-effect contrasts between different groups when a GROUP= variable appears in
the RANDOM statement. By default, ESTIMATE statement coefficients on random effects
are distributed equally across groups. If you enter a multirow estimate, you can also enter
multiple rows for the GROUP coefficients. If the number of GROUP coefficients is less
than the number of contrasts in the ESTIMATE statement, the GLIMMIX procedure cycles
through the GROUP coefficients. For example, the following two statements are equivalent:
estimate ’Trt
’Trt
’Trt
’Trt
1
1
1
1
vs 2 @ x=0.4’
vs 3 @ x=0.4’
vs 2 @ x=0.5’
vs 3 @ x=0.5’
group 1 -1, 1
trt 1 -1 0 | x 0.4,
trt 1 0 -1 | x 0.4,
trt 1 -1 0 | x 0.5,
trt 1 0 -1 | x 0.5 /
0 -1, 1 -1, 1 0 -1;
estimate ’Trt
’Trt
’Trt
’Trt
1
1
1
1
vs 2 @ x=0.4’
vs 3 @ x=0.4’
vs 2 @ x=0.5’
vs 3 @ x=0.5’
group 1 -1, 1
trt 1 -1 0 | x
trt 1 0 -1 | x
trt 1 -1 0 | x
trt 1 0 -1 | x
0 -1;
0.4,
0.4,
0.5,
0.5 /
ILINK
requests that the estimate and its standard error are also reported on the scale of the mean (the
inverse linked scale). PROC GLIMMIX computes the value on the mean scale by applying
the inverse link to the estimate. The interpretation of this quantity depends on the fixedeffect values and random-effect values specified in your ESTIMATE statement and on the
link function. In a model for binary data with logit link, for example, the following statements
2136 F Chapter 38: The GLIMMIX Procedure
compute
1
1 C expf .˛1
˛2 /g
where ˛1 and ˛2 are the fixed-effects solutions associated with the first two levels of the
classification effect A:
proc glimmix;
class A;
model y = A / dist=binary link=logit;
estimate ’A one vs. two’ A 1 -1 / ilink;
run;
This quantity is not the difference of the probabilities associated with the two levels,
1
2 D
1
1
1 C expf ˇ0
˛1 g
1 C expf ˇ0
˛2 g
The standard error of the inversely linked estimate is based on the delta method. If you also
specify the CL option, the GLIMMIX procedure computes confidence limits for the estimate
on the mean scale. In multinomial models for nominal data, the limits are obtained by the
delta method. In other models they are obtained from the inverse link transformation of the
confidence limits for the estimate. The ILINK option is specific to an ESTIMATE statement.
LOWER
LOWERTAILED
requests that the p-value for the t test be based only on values less than the test statistic. A
two-tailed test is the default. A lower-tailed confidence limit is also produced if you specify
the CL or ALPHA= option.
Note that for ADJUST=SCHEFFE the one-sided adjusted confidence intervals and one-sided
adjusted p-values are the same as the corresponding two-sided statistics, because this adjustment is based on only the right tail of the F distribution.
SINGULAR=number
tunes the estimability checking as documented for the CONTRAST statement..
STEPDOWN< (step-down-options) >
requests that multiplicity adjustments for the p-values of estimates be further adjusted in
a step-down fashion. Step-down methods increase the power of multiple testing procedures by taking advantage of the fact that a p-value will never be declared significant unless all smaller p-values are also declared significant. Note that the STEPDOWN adjustment
combined with ADJUST=BON corresponds to the methods of Holm (1979) and Shaffer’s
“Method 2” (1986); this is the default. Using step-down-adjusted p-values combined with
ADJUST=SIMULATE corresponds to the method of Westfall (1997).
If the degrees-of-freedom method is DDFM=KENWARDROGER or DDFM=SATTERTHWAITE,
then step-down-adjusted p-values are produced only if the ADJDFE=ROW option is in effect.
Also, the STEPDOWN option affects only p-values, not confidence limits.
For
ADJUST=SIMULATE, the generalized least squares hybrid approach of Westfall (1997)
is employed to increase Monte Carlo accuracy.
ESTIMATE Statement F 2137
You can specify the following step-down-options in parentheses after the STEPDOWN option.
MAXTIME=n
specifies the time (in seconds) to spend computing the maximal logically consistent
sequential subsets of equality hypotheses for TYPE=LOGICAL. The default is MAXTIME=60. If the MAXTIME value is exceeded, the adjusted tests are not computed.
When this occurs, you can try increasing the MAXTIME value. However, note that
there are common multiple comparisons problems for which this computation requires
a huge amount of time—for example, all pairwise comparisons between more than 10
groups. In such cases, try to use TYPE=FREE (the default) or TYPE=LOGICAL(n)
for small n.
ORDER=PVALUE
ORDER=ROWS
specifies the order in which the step-down tests are performed. ORDER=PVALUE is
the default, with estimates being declared significant only if all estimates with smaller
(unadjusted) p-values are significant. If you specify ORDER=ROWS, then significances are evaluated in the order in which they are specified in the syntax.
REPORT
specifies that a report on the step-down adjustment be displayed, including a listing of
the sequential subsets (Westfall 1997) and, for ADJUST=SIMULATE, the step-down
simulation results.
TYPE=LOGICAL< (n) >
TYPE=FREE
If you specify TYPE=LOGICAL, the step-down adjustments are computed by using
maximal logically consistent sequential subsets of equality hypotheses (Shaffer 1986,
Westfall 1997). Alternatively, for TYPE=FREE, sequential subsets are computed ignoring logical constraints. The TYPE=FREE results are more conservative than those
for TYPE=LOGICAL, but they can be much more efficient to produce for many estimates. For example, it is not feasible to take logical constraints between all pairwise
comparisons of more than about 10 groups. For this reason, TYPE=FREE is the default.
However, you can reduce the computational complexity of taking logical constraints
into account by limiting the depth of the search tree used to compute them, specifying
the optional depth parameter as a number n in parentheses after TYPE=LOGICAL.
As with TYPE=FREE, results for TYPE=LOGICAL(n) are conservative relative to the
true TYPE=LOGICAL results, but even for TYPE=LOGICAL(0) they can be appreciably less conservative than TYPE=FREE and they are computationally feasible for
much larger numbers of estimates. If you do not specify n or if n D 1, the full search
tree is used.
SUBJECT coeffs
sets up random-effect contrasts between different subjects when a SUBJECT= variable appears in the RANDOM statement. By default, ESTIMATE statement coefficients on random
effects are distributed equally across subjects. Listing subject coefficients for an ESTIMATE
statement with multiple rows follows the same rules as for GROUP coefficients.
2138 F Chapter 38: The GLIMMIX Procedure
UPPER
UPPERTAILED
requests that the p-value for the t test be based only on values greater than the test statistic. A
two-tailed test is the default. An upper-tailed confidence limit is also produced if you specify
the CL or ALPHA= option.
Note that for ADJUST=SCHEFFE the one-sided adjusted confidence intervals and one-sided
adjusted p-values are the same as the corresponding two-sided statistics, because this adjustment is based on only the right tail of the F distribution.
FREQ Statement
FREQ variable ;
The variable in the FREQ statement identifies a numeric variable in the data set or one computed
through PROC GLIMMIX programming statements that contains the frequency of occurrence for
each observation. PROC GLIMMIX treats each observation as if it appears f times, where f is
the value of the FREQ variable for the observation. If it is not an integer, the frequency value is
truncated to an integer. If the frequency value is less than 1 or missing, the observation is not used
in the analysis. When the FREQ statement is not specified, each observation is assigned a frequency
of 1.
The analysis produced by using a FREQ statement reflects the expanded number of observations.
For an example of a FREQ statement in a model with random effects, see Example 38.11 in this
chapter.
ID Statement
ID variables ;
The ID statement specifies which quantities to include in the OUT= data set from the OUTPUT
statement in addition to any statistics requested in the OUTPUT statement. If no ID statement is
given, the GLIMMIX procedure includes all variables from the input data set in the OUT= data set.
Otherwise, only the variables listed in the ID statement are included. Automatic variables such as
_LINP_, _MU_, _VARIANCE_, etc. are not transferred to the OUT= data set unless they are listed
in the ID statement.
The ID statement can be used to transfer computed quantities that depend on the model to an output
data set. In the following example, two sets of Hessian weights are computed in a gamma regression
with a noncanonical link. The covariance matrix for the fixed effects can be constructed as the
inverse of X0 WX. W is a diagonal matrix of the wei or woi , depending on whether the expected or
observed Hessian matrix is desired, respectively.
LSMEANS Statement F 2139
proc glimmix;
class group age;
model cost = group age / s error=gamma link=pow(0.5);
output out=gmxout pred=pred;
id _variance_ wei woi;
vpmu = 2*_mu_;
if (_mu_ > 1.0e-8) then do;
gpmu = 0.5 * (_mu_**(-0.5));
gppmu = -0.25 * (_mu_**(-1.5));
wei
= 1/(_phi_*_variance_*gpmu*gpmu);
woi
= wei + (cost-_mu_) *
(_variance_*gppmu + vpmu*gpmu) /
(_variance_*_variance_*gpmu*gpmu*gpmu*_phi_);
end;
run;
The variables _VARIANCE_ and _MU_ and other symbols are predefined by PROC GLIMMIX and
can be used in programming statements. For rules and restrictions, see the section “Programming
Statements” on page 2205.
LSMEANS Statement
LSMEANS fixed-effects < / options > ;
The LSMEANS statement computes least squares means (LS-means) of fixed effects. As in the
GLM and the MIXED procedures, LS-means are predicted population margins—that is, they estimate the marginal means over a balanced population. In a sense, LS-means are to unbalanced
designs as class and subclass arithmetic means are to balanced designs. The L matrix constructed
to compute them is the same as the L matrix formed in PROC GLM; however, the standard errors
are adjusted for the covariance parameters in the model. Least squares means computations are not
supported for multinomial models.
Each LS-mean is computed as Lb̌, where L is the coefficient matrix associated with the least squares
mean and b̌ is the estimate of the fixed-effects parameter vector. The approximate standard error
c b̌L0 . The approximate variance matrix of
for the LS-mean is computed as the square root of LVarŒ
the fixed-effects estimates depends on the estimation method.
LS-means are constructed on the linked scale—that is, the scale on which the model effects are
additive. For example, in a binomial model with logit link, the least squares means are predicted
population margins of the logits.
LS-means can be computed for any effect in the MODEL statement that involves only CLASS
variables. You can specify multiple effects in one LSMEANS statement or in multiple LSMEANS
statements, and all LSMEANS statements must appear after the MODEL statement. As in the
ESTIMATE statement, the L matrix is tested for estimability, and if this test fails, PROC GLIMMIX
displays “Non-est” for the LS-means entries.
Assuming the LS-mean is estimable, PROC GLIMMIX constructs an approximate t test to test the
2140 F Chapter 38: The GLIMMIX Procedure
null hypothesis that the associated population quantity equals zero. By default, the denominator
degrees of freedom for this test are the same as those displayed for the effect in the “Type III Tests
of Fixed Effects” table. If the DDFM=SATTERTHWAITE or DDFM=KENWARDROGER option
is specified in the MODEL statement, PROC GLIMMIX determines degrees of freedom separately
for each test, unless the DDF= option overrides it for a particular effect. See the DDFM= option for
more information.
Table 38.5 summarizes important options in the LSMEANS statement. All LSMEANS options are
subsequently discussed in alphabetical order.
Table 38.5
Summary of Important LSMEANS Statement Options
Option
Description
Construction and Computation of LS-Means
AT
modifies covariate value in computing LS-means
BYLEVEL
computes separate margins
DIFF
requests differences of LS-means
OM
specifies weighting scheme for LS-mean computation as determined by the input data set
SINGULAR=
tunes estimability checking
SLICE=
partitions F tests (simple effects)
SLICEDIFF=
requests simple effects differences
SLICEDIFFTYPE
determines the type of simple difference
Degrees of Freedom and P-values
ADJDFE=
determines whether to compute row-wise denominator degrees of freedom with DDFM=SATTERTHWAITE or
DDFM=KENWARDROGER
ADJUST=
determines the method for multiple comparison adjustment of LSmean differences
ALPHA=˛
determines the confidence level (1 ˛)
DF=
assigns specific value to degrees of freedom for tests and confidence limits
STEPDOWN
adjusts multiple comparison p-values further in a step-down
fashion
Statistical Output
CL
CORR
COV
E
ILINK
LINES
ODDS
ODDSRATIO
PLOTS=
constructs confidence limits for means and or mean differences
displays correlation matrix of LS-means
displays covariance matrix of LS-means
prints the L matrix
computes and displays estimates and standard errors of LS-means
(not differences) on the inverse linked scale
produces “Lines” display for pairwise LS-mean differences
reports odds of levels of fixed effects if permissible by the link
function
reports (simple) differences of least squares means in terms of odds
ratios if permissible by the link function
requests ODS statistical graphics of means and mean comparisons
LSMEANS Statement F 2141
You can specify the following options in the LSMEANS statement after a slash (/).
ADJDFE=ROW
ADJDFE=SOURCE
specifies how denominator degrees of freedom are determined when p-values and confidence
limits are adjusted for multiple comparisons with the ADJUST= option. When you do not
specify the ADJDFE= option, or when you specify ADJDFE=SOURCE, the denominator
degrees of freedom for multiplicity-adjusted results are the denominator degrees of freedom
for the LS-mean effect in the “Type III Tests of Fixed Effects” table. When you specify ADJDFE=ROW, the denominator degrees of freedom for multiplicity-adjusted results correspond
to the degrees of freedom displayed in the DF column of the “Differences of Least Squares
Means” table.
The ADJDFE=ROW setting is particularly useful if you want multiplicity adjustments to
take into account that denominator degrees of freedom are not constant across LS-mean
differences. This can be the case, for example, when the DDFM=SATTERTHWAITE or
DDFM=KENWARDROGER degrees-of-freedom method is in effect.
In one-way models with heterogeneous variance, combining certain ADJUST= options with
the ADJDFE=ROW option corresponds to particular methods of performing multiplicity adjustments in the presence of heteroscedasticity. For example, the following statements fit a
heteroscedastic one-way model and perform Dunnett’s T3 method (Dunnett 1980), which is
based on the studentized maximum modulus (ADJUST=SMM):
proc glimmix;
class A;
model y = A / ddfm=satterth;
random _residual_ / group=A;
lsmeans A / adjust=smm adjdfe=row;
run;
If you combine the ADJDFE=ROW option with ADJUST=SIDAK, the multiplicity adjustment corresponds to the T2 method of Tamhane (1979), while ADJUST=TUKEY
corresponds to the method of Games-Howell (Games and Howell 1976). Note that
ADJUST=TUKEY gives the exact results for the case of fractional degrees of freedom in
the one-way model, but it does not take into account that the degrees of freedom are subject to variability. A more conservative method, such as ADJUST=SMM, might protect the
overall error rate better.
Unless the ADJUST= option is specified in the LSMEANS statement, the ADJDFE= option
has no effect.
2142 F Chapter 38: The GLIMMIX Procedure
ADJUST=BON
ADJUST=DUNNETT
ADJUST=NELSON
ADJUST=SCHEFFE
ADJUST=SIDAK
ADJUST=SIMULATE< (simoptions) >
ADJUST=SMM | GT2
ADJUST=TUKEY
requests a multiple comparison adjustment for the p-values and confidence limits for the
differences of LS-means. The adjusted quantities are produced in addition to the unadjusted
quantities. By default, PROC GLIMMIX performs all pairwise differences. If you specify
ADJUST=DUNNETT, the procedure analyzes all differences with a control level. If you
specify ADJUST=NELSON, ANOM differences are taken. The ADJUST= option implies
the DIFF option, unless the SLICEDIFF= option is specified.
The BON (Bonferroni) and SIDAK adjustments involve correction factors described in
Chapter 39, “The GLM Procedure,” and Chapter 58, “The MULTTEST Procedure,” of the
SAS/STAT User’s Guide; also see Westfall and Young (1993) and Westfall et al. (1999). When
you specify ADJUST=TUKEY and your data are unbalanced, PROC GLIMMIX uses the approximation described in Kramer (1956) and identifies the adjustment as “Tukey-Kramer”
in the results. Similarly, when you specify ADJUST=DUNNETT or ADJUST=NELSON
and the LS-means are correlated, the GLIMMIX procedure uses the factor-analytic covariance approximation described in Hsu (1992) and identifies the adjustment in the results as
“Dunnett-Hsu” or “Nelson-Hsu,” respectively. The approximation derives an approximate
“effective sample sizes” for which exact critical values are computed. Note that computing
the exact adjusted p-values and critical values for unbalanced designs can be computationally
intensive, in particular for ADJUST=NELSON. A simulation-based approach, as specified by
the ADJUST=SIM option, while nondeterministic, can provide inferences that are sufficiently
accurate in much less time. The preceding references also describe the SCHEFFE and SMM
adjustments.
Nelson’s adjustment applies only to the analysis of means (Ott 1967; Nelson 1982, 1991,
1993), where LS-means are compared against an average LS-mean. It does not apply to all
pairwise differences of least squares means, or to slice differences that you specify with the
SLICEDIFF= option. See the DIFF=ANOM option for more details regarding the analysis of
means with the GLIMMIX procedure.
The SIMULATE adjustment computes adjusted p-values and confidence limits from the simulated distribution of the maximum or maximum absolute value of a multivariate t random
vector. All covariance parameters, except the residual scale parameter, are fixed at their estimated values throughout the simulation, potentially resulting in some underdispersion. The
simulation estimates q, the true .1 ˛/th quantile, where 1 ˛ is the confidence coefficient. The default ˛ is 0.05, and you can change this value with the ALPHA= option in the
LSMEANS statement.
The number of samples is set so that the tail area for the simulated q is within of 1
100.1 /% confidence. In equation form,
Pr.jF .b
q/
.1
˛/j / D 1
˛ with
LSMEANS Statement F 2143
where qO is the simulated q and F is the true distribution function of the maximum; see Edwards and Berry (1987) for details. By default, = 0.005 and = 0.01, placing the tail area of
qO within 0.005 of 0.95 with 99% confidence. The ACC= and EPS= simoptions reset and ,
respectively, the NSAMP= simoption sets the sample size directly, and the SEED= simoption
specifies an integer used to start the pseudo-random number generator for the simulation. If
you do not specify a seed, or if you specify a value less than or equal to zero, the seed is
generated from reading the time of day from the computer clock. For additional descriptions
of these and other simulation options, see the section “LSMEANS Statement” on page 2456
in Chapter 39, “The GLM Procedure.”
If the STEPDOWN option is in effect, the p-values are further adjusted in a step-down fashion.
For certain options and data, this adjustment is exact under an iid N.0; 2 / model for the
dependent variable, in particular for the following:
for ADJUST=DUNNETT when the means are uncorrelated
for ADJUST=TUKEY with STEPDOWN(TYPE=LOGICAL) when the means are balanced and uncorrelated.
The first case is a consequence of the nature of the successive step-down hypotheses for
comparisons with a control; the second employs an extension of the maximum studentized range distribution appropriate for partition hypotheses (Royen 1989). Finally, for
STEPDOWN(TYPE=FREE), ADJUST=TUKEY employs the Royen (1989) extension in
such a way that the resulting p-values are conservative.
ALPHA=number
requests that a t-type confidence interval be constructed for each of the LS-means with confidence level 1 number. The value of number must be between 0 and 1; the default is
0.05.
AT variable=value
AT (variable-list)=(value-list)
AT MEANS
enables you to modify the values of the covariates used in computing LS-means. By default,
all covariate effects are set equal to their mean values for computation of standard LS-means.
The AT option enables you to assign arbitrary values to the covariates. Additional columns in
the output table indicate the values of the covariates.
If there is an effect containing two or more covariates, the AT option sets the effect equal
to the product of the individual means rather than the mean of the product (as with standard
LS-means calculations). The AT MEANS option sets covariates equal to their mean values
(as with standard LS-means) and incorporates this adjustment to crossproducts of covariates.
As an example, consider the following invocation of PROC GLIMMIX:
proc glimmix;
class A;
model Y = A
lsmeans A;
lsmeans A /
lsmeans A /
lsmeans A /
run;
x1 x2 x1*x2;
at means;
at x1=1.2;
at (x1 x2)=(1.2 0.3);
2144 F Chapter 38: The GLIMMIX Procedure
For the first two LSMEANS statements, the LS-means coefficient for x1 is x 1 (the mean of x1)
and for x2 is x 2 (the mean of x2). However, for the first LSMEANS statement, the coefficient
for x1*x2 is x1 x2 , but for the second LSMEANS statement, the coefficient is x 1 x 2 . The
third LSMEANS statement sets the coefficient for x1 equal to 1:2 and leaves it at x 2 for x2,
and the final LSMEANS statement sets these values to 1:2 and 0:3, respectively.
Even if you specify a WEIGHT variable, the unweighted covariate means are used for the
covariate coefficients if there is no AT specification. If you specify the AT option, WEIGHT
or FREQ variables are taken into account as follows. The weighted covariate means are
then used for the covariate coefficients for which no explicit AT values are given, or if you
specify AT MEANS. Observations that do not contribute to the analysis because of a missing
dependent variable are included in computing the covariate means. You should use the E
option in conjunction with the AT option to check that the modified LS-means coefficients
are the ones you want.
The AT option is disabled if you specify the BYLEVEL option.
BYLEVEL
requests that separate margins be computed for each level of the LSMEANS effect.
The standard LS-means have equal coefficients across classification effects. The BYLEVEL
option changes these coefficients to be proportional to the observed margins. This adjustment
is reasonable when you want your inferences to apply to a population that is not necessarily
balanced but has the margins observed in the input data set. In this case, the resulting LSmeans are actually equal to raw means for fixed-effects models and certain balanced randomeffects models, but their estimated standard errors account for the covariance structure that
you have specified. If a WEIGHT statement is specified, PROC GLIMMIX uses weighted
margins to construct the LS-means coefficients.
If the AT option is specified, the BYLEVEL option disables it.
CL
requests that t-type confidence limits be constructed for each of the LS-means. If
DDFM=NONE, then PROC GLIMMIX uses infinite degrees of freedom for this test, essentially computing a z interval. The confidence level is 0.95 by default; this can be changed
with the ALPHA= option. If you specify an ADJUST= option, then the confidence limits
are adjusted for multiplicity, but if you also specify STEPDOWN, then only p-values are
step-down adjusted, not the confidence limits.
CORR
displays the estimated correlation matrix of the least squares means as part of the “Least
Squares Means” table.
COV
displays the estimated covariance matrix of the least squares means as part of the “Least
Squares Means” table.
DF=number
specifies the degrees of freedom for the t test and confidence limits. The default is the denominator degrees of freedom taken from the “Type III Tests of Fixed Effects” table corresponding
to the LS-means effect.
LSMEANS Statement F 2145
DIFF< =difftype >
PDIFF< =difftype >
requests that differences of the LS-means be displayed. The optional difftype specifies which
differences to produce, with possible values ALL, ANOM, CONTROL, CONTROLL, and
CONTROLU. The ALL value requests all pairwise differences, and it is the default. The
CONTROL difftype requests differences with a control, which, by default, is the first level of
each of the specified LSMEANS effects.
The ANOM value requests differences between each LS-mean and the average LS-mean, as
in the analysis of means (Ott 1967). The average is computed as a weighted mean of the
LS-means, the weights being inversely proportional to the diagonal entries of the
L X0 X L0
matrix. If LS-means are nonestimable, this design-based weighted mean is replaced with
an equally weighted mean. Note that the ANOM procedure in SAS/QC software implements both tables and graphics for the analysis of means with a variety of response types.
For one-way designs and normal data with identity link, the DIFF=ANOM computations are
equivalent to the results of PROC ANOM. If the LS-means being compared are uncorrelated,
exact adjusted p-values and critical values for confidence limits can be computed in the analysis of means; see Nelson (1982, 1991, 1993) and Guirguis and Tobias (2004) as well as the
documentation for the ADJUST=NELSON option.
To specify which levels of the effects are the controls, list the quoted formatted values in
parentheses after the CONTROL keyword. For example, if the effects A, B, and C are classification variables, each having two levels, 1 and 2, the following LSMEANS statement
specifies the (1,2) level of A*B and the (2,1) level of B*C as controls:
lsmeans A*B B*C / diff=control(’1’ ’2’ ’2’ ’1’);
For multiple effects, the results depend upon the order of the list, and so you should check the
output to make sure that the controls are correct.
Two-tailed tests and confidence limits are associated with the CONTROL difftype. For onetailed results, use either the CONTROLL or CONTROLU difftype. The CONTROLL difftype
tests whether the noncontrol levels are significantly smaller than the control; the upper confidence limits for the control minus the noncontrol levels are considered to be infinity and
are displayed as missing. Conversely, the CONTROLU difftype tests whether the noncontrol
levels are significantly larger than the control; the upper confidence limits for the noncontrol
levels minus the control are considered to be infinity and are displayed as missing.
If you want to perform multiple comparison adjustments on the differences of LS-means, you
must specify the ADJUST= option.
The differences of the LS-means are displayed in a table titled “Differences of Least Squares
Means.”
E
requests that the L matrix coefficients for the LSMEANS effects be displayed.
ILINK
requests that estimates and their standard errors in the “Least Squares Means” table also be
2146 F Chapter 38: The GLIMMIX Procedure
reported on the scale of the mean (the inverse linked scale). This enables you to obtain
estimates of predicted probabilities and their standard errors in logistic models, for example.
The option is specific to an LSMEANS statement. If you also specify the CL option, the
GLIMMIX procedure computes confidence intervals for the predicted means by applying the
inverse link transform to the confidence limits on the linked (linear) scale. Standard errors on
the inverse linked scale are computed by the delta method.
LINES
presents results of comparisons between all pairs of least squares means by listing the means
in descending order and indicating nonsignificant subsets by line segments beside the corresponding LS-means. When all differences have the same variance, these comparison lines
are guaranteed to accurately reflect the inferences based on the corresponding tests, made
by comparing the respective p-values to the value of the ALPHA= option (0.05 by default).
However, equal variances might not be the case for differences between LS-means. If the
variances are not all the same, then the comparison lines might be conservative, in the sense
that if you base your inferences on the lines alone, you will detect fewer significant differences than the tests indicate. If there are any such differences, PROC GLIMMIX lists the
pairs of means that are inferred to be significantly different by the tests but not by the comparison lines. Note, however, that in many cases, even though the variances are unequal, they
are similar enough that the comparison lines accurately reflect the test inferences.
ODDS
requests that in models with logit, cumulative logit, and generalized logit link function the
odds of the levels of the fixed effects are reported. If you specify the CL or ALPHA= option,
confidence intervals for the odds are also computed. See the section “Odds and Odds Ratio
Estimation” on page 2254 for further details about computation and interpretation of odds
and odds ratios with the GLIMMIX procedure.
ODDSRATIO
OR
requests that LS-mean differences (DIFF, ADJUST= options) and simple effect comparisons
(SLICEDIFF option) are also reported in terms of odds ratios. The ODDSRATIO option is
ignored unless you use either the logit, cumulative logit, or generalized logit link function.
If you specify the CL or ALPHA= option, confidence intervals for the odds ratios are also
computed. These intervals are adjusted for multiplicity when you specify the ADJUST=
option. See the section “Odds and Odds Ratio Estimation” on page 2254 for further details
about computation and interpretation of odds and odds ratios with the GLIMMIX procedure.
OBSMARGINS
OM
specifies a potentially different weighting scheme for the computation of LS-means coefficients. The standard LS-means have equal coefficients across classification effects; however,
the OM option changes these coefficients to be proportional to those found in the input data
set. This adjustment is reasonable when you want your inferences to apply to a population
that is not necessarily balanced but has the margins observed in your data.
In computing the observed margins, PROC GLIMMIX uses all observations for which there
are no missing or invalid independent variables, including those for which there are missing
dependent variables. Also, if you use a WEIGHT statement, PROC GLIMMIX computes
LSMEANS Statement F 2147
weighted margins to construct the LS-means coefficients. If your data are balanced, the LSmeans are unchanged by the OM option.
The BYLEVEL option modifies the observed-margins LS-means. Instead of computing the
margins across all of the input data set, PROC GLIMMIX computes separate margins for each
level of the LSMEANS effect in question. In this case the resulting LS-means are actually
equal to raw means for fixed-effects models and certain balanced random-effects models, but
their estimated standard errors account for the covariance structure that you have specified.
You can use the E option in conjunction with either the OM or BYLEVEL option to check
that the modified LS-means coefficients are the ones you want. It is possible that the modified
LS-means are not estimable when the standard ones are estimable, or vice versa.
PDIFF
is the same as the DIFF option. See the description of the DIFF option on page 2144.
PLOT | PLOTS< =plot-request< (options) > >
PLOT | PLOTS< =(plot-request< (options) > < . . . plot-request< (options) > >) >
requests that least squares means related graphics are produced via ODS Graphics, provided
that the ODS GRAPHICS statement has been specified and the plot request does not conflict
with other options in the LSMEANS statement. For general information about ODS Graphics, see Chapter 21, “Statistical Graphics Using ODS.” For examples of the basic statistical
graphics for least squares means and aspects of their computation and interpretation, see the
section “Graphics for LS-Mean Comparisons” on page 2286 in this chapter.
The options for a specific plot request (and their suboptions) of the LSMEANS statement
include those for the PLOTS= option in the PROC GLIMMIX statement. You can specify
classification effects in the MEANPLOT request of the LSMEANS statement to control the
display of interaction means with the PLOTBY= and SLICEBY= suboptions; these are not
available in the PLOTS= option in the PROC GLIMMIX statement. Options specified in
the LSMEANS statement override those in the PLOTS= option in the PROC GLIMMIX
statement.
The available options and suboptions are as follows.
ALL
requests that the default plots corresponding to this LSMEANS statement be produced.
The default plot depends on the options in the statement.
ANOMPLOT
ANOM
requests an analysis of means display in which least squares means are compared to
an average least squares mean. Least squares mean ANOM plots are produced only
for those model effects listed in LSMEANS statements that have options that do not
contradict with the display. For example, the following statements produce analysis of
mean plots for effects A and C:
lsmeans A / diff=anom plot=anom;
lsmeans B / diff
plot=anom;
lsmeans C /
plot=anom;
2148 F Chapter 38: The GLIMMIX Procedure
The DIFF option in the second LSMEANS statement implies all pairwise differences.
CONTROLPLOT
CONTROL
requests a display in which least squares means are visually compared against a reference level. These plots are produced only for statements with options that are compatible with control differences. For example, the following statements produce control
plots for effects A and C:
lsmeans A / diff=control(’1’) plot=control;
lsmeans B / diff
plot=control;
lsmeans C
plot=control;
The DIFF option in the second LSMEANS statement implies all pairwise differences.
DIFFPLOT< (diffplot-options) >
DIFFOGRAM< (diffplot-options) >
DIFF< (diffplot-options) >
requests a display of all pairwise least squares mean differences and their significance.
The display is also known as a “mean-mean scatter plot” when it is based on arithmetic
means (Hsu 1996; Hsu and Peruggia 1994). For each comparison a line segment,
centered at the LS-means in the pair, is drawn. The length of the segment corresponds
to the projected width of a confidence interval for the least squares mean difference.
Segments that fail to cross the 45-degree reference line correspond to significant least
squares mean differences.
LS-mean difference plots are produced only for statements with options that are compatible with the display. For example, the following statements request differences
against a control level for the A effect, all pairwise differences for the B effect, and the
least squares means for the C effect:
lsmeans A / diff=control(’1’) plot=diff;
lsmeans B / diff
plot=diff;
lsmeans C
plot=diff;
The DIFF= type in the first statement is incompatible with a display of all pairwise
differences.
You can specify the following diffplot-options. The ABS and NOABS options determine the positioning of the line segments in the plot. When the ABS option is in effect,
and this is the default, all line segments are shown on the same side of the reference line.
The NOABS option separates comparisons according to the sign of the difference. The
CENTER option marks the center point for each comparison. This point corresponds
to the intersection of two least squares means. The NOLINES option suppresses the
display of the line segments that represent the confidence bounds for the differences
of the least squares means. The NOLINES option implies the CENTER option. The
default is to draw line segments in the upper portion of the plot area without marking
the center point.
MEANPLOT< (meanplot-options) >
requests displays of the least squares means.
LSMEANS Statement F 2149
The following meanplot-options control the display of the least squares means.
ASCENDING
displays the least squares means in ascending order. This option has no effect if
means are sliced or displayed in separate plots.
CL
displays upper and lower confidence limits for the least squares means. By
default, 95% limits are drawn. You can change the confidence level with the
ALPHA= option. Confidence limits are drawn by default if the CL option is
specified in the LSMEANS statement.
CLBAND
displays confidence limits as bands. This option implies the JOIN option.
DESCENDING
displays the least squares means in descending order. This option has no effect if
means are sliced or displayed in separate plots.
ILINK
requests that means (and confidence limits) are displayed on the inverse linked
scale.
JOIN
CONNECT
connects the least squares means with lines. This option is implied by the
CLBAND option. If the effect contains nested variables, and a SLICEBY= effect contains classification variables that appear as crossed effects, this option is
ignored.
SLICEBY=fixed-effect
specifies an effect by which to group the means in a single plot. For example,
the following statement requests a plot in which the levels of A are placed on the
horizontal axis and the means that belong to the same level of B are joined by
lines:
lsmeans A*B / plot=meanplot(sliceby=b join);
Unless the LS-mean effect contains at least two classification variables, the
SLICEBY= option has no effect. The SLICEBY= effect does not have to be
an effect in your MODEL statement, but it must consist entirely of classification
variables.
PLOTBY=fixed-effect
specifies an effect by which to break interaction plots into separate displays. For
example, the following statement requests for each level of C one plot of the A*B
cell means that are associated with that level of C:
lsmeans A*B*C / plot=meanplot(sliceby=b plotby=c clband);
2150 F Chapter 38: The GLIMMIX Procedure
In each plot, levels of A are displayed on the horizontal axis, and confidence
bands are drawn around the means that share the same level of B.
The PLOTBY= option has no effect unless the LS-mean effect contains at least
three classification variables. The PLOTBY= effect does not have to be an effect
in the MODEL statement, but it must consist entirely of classification variables.
NONE
requests that no plots be produced.
When LS-mean calculations are adjusted for multiplicity by using the ADJUST= option, the
plots are adjusted accordingly.
SINGULAR=number
tunes the estimability checking as documented for the CONTRAST statement.
SLICE=fixed-effect
SLICE=(fixed-effects)
specifies effects by which to partition interaction LSMEANS effects. This can produce what
are known as tests of simple effects (Winer 1971). For example, suppose that A*B is significant, and you want to test the effect of A for each level of B. The appropriate LSMEANS
statement is
lsmeans A*B / slice=B;
This statement tests for the simple main effects of A for B, which are calculated by extracting
the appropriate rows from the coefficient matrix for the A*B LS-means and by using them to
form an F test.
The SLICE option produces F tests that test the simultaneous equality of cell means at a
fixed level of the slice effect (Schabenberger, Gregoire, and Kong 2000). You can request
differences of the least squares means while holding one or more factors at a fixed level with
the SLICEDIFF= option.
The SLICE option produces a table titled “Tests of Effect Slices.”
SLICEDIFF=fixed-effect
SLICEDIFF=(fixed-effects)
SIMPLEDIFF=fixed-effect
SIMPLEDIFF=(fixed-effects)
requests that differences of simple effects be constructed and tested against zero. Whereas
the SLICE option extracts multiple rows of the coefficient matrix and forms an F test, the
SLICEDIFF option tests pairwise differences of these rows. This enables you to perform
multiple comparisons among the levels of one factor at a fixed level of the other factor. For
example, assume that, in a balanced design, factors A and B have a D 4 and b D 3 levels,
respectively. Consider the following statements:
proc glimmix;
class a b;
model y = a b a*b;
lsmeans a*b / slice=a;
lsmeans a*b / slicediff=a;
run;
LSMEANS Statement F 2151
The first LSMEANS statement produces four F tests, one per level of A. The first of these
tests is constructed by extracting the three rows corresponding to the first level of A from the
.1/ .2/
.3/
coefficient matrix for the A*B interaction. Call this matrix La1 and its rows la1 , la1 , and la1 .
The SLICE tests the two-degrees-of-freedom hypothesis
8 < l.1/ l.2/ ˇ D 0
a1
a1
H W .1/ .3/ : l
la1 ˇ D 0
a1
In a balanced design, where ij denotes the mean response if A is at level i and B is at level
j , this hypothesis is equivalent to H W 11 D 12 D 13 . The SLICEDIFF option considers
the three rows of La1 in turn and performs tests of the difference between pairs of rows. How
these differences are constructed depends on the SLICEDIFFTYPE= option. By default, all
pairwise differences within the subset of L are considered; in the example this corresponds to
tests of the form
.1/
.2/
H W la1 la1 ˇ D 0
.1/
.3/
H W la1 la1 ˇ D 0
.2/
.3/
H W la1 la1 ˇ D 0
In the example, with a D 4 and b D 3, the second LSMEANS statement produces four sets
of least squares means differences. Within each set, factor A is held fixed at a particular level
and each set consists of three comparisons.
When the ADJUST= option is specified, the GLIMMIX procedure also adjusts the tests for
multiplicity. The adjustment is based on the number of comparisons within each level of
the SLICEDIFF= effect; see the SLICEDIFFTYPE= option. The Nelson adjustment is not
available for slice differences.
SLICEDIFFTYPE< =difftype >
SIMPLEDIFFTYPE< =difftype >
determines the type of simple effect differences produced with the SLICEDIFF= option.
The possible values for the difftype are ALL, CONTROL, CONTROLL, and CONTROLU.
The difftype ALL requests all simple effects differences, and it is the default. The difftype
CONTROL requests the differences with a control, which, by default, is the first level of each
of the specified LSMEANS effects.
To specify which levels of the effects are the controls, list the quoted formatted values in
parentheses after the keyword CONTROL. For example, if the effects A, B, and C are classification variables, each having three levels (1, 2, and 3), the following LSMEANS statement
specifies the (1,3) level of A*B as the control:
lsmeans A*B / slicediff=(A B)
slicedifftype=control(’1’ ’3’);
This LSMEANS statement first produces simple effects differences holding the levels of A
fixed, and then it produces simple effects differences holding the levels of B fixed. In the
former case, level ’3’ of B serves as the control level. In the latter case, level ’1’ of A serves
as the control.
2152 F Chapter 38: The GLIMMIX Procedure
For multiple effects, the results depend upon the order of the list, and so you should check the
output to make sure that the controls are correct.
Two-tailed tests and confidence limits are associated with the CONTROL difftype. For onetailed results, use either the CONTROLL or CONTROLU difftype. The CONTROLL difftype
tests whether the noncontrol levels are significantly smaller than the control; the upper confidence limits for the control minus the noncontrol levels are considered to be infinity and
are displayed as missing. Conversely, the CONTROLU difftype tests whether the noncontrol
levels are significantly larger than the control; the upper confidence limits for the noncontrol
levels minus the control are considered to be infinity and are displayed as missing.
STEPDOWN< (step-down options) >
requests that multiple comparison adjustments for the p-values of LS-mean differences be
further adjusted in a step-down fashion. Step-down methods increase the power of multiple
comparisons by taking advantage of the fact that a p-value will never be declared significant
unless all smaller p-values are also declared significant. Note that the STEPDOWN adjustment combined with ADJUST=BON corresponds to the methods of Holm (1979) and Shaffer’s “Method 2” (1986); this is the default. Using step-down-adjusted p-values combined
with ADJUST=SIMULATE corresponds to the method of Westfall (1997).
If the degrees-of-freedom method is DDFM=KENWARDROGER or DDFM=SATTERTHWAITE, then step-down-adjusted p-values are produced only if the ADJDFE=ROW option is
in effect.
Also, STEPDOWN affects only p-values, not confidence limits. For ADJUST=SIMULATE,
the generalized least squares hybrid approach of Westfall (1997) is employed to increase
Monte Carlo accuracy.
You can specify the following step-down options in parentheses:
MAXTIME=n
specifies the time (in seconds) to spend computing the maximal logically consistent
sequential subsets of equality hypotheses for TYPE=LOGICAL. The default is MAXTIME=60. If the MAXTIME value is exceeded, the adjusted tests are not computed.
When this occurs, you can try increasing the MAXTIME value. However, note that
there are common multiple comparisons problems for which this computation requires
a huge amount of time—for example, all pairwise comparisons between more than 10
groups. In such cases, try to use TYPE=FREE (the default) or TYPE=LOGICAL(n)
for small n.
REPORT
specifies that a report on the step-down adjustment should be displayed, including a
listing of the sequential subsets (Westfall 1997) and, for ADJUST=SIMULATE, the
step-down simulation results.
TYPE=LOGICAL< (n) >
TYPE=FREE
If you specify TYPE=LOGICAL, the step-down adjustments are computed by using
maximal logically consistent sequential subsets of equality hypotheses (Shaffer 1986,
LSMESTIMATE Statement F 2153
Westfall 1997). Alternatively, for TYPE=FREE, sequential subsets are computed ignoring logical constraints. The TYPE=FREE results are more conservative than those
for TYPE=LOGICAL, but they can be much more efficient to produce for many comparisons. For example, it is not feasible to take logical constraints between all pairwise
comparisons of more than 10 groups. For this reason, TYPE=FREE is the default.
However, you can reduce the computational complexity of taking logical constraints
into account by limiting the depth of the search tree used to compute them, specifying
the optional depth parameter as a number n in parentheses after TYPE=LOGICAL.
As with TYPE=FREE, results for TYPE=LOGICAL(n) are conservative relative to the
true TYPE=LOGICAL results, but even for TYPE=LOGICAL(0) they can be appreciably less conservative than TYPE=FREE and they are computationally feasible for
much larger numbers of comparisons. If you do not specify n or if n D 1, the full
search tree is used.
LSMESTIMATE Statement
LSMESTIMATE fixed-effect < ’label’ > values < divisor =n >
< , < ’label’ > values < divisor =n > > < , . . . >
< / options > ;
The LSMESTIMATE statement provides a mechanism for obtaining custom hypothesis tests among
the least squares means. In contrast to the hypotheses tested with the ESTIMATE or CONTRAST
statements, the LSMESTIMATE statement enables you to form linear combinations of the least
squares means, rather than linear combination of fixed-effects parameter estimates and/or randomeffects solutions. Multiple-row sets of coefficients are permitted.
The computation of an LSMESTIMATE involves two coefficient matrices. Suppose that the fixedeffect has nl levels. Then the LS-means are formed as L1b̌, where L1 is a .nl p/ coefficient
matrix. The .k nl / coefficient matrix K is formed from the values that you supply in the k rows of
the LSMESTIMATE statement. The least squares means estimates then represent the .k 1/ vector
KL1 ˇ D Lˇ
The GLIMMIX procedure supports nonpositional syntax for the coefficients (values) in the LSMESTIMATE statement. For details see the section “Positional and Nonpositional Syntax for Contrast
Coefficients” on page 2262.
PROC GLIMMIX produces a t test for each row of coefficients specified in the LSMESTIMATE
statement. You can adjust p-values and confidence intervals for multiplicity with the ADJUST=
option. You can obtain an F test of single-row or multirow LSMESTIMATEs with the FTEST
option.
Note that in contrast to a multirow estimate in the ESTIMATE statement, you specify only a single
fixed effect in the LSMESTIMATE statement. The row labels are optional and follow the effects
specification. For example, the following statements fit a split-split-plot design and compare the
average of the third and fourth LS-mean of the whole-plot factor A to the first LS-mean of the
factor:
2154 F Chapter 38: The GLIMMIX Procedure
proc glimmix;
class a b block;
model y = a b a*b / s;
random int a / sub=block;
lsmestimate A ’a1 vs avg(a3,a4)’ 2 0 -1 -1 divisor=2;
run;
The order in which coefficients are assigned to the least squares means corresponds to the order in
which they are displayed in the “Least Squares Means” table. You can use the ELSM option to see
how coefficients are matched to levels of the fixed-effect.
The optional divisor=n specification enables you to assign a separate divisor to each row of the
LSMESTIMATE. You can also assign divisor values through the DIVISOR= option. See the documentation that follows for the interaction between the two ways of specifying divisors.
Many options of the LSMESTIMATE statement affect the computation of least squares means—
for example, the AT=, BYLEVEL, and OM options. See the documentation for the LSMEANS
statement for details.
You can specify the following options in the LSMESTIMATE statement after a slash (/).
ADJDFE=SOURCE
ADJDFE=ROW
specifies how denominator degrees of freedom are determined when p-values and confidence
limits are adjusted for multiple comparisons with the ADJUST= option. When you do not
specify the ADJDFE= option, or when you specify ADJDFE=SOURCE, the denominator
degrees of freedom for multiplicity-adjusted results are the denominator degrees of freedom
for the LS-mean effect in the “Type III Tests of Fixed Effects” table.
The ADJDFE=ROW setting is useful if you want multiplicity adjustments to take into account
that denominator degrees of freedom are not constant across estimates. This can be the case,
for example, when DDFM=SATTERTHWAITE or DDFM=KENWARDROGER is specified
in the MODEL statement.
ADJUST=BON
ADJUST=SCHEFFE
ADJUST=SIDAK
ADJUST=SIMULATE< (simoptions) >
ADJUST=T
requests a multiple comparison adjustment for the p-values and confidence limits for the LSmean estimates. The adjusted quantities are produced in addition to the unadjusted p-values
and confidence limits. Adjusted confidence limits are produced if the CL or ALPHA= option
is in effect. For a description of the adjustments, see Chapter 39, “The GLM Procedure,”
and Chapter 58, “The MULTTEST Procedure,” of the SAS/STAT User’s Guide as well as the
documentation for the ADJUST= option in the LSMEANS statement.
Note that not all adjustment methods of the LSMEANS statement are available for the LSMESTIMATE statement. Multiplicity adjustments in the LSMEANS statement are designed
specifically for differences of least squares means.
LSMESTIMATE Statement F 2155
If you specify the STEPDOWN option, the p-values are further adjusted in a step-down fashion.
ALPHA=number
requests that a t-type confidence interval be constructed for each of the LS-means with confidence level 1 number. The value of number must be between 0 and 1; the default is
0.05.
AT variable=value
AT (variable-list)=(value-list)
AT MEANS
enables you to modify the values of the covariates used in computing LS-means. See the AT
option in the LSMEANS statement for details.
BYLEVEL
requests that PROC GLIMMIX compute separate margins for each level of the LSMEANS
effect.
The standard LS-means have equal coefficients across classification effects. The BYLEVEL
option changes these coefficients to be proportional to the observed margins. This adjustment
is reasonable when you want your inferences to apply to a population that is not necessarily
balanced but has the margins observed in the input data set. In this case, the resulting LSmeans are actually equal to raw means for fixed-effects models and certain balanced randomeffects models, but their estimated standard errors account for the covariance structure that
you have specified. If a WEIGHT statement is specified, PROC GLIMMIX uses weighted
margins to construct the LS-means coefficients.
If the AT option is specified, the BYLEVEL option disables it.
CHISQ
requests that chi-square tests be performed in addition to F tests, when you request an F test
with the FTEST option.
CL
requests that t-type confidence limits be constructed for each of the LS-means. If
DDFM=NONE, then PROC GLIMMIX uses infinite degrees of freedom for this test, essentially computing a z interval. The confidence level is 0.95 by default; this can be changed
with the ALPHA= option.
CORR
displays the estimated correlation matrix of the linear combination of the least squares means.
COV
displays the estimated covariance matrix of the linear combination of the least squares means.
DF=number
specifies the degrees of freedom for the t test and confidence limits. The default is the denominator degrees of freedom taken from the “Type III Tests of Fixed Effects” table corresponding
to the LS-means effect.
2156 F Chapter 38: The GLIMMIX Procedure
DIVISOR=value-list
specifies a list of values by which to divide the coefficients so that fractional coefficients can
be entered as integer numerators. If you do not specify value-list, a default value of 1.0 is
assumed. Missing values in the value-list are converted to 1.0.
If the number of elements in value-list exceeds the number of rows of the estimate, the extra
values are ignored. If the number of elements in value-list is less than the number of rows of
the estimate, the last value in value-list is carried forward.
If you specify a row-specific divisor as part of the specification of the estimate row, this value
multiplies the corresponding value in the value-list. For example, the following statement
divides the coefficients in the first row by 8, and the coefficients in the third and fourth row
by 3:
lsmestimate A ’One
’One
’One
’One
vs.
vs.
vs.
vs.
two’
three’
four’
five’
8 -8
divisor=2,
1 0 -1
,
3 0 0 -3
,
3 0 0 0 -3 / divisor=4,.,3;
Coefficients in the second row are not altered.
E
requests that the L coefficients of the estimable function be displayed. These are the coefficients that apply to the fixed-effect parameter estimates. The E option displays the coefficients
that you would need to enter in an equivalent ESTIMATE statement.
ELSM
requests that the K matrix coefficients be displayed. These are the coefficients that apply to
the LS-means. This option is useful to ensure that you assigned the coefficients correctly to
the LS-means.
EXP
requests exponentiation of the least squares means estimate. When you model data with the
logit link function and the estimate represents a log odds ratio, the EXP option produces an
odds ratio. See the section “Odds and Odds Ratio Estimation” on page 2254 for important
details concerning the computation and interpretation of odds and odds ratio results with the
GLIMMIX procedure. If you specify the CL or ALPHA= option, the (adjusted) confidence
limits for the estimate are also exponentiated.
FTEST< (LABEL=’label’) >
produces an F test that jointly tests the rows of the LSMESTIMATE against zero. You can
specify the optional label to identify the results from that test in the “LSMFtest” table.
ILINK
requests that the estimate and its standard error also be reported on the scale of the mean (the
inverse linked scale). PROC GLIMMIX computes the value on the mean scale by applying
the inverse link to the estimate. The interpretation of this quantity depends on the coefficients
specified in your LSMESTIMATE statement and the link function. In a model for binary data
with logit link, for example, the following LSMESTIMATE statement computes
qD
1
1 C expf .1
2 /g
LSMESTIMATE Statement F 2157
where 1 and 2 are the least squares means associated with the first two levels of the classification effect A:
proc glimmix;
class A;
model y = A / dist=binary link=logit;
lsmestimate A 1 -1 / ilink;
run;
The quantity q is not the difference of the probabilities associated with the two levels,
1
2 D
1
1 C expf 1 g
1
1 C expf 2 g
The standard error of the inversely linked estimate is based on the delta method. If you also
specify the CL or ALPHA= option, the GLIMMIX procedure computes confidence intervals
for the inversely linked estimate. These intervals are obtained by applying the inverse link to
the confidence intervals on the linked scale.
LOWER
LOWERTAILED
requests that the p-value for the t test be based only on values less than the test statistic. A
two-tailed test is the default. A lower-tailed confidence limit is also produced if you specify
the CL or ALPHA= option.
Note that for ADJUST=SCHEFFE the one-sided adjusted confidence intervals and one-sided
adjusted p-values are the same as the corresponding two-sided statistics, because this adjustment is based on only the right tail of the F distribution.
OBSMARGINS
OM
specifies a potentially different weighting scheme for the computation of LS-means coefficients. The standard LS-means have equal coefficients across classification effects; however,
the OM option changes these coefficients to be proportional to those found in the input data
set. See the OBSMARGINS option in the LSMEANS statement for further details.
SINGULAR=number
tunes the estimability checking as documented for the CONTRAST statement.
STEPDOWN< (step-down-options) >
requests that multiplicity adjustments for the p-values of LS-mean estimates be further adjusted in a step-down fashion. Step-down methods increase the power of multiple testing
procedures by taking advantage of the fact that a p-value will never be declared significant
unless all smaller p-values are also declared significant. Note that the STEPDOWN adjustment combined with ADJUST=BON corresponds to the methods of Holm (1979) and Shaffer’s “Method 2” (1986); this is the default. Using step-down-adjusted p-values combined
with ADJUST=SIMULATE corresponds to the method of Westfall (1997).
If the degrees-of-freedom method is DDFM=KENWARDROGER or DDFM=SATTERTHWAITE,
then step-down-adjusted p-values are produced only if the ADJDFE=ROW option is in effect.
2158 F Chapter 38: The GLIMMIX Procedure
Also, the STEPDOWN option affects only p-values, not confidence limits.
For
ADJUST=SIMULATE, the generalized least squares hybrid approach of Westfall (1997)
is employed to increase Monte Carlo accuracy.
You can specify the following step-down-options in parentheses:
MAXTIME=n
specifies the time (in seconds) to spend computing the maximal logically consistent
sequential subsets of equality hypotheses for TYPE=LOGICAL. The default is MAXTIME=60. If the MAXTIME value is exceeded, the adjusted tests are not computed.
When this occurs, you can try increasing the MAXTIME value. However, note that
there are common multiple comparisons problems for which this computation requires
a huge amount of time—for example, all pairwise comparisons between more than 10
groups. In such cases, try to use TYPE=FREE (the default) or TYPE=LOGICAL(n)
for small n.
ORDER=PVALUE
ORDER=ROWS
specifies the order in which the step-down tests are performed. ORDER=PVALUE is
the default, with LS-mean estimates being declared significant only if all LS-mean
estimates with smaller (unadjusted) p-values are significant. If you specify ORDER=ROWS, then significances are evaluated in the order in which they are specified.
REPORT
specifies that a report on the step-down adjustment be displayed, including a listing of
the sequential subsets (Westfall 1997) and, for ADJUST=SIMULATE, the step-down
simulation results.
TYPE=LOGICAL< (n) >
TYPE=FREE
If you specify TYPE=LOGICAL, the step-down adjustments are computed by using
maximal logically consistent sequential subsets of equality hypotheses (Shaffer 1986,
Westfall 1997). Alternatively, for TYPE=FREE, sequential subsets are computed ignoring logical constraints. The TYPE=FREE results are more conservative than those
for TYPE=LOGICAL, but they can be much more efficient to produce for many estimates. For example, it is not feasible to take logical constraints between all pairwise
comparisons of more than about 10 groups. For this reason, TYPE=FREE is the default.
However, you can reduce the computational complexity of taking logical constraints
into account by limiting the depth of the search tree used to compute them, specifying
the optional depth parameter as a number n in parentheses after TYPE=LOGICAL.
As with TYPE=FREE, results for TYPE=LOGICAL(n) are conservative relative to the
true TYPE=LOGICAL results, but even for TYPE=LOGICAL(0), they can be appreciably less conservative than TYPE=FREE, and they are computationally feasible for
much larger numbers of estimates. If you do not specify n or if n D 1, the full search
tree is used.
MODEL Statement F 2159
UPPER
UPPERTAILED
requests that the p-value for the t test be based only on values greater than the test statistic. A
two-tailed test is the default. An upper-tailed confidence limit is also produced if you specify
the CL or ALPHA= option.
Note that for ADJUST=SCHEFFE the one-sided adjusted confidence intervals and one-sided
adjusted p-values are the same as the corresponding two-sided statistics, because this adjustment is based on only the right tail of the F distribution.
MODEL Statement
MODEL response < (response-options) > = < fixed-effects > < / model-options > ;
MODEL events/trials = < fixed-effects > < / model-options > ;
The MODEL statement is required and names the dependent variable and the fixed effects. The
fixed-effects determine the X matrix of the model (see the section “Notation for the Generalized
Linear Mixed Model” for details). The specification of effects is the same as in the GLM or MIXED
procedure. In contrast to PROC GLM, you do not specify random effects in the MODEL statement.
However, in contrast to PROC GLM and PROC MIXED, continuous variables on the left and right
side of the MODEL statement can be computed through PROC GLIMMIX programming statements.
An intercept is included in the fixed-effects model by default. It can be removed with the NOINT
option.
The dependent variable can be specified by using either the response syntax or the events/trials
syntax. The events/trials syntax is specific to models for binomial data. A binomial(n,) variable
is the sum of n independent Bernoulli trials with event probability . Each Bernoulli trial results in
either an event or a nonevent (with probability 1 ). You use the events/trials syntax to indicate
to the GLIMMIX procedure that the Bernoulli outcomes are grouped. The value of the second
variable, trials, gives the number n of Bernoulli trials. The value of the first variable, events, is
the number of events out of n. The values of both events and (trials events) must be nonnegative
and the value of trials must be positive. Observations for which these conditions are not met are
excluded from the analysis. If the events/trials syntax is used, the GLIMMIX procedure defaults to
the binomial distribution. The response is then the events variable. The trials variable is accounted
in model fitting as an additional weight. If you use the response syntax, the procedure defaults to
the normal distribution.
There are two sets of options in the MODEL statement. The response-options determine how
the GLIMMIX procedure models probabilities for binary and multinomial data. The model-options
control other aspects of model formation and inference. Table 38.6 summarizes important responseoptions and model-options. These are subsequently discussed in detail in alphabetical order by
option category.
2160 F Chapter 38: The GLIMMIX Procedure
Table 38.6
Summary of Important MODEL Statement Options
Option
Description
Response Variable Options
DESCENDING
reverses the order of response categories
EVENT=
specifies the event category in binary models
ORDER=
specifies the sort order for the response variable
REFERENCE=
specifies the reference category in generalized logit models
Model Building
DIST=
LINK=
NOINT
OFFSET=
Statistical Computations
ALPHA=˛
CHISQ
DDF=
DDFM=
HTYPE=
NOCENTER
ZETA=
Statistical Output
CL
CORRB
COVB
COVBI
E, E1, E2, E3
INTERCEPT
ODDSRATIO
SOLUTION
STDCOEF
specifies the response distribution
specifies the link function
excludes fixed-effect intercept from model
specifies the offset variable for linear predictor
determines the confidence level (1 ˛)
requests chi-square tests
specifies the denominator degrees of freedom (list)
specifies the method for computing denominator degrees of freedom
selects the type of hypothesis test
suppresses centering and scaling of X columns during the estimation phase
tunes sensitivity in computing Type III functions
displays confidence limits for fixed-effects parameter estimates
displays the correlation matrix of fixed-effects parameter estimates
displays the covariance matrix of fixed-effects parameter estimates
displays the inverse covariance matrix of fixed-effects parameter
estimates
displays the L matrix coefficients
adds a row for the intercept to test tables
displays odds ratios and confidence limits
displays fixed-effects parameter estimates (and scale parameter in
GLM models)
displays standardized coefficients
MODEL Statement F 2161
Response Variable Options
Response variable options determine how the GLIMMIX procedure models probabilities for binary
and multinomial data.
You can specify the following options by enclosing them in parentheses after the response variable.
See the section “Response-Level Ordering and Referencing” on page 2265 for more detail and
examples.
DESCENDING
DESC
reverses the order of the response categories. If both the DESCENDING and ORDER=
options are specified, PROC GLIMMIX orders the response categories according to the
ORDER= option and then reverses that order.
EVENT=’category ’ | keyword
specifies the event category for the binary response model. PROC GLIMMIX models the
probability of the event category. The EVENT= option has no effect when there are more
than two response categories. You can specify the value (formatted, if a format is applied) of
the event category in quotes, or you can specify one of the following keywords:
FIRST
designates the first ordered category as the event. This is the default.
LAST
designates the last ordered category as the event.
ORDER=DATA | FORMATTED | FREQ | INTERNAL
specifies the sort order for the levels of the response variable. When ORDER=FORMATTED
(the default) for numeric variables for which you have supplied no explicit format (that is, for
which there is no corresponding FORMAT statement in the current PROC GLIMMIX run or
in the DATA step that created the data set), the levels are ordered by their internal (numeric)
value. If you specify the ORDER= option in the MODEL statement and the ORDER= option
in the PROC GLIMMIX statement, the former takes precedence. The following table shows
the interpretation of the ORDER= values:
Value of ORDER=
Levels Sorted By
DATA
order of appearance in the input data set
FORMATTED
external formatted value, except for numeric variables
with no explicit format, which are sorted by their unformatted (internal) value
FREQ
descending frequency count; levels with the most observations come first in the order
INTERNAL
unformatted value
2162 F Chapter 38: The GLIMMIX Procedure
By default, ORDER=FORMATTED. For the FORMATTED and INTERNAL values, the sort
order is machine dependent.
For more information about sort order, see the chapter on the SORT procedure in the Base SAS
Procedures Guide and the discussion of BY-group processing in SAS Language Reference:
Concepts.
REFERENCE=’category ’ | keyword
REF=’category ’ | keyword
specifies the reference category for the generalized logit model and the binary response
model. For the generalized logit model, each nonreference category is contrasted with the
reference category. For the binary response model, specifying one response category as the
reference is the same as specifying the other response category as the event category. You can
specify the value (formatted if a format is applied) of the reference category in quotes, or you
can specify one of the following keywords:
FIRST
designates the first ordered category as the reference category.
LAST
designates the last ordered category as the reference category. This is the default.
Model Options
ALPHA=number
requests that a t-type confidence interval be constructed for each of the fixed-effects parameters with confidence level 1 number. The value of number must be between 0 and 1; the
default is 0.05.
CHISQ
requests that chi-square tests be performed for all specified effects in addition to the F tests.
Type III tests are the default; you can produce the Type I and Type II tests by using the
HTYPE= option.
CL
requests that t-type confidence limits be constructed for each of the fixed-effects parameter
estimates. The confidence level is 0.95 by default; this can be changed with the ALPHA=
option.
CORRB
produces the correlation matrix from the approximate covariance matrix of the fixed-effects
parameter estimates.
COVB< (DETAILS) >
produces the approximate variance-covariance matrix of the fixed-effects parameter estimates
b̌. In a generalized linear mixed model this matrix typically takes the form .X0 b
V 1 X/
and can be obtained by sweeping the mixed model equations; see the section “Estimated
Precision of Estimates” on page 2219. In a model without random effects, it is obtained
from the inverse of the observed or expected Hessian matrix. Which Hessian is used in
MODEL Statement F 2163
the computation depends on whether the procedure is in scoring mode (see the SCORING=
option in the PROC GLIMMIX statement) and whether the EXPHESSIAN option is in effect.
Note that if you use EMPIRICAL= or DDFM=KENWARDROGER, the matrix displayed by
the COVB option is the empirical (sandwich) estimator or the adjusted estimator, respectively.
The DETAILS suboption of the COVB option enables you to obtain a table of statistics about
the covariance matrix of the fixed effects. If an adjusted estimator is used because of the
EMPIRICAL= or DDFM=KENWARDROGER option, the GLIMMIX procedure displays
statistics for the adjusted and unadjusted estimators as well as statistics comparing them. This
enables you to diagnose, for example, changes in rank (because of an insufficient number of
subjects for the empirical estimator) and to assess the extent of the covariance adjustment. In
addition, the GLIMMIX procedure then displays the unadjusted (=model-based) covariance
matrix of the fixed-effects parameter estimates. For more details, see the section “Exploring
and Comparing Covariance Matrices” on page 2244.
COVBI
produces the inverse of the approximate covariance matrix of the fixed-effects parameter estimates.
DDF=value-list
DF=value-list
enables you to specify your own denominator degrees of freedom for the fixed effects. The
value-list specification is a list of numbers or missing values (.) separated by commas. The
degrees of freedom should be listed in the order in which the effects appear in the “Type
III Tests of Fixed Effects” table. If you want to retain the default degrees of freedom for a
particular effect, use a missing value for its location in the list. For example, the statement
assigns 3 denominator degrees of freedom to A and 4.7 to A*B, while those for B remain the
same:
model Y = A B A*B / ddf=3,.,4.7;
If you select a degrees-of-freedom method with the DDFM= option, then nonmissing, positive
values in value-list override the degrees of freedom for the particular effect. For example, the
statement assigns 3 and 6 denominator degrees of freedom in the test of the A main effect and
the A*B interaction, respectively:
model Y = A B A*B / ddf=3,.,6
ddfm=Satterth;
The denominator degrees of freedom for the test for the B effect are determined from a Satterthwaite approximation.
Note that the DDF= and DDFM= options determine the degrees of freedom in the “Type I
Tests of Fixed Effects,” “Type II Tests of Fixed Effects,” and “Type III Tests of Fixed Effects” tables. These degrees of freedom are also used in determining the degrees of freedom in tests and confidence intervals from the CONTRAST, ESTIMATE, LSMEANS, and
LSMESTIMATE statements. Exceptions from this rule are noted in the documentation for
the respective statements.
2164 F Chapter 38: The GLIMMIX Procedure
DDFM=BETWITHIN
DDFM=CONTAIN
DDFM=KENWARDROGER< (FIRSTORDER) >
DDFM=NONE
DDFM=RESIDUAL
DDFM=SATTERTHWAITE
specifies the method for computing the denominator degrees of freedom for the tests
of fixed effects resulting from the MODEL, CONTRAST, ESTIMATE, LSMEANS, and
LSMESTIMATE statements.
Table 38.7 table lists syntax aliases for the degrees-of-freedom methods.
Table 38.7
Aliases for the DDFM= Option
DDFM= Option
Alias
BETWITHIN
CONTAIN
KENWARDROGER
RESIDUAL
SATTERTHWAITE
BW
CON
KENROG, KR
RES
SATTERTH, SAT
The DDFM=BETWITHIN option divides the residual degrees of freedom into betweensubject and within-subject portions. PROC GLIMMIX then determines whether a fixed effect
changes within any subject. If the GLIMMIX procedure does not process the data by subjects,
the DDFM=BETWITHIN option has no effect. See the section “Processing by Subjects” on
page 2245 for details. If so, it assigns within-subject degrees of freedom to the effect; otherwise, it assigns the between-subject degrees of freedom to the effect (see Schluchter and
Elashoff 1990). If there are multiple within-subject effects containing classification variables, the within-subject degrees of freedom are partitioned into components corresponding
to the subject-by-effect interactions.
One exception to the preceding method is the case where you model only R-side covariation
with an unstructured covariance matrix (TYPE=UN option). In this case, all fixed effects
are assigned the between-subject degrees of freedom to provide for better small-sample approximations to the relevant sampling distributions. The DDFM=BETWITHIN method is the
default for models with only R-side random effects and a SUBJECT= option.
The DDFM=CONTAIN option invokes the containment method to compute denominator degrees of freedom, and this method is the default when the model contains G-side random
effects. The containment method is carried out as follows: Denote the fixed effect in question
A and search the G-side random effect list for the effects that syntactically contain A. For
example, the effect B(A) contains A, but the effect C does not, even if it has the same levels as
B(A).
Among the random effects that contain A, compute their rank contributions to the ŒX Z
matrix (in order). The denominator degrees of freedom assigned to A is the smallest of these
rank contributions. If no effects are found, the denominator df for A is set equal to the residual
degrees of freedom, n rankŒX Z. This choice of degrees of freedom is the same as for
MODEL Statement F 2165
the tests performed for balanced split-plot designs and should be adequate for moderately
unbalanced designs.
C AUTION : If you have a Z matrix with a large number of columns, the overall memory
requirements and the computing time after convergence can be substantial for the containment
method. If it is too large, you might want to use a different degrees-of-freedom method, such
as DDFM=RESIDUAL, DDFM=NONE, or DDFM=BETWITHIN.
DDFM=NONE specifies that no denominator degrees of freedom be applied. PROC GLIMMIX then essentially assumes that infinite degrees of freedom are available in the calculation
of p-values. The p-values for t tests are then identical to p-values derived from the standard
normal distribution. In the case of F tests, the p-values equal those of chi-square tests determined as follows: if Fobs is the observed value of the F test with l numerator degrees of
freedom, then
p D PrfFl;1 > Fobs g D Prf2l > lFobs g
Regardless of the DDFM= method, you can obtain these chi-square p-values with the CHISQ
option in the MODEL statement.
The DDFM=RESIDUAL option performs all tests by using the residual degrees of freedom,
n rank.X/, where n is the sum of the frequencies used. It is the default degrees of freedom
method for GLMs and overdispersed GLMs.
The DDFM=KENWARDROGER option applies the (prediction) standard error and degreesof-freedom correction detailed by Kenward and Roger (1997). This approximation involves inflating the estimated variance-covariance matrix of the fixed and random effects
in a manner similar to that of Prasad and Rao (1990), Harville and Jeske (1992), and
Kackar and Harville (1984). Satterthwaite-type degrees of freedom are then computed
based on this adjustment. By default, the observed information matrix of the covariance parameter estimates is used in the calculations. For covariance structures that have
nonzero second derivatives with respect to the covariance parameters, the Kenward-Roger
covariance matrix adjustment includes a second-order term. This term can result in standard error shrinkage. Also, the resulting adjusted covariance matrix can then be indefinite and is not invariant under reparameterization. The FIRSTORDER suboption of the
DDFM=KENWARDROGER option eliminates the second derivatives from the calculation of the covariance matrix adjustment. For the case of scalar estimable functions, the
resulting estimator is referred to as the Prasad-Rao estimator m
[email protected] in Harville and Jeske
(1992). You can use the COVB(DETAILS) option to diagnose the adjustments made to the
covariance matrix of fixed-effects parameter estimates by the GLIMMIX procedure. An
application with DDFM=KENWARDROGER is presented in Example 38.8. The following
are examples of covariance structures that generally lead to nonzero second derivatives:
TYPE=ANTE(1), TYPE=AR(1), TYPE=ARH(1), TYPE=ARMA(1,1), TYPE=CHOL,
TYPE=CSH, TYPE=FA0(q), TYPE=TOEPH, TYPE=UNR, and all TYPE=SP() structures.
The DDFM=SATTERTHWAITE option performs a general Satterthwaite approximation for
the denominator degrees of freedom in a generalized linear mixed model. This method is
a generalization of the techniques described in Giesbrecht and Burns (1985), McLean and
Sanders (1988), and Fai and Cornelius (1996). The method can also include estimated random effects. The calculations require extra memory to hold q matrices that are the size of
2166 F Chapter 38: The GLIMMIX Procedure
the mixed model equations, where q is the number of covariance parameters. Extra computing time is also required to process these matrices. The Satterthwaite method implemented
is intended to produce an accurate F approximation; however, the results can differ from
those produced by PROC GLM. Also, the small sample properties of this approximation have
not been extensively investigated for the various models available with PROC GLIMMIX.
Computational details can be found in the section “Satterthwaite Degrees of Freedom Approximation” on page 2239.
When the asymptotic variance matrix of the covariance parameters is found to be singular, a generalized inverse is used. Covariance parameters with zero variance then
do not contribute to the degrees of freedom adjustment for DDFM=SATTERTH and
DDFM=KENWARDROGER, and a message is written to the log.
DISTRIBUTION=keyword
DIST=keyword
D=keyword
ERROR=keyword
E=keyword
specifies the built-in (conditional) probability distribution of the data. If you specify the
DIST= option and you do not specify a user-defined link function, a default link function is
chosen according to the following table. If you do not specify a distribution, the GLIMMIX
procedure defaults to the normal distribution for continuous response variables and to the
multinomial distribution for classification or character variables, unless the events/trial syntax is used in the MODEL statement. If you choose the events/trial syntax, the GLIMMIX
procedure defaults to the binomial distribution.
Table 38.8 lists the values of the DIST= option and the corresponding default link functions.
For the case of generalized linear models with these distributions, you can find expressions
for the log-likelihood functions in the section “Maximum Likelihood” on page 2210.
Table 38.8
Keyword Values of the DIST= Option
DIST=
Distribution
BETA
BINARY
BINOMIAL | BIN | B
EXPONENTIAL | EXPO
GAMMA | GAM
GAUSSIAN | G | NORMAL | N
GEOMETRIC | GEOM
INVGAUSS | IGAUSSIAN | IG
beta
binary
binomial
exponential
gamma
normal
geometric
inverse Gaussian
LOGNORMAL | LOGN
MULTINOMIAL | MULTI | MULT
NEGBINOMIAL | NEGBIN | NB
POISSON | POI | P
TCENTRAL | TDIST | T
lognormal
multinomial
negative binomial
Poisson
t
BYOBS(variable)
multivariate
Default Link
Function
Numeric
Value
logit
logit
logit
log
log
identity
log
inverse squared
(power( 2) )
identity
cumulative logit
log
log
identity
12
4
3
9
5
1
8
6
11
NA
7
2
10
varied
NA
MODEL Statement F 2167
Note that the PROC GLIMMIX default link for the gamma or exponential distribution is not
the canonical link (the reciprocal link).
The numeric value in the last column of Table 38.8 can be used in combination with
DIST=BYOBS. The BYOBS(variable) syntax designates a variable whose value identifies
the distribution to which an observation belongs. If the variable is numeric, its values must
match values in the last column of Table 38.8. If the variable is not numeric, an observation’s distribution is identified by the first four characters of the distribution’s name in the
leftmost column of the table. Distributions whose numeric value is “NA” cannot be used with
DIST=BYOBS.
If the variable in BYOBS(variable) is a data set variable, it can also be used in the CLASS
statement of the GLIMMIX procedure. For example, this provides a convenient method to
model multivariate data jointly while varying fixed-effects components across outcomes. Assume that, for example, for each patient, a count and a continuous outcome were observed; the
count data are modeled as Poisson data and the continuous data are modeled as gamma variates. The following statements fit a Poisson and a gamma regression model simultaneously:
proc sort data=yourdata;
by patient;
run;
data yourdata;
set yourdata;
by patient;
if first.patient then dist=’POIS’ else dist=’GAMM’;
run;
proc glimmix data=yourdata;
class treatment dist;
model y = dist treatment*dist / dist=byobs(dist);
run;
The two models have separate intercepts and treatment effects. To correlate the outcomes,
you can share a random effect between the observations from the same patient:
proc glimmix data=yourdata;
class treatment dist patient;
model y = dist treatment*dist / dist=byobs(dist);
random intercept / subject=patient;
run;
Or, you could use an R-side correlation structure:
proc glimmix data=yourdata;
class treatment dist patient;
model y = dist treatment*dist / dist=byobs(dist);
random _residual_ / subject=patient type=un;
run;
Although DIST=BYOBS(variable) is used to model multivariate data, you only need a single
response variable in PROC GLIMMIX. The responses are in “univariate” form. This allows,
for example, different missing value patterns across the responses. It does, however, require
that all response variables be numeric.
2168 F Chapter 38: The GLIMMIX Procedure
The default links that are assigned when DIST=BYOBS is in effect correspond to the respective default links in Table 38.8.
When you choose DIST=LOGNORMAL, the GLIMMIX procedure models the logarithm
of the response variable as a normal random variable. That is, the mean and variance are
estimated on the logarithmic scale, assuming a normal distribution, logfY g N.; 2 /.
This enables you to draw on options that require a distribution in the exponential family—for
example, by using a scoring algorithm in a GLM. To convert means and variances for logfY g
into those of Y , use the relationships
p
EŒY  D expfg !
VarŒY  D expf2g!.!
1/
! D expf 2 g
The DIST=T option models the data as a shifted and scaled central t variable. This enables
you to model data with heavy-tailed distributions. If Y denotes the response and X has a t
distribution with degrees of freedom, then PROC GLIMMIX models
p
Y D C X
In this parameterization, Y has mean and variance =.
2/.
By default, D 3. You can supply different degrees of freedom for the t variate as in the
following statements:
proc glimmix;
class a b;
model y = b x b*x / dist=tcentral(9.6);
random a;
run;
The GLIMMIX procedure does not accept values for the degrees of freedom parameter less
than 3.0. If the t distribution is used with the DIST=BYOBS(variable) specification, the degrees of freedom are fixed at D 3. For mixed models where parameters are estimated based
on linearization, choosing DIST=T instead of DIST=NORMAL affects only the residual variance, which decreases by the factor =. 2/.
q
Note that in SAS 9.1, the GLIMMIX procedure modeled Y D C 2 X. The scale
parameter of the parameterizations are related as D . 2/=.
The DIST=BETA option implements the parameterization of the beta distribution in Ferrari
and Cribari-Neto (2004). If Y has a beta.˛; ˇ/ density, so that EŒY  D D ˛=.˛ C ˇ/, this
parameterization uses the variance function a./ D .1 / and VarŒY  D a./=.1 C /.
See the section “Maximum Likelihood” on page 2210 for the log likelihoods of the distributions fitted by the GLIMMIX procedure.
E
requests that Type I, Type II, and Type III L matrix coefficients be displayed for all specified
effects.
MODEL Statement F 2169
E1 | EI
requests that Type I L matrix coefficients be displayed for all specified effects.
E2 | EII
requests that Type II L matrix coefficients be displayed for all specified effects.
E3 | EIII
requests that Type III L matrix coefficients be displayed for all specified effects.
HTYPE=value-list
indicates the type of hypothesis test to perform on the fixed effects. Valid entries for values in
the value-list are 1, 2, and 3, corresponding to Type I, Type II, and Type III tests. The default
value is 3. You can specify several types by separating the values with a comma or a space.
The ODS table names are “Tests1,” “Tests2,” and “Tests3” for the Type I, Type II, and Type
III tests, respectively.
INTERCEPT
adds a row to the tables for Type I, II, and III tests corresponding to the overall intercept.
LINK=keyword
specifies the link function in the generalized linear mixed model. The keywords and their
associated built-in link functions are shown in Table 38.9.
Table 38.9
Built-in Link Functions of the GLIMMIX Procedure
LINK=
CUMCLL | CCLL
Link
Function
CUMLOGIT | CLOGIT
CUMLOGLOG
CUMPROBIT | CPROBIT
CLOGLOG | CLL
GLOGIT | GENLOGIT
IDENTITY | ID
LOG
LOGIT
LOGLOG
PROBIT
cumulative
complementary log-log
cumulative logit
cumulative log-log
cumulative probit
complementary log-log
generalized logit
identity
log
logit
log-log
probit
POWER() | POW()
power with exponent = number
POWERMINUS2
RECIPROCAL | INVERSE
power with exponent -2
reciprocal
BYOBS(variable)
varied
g./ D D
Numeric
Value
//
NA
log.=.1 //
log. log.//
ˆ 1 ./
log. log.1 //
NA
NA
NA
5
NA
1
4
2
6
3
log. log.1
log./
log.=.1 //
log. log.//
1
ˆ ./
if 6D 0
log./ if D 0
1=2
1=
varied
NA
8
7
NA
For the probit and cumulative probit links, ˆ 1 ./ denotes the quantile function of the standard normal distribution. For the other cumulative links, denotes a cumulative category
probability. The cumulative and generalized logit link functions are appropriate only for the
2170 F Chapter 38: The GLIMMIX Procedure
multinomial distribution. When you choose a cumulative link function, PROC GLIMMIX assumes that the data are ordinal. When you specify LINK=GLOGIT, the GLIMMIX procedure
assumes that the data are nominal (not ordered).
The numeric value in the rightmost column of Table 38.9 can be used in conjunction with
LINK=BYOBS(variable). This syntax designates a variable whose values identify the link
function associated with an observation. If the variable is numeric, its values must match
those in the last column of Table 38.9. If the variable is not numeric, an observation’s link
function is determined by the first four characters of the link’s name in the first column. Those
link functions whose numeric value is “NA” cannot be used with LINK=BYOBS(variable).
You can define your own link function through programming statements. See the section
“User-Defined Link or Variance Function” on page 2206 for more information about how to
specify a link function. If a user-defined link function is in effect, the specification in the
LINK= option is ignored. If you specify neither the LINK= option nor a user-defined link
function, then the default link function is chosen according to Table 38.8.
LWEIGHT=FIRSTORDER | FIRO
LWEIGHT=NONE
LWEIGHT=VAR
determines how weights are used in constructing the coefficients for Type I through Type
III L matrices. The default is LWEIGHT=VAR, and the values of the WEIGHT variable
are used in forming crossproduct matrices. If you specify LWEIGHT=FIRO, the weights
incorporate the WEIGHT variable as well as the first-order weights of the linearized model.
For LWEIGHT=NONE, the L matrix coefficients are based on the raw crossproduct matrix,
whether a WEIGHT variable is specified or not.
NOCENTER
requests that the columns of the X matrix are not centered and scaled. By default, the columns
of X are centered and scaled. Unless the NOCENTER option is in effect, X is replaced by X
during estimation. The columns of X are computed as follows:
In models with an intercept, the intercept column remains the same and the j th entry in
row i of X is
xij
xij
D qP
n
xj
i D1 .xij
x j /2
In models without intercept, no centering takes place and the j th entry in row i of X is
xij
D qP
n
xij
i D1 .xij
x j /2
The effects of centering and scaling are removed when results are reported. For example, if
the covariance matrix of the fixed effects is printed with the COVB option of the MODEL
statement, the covariances are reported in terms of the original parameters, not the centered
and scaled versions. If you specify the STDCOEF option, fixed-effects parameter estimates
and their standard errors are reported in terms of the standardized (scaled and/or centered)
coefficients in addition to the usual results in noncentered form.
MODEL Statement F 2171
NOINT
requests that no intercept be included in the fixed-effects model. An intercept is included by
default.
ODDSRATIO< (odds-ratio-options) >
OR< (odds-ratio-options) >
requests estimates of odds ratios and their confidence limits, provided the link function is the
logit, cumulative logit, or generalized logit. Odds ratios are produced for the following:
classification main effects, if they appear in the MODEL statement
continuous variables in the MODEL statement, unless they appear in an interaction with
a classification effect
continuous variables in the MODEL statement at fixed levels of a classification effect,
if the MODEL statement contains an interaction of the two
continuous variables in the MODEL statement, if they interact with other continuous
variables
You can specify the following odds-ratio-options to create customized odds ratio results.
AT var-list=value-list
specifies the reference values for continuous variables in the model. By default, the
average value serves as the reference. Consider, for example, the following statements:
proc glimmix;
class A;
model y = A x A*x / dist=binary oddsratio;
run;
Odds ratios for A are based on differences of least squares means for which x is set to
its mean. Odds ratios for x are computed by differencing two sets of least squares mean
for the A factor. One set is computed at x = x C 1, and the second set is computed at x
= x. The following MODEL statement changes the reference value for x to 3:
model y = A x A*x / dist=binary
oddsratio(at x=3);
DIFF< =difftype >
controls the type of differences for classification main effects. By default, odds ratios
compare the odds of a response for level j of a factor to the odds of the response
for the last level of that factor (DIFF=LAST). The DIFF=FIRST option compares the
levels against the first level, DIFF=ALL produces odds ratios based on all pairwise
differences, and DIFF=NONE suppresses odds ratios for classification main effects.
LABEL
displays a label in the “Odds Ratio Estimates” table. The table describes the comparison
associated with the table row.
2172 F Chapter 38: The GLIMMIX Procedure
UNIT var-list=value-list
specifies the units in which the effects of continuous variable in the model are assessed.
By default, odds ratios are computed for a change of one unit from the average. Consider a model with a classification factor A with 4 levels. The following statements
produce an “Odds Ratio Estimates” table with 10 rows:
proc glimmix;
class A;
model y = A x A*x / dist=binary
oddsratio(diff=all unit x=2);
run;
The first 4 3=2 D 6 rows correspond to pairwise differences of levels of A. The
underlying log odds ratios are computed as differences of A least squares means. In the
least squares mean computation the covariate x is set to x. The next four rows compare
least squares means for A at x = x C 2 and at x = x. You can combine the AT and UNIT
options to produce custom odds ratios. For example, the following statements produce
an “Odds Ratio Estimates” table with 8 rows:
proc glimmix;
class A;
model y = A x x*z / dist=binary
oddsratio(diff=all
at
x
= 3
unit x z = 2 4);
run;
The first 4 3=2 D 6 rows correspond to pairwise differences of levels of A. The
underlying log odds ratios are computed as differences of A least squares means. In the
least squares mean computation, the covariate x is set to 3, and the covariate x*z is set to
3z. The next odds ratio measures the effect of a change in x. It is based on differencing
the linear predictor for x = 3 C 2 and x*z = .3 C 2/z with the linear predictor for x =
3 and x*z = 3z. The last odds ratio expresses a change in z by contrasting the linear
predictors based on x = 3 and x*z = 3.z C 4/ with the predictor based on x = 3 and x*z
= 3z.
To compute odds and odds ratios for general estimable functions and least squares means, see
the ODDSRATIO option in the LSMEANS statement and the EXP options in the ESTIMATE
and LSMESTIMATE statements.
For important details concerning interpretation and computation of odds ratios with the
GLIMMIX procedure, see the section “Odds and Odds Ratio Estimation” on page 2254.
OFFSET=variable
specifies a variable to be used as an offset for the linear predictor. An offset plays the role of a
fixed effect whose coefficient is known to be 1. You can use an offset in a Poisson model, for
example, when counts have been obtained in time intervals of different lengths. With a log
link function, you can model the counts as Poisson variables with the logarithm of the time
interval as the offset variable. The offset variable cannot appear in the CLASS statement or
elsewhere in the MODEL or RANDOM statement.
NLOPTIONS Statement F 2173
REFLINP=r
specifies a value for the linear predictor of the reference level in the generalized logit model
for nominal data. By default r D 0.
SOLUTION
S
requests that a solution for the fixed-effects parameters be produced. Using notation from the
section “Notation for the Generalized Linear Mixed Model” on page 2081, the fixed-effects
parameter estimates are b̌, and their (approximate) estimated standard errors are the square
c b̌. This matrix commonly is of the form .X0 b
V 1 X/
roots of the diagonal elements of VarŒ
in GLMMs. You can output this approximate variance matrix with the COVB option. See the
V in the various
section “Details: GLIMMIX Procedure” on page 2210 on the construction of b
models.
Along with the estimates and their approximate standard errors, a t statistic is computed as
the estimate divided by its standard error. The degrees of freedom for this t statistic matches
the one appearing in the “Type III Tests of Fixed Effects” table under the effect containing
the parameter. If DDFM=KENWARDROGER or DDFM=SATTERTHWAITE, the degrees
of freedom are computed separately for each fixed-effect estimate, unless you override the
value for any specific effect with the DDF=value-list option. The “Pr > |t|” column contains
the two-tailed p-value corresponding to the t statistic and associated degrees of freedom. You
can use the CL option to request confidence intervals for the fixed-effects parameters; they
are constructed around the estimate by using a radius of the standard error times a percentage
point from the t distribution.
STDCOEF
reports solutions for fixed effects in terms of the standardized (scaled and/or centered) coefficients. This option has no effect when the NOCENTER option is specified or in models for
multinomial data.
ZETA=number
tunes the sensitivity in forming Type III functions. Any element in the estimable function
basis with an absolute value less than number is set to 0. The default is 1E 8.
NLOPTIONS Statement
NLOPTIONS < options > ;
Most models fit with the GLIMMIX procedure typically have one or more nonlinear parameters.
Estimation requires nonlinear optimization methods. You can control the optimization through
options in the NLOPTIONS statement.
Several estimation methods of the GLIMMIX procedure (METHOD=RSPL, MSPL, RMPL,
MMPL) are doubly iterative in the following sense. The generalized linear mixed model is approximated by a linear mixed model based on current values of the covariance parameter estimates.
The resulting linear mixed model is then fit, which is itself an iterative process (with some exceptions). On convergence, new covariance parameters and fixed-effects estimates are obtained and
2174 F Chapter 38: The GLIMMIX Procedure
the approximated linear mixed model is updated. Its parameters are again estimated iteratively. It
is thus reasonable to refer to outer and inner iterations. The outer iterations involve the repeated
updates of the linear mixed models, and the inner iterations are the iterative steps that lead to parameter estimates in any given linear mixed model. The NLOPTIONS statement controls the inner
iterations. The outer iteration behavior can be controlled with options in the PROC GLIMMIX
statement, such as the MAXLMMUPDATE=, PCONV=, and ABSPCONV= options. If the estimation method involves a singly iterative approach, then there is no need for the outer cycling and
the model is fit in a single optimization controlled by the NLOPTIONS statement (see the section
“Singly or Doubly Iterative Fitting” on page 2269).
The syntax and options of the NLOPTIONS statement are described in the section “Nonlinear Optimization: The NLOPTIONS Statement” on page 391 of Chapter 18, “Shared Concepts and Topics.”
Note that in a GLMM with pseudo-likelihood estimation, specifying TECHNIQUE=NONE has
the same effect as specifying the NOITER option in the PARMS statement. If you estimate the
parameters by METHOD=LAPLACE or METHOD=QUAD, TECHNIQUE=NONE applies to the
optimization after starting values have been determined.
The GLIMMIX procedure applies the default optimization technique shown in Table 38.10, depending on your model.
Table 38.10
Default Techniques
Model Family
Setting
TECHNIQUE=
GLM
DIST=NORMAL
LINK=IDENTITY
NONE
GLM
otherwise
NEWRAP
GLMM
PARMS NOITER, PL
NONE
GLMM
binary data, PL
NRRIDG
GLMM
otherwise
QUANEW
OUTPUT Statement
OUTPUT < OUT=SAS-data-set >
< keyword< (keyword-options) > < =name > >. . .
< keyword< (keyword-options) > < =name > > < / options > ;
The OUTPUT statement creates a data set that contains predicted values and residual diagnostics,
computed after fitting the model. By default, all variables in the original data set are included in the
output data set.
You can use the ID statement to select a subset of the variables from the input data set as well
as computed variables for adding to the output data set. If you reassign a data set variable through
programming statements, the value of the variable from the input data set supersedes the recomputed
value when observations are written to the output data set. If you list the variable in the ID statement,
OUTPUT Statement F 2175
however, PROC GLIMMIX saves the current value of the variable after the programming statements
have been executed.
For example, suppose that data set Scores contains the variables score, machine, and person. The
following statements fit a model with fixed machine and random person effects. The variable score
divided by 100 is assumed to follow an inverse Gaussian distribution. The (conditional) mean and
residuals are saved to the data set igausout. Because no ID statement is given, the variable score in
the output data set contains the values from the input data set.
proc glimmix;
class machine person;
score = score/100;
p = 4*_linp_;
model score = machine / dist=invgauss;
random int / sub=person;
output out=igausout pred=p resid=r;
run;
On the contrary, the following statements list explicitly which variables to save to the OUTPUT
data set. Because the variable score is listed in the ID statement, and is (re-)assigned through
programming statements, the values of score saved to the OUTPUT data set are the input values
divided by 100.
proc glimmix;
class machine person;
score = score / 100;
model score = machine / dist=invgauss;
random int / sub=person;
output out=igausout pred=p resid=r;
id machine score _xbeta_ _zgamma_;
run;
You can specify the following syntax elements in the OUTPUT statement before the slash (/).
OUT=SAS-data-set
DATA=SAS-data-set
specifies the name of the output data set. If the OUT= (or DATA=) option is omitted, the
procedure uses the DATAn convention to name the output data set.
keyword< (keyword-options) > < =name >
specifies a statistic to include in the output data set and optionally assigns the variable the
name name. You can use the keyword-options to control which type of a particular statistic to
compute. The keyword-options can take on the following values:
BLUP
uses the predictors of the random effects in computing the statistic.
ILINK
computes the statistic on the scale of the data.
NOBLUP
does not use the predictors of the random effects in computing the statistic.
NOILINK
computes the statistic on the scale of the link function.
2176 F Chapter 38: The GLIMMIX Procedure
The default is to compute statistics by using BLUPs on the scale of the link function (the
linearized scale). For example, the following OUTPUT statements are equivalent:
output out=out1
pred=predicted lcl=lower;
output out=out1
pred(blup noilink)=predicted
lcl (blup noilink)=lower;
If a particular combination of keyword and keyword options is not supported, the statistic is
not computed and a message is produced in the SAS log.
A keyword can appear multiple times in the OUTPUT statement. Table 38.11 lists the keywords and the default names assigned by the GLIMMIX procedure if you do not specify a
name. In this table, y denotes the observed response, and p denotes the linearized pseudodata. See the section “Pseudo-likelihood Estimation Based on Linearization” on page 2218
for details on notation and the section “Notes on Output Statistics” on page 2276 for further
details regarding the output statistics.
Table 38.11
Keywords for Output Statistics
Keyword
Options
Description
PREDICTED
Default
NOBLUP
ILINK
NOBLUP ILINK
STDERR
Default
NOBLUP
ILINK
NOBLUP ILINK
RESIDUAL
PEARSON
STUDENT
Default
NOBLUP
ILINK
NOBLUP ILINK
Expression
Name
Linear predictor
Marginal linear predictor
Predicted mean
Marginal mean
b
D x0b̌ C z0b
0
b̌
b
m D x
Pred
PredPA
Standard deviation of
linear predictor
Standard deviation of
marginal linear predictor
Standard deviation of
mean
Standard deviation of
marginal mean
p
VarŒb
Residual
Marginal residual
Residual on mean scale
Marginal residual on
mean scale
Default
Pearson-type residual
NOBLUP
Marginal
residual
Pearson-type
ILINK
Conditional
Pearsontype mean residual
Default
Studentized residual
g 1 .b
/
1
g .b
m /
p
PredMu
PredMuPA
z0 
VarŒb
m 
p
VarŒg
1 .b
p
VarŒg
StdErr
StdErrPA
z0 /
1 .b
m /
r Dp b
rm D pm b
m
1
ry D y g .b
/
ry m D y g 1 .b
m /
StdErr
StdErrMuPA
Resid
ResidPA
ResidMu
ResidMuPA
q
c
r= VarŒpj
q
c m
rm = VarŒp
PearsonPA
q
c j
ry = VarŒY
PearsonMu
q
c
r= VarŒr
Pearson
Student
OUTPUT Statement F 2177
Table 38.11
Keyword
LCL
continued
Options
Description
NOBLUP
Studentized
residual
Default
Lower prediction limit
for linear predictor
Lower confidence limit
for marginal linear predictor
Lower prediction limit
for mean
Lower confidence limit
for marginal mean
LCL
Upper prediction limit
for linear predictor
Upper confidence limit
for marginal linear predictor
Upper prediction limit
for mean
Upper confidence limit
for marginal mean
UCL
NOBLUP
ILINK
NOBLUP ILINK
UCL
Default
NOBLUP
ILINK
NOBLUP ILINK
VARIANCE
Default
NOBLUP
ILINK
NOBLUP ILINK
marginal
Conditional variance of
pseudo-data
Marginal variance of
pseudo-data
Conditional variance of
response
Marginal variance of response
Expression
q
c m
rm = VarŒr
Name
StudentPA
LCLPA
LCLMu
LCLMuPA
UCLPA
UCLMu
UCLMuPA
c
VarŒpj
Variance
c m
VarŒp
VariancePA
c j
VarŒY
Variance_Dep
c 
VarŒY
Variance_DepPA
Studentized residuals are computed only on the linear scale (scale of the link), unless the
link is the identity, in which case the two scales are equal. The keywords RESIDUAL,
PEARSON, STUDENT, and VARIANCE are not available with the multinomial distribution. You can use the following shortcuts to request statistics: PRED for PREDICTED,
STD for STDERR, RESID for RESIDUAL, and VAR for VARIANCE. Output statistics that
depend on the marginal variance VarŒYi  are not available with METHOD=LAPLACE or
METHOD=QUAD.
You can specify the following options in the OUTPUT statement after a slash (/).
ALLSTATS
requests that all statistics are computed. If you do not use a keyword to assign a name, the
GLIMMIX procedure uses the default name.
2178 F Chapter 38: The GLIMMIX Procedure
ALPHA=number
determines the coverage probability for two-sided confidence and prediction intervals. The
coverage probability is computed as 1 number. The value of number must be between 0
and 1; the default is 0.05.
DERIVATIVES
DER
adds derivatives of model quantities to the output data set. If, for example, the model fit
requires the (conditional) log likelihood of the data, then the DERIVATIVES option writes
for each observation the evaluations of the first and second derivatives of the log likelihood
with respect to _LINP_ and _PHI_ to the output data set. The particular derivatives produced
by the GLIMMIX procedure depend on the type of model and the estimation method.
NOMISS
requests that records be written to the output data only for those observations that were used
in the analysis. By default, the GLIMMIX procedure produces output statistics for all observations in the input data set.
NOUNIQUE
requests that names not be made unique in the case of naming conflicts. By default, the
GLIMMIX procedure avoids naming conflicts by assigning a unique name to each output
variable. If you specify the NOUNIQUE option, variables with conflicting names are not
renamed. In that case, the first variable added to the output data set takes precedence.
NOVAR
requests that variables from the input data set not be added to the output data set. This option
does not apply to variables listed in the BY statement or to computed variables listed in the
ID statement.
OBSCAT
requests that in models for multinomial data statistics be written to the output data set only
for the response level that corresponds to the observed level of the observation.
SYMBOLS
SYM
adds to the output data set computed variables that are defined or referenced in the program.
PARMS Statement F 2179
PARMS Statement
PARMS < (value-list) > . . . < / options > ;
The PARMS statement specifies initial values for the covariance or scale parameters, or it requests
a grid search over several values of these parameters in generalized linear mixed models.
The value-list specification can take any of several forms:
m
a single value
m1 ; m2 ; : : : ; mn
several values
m to n
a sequence where m equals the starting value, n equals the ending value, and the
increment equals 1
m to n by i
a sequence where m equals the starting value, n equals the ending value, and the
increment equals i
m1 ; m2 to m3
mixed values and sequences
Using the PARMS Statement with a GLM
If you are fitting a GLM or a GLM with overdispersion, the scale parameters are listed at the end of
the “Parameter Estimates” table in the same order as value-list. If you specify more than one set of
initial values, PROC GLIMMIX uses only the first value listed for each parameter. Grid searches
by using scale parameters are not possible for these models, because the fixed effects are part of the
optimization.
Using the PARMS Statement with a GLMM
If you are fitting a GLMM, the value-list corresponds to the parameters as listed in the “Covariance Parameter Estimates” table. Note that this order can change depending on whether a residual
variance is profiled or not; see the NOPROFILE option in the PROC GLIMMIX statement.
If you specify more than one set of initial values, PROC GLIMMIX performs a grid search of the
objective function surface and uses the best point on the grid for subsequent analysis. Specifying a
large number of grid points can result in long computing times.
Options in the PARMS Statement
You can specify the following options in the PARMS statement after a slash (/).
HOLD=value-list
specifies which parameter values PROC GLIMMIX should hold equal to the specified values.
For example, the following statement constrains the first and third covariance parameters to
equal 5 and 2, respectively:
2180 F Chapter 38: The GLIMMIX Procedure
parms (5) (3) (2) (3) / hold=1,3;
Covariance or scale parameters that are held fixed with the HOLD= option are treated as
constrained parameters in the optimization. This is different from evaluating the objective function, gradient, and Hessian matrix at known values of the covariance parameters.
A constrained parameter introduces a singularity in the optimization process. The covariance matrix of the covariance parameters (see the ASYCOV option of the PROC GLIMMIX statement) is then based on the projected Hessian matrix. As a consequence, the variance of parameters subjected to a HOLD= is zero. Such parameters do not contribute to
the computation of denominator degrees of freedom with the DDFM=KENWARDROGER
and DDFM=SATTERTHWAITE methods, for example. If you want to treat the covariance
parameters as known, without imposing constraints on the optimization, you should use the
NOITER option.
When you place a hold on all parameters (or when you specify the NOITER) option in a
GLMM, you might notice that PROC GLIMMIX continues to produce an iteration history.
Unless your model is a linear mixed model, several recomputations of the pseudo-response
might be required in linearization-based methods to achieve agreement between the pseudodata and the covariance matrix. In other words, the GLIMMIX procedure continues to update
the fixed-effects estimates (and random-effects solutions) until convergence is achieved.
In certain models, placing a hold on covariance parameters implies that the procedure processes the parameters in the same order as if the NOPROFILE were in effect. This can
change the order of the covariance parameters when you place a hold on one or more parameters. Models that are subject to this reordering are those with R-side covariance structures whose scale parameter could be profiled. This includes the TYPE=CS, TYPE=SP,
TYPE=AR, TYPE=TOEP, and TYPE=ARMA covariance structures.
LOWERB=value-list
enables you to specify lower boundary constraints for the covariance or scale parameters.
The value-list specification is a list of numbers or missing values (.) separated by commas.
You must list the numbers in the same order that PROC GLIMMIX uses for the value-list
in the PARMS statement, and each number corresponds to the lower boundary constraint.
A missing value instructs PROC GLIMMIX to use its default constraint, and if you do not
specify numbers for all of the covariance parameters, PROC GLIMMIX assumes that the
remaining ones are missing.
This option is useful, for example, when you want to constrain the G matrix to be positive
definite in order to avoid the more computationally intensive algorithms required when G becomes singular. The corresponding statements for a random coefficients model are as follows:
proc glimmix;
class person;
model y = time;
random int time / type=chol sub=person;
parms / lowerb=1e-4,.,1e-4;
run;
PARMS Statement F 2181
Here, the TYPE=CHOL structure is used in order to specify a Cholesky root parameterization
for the 2 2 unstructured blocks in G. This parameterization ensures that the G matrix is
nonnegative definite, and the PARMS statement then ensures that it is positive definite by
constraining the two diagonal terms to be greater than or equal to 1E 4.
NOBOUND
requests the removal of boundary constraints on covariance and scale parameters in mixed
models. For example, variance components have a default lower boundary constraint of 0,
and the NOBOUND option allows their estimates to be negative. See the NOBOUND option
in the PROC GLIMMIX statement for further details.
NOITER
requests that no optimization of the covariance parameters be performed. This option has no
effect in generalized linear models.
If you specify the NOITER option, PROC GLIMMIX uses the values for the covariance
parameters given in the PARMS statement to perform statistical inferences. Note that the
NOITER option is not equivalent to specifying a HOLD= value for all covariance parameters.
If you use the NOITER option, covariance parameters are not constrained in the optimization.
This prevents singularities that might otherwise occur in the optimization process.
If a residual variance is profiled, the parameter estimates can change from the initial values
you provide as the residual variance is recomputed. To prevent an update of the residual
variance, combine the NOITER option with the NOPROFILE option in the PROC GLIMMIX
statements, as in the following code:
proc glimmix noprofile;
class A B C rep mp sp;
model y = A | B | C;
random rep mp sp;
parms (180) (200) (170) (1000) / noiter;
run;
When you specify the NOITER option in a model where parameters are estimated by pseudolikelihood techniques, you might notice that the GLIMMIX procedure continues to produce
an iteration history. Unless your model is a linear mixed model, several recomputations
of the pseudo-response might be required in linearization-based methods to achieve agreement between the pseudo-data and the covariance matrix. In other words, the GLIMMIX
procedure continues to update the profiled fixed-effects estimates (and random-effects solutions) until convergence is achieved. To prevent these updates, use the MAXLMMUPDATE=
option in the PROC GLIMMIX statement. Specifying the NOITER option in the PARMS
statement of a GLMM with pseudo-likelihood estimation has the same effect as choosing
TECHNIQUE=NONE in the NLOPTIONS statement.
If you want to base initial fixed-effects estimates on the results of fitting a generalized linear
model, then you can combine the NOITER option with the TECHNIQUE= option. For example, the following statements determine the starting values for the fixed effects by fitting a
logistic model (without random effects) with the Newton-Raphson algorithm:
2182 F Chapter 38: The GLIMMIX Procedure
proc glimmix startglm inititer=10;
class clinic A;
model y/n = A / link=logit dist=binomial;
random clinic;
parms (0.4) / noiter;
nloptions technique=newrap;
run;
The initial GLM fit stops at convergence or after at most 10 iterations, whichever comes
first. The pseudo-data for the linearized GLMM is computed from the GLM estimates. The
variance of the Clinic random effect is held constant at 0.4 during subsequent iterations that
update the fixed effects only.
If you also want to combine the GLM fixed-effects estimates with known and fixed covariance
parameter values without updating the fixed effects, you can add the MAXLMMUPDATE=0
option:
proc glimmix startglm inititer=10 maxlmmupdate=0;
class clinic A;
model y/n = A / link=logit dist=binomial;
random clinic;
parms (0.4) / noiter;
nloptions technique=newrap;
run;
In a GLMM with parameter estimation by METHOD=LAPLACE or METHOD=QUAD the
NOITER option also leads to an iteration history, since the fixed-effects estimates are part of
the optimization and the PARMS statement places restrictions on only the covariance parameters.
Finally, the NOITER option can be useful if you want to obtain minimum variance quadratic
unbiased estimates (with 0 priors), also known as MIVQUE0 estimates (Goodnight 1978b).
Because MIVQUE0 estimates are starting values for covariance parameters—unless you provide (value-list) in the PARMS statement—the following statements produce MIVQUE0
mixed model estimates:
proc glimmix noprofile;
class A B;
model y = A;
random int / subject=B;
parms / noiter;
run;
PARMSDATA=SAS-data-set
PDATA=SAS-data-set
reads in covariance parameter values from a SAS data set. The data set should contain the
numerical variable ESTIMATE or the numerical variables Covp1–Covpq, where q denotes the
number of covariance parameters.
If the PARMSDATA= data set contains multiple sets of covariance parameters, the GLIMMIX
procedure evaluates the initial objective function for each set and commences the optimization
PARMS Statement F 2183
step by using the set with the lowest function value as the starting values. For example, the
following SAS statements request that the objective function be evaluated for three sets of
initial values:
data data_covp;
input covp1-covp4;
datalines;
180 200 170 1000
170 190 160 900
160 180 150 800
;
proc glimmix;
class A B C rep mainEU smallEU;
model yield = A|B|C;
random rep mainEU smallEU;
parms / pdata=data_covp;
run;
Each set comprises four covariance parameters.
The order of the observations in a data set with the numerical variable Estimate corresponds
to the order of the covariance parameters in the “Covariance Parameter Estimates” table. In
a GLM, the PARMSDATA= option can be used to set the starting value for the exponential
family scale parameter. A grid search is not conducted for GLMs if you specify multiple
values.
The PARMSDATA= data set must not contain missing values.
If the GLIMMIX procedure is processing the input data set in BY groups, you can add the
BY variables to the PARMSDATA= data set. If this data set is sorted by the BY variables, the
GLIMMIX procedure matches the covariance parameter values to the current BY group. If
the PARMSDATA= data set does not contain all BY variables, the data set is processed in its
entirety for every BY group and a message is written to the log. This enables you to provide
a single set of starting values across BY groups, as in the following statements:
data data_covp;
input covp1-covp4;
datalines;
180 200 170 1000
;
proc glimmix;
class A B C rep mainEU smallEU;
model yield = A|B|C;
random rep mainEU smallEU;
parms / pdata=data_covp;
by year;
run;
The same set of starting values is used for each value of the year variable.
UPPERB=value-list
enables you to specify upper boundary constraints on the covariance parameters. The valuelist specification is a list of numbers or missing values (.) separated by commas. You must list
the numbers in the same order that PROC GLIMMIX uses for the value-list in the PARMS
2184 F Chapter 38: The GLIMMIX Procedure
statement, and each number corresponds to the upper boundary constraint. A missing value
instructs PROC GLIMMIX to use its default constraint. If you do not specify numbers for all
of the covariance parameters, PROC GLIMMIX assumes that the remaining ones are missing.
RANDOM Statement
RANDOM random-effects < / options > ;
Using notation from “Notation for the Generalized Linear Mixed Model” on page 2081, the RANDOM statement defines the Z matrix of the mixed model, the random effects in the vector, the
structure of G, and the structure of R.
The Z matrix is constructed exactly like the X matrix for the fixed effects, and the G matrix is
constructed to correspond to the effects constituting Z. The structures of G and R are defined
by using the TYPE= option described on page 2191. The random effects can be classification or
continuous effects, and multiple RANDOM statements are possible.
Some reserved keywords have special significance in the random-effects list. You can specify INTERCEPT (or INT) as a random effect to indicate the intercept. PROC GLIMMIX does not include
the intercept in the RANDOM statement by default as it does in the MODEL statement. You can
specify the _RESIDUAL_ keyword (or RESID, RESIDUAL, _RESID_) before the option slash
(/) to indicate a residual-type (R-side) random component that defines the R matrix. Basically, the
_RESIDUAL_ keyword takes the place of the random-effect if you want to specify R-side variances
and covariance structures. These keywords take precedence over variables in the data set with the
same name. If your data or the covariance structure requires that an effect is specified, you can
use the RESIDUAL option to instruct the GLIMMIX procedure to model the R-side variances and
covariances.
In order to add an overdispersion component to the variance function, simply specify a single residual random component. For example, the following statements fit a polynomial Poisson regression
model with overdispersion. The variance function a./ D is replaced by a./:
proc glimmix;
model count = x x*x / dist=poisson;
random _residual_;
run;
Table 38.12 summarizes important options in the RANDOM statement. All options are subsequently discussed in alphabetical order.
Table 38.12
Option
Summary of Important RANDOM Statement Options
Description
Construction of Covariance Structure
GCOORD=
determines coordinate association for G-side spatial structures
with repeat levels
GROUP=
varies covariance parameters by groups
LDATA=
specifies a data set with coefficient matrices for TYPE= LIN
RANDOM Statement F 2185
Table 38.12
continued
Option
Description
NOFULLZ
RESIDUAL
SUBJECT=
TYPE=
eliminates columns in Z corresponding to missing values
designates a covariance structure as R-side
identifies the subjects in the model
specifies the covariance structure
Mixed Model Smoothing
KNOTINFO
displays spline knots
KNOTMAX=
specifies the upper limit for knot construction
KNOTMETHOD
specifies the method for constructing knots for radial smoother and
penalized B-splines
KNOTMIN=
specifies the lower limit for knot construction
Statistical Output
ALPHA=˛
CL
G
GC
GCI
GCORR
GI
SOLUTION
V
VC
VCI
VCORR
VI
determines the confidence level (1 ˛)
requests confidence limits for predictors of random effects
displays the estimated G matrix
displays the Cholesky root (lower) of the estimated G matrix
displays the inverse Cholesky root (lower) of the estimated G matrix
displays the correlation matrix that corresponds to the estimated G
matrix
displays the inverse of the estimated G matrix
displays solutions b
of the G-side random effects
displays blocks of the estimated V matrix
displays the lower-triangular Cholesky root of blocks of the estimated V matrix
displays the inverse Cholesky root of blocks of the estimated V
matrix
displays the correlation matrix corresponding to blocks of the estimated V matrix
displays the inverse of the blocks of the estimated V matrix
You can specify the following options in the RANDOM statement after a slash (/).
ALPHA=number
requests that a t-type confidence interval with confidence level 1 number be constructed
for the predictors of G-side random effects in this statement. The value of number must be
between 0 and 1; the default is 0.05. Specifying the ALPHA= option implies the CL option.
CL
requests that t-type confidence limits be constructed for each of the predictors of G-side random effects in this statement. The confidence level is 0.95 by default; this can be changed
with the ALPHA= option. The CL option implies the SOLUTION option.
2186 F Chapter 38: The GLIMMIX Procedure
G
requests that the estimated G matrix be displayed for G-side random effects associated with
this RANDOM statement. PROC GLIMMIX displays blanks for values that are 0.
GC
displays the lower-triangular Cholesky root of the estimated G matrix for G-side random
effects.
GCI
displays the inverse Cholesky root of the estimated G matrix for G-side random effects.
GCOORD=LAST
GCOORD=FIRST
GCOORD=MEAN
determines how the GLIMMIX procedure associates coordinates for TYPE=SP() covariance
structures with effect levels for G-side random effects. In these covariance structures, you
specify one or more variables that identify the coordinates of a data point. The levels of
classification variables, on the other hand, can occur multiple times for a particular subject.
For example, in the following statements the same level of A can occur multiple times, and
the associated values of x might be different:
proc glimmix;
class A B;
model y = B;
random A / type=sp(pow)(x);
run;
The GCOORD=LAST option determines the coordinates for a level of the random effect
from the last observation associated with the level. Similarly, the GCOORD=FIRST and
GCOORD=MEAN options determine the coordinate from the first observation and from the
average of the observations. Observations not used in the analysis are not considered in
determining the first, last, or average coordinate. The default is GCOORD=LAST.
GCORR
displays the correlation matrix that corresponds to the estimated G matrix for G-side random
effects.
GI
displays the inverse of the estimated G matrix for G-side random effects.
GROUP=effect
GRP=effect
identifies groups by which to vary the covariance parameters. Each new level of the grouping
effect produces a new set of covariance parameters. Continuous variables and computed
variables are permitted as group effects. PROC GLIMMIX does not sort by the values of
the continuous variable; rather, it considers the data to be from a new group whenever the
value of the continuous variable changes from the previous observation. Using a continuous
variable decreases execution time for models with a large number of groups and also prevents
the production of a large “Class Levels Information” table.
RANDOM Statement F 2187
Specifying a GROUP effect can greatly increase the number of estimated covariance parameters, which can adversely affect the optimization process.
KNOTINFO
displays the number and coordinates of the knots as determined by the KNOTMETHOD=
option.
KNOTMAX=number-list
provides upper limits for the values of random effects used in the construction of knots for
TYPE=RSMOOTH. The items in number-list correspond to the random effects of the radial
smooth. If the KNOTMAX= option is not specified, or if the value associated with a particular random effect is set to missing, the maximum is based on the values in the data set for
KNOTMETHOD=EQUAL or KNOTMETHOD=KDTREE, and is based on the values in the
knot data set for KNOTMETHOD=DATA.
KNOTMETHOD=KDTREE< (tree-options) >
KNOTMETHOD=EQUAL< (number-list) >
KNOTMETHOD=DATA(SAS-data-set)
determines the method of constructing knots for the radial smoother fit with the
TYPE=RSMOOTH covariance structure and the TYPE=PSPLINE covariance structure.
Unless you select the TYPE=RSMOOTH or TYPE=PSPLINE covariance structure, the
KNOTMETHOD= option has no effect. The default for TYPE=RSMOOTH is KNOTMETHOD=KDTREE. For TYPE=PSPLINE, only equally spaced knots are used and you can
use the optional numberlist argument of KNOTMETHOD=EQUAL to determine the number
of interior knots for TYPE=PSPLINE.
Knot Construction for TYPE=RSMOOTH
PROC GLIMMIX fits a low-rank smoother, meaning that the number of knots is considerably
less than the number of observations. By default, PROC GLIMMIX determines the knot
locations based on the vertices of a k-d tree (Friedman, Bentley, and Finkel 1977; Cleveland
and Grosse 1991). The k-d tree is a tree data structure that is useful for efficiently determining
the m nearest neighbors of a point. The k-d tree also can be used to obtain a grid of points
that adapts to the configuration of the data. The process starts with a hypercube that encloses
the values of the random effects. The space is then partitioned recursively by splitting cells at
the median of the data in the cell for the random effect. The procedure is repeated for all cells
that contain more than a specified number of points, b. The value b is called the bucket size.
The k-d tree is thus a division of the data into cells such that cells representing leaf nodes
contain at most b values. You control the building of the k-d tree through the BUCKET=
tree-option. You control the construction of knots from the cell coordinates of the tree with
the other options as follows.
BUCKET=number
determines the bucket size b. A larger bucket size will result in fewer knots. For k-d
trees in more than one dimension, the correspondence between bucket size and number
of knots is difficult to determine. It depends on the data configuration and on other
2188 F Chapter 38: The GLIMMIX Procedure
suboptions. In the multivariate case, you might need to try out different bucket sizes
to obtain the desired number of knots. The default value of number is 4 for univariate
trees (a single random effect) and b0:1nc in the multidimensional case.
KNOTTYPE=type
specifies whether the knots are based on vertices of the tree cells or the centroid.
The two possible values of type are VERTEX and CENTER. The default is KNOTTYPE=VERTEX. For multidimensional smoothing, such as smoothing across irregularly shaped spatial domains, the KNOTTYPE=CENTER option is useful to move knot
locations away from the bounding hypercube toward the convex hull.
NEAREST
specifies that knot coordinates are the coordinates of the nearest neighbor of either the
centroid or vertex of the cell, as determined by the KNOTTYPE= suboption.
TREEINFO
displays details about the construction of the k-d tree, such as the cell splits and the
split values.
See the section “Knot Selection” on page 2249 for a detailed example of how the specification
of the bucket size translates into the construction of a k-d tree and the spline knots.
The KNOTMETHOD=EQUAL option enables you to define a regular grid of knots. By
default, PROC GLIMMIX constructs 10 knots for one-dimensional smooths and 5 knots in
each dimension for smoothing in higher dimensions. You can specify a different number
of knots with the optional number-list. Missing values in the number-list are replaced with
the default values. A minimum of two knots in each dimension is required. For example,
the following statements use a rectangular grid of 35 knots, five knots for x1 combined with
seven knots for x2:
proc glimmix;
model y=;
random x1 x2 / type=rsmooth knotmethod=equal(5 7);
run;
When you use the NOFIT option in the PROC GLIMMIX statement, the GLIMMIX procedure computes the knots but does not fit the model. This can be useful if you want to compare
knot selections with different suboptions of KNOTMETHOD=KDTREE. Suppose you want
to determine the number of knots based on a particular bucket size. The following statements
compute and display the knots in a bivariate smooth, constructed from nearest neighbors of
the vertices of a k-d tree with bucket size 10:
proc glimmix nofit;
model y = Latitude Longitude;
random Latitude Longitude / type=rsmooth
knotmethod=kdtree(knottype=vertex
nearest bucket=10) knotinfo;
run;
You can specify a data set that contains variables whose values give the knot coordinates
with the KNOTMETHOD=DATA option. The data set must contain numeric variables with
the same name as the radial smoothing random-effects. PROC GLIMMIX uses only the
RANDOM Statement F 2189
unique knot coordinates in the knot data set. This option is useful to provide knot coordinates
different from those that can be produced from a k-d tree. For example, in spatial problems
where the domain is irregularly shaped, you might want to determine knots by a space-filling
algorithm. The following SAS statements invoke the OPTEX procedure to compute 45 knots
that uniformly cover the convex hull of the data locations (see Chapter 30, “Introduction to
the OPTEX Procedure,” (SAS/QC User’s Guide) and Chapter 31, “Details of the OPTEX
Procedure,” (SAS/QC User’s Guide) for details about the OPTEX procedure).
proc optex coding=none;
model latitude longitude / noint;
generate n=45 criterion=u method=m_fedorov;
output out=knotdata;
run;
proc glimmix;
model y = Latitude Longitude;
random Latitude Longitude / type=rsmooth
knotmethod=data(knotdata);
run;
Knot Construction for TYPE=PSPLINE
Only evenly spaced knots are supported when you fit penalized B-splines with the GLIMMIX
procedure. For the TYPE=PSPLINE covariance structure, the numberlist argument specifies
the number m of interior knots, the default is m D 10. Suppose that x.1/ and x.n/ denote
the smallest and largest values, respectively. For a B-spline of degree d (de Boor 2001),
the interior knots are supplemented with d exterior knots below x.1/ and maxf1; d g exterior
knots above x.n/ . PROC GLIMMIX computes the location of these m C d C maxf1; d g knots
as follows. Let ıx D .x.n/ x.1/ /=.m C 1/, then interior knots are placed at
x.1/ C j ıx ;
j D 1; ; m
The exterior knots are also evenly spaced with step size ıx and start at x.1/ ˙ 100 times the
machine epsilon. At least one interior knot is required.
KNOTMIN=number-list
provides lower limits for the values of random effects used in the construction of knots for
TYPE=RSMOOTH. The items in number-list correspond to the random effects of the radial
smooth. If the KNOTMIN= option is not specified, or if the value associated with a particular random effect is set to missing, the minimum is based on the values in the data set for
KNOTMETHOD=EQUAL or KNOTMETHOD=KDTREE, and is based on the values in the
knot data set for KNOTMETHOD=DATA.
LDATA=SAS-data-set
reads the coefficient matrices A1 ; ; Aq for the TYPE=LIN(q) option. You can specify the
LDATA= data set in a sparse or dense form. In the sparse form the data set must contain
the numeric variables Parm, Row, Col, and Value. The Parm variable contains the indices
i D 1; ; q of the Ai matrices. The Row and Col variables identify the position within a
matrix and the Value variable contains the matrix element. Values not specified for a particular
row and column are set to zero. Missing values are allowed in the Value column of the
2190 F Chapter 38: The GLIMMIX Procedure
LDATA= data set; these values are also replaced by zeros. The sparse form is particularly
useful if the A matrices have only a few nonzero elements.
In the dense form the LDATA= data set contains the numeric variables Parm and Row (with
the same function as above), in addition to the numeric variables Col1–Colq. If you omit one
or more of the Col1–Colq variables from the data set, zeros are assumed for the respective
rows and columns of the A matrix. Missing values for Col1–Colq are ignored in the dense
form.
The GLIMMIX procedure assumes that the matrices A1 ; ; Aq are symmetric. In the sparse
LDATA= form you do not need to specify off-diagonal elements in position .i; j / and .j; i /.
One of them is sufficient. Row-column indices are converted in both storage forms into
positions in lower triangular storage. If you specify multiple values in row .maxfi; j g and
column minfi; j g/ of a particular matrix, only the last value is used. For example, assume
you are specifying elements of a 4 4 matrix. The lower triangular storage of matrix A3
defined by
data ldata;
input parm row col value;
datalines;
3 2 1 2
3 1 2 5
;
is
2
3
0
6 5 0
7
6
7
4 0 0 0
5
0 0 0 0
NOFULLZ
eliminates the columns in Z corresponding to missing levels of random effects involving
CLASS variables. By default, these columns are included in Z. It is sufficient to specify the
NOFULLZ option on any G-side RANDOM statement.
RESIDUAL
RSIDE
specifies that the random effects listed in this statement be R-side effects. You use the RESIDUAL option in the RANDOM statement if the nature of the covariance structure requires you
to specify an effect. For example, if it is necessary to order the columns of the R-side AR(1)
covariance structure by the time variable, you can use the RESIDUAL option as in the following statements:
class time id;
random time / subject=id type=ar(1) residual;
SOLUTION
S
requests that the solution b
for the random-effects parameters be produced, if the statement
defines G-side random effects.
RANDOM Statement F 2191
The numbers displayed in the Std Err Pred column of the “Solution for Random Effects”
table are not the standard errors of the b
displayed in the Estimate column; rather, they are the
square roots of the prediction errors b
i i , where b
i is the predictor of the i th random effect
and i is the ith random effect. In pseudo-likelihood methods that are based on linearization,
these EBLUPs are the estimated best linear unbiased predictors in the linear mixed pseudomodel. In models fit by maximum likelihood by using the Laplace approximation or by using
adaptive quadrature, the SOLUTION option displays the empirical Bayes estimates (EBE) of
i .
SUBJECT=effect
SUB=effect
identifies the subjects in your generalized linear mixed model. Complete independence is
assumed across subjects. Specifying a subject effect is equivalent to nesting all other effects
in the RANDOM statement within the subject effect.
Continuous variables and computed variables are permitted with the SUBJECT= option.
PROC GLIMMIX does not sort by the values of the continuous variable but considers the
data to be from a new subject whenever the value of the continuous variable changes from the
previous observation. Using a continuous variable can decrease execution time for models
with a large number of subjects and also prevents the production of a large “Class Levels
Information” table.
TYPE=covariance-structure
specifies the covariance structure of G for G-side effects and the covariance structure of R for
R-side effects.
Although a variety of structures are available, many applications call for either simple diagonal (TYPE=VC) or unstructured covariance matrices. The TYPE=VC (variance components)
option is the default structure, and it models a different variance component for each random effect. It is recommended to model unstructured covariance matrices in terms of their
Cholesky parameterization (TYPE=CHOL) rather than TYPE=UN.
If you want different covariance structures in different parts of G, you must use multiple
RANDOM statements with different TYPE= options.
Valid values for covariance-structure are as follows. Examples are shown in Table 38.14.
The variances and covariances in the formulas that follow in the TYPE= descriptions are
expressed in terms of generic random variables i and j . They represent the G-side random
effects or the residual random variables for which the G or R matrices are constructed.
ANTE(1)
specifies a first-order ante-dependence structure (Kenward 1987; Patel 1991) parameterized in terms of variances and correlation parameters. If t ordered random variables
1 ; ; t have a first-order ante-dependence structure, then each j , j > 1, is independent of all other k ; k < j , given j 1 . This Markovian structure is characterized
by its inverse variance matrix, which is tridiagonal. Parameterizing an ANTE(1) structure for a random vector of size t requires 2t 1 parameters: variances 12 ; ; t2 and
t 1 correlation parameters 1 ; ; t 1 . The covariances among random variables i
2192 F Chapter 38: The GLIMMIX Procedure
and j are then constructed as
jY1
q
k
Cov i ; j D i2 j2
kDi
PROC GLIMMIX constrains the correlation parameters to satisfy jk j < 1; 8k. For
variable-order ante-dependence models see Macchiavelli and Arnold (1994).
AR(1)
specifies a first-order autoregressive structure,
Cov i ; j D 2 ji j j
The values i and j are derived for the i th and j th observations, respectively, and are
not necessarily the observation numbers. For example, in the following statements the
values correspond to the class levels for the time effect of the i th and j th observation
within a particular subject:
proc glimmix;
class time patient;
model y = x x*x;
random time / sub=patient type=ar(1);
run;
PROC GLIMMIX imposes the constraint jj < 1 for stationarity.
ARH(1)
specifies a heterogeneous first-order autoregressive structure,
q
Cov i ; j D i2 j2 ji j j
with jj < 1. This covariance structure has the same correlation pattern as the
TYPE=AR(1) structure, but the variances are allowed to differ.
ARMA(1,1)
specifies the first-order autoregressive moving-average structure,
2
i Dj
Cov i ; j D
2 ji j j 1 i 6D j
Here, is the autoregressive parameter, models a moving-average component, and
2 is a scale parameter. In the notation of Fuller (1976, p. 68), D 1 and
D
.1 C b1 1 /.1 C b1 /
1 C b12 C 2b1 1
The example in Table 38.14 and jb1 j < 1 imply that
p
ˇ
ˇ 2 4˛ 2
b1 D
2˛
RANDOM Statement F 2193
where ˛ D and ˇ D 1 C 2 2. PROC GLIMMIX imposes the constraints
jj < 1 and jj < 1 for stationarity, although for some values of and in this region
the resulting covariance matrix is not positive definite. When the estimated value of becomes negative, the computed covariance is multiplied by cos.dij / to account for
the negativity.
CHOL< (q) >
specifies an unstructured variance-covariance matrix parameterized through its
Cholesky root. This parameterization ensures that the resulting variance-covariance
matrix is at least positive semidefinite. If all diagonal values are nonzero, it is positive
definite. For example, a 2 2 unstructured covariance matrix can be written as
1 12
VarŒ D
12 2
Without imposing constraints on the three parameters, there is no guarantee that the
estimated variance matrix is positive definite. Even if 1 and 2 are nonzero, a large
value for 12 can lead to a negative eigenvalue of VarŒ. The Cholesky root of a
positive definite matrix A is a lower triangular matrix C such that CC0 D A. The
Cholesky root of the above 2 2 matrix can be written as
˛1 0
CD
˛12 ˛2
The elements of the unstructured variance matrix are then simply 1 D ˛12 , 12 D
2
˛1 ˛12 , and 2 D ˛12
C ˛22 . Similar operations yield the generalization to covariance
matrices of higher orders.
For example, the following statements model the covariance matrix of each subject as
an unstructured matrix:
proc glimmix;
class sub;
model y = x;
random _residual_ / subject=sub type=un;
run;
The next set of statements accomplishes the same, but the estimated R matrix is guaranteed to be nonnegative definite:
proc glimmix;
class sub;
model y = x;
random _residual_ / subject=sub type=chol;
run;
The GLIMMIX procedure constrains the diagonal elements of the Cholesky root to be
positive. This guarantees a unique solution when the matrix is positive definite.
The optional order parameter q > 0 determines how many bands below the diagonal
are modeled. Elements in the lower triangular portion of C in bands higher than q are
2194 F Chapter 38: The GLIMMIX Procedure
set to zero. If you consider the resulting covariance matrix A D CC0 , then the order
parameter has the effect of zeroing all off-diagonal elements that are at least q positions
away from the diagonal.
Because of its good computational and statistical properties, the Cholesky root parameterization is generally recommended over a completely unstructured covariance matrix
(TYPE=UN). However, it is computationally slightly more involved.
CS
specifies the compound-symmetry structure, which has constant variance and constant
covariance
C i Dj
Cov i ; j D
i 6D j
The compound symmetry structure arises naturally with nested random effects, such as
when subsampling error is nested within experimental error. The models constructed
with the following two sets of GLIMMIX statements have the same marginal variance
matrix, provided is positive:
proc glimmix;
class block A;
model y = block A;
random block*A / type=vc;
run;
proc glimmix;
class block A;
model y = block A;
random _residual_ / subject=block*A
type=cs;
run;
In the first case, the block*A random effect models the G-side experimental error.
Because the distribution defaults to the normal, the R matrix is of form I (see
Table 38.15), and is the subsampling error variance. The marginal variance for the
2
data from a particular experimental unit is thus ba
JCI. This matrix is of compound
symmetric form.
Hierarchical random assignments or selections, such as subsampling or split-plot designs, give rise to compound symmetric covariance structures. This implies exchangeability of the observations on the subunit, leading to constant correlations between the
observations. Compound symmetric structures are thus usually not appropriate for processes where correlations decline according to some metric, such as spatial and temporal processes.
Note that R-side compound-symmetry structures do not impose any constraint on .
You can thus use an R-side TYPE=CS structure to emulate a variance-component
model with unbounded estimate of the variance component.
RANDOM Statement F 2195
CSH
specifies the heterogeneous compound-symmetry structure, which is an equicorrelation structure but allows for different variances
8 q
< i2 j2 i D j
q
Cov i ; j D
: 2 2 i 6D j
i
j
FA(q)
specifies the factor-analytic structure with q factors (Jennrich and Schluchter 1986).
This structure is of the form ƒƒ0 C D, where ƒ is a t q rectangular matrix and D is
a t t diagonal matrix with t different parameters. When q > 1, the elements of ƒ in
its upper-right corner (that is, the elements in the i th row and j th column for j > i )
are set to zero to fix the rotation of the structure.
FA0(q)
specifies a factor-analytic structure with q factors of the form VarŒ D ƒƒ0 , where ƒ
is a t q rectangular matrix and t is the dimension of Y. When q > 1, ƒ is a lower
triangular matrix. When q < t—that is, when the number of factors is less than the
dimension of the matrix—this structure is nonnegative definite but not of full rank. In
this situation, you can use it to approximate an unstructured covariance matrix.
HF
specifies a covariance structure that satisfies the general Huynh-Feldt condition (Huynh
and Feldt 1970). For a random vector with t elements, this structure has t C 1 positive
parameters and covariances
2
i
i Dj
Cov i ; j D
0:5.i2 C j2 / i 6D j
A covariance matrix † generally satisfies the Huynh-Feldt condition if it can be written
as † D 10 C1 0 CI. The preceding parameterization chooses i D 0:5.i2 /. Several simpler covariance structures give rise to covariance matrices that also satisfy the
Huynh-Feldt condition. For example, TYPE=CS, TYPE=VC, and TYPE=UN(1) are
nested within TYPE=HF. You can use the COVTEST statement to test the HF structure
against one of these simpler structures. Note also that the HF structure is nested within
an unstructured covariance matrix.
The TYPE=HF covariance structure can be sensitive to the choice of starting values and
the default MIVQUE(0) starting values can be poor for this structure; you can supply
your own starting values with the PARMS statement.
LIN(q)
specifies a general linear covariance structure with q parameters. This structure consists
of a linear combination of known matrices that you input with the LDATA= option.
Suppose that you want to model the covariance of a random vector of length t , and
further suppose that A1 ; ; Aq are symmetric .t t) matrices constructed from the
information in the LDATA= data set. Then,
q
X
Cov i ; j D
k ŒAk ij
kD1
2196 F Chapter 38: The GLIMMIX Procedure
where ŒAk ij denotes the element in row i , column j of matrix Ak .
Linear structures are very flexible and general. You need to exercise caution to ensure
that the variance matrix is positive definite. Note that PROC GLIMMIX does not impose boundary constraints on the parameters 1 ; ; k of a general linear covariance
structure. For example, if classification variable A has 6 levels, the following statements
fit a variance component structure for the random effect without boundary constraints:
data ldata;
retain parm 1 value 1;
do row=1 to 6; col=row; output; end;
run;
proc glimmix data=MyData;
class A B;
model Y = B;
random A / type=lin(1) ldata=ldata;
run;
PSPLINE< (options) >
requests that PROC GLIMMIX form a B-spline basis and fits a penalized B-spline
(P-spline, Eilers and Marx 1996) with random spline coefficients. This covariance
structure is available only for G-side random effects and only a single continuous random effect can be specified with TYPE=PSPLINE. As for TYPE=RSMOOTH, PROC
GLIMMIX forms a modified Z matrix and fits a mixed model in which the random
variables associated with the columns of Z are independent with a common variance.
The Z matrix is constructed as follows.
Denote as e
Z the .n K/ matrix of B-splines of degree d and denote as Dr the .K
r K/ matrix of rth-order differences. For example, for K D 5,
2
1
6 0
D1 D 6
4 0
0
2
1
D2 D 4 0
0
1
D3 D
0
3
0
0 7
7
0 5
1
3
1
1
0
0
0
1
1
0
0
0
1
1
2
1
0
1
2
1
3
1
3
3
0 0
1 0 5
2 1
1
0
3
1
Then, the Z matrix used in fitting the mixed model is the .n K
r/ matrix
ZDe
Z.D0r Dr / D0r
The construction of the B-spline knots is controlled with the KNOTMETHOD=
EQUAL(m) option and the DEGREE=d suboption of TYPE=PSPLINE. The total number of knots equals the number m of equally spaced interior knots plus d knots at the
RANDOM Statement F 2197
low end and maxf1; d g knots at the high end. The number of columns in the B-spline
basis equals K D m C d C 1. By default, the interior knots exclude the minimum and
maximum of the random-effect values and are based on m 1 equally spaced intervals.
Suppose x.1/ and x.n/ are the smallest and largest random-effect values; then interior
knots are placed at
x.1/ C j.x.n/
x.1/ /=.m C 1/;
j D 1; ; m
In addition, d evenly spaced exterior knots are placed below x.1/ and maxfd; 1g exterior knots are placed above x.m/ . The exterior knots are evenly spaced and start at
x.1/ ˙ 100 times the machine epsilon. For example, based on the defaults d D 3,
r D 3, the following statements lead to 26 total knots and 21 columns in Z, m D 20,
K D m C d C 1 D 24, K r D 21:
proc glimmix;
model y = x;
random x / type=pspline knotmethod=equal(20);
run;
Details about the computation and properties of B-splines can be found in de Boor
(2001).
You can extend or limit the range of the knots with the KNOTMIN= and KNOTMAX=
options. Table 38.13 lists some of the parameters that control this covariance type and
their relationships.
Table 38.13
P-Spline Parameters
Parameter
Description
d
r
m
m C d C maxf1; d g
K DmCd C1
K r
Degree of B-spline, default d D 3
Order of differencing in construction of Dr , default r D 3
Number of interior knots, default m D 10
Total number of knots
Number of columns in B-spline basis
Number of columns in Z
You can specify the following options for TYPE=PSPLINE:
DEGREE=d
specifies the degree of the B-spline. The default is d D 3.
DIFFORDER=r
specifies the order of the differencing matrix Dr . The default and
maximum is r D 3.
RSMOOTH< (m | NOLOG) >
specifies a radial smoother covariance structure for G-side random effects. This results in an approximate low-rank thin-plate spline where the smoothing parameter is
obtained by the estimation method selected with the METHOD= option of the PROC
GLIMMIX statement. The smoother is based on the automatic smoother in Ruppert,
Wand, and Carroll (2003, Chapter 13.4–13.5), but with a different method of selecting the spline knots. See the section “Radial Smoothing Based on Mixed Models”
2198 F Chapter 38: The GLIMMIX Procedure
on page 2248 for further details about the construction of the smoother and the knot
selection.
Radial smoothing is possible in one or more dimensions. A univariate smoother is
obtained with a single random effect, while multiple random effects in a RANDOM
statement yield a multivariate smoother. Only continuous random effects are permitted
with this covariance structure. If nr denotes the number of continuous random effects
in the RANDOM statement, then the covariance structure of the random effects is
determined as follows. Suppose that zi denotes the vector of random effects for the i th
observation. Let k denote the .nr 1/ vector of knot coordinates, k D 1; ; K, and
K is the total number of knots. The Euclidean distance between the knots is computed
as
v
uX
u nr
dkp D jjk p jj D t
.j k jp /2
j D1
and the distance between knots and effects is computed as
v
uX
u nr
.zij j k /2
hik D jjzi k jj D t
j D1
The Z matrix for the GLMM is constructed as
ZDe
Z
1=2
where the .n K/ matrix e
Z has typical element
p
hik
nr odd
Œe
Zik D
p
hik logfhi k g nr even
and the .K K/ matrix  has typical element
( p
nr odd
dkp
Œkp D
p
dkp logfdkp g nr even
The exponent in these expressions equals p D 2m nr , where the optional value m
corresponds to the derivative penalized in the thin-plate spline. A larger value of m will
yield a smoother fit. The GLIMMIX procedure requires p > 0 and chooses by default
m D 2 if nr < 3 and m D .nr C 1/=2 otherwise. The NOLOG option removes the
logfhik g and logfdkp g terms from the computation of the e
Z and  matrices when nr
is even; this yields invariance under rescaling of the coordinates.
Finally, the components of are assumed to have equal variance r2 . The “smoothing
parameter” of the low-rank spline is related to the variance components in the model,
2 D f .; r2 /. See Ruppert, Wand, and Carroll (2003) for details. If the conditional
distribution does not provide a scale parameter , you can add a single R-side residual
parameter.
The knot selection is controlled with the KNOTMETHOD= option. The GLIMMIX
procedure selects knots automatically based on the vertices of a k-d tree or reads knots
from a data set that you supply. See the section “Radial Smoothing Based on Mixed
Models” on page 2248 for further details on radial smoothing in the GLIMMIX procedure and its connection to a mixed model formulation.
RANDOM Statement F 2199
SIMPLE
is an alias for TYPE=VC.
SP(EXP)(c-list)
models an exponential spatial or temporal covariance structure, where the covariance
between two observations depends on a distance metric dij . The c-list contains the
names of the numeric variables used as coordinates to determine distance. For a
stochastic process in Rk , there are k elements in c-list. If the .k 1/ vectors of coordinates for observations i and j are ci and cj , then PROC GLIMMIX computes the
Euclidean distance
v
u k
uX
dij D jjci cj jj D t
.cmi cmj /2
mD1
The covariance between two observations is then
Cov i ; j D 2 expf dij =˛g
The parameter ˛ is not what is commonly referred to as the range parameter in geostatistical applications. The practical range of a (second-order stationary) spatial process is
the distance d .p/ at which the correlations fall below 0.05. For the SP(EXP) structure,
this distance is d .p/ D 3˛. PROC GLIMMIX constrains ˛ to be positive.
SP(GAU)(c-list)
models a gaussian covariance structure,
Cov i ; j D 2 expf dij2 =˛ 2 g
See TYPE=SP(EXP) for the computation of the distance dij . The parameter ˛ is related
.p/ is defined as the disto the range of the process as follows. If the practical range dp
tance at which the correlations fall below 0.05, then d .p/ D 3˛. PROC GLIMMIX
constrains ˛ to be positive. See TYPE=SP(EXP) for the computation of the distance
dij from the variables specified in c-list.
SP(MAT)(c-list)
models a covariance structure in the Matérn class of covariance functions (Matérn
1986). The covariance is expressed in the parameterization of Handcock and Stein
(1993) and Handcock and Wallis (1994); it can be written as
p p dij 2dij 2 1
Cov i ; j D 2K
€./
The function K is the modified Bessel function of the second kind of (real) order
> 0. The smoothness (continuity) of a stochastic process with covariance function
in the Matérn class increases with . This class thus enables data-driven estimation of
the smoothness properties of the process. The covariance is identical to the exponential
model for D 0:5 (TYPE=SP(EXP)(c-list )), while for D 1 the model advocated
by Whittle (1954) results. As ! 1, the model approaches the gaussian covariance
structure (TYPE=SP(GAU)(c-list )).
2200 F Chapter 38: The GLIMMIX Procedure
Note that the MIXED procedure offers covariance structures in the Matérn class
in two parameterizations, TYPE=SP(MATERN) and TYPE=SP(MATHSW). The
TYPE=SP(MAT) in the GLIMMIX procedure is equivalent to TYPE=SP(MATHSW)
in the MIXED procedure.
Computation of the function K and its derivatives is numerically demanding; fitting
models with Matérn covariance structures can be time-consuming. Good starting values
are essential.
SP(POW)(c-list)
models a power covariance structure,
Cov i ; j D 2 dij
where 0.
This is a reparameterization of the exponential structure,
TYPE=SP(EXP). Specifically, logfg D 1=˛. See TYPE=SP(EXP) for the computation of the distance dij from the variables specified in c-list. When the estimated value
of becomes negative, the computed covariance is multiplied by cos.dij / to account
for the negativity.
SP(POWA)(c-list)
models an anisotropic power covariance structure in k dimensions, provided that the
coordinate list c-list has k elements. If ci m denotes the coordinate for the i th observation of the mth variable in c-list, the covariance between two observations is given
by
jc
Cov i ; j D 2 1 i1
cj1 j jci 2 cj 2 j
jc
c j
2
: : : k i k j k
Note that for k D 1, TYPE=SP(POWA) is equivalent to TYPE=SP(POW), which is itself a reparameterization of TYPE=SP(EXP). When the estimated value of m becomes
negative, the computed covariance is multiplied by cos.jci m cj m j/ to account for
the negativity.
SP(SPH)(c-list)
models a spherical covariance structure,
8
< 2 1 3dij C
2˛
Cov i ; j D
:
0
1
2
dij
˛
3 dij ˛
dij > ˛
The spherical covariance structure has a true range parameter. The covariances between
observations are exactly zero when their distance exceeds ˛. See TYPE=SP(EXP) for
the computation of the distance dij from the variables specified in c-list.
TOEP
models a Toeplitz covariance structure. This structure can be viewed as an autoregressive structure with order equal to the dimension of the matrix,
2
i Dj
Cov i ; j D
ji j j i 6D j
RANDOM Statement F 2201
TOEP(q)
specifies a banded Toeplitz structure,
2
i Dj
Cov i ; j D
ji j j ji j j < q
This can be viewed as a moving-average structure with order equal to q 1. The
specification TYPE=TOEP(1) is the same as 2 I, and it can be useful for specifying
the same variance component for several effects.
TOEPH< (q) >
models a Toeplitz covariance structure. The correlations of this structure are banded as
the TOEP or TOEP(q) structures, but the variances are allowed to vary:
( 2
i Dj
i
q
Cov i ; j D
ji j j i2 j2 i 6D j
The correlation parameters satisfy jji j j j < 1. If you specify the optional value q, the
correlation parameters with ji j j q are set to zero, creating a banded correlation
structure. The specification TYPE=TOEPH(1) results in a diagonal covariance matrix
with heterogeneous variances.
UN< (q) >
specifies a completely general (unstructured) covariance matrix parameterized directly
in terms of variances and covariances,
Cov i ; j D ij
The variances are constrained to be nonnegative, and the covariances are unconstrained.
This structure is not constrained to be nonnegative definite in order to avoid nonlinear
constraints; however, you can use the TYPE=CHOL structure if you want this constraint to be imposed by a Cholesky factorization. If you specify the order parameter q,
then PROC GLIMMIX estimates only the first q bands of the matrix, setting elements
in all higher bands equal to 0.
UNR< (q) >
specifies a completely general (unstructured) covariance matrix parameterized in terms
of variances and correlations,
Cov i ; j D i j ij
where i denotes the standard deviation and the correlation ij is zero when i D j
and when ji j j q, provided the order parameter q is given. This structure fits the
same model as the TYPE=UN(q) option, but with a different parameterization. The i th
variance parameter is i2 . The parameter ij is the correlation between the i th and j th
measurements; it satisfies jij j < 1. If you specify the order parameter q, then PROC
GLIMMIX estimates only the first q bands of the matrix, setting all higher bands equal
to zero.
2202 F Chapter 38: The GLIMMIX Procedure
VC
specifies standard variance components and is the default structure for both G-side and
R-side covariance structures. In a G-side covariance structure, a distinct variance component is assigned to each effect. In an R-side structure TYPE=VC is usually used only
to add overdispersion effects or with the GROUP= option to specify a heterogeneous
variance model.
Table 38.14
Covariance Structure Examples
Description
Structure
Variance
Components
VC (default)
Compound
Symmetry
CS
Heterogeneous
CS
CSH
First-Order
Autoregressive
AR(1)
Heterogeneous
AR(1)
ARH(1)
Unstructured
UN
Banded Main
Diagonal
UN(1)
Unstructured
Correlations
UNR
Example
3
2 2
0
0
B 0
6 0 2
0
0 7
B
7
6
2
40
0 5
0 AB
2
0
0
0
AB
2
3
C
6 C
7
6
7
4 C
5
C
2 2
3
1
1 2 1 3 1 4 62 1 22
2 3 2 4 7
6
7
43 1 3 2 32
3 4 5
4 1 4 2 4 3 42
2
3
1 2 3
6 1 2 7
7
2 6
4 2 1 5
3 2 1
2
3
12
1 2 1 3 2 1 4 3
6 2 1 22
2 3 2 4 2 7
6
7
43 1 2 3 2 32
3 4 5
4 1 3 4 2 4 3 42
2 2
3
1 21 31 41
621 2 32 42 7
2
6
7
431 32 2 43 5
3
41 42 43 42
2 2
3
1 0
0
0
6 0 2 0
07
2
6
7
2
40
0 3 0 5
0
0
0 42
2
3
12
1 2 21 1 3 31 1 4 41
62 1 21
22
2 3 32 2 4 42 7
6
7
43 1 31 3 2 32
32
3 4 43 5
4 1 41 4 2 42 4 3 43
42
RANDOM Statement F 2203
Table 38.14
continued
Description
Structure
Toeplitz
TOEP
Toeplitz with
Two Bands
TOEP(2)
Heterogeneous
Toeplitz
TOEPH
Spatial
Power
SP(POW)(c-list )
First-Order
Autoregressive
Moving-Average
ARMA(1,1)
First-Order
Factor
Analytic
FA(1)
Huynh-Feldt
HF
First-Order
Ante-dependence
ANTE(1)
Example
2 2
1
6 1 2
6
4 2 1
3 2
2 2
1
6 1 2
6
4 0 1
0
0
2
2
1
62 1 1
6
43 1 2
4 1 3
2
1
d
6
21
2 6
4d31
d41
2
1
6
2 6
4 2
2 2
1 C d1
6 2 1
6
4 3 1
4 1
2
2
6 2 C 21
6 2 1
4 2
3
2 3
1 2 7
7
2 1 5
1 2
3
0
0
1 0 7
7
2 1 5
1 2
3
1 2 1 1 3 2 1 4 3
2 3 1 2 4 2 7
22
7
3 4 1 5
3 2 1
32
4 2 2 4 3 1
42
3
d12 d13 d14
1
d23 d24 7
7
d
32
1
d34 5
d42 d43
1
3
2
1 7
7
1
5
1
3
1 2
1 3
1 4
22 C d2
2 3
2 4 7
7
2
3 2
3 C d3
3 4 5
4 2
4 3
24 C d4
3
12 C22
12 C32
2
2
7
22 C32
7
2
2
5
2
2
2
2
2
3 C1
3 C2
2
3
2
2
2
3
2
1
1 2 1 1 3 1 2
4 2 1 1
22
2 3 2 5
3 1 2 1 3 2 2
32
V< =value-list >
requests that blocks of the estimated marginal variance-covariance matrix V.b
/ be displayed
in generalized linear mixed models. This matrix is based on the last linearization as described
in the section “The Pseudo-model” on page 2218. You can use the value-list to select the
subjects for which the matrix is displayed. If value-list is not specified, the V matrix for the
first subject is chosen.
Note that the value-list refers to subjects as the processing units in the “Dimensions” table.
For example, the following statements request that the estimated marginal variance matrix for
2204 F Chapter 38: The GLIMMIX Procedure
the second subject be displayed:
proc glimmix;
class A B;
model y = B;
random int / subject=A;
random int / subject=A*B v=2;
run;
The subject effect for processing in this case is the A effect, because it is contained in the A*B
interaction. If there is only a single subject as per the “Dimensions” table, then the V option
displays an .n n/ matrix.
See the section “Processing by Subjects” on page 2245 for how the GLIMMIX procedure
determines the number of subjects in the “Dimensions” table.
The GLIMMIX procedure displays blanks for values that are 0.
VC< =value-list >
displays the lower-triangular Cholesky root of the blocks of the estimated V.b
/ matrix. See
the V option for the specification of value-list.
VCI< =value-list >
displays the inverse Cholesky root of the blocks of the estimated V.b
/ matrix. See the V
option for the specification of value-list.
VCORR< =value-list >
displays the correlation matrix corresponding to the blocks of the estimated V.b
/ matrix. See
the V option for the specification of value-list.
VI< =value-list >
displays the inverse of the blocks of the estimated V.b
/ matrix. See the V option for the
specification of value-list.
WEIGHT Statement
WEIGHT variable ;
The WEIGHT statement replaces R with W 1=2 RW 1=2 , where W is a diagonal matrix containing
the weights. Observations with nonpositive or missing weights are not included in the resulting
PROC GLIMMIX analysis. If a WEIGHT statement is not included, all observations used in the
analysis are assigned a weight of 1.
Programming Statements F 2205
Programming Statements
This section lists the programming statements available in PROC GLIMMIX to compute various
aspects of the generalized linear mixed model or output quantities. For example, you can compute
model effects, weights, frequency, subject, group, and other variables. You can use programming
statements to define the mean and variance functions. This section also documents the differences
between programming statements in PROC GLIMMIX and programming statements in the SAS
DATA step. The syntax of programming statements used in PROC GLIMMIX is identical to that
used in the NLMIXED procedure (see Chapter 61, “The NLMIXED Procedure,” of the SAS/STAT
User’s Guide) and the MODEL procedure (see the SAS/ETS User’s Guide). Most of the programming statements that can be used in the DATA step can also be used in the GLIMMIX procedure.
Refer to SAS Language Reference: Dictionary for a description of SAS programming statements.
The following are valid statements:
ABORT;
CALL name [ ( expression [, expression . . . ] ) ];
DELETE;
DO [ variable = expression
[ TO expression] [ BY expression]
[, expression [ TO expression] [ BY expression ] . . . ]
]
[ WHILE expression ] [ UNTIL expression ];
END;
GOTO statement_label;
IF expression;
IF expression THEN program_statement;
ELSE program_statement;
variable = expression;
variable + expression;
LINK statement_label;
PUT [ variable] [=] [...];
RETURN;
SELECT[(expression )];
STOP;
SUBSTR( variable, index, length )= expression;
WHEN (expression) program_statement;
OTHERWISE program_statement;
For the most part, the SAS programming statements work the same as they do in the SAS DATA
step, as documented in SAS Language Reference: Concepts. However, there are several differences:
The ABORT statement does not allow any arguments.
The DO statement does not allow a character index variable. Thus
do i = 1,2,3;
is supported; however, the following statement is not supported:
do i = ’A’,’B’,’C’;
2206 F Chapter 38: The GLIMMIX Procedure
The LAG function is not supported with PROC GLIMMIX.
The PUT statement, used mostly for program debugging in PROC GLIMMIX, supports only
some of the features of the DATA step PUT statement, and it has some features not available
with the DATA step PUT statement:
– The PROC GLIMMIX PUT statement does not support line pointers, factored lists,
iteration factors, overprinting, _INFILE_, the colon (:) format modifier, or “$”.
– The PROC GLIMMIX PUT statement does support expressions, but the expression
must be enclosed in parentheses. For example, the following statement displays the
square root of x:
put (sqrt(x));
– The PROC GLIMMIX PUT statement supports the item _PDV_ to display a formatted
listing of all variables in the program. For example:
put _pdv_;
The WHEN and OTHERWISE statements enable you to specify more than one target statement. That is, DO/END groups are not necessary for multiple statement WHENs. For example, the following syntax is valid:
select;
when (exp1) stmt1;
stmt2;
when (exp2) stmt3;
stmt4;
end;
The LINK statement is used in a program to jump immediately to the label statement_label and to
continue program execution at that point. It is not used to specify a user-defined link function.
When coding your programming statements, you should avoid defining variables that begin with an
underscore (_), because they might conflict with internal variables created by PROC GLIMMIX.
User-Defined Link or Variance Function
Implied Variance Functions
While link functions are not unique for each distribution (see Table 38.9 for the default link functions), the distribution does determine the variance function a./. This function expresses the variance of an observation as a function of the mean, apart from weights, frequencies, and additional
scale parameters. The implied variance functions a./ of the GLIMMIX procedure are shown in
Table 38.15 for the supported distributions. For the binomial distribution, n denotes the number of
trials in the events/trials syntax. For the negative binomial distribution, k denotes the scale parameter. The multiplicative scale parameter is not included for the other distributions. The last column
of the table indicates whether has a value equal to 1:0 for the particular distribution.
User-Defined Link or Variance Function F 2207
Table 38.15
Variance Functions in PROC GLIMMIX
DIST=
Distribution
BETA
BINARY
BINOMIAL | BIN | B
EXPONENTIAL | EXPO
GAMMA | GAM
GAUSSIAN | G | NORMAL | N
GEOMETRIC | GEOM
INVGAUSS | IGAUSSIAN | IG
LOGNORMAL | LOGN
NEGBINOMIAL | NEGBIN | NB
POISSON | POI | P
TCENTRAL | TDIST | T
beta
binary
binomial
exponential
gamma
normal
geometric
inverse gaussian
lognormal
negative binomial
Poisson
t
Variance function
a./
/=.1 C /
.1 /
.1 /=n
2
2
1
C 2
3
1
C k2
=. 2/
.1
1
No
Yes
Yes
Yes
No
No
Yes
No
No
Yes
Yes
No
To change the variance function, you can use SAS programming statements and the predefined
automatic variables, as outlined in the following section. Your definition of a variance function will
override the DIST= option and its implied variance function. This has the following implication
for parameter estimation with the GLIMMIX procedure. When a user-defined link is available, the
distribution of the data is determined from the DIST= option, or the respective default for the type
of response. In a GLM, for example, this enables maximum likelihood estimation. If a user-defined
variance function is provided, the DIST= option is not honored and the distribution of the data
is assumed unknown. In a GLM framework, only quasi-likelihood estimation is then available to
estimate the model parameters.
Automatic Variables
To specify your own link or variance function you can use SAS programming statements and draw
on the following automatic variables:
_LINP_
is the current value of the linear predictor. It equals either b
D x0b̌ C z0b
C o or
0
b̌
b
D x C o, where o is the value of the offset variable, or 0 if no offset is specified. The estimated random effects solutions b
are used in the calculation of the
linear predictor during the model fitting phase, if a linearization expands about
the current values of . During the computation of output statistics, the EBLUPs
are used if statistics depend on them. For example, the following statements add
the variable p to the output data set glimmixout:
proc glimmix;
model y = x / dist=binary;
random int / subject=b;
p = 1/(1+exp(-_linp_);
output out=glimmixout;
id p;
run;
2208 F Chapter 38: The GLIMMIX Procedure
Because no output statistics are requested in the OUTPUT statement that depend on the random-effects solutions (BLUPs, EBEs), the value of _LINP_ in
this example equals x0b̌. On the contrary, the following statements also request
conditional residuals on the logistic scale:
proc glimmix;
model y = x / dist=binary;
random int / subject=b;
p = 1/(1+exp(-_linp_);
output out=glimmixout resid(blup)=r;
id p;
run;
The value of _LINP_ when computing the variable p is x0b̌ C z0b
. To ensure
that computed statistics are formed from x0b̌ and z0b
terms as needed, it is recommended that you use the automatic variables _XBETA_ and _ZGAMMA_
instead of _LINP_.
_MU_
expresses the mean of an observation as a function of the linear predictor, b
D
1
g .b
/.
_N_
is the observation number in the sequence of the data read.
_VARIANCE_
_XBETA_
is the estimate of the variance function, a.b
/.
equals x0b̌.
_ZGAMMA_
equals z0b
.
The automatic variable _N_ is incremented whenever the procedure reads an observation from the
data set. Observations that are not used in the analysis—for example, because of missing values or
invalid weights—are counted. The counter is reset to 1 at the start of every new BY group. Only
in some circumstances will _N_ equal the actual observation number. The symbol should thus be
used sparingly to avoid unexpected results.
You must observe the following syntax rules when you use the automatic variables. The _LINP_
symbol cannot appear on the left side of programming statements; you cannot make an assignment
to the _LINP_ variable. The value of the linear predictor is controlled by the CLASS, MODEL, and
RANDOM statements as well as the current parameter estimates and solutions. You can, however,
use the _LINP_ variable on the right side of other operations. Suppose, for example, that you want
to transform the linear predictor prior to applying the inverse log link. The following statements are
not valid because the linear predictor appears in an assignment:
proc glimmix;
_linp_ = sqrt(abs(_linp_));
_mu_
= exp(_linp_);
model count = logtstd / dist=poisson;
run;
The next statements achieve the desired result:
proc glimmix;
_mu_ = exp(sqrt(abs(_linp_)));
model count = logtstd / dist=poisson;
run;
User-Defined Link or Variance Function F 2209
If the value of the linear predictor is altered in any way through programming statements, you need
to ensure that an assignment to _MU_ follows. The assignment to variable P in the next set of
GLIMMIX statements is without effect:
proc glimmix;
p = _linp_ + rannor(454);
model count = logtstd / dist=poisson;
run;
A user-defined link function is implied by expressing _MU_ as a function of _LINP_. That is,
if D g 1 ./, you are providing an expression for the inverse link function with programming
statements. It is neither necessary nor possible to give an expression for the inverse operation,
D g./. The variance function is determined by expressing _VARIANCE_ as a function of
_MU_. If the _MU_ variable appears in an assignment statement inside PROC GLIMMIX, the
LINK= option of the MODEL statement is ignored. If the _VARIANCE_ function appears in an
assignment statement, the DIST= option is ignored. Furthermore, the associated variance function
per Table 38.15 is not honored. In short, user-defined expressions take precedence over built-in
defaults.
If you specify your own link and variance function, the assignment to _MU_ must precede an
assignment to the variable _VARIANCE_.
The following two sets of GLIMMIX statements yield the same parameter estimates, but the models
differ statistically:
proc glimmix;
class block entry;
model y/n = block entry / dist=binomial link=logit;
run;
proc glimmix;
class block entry;
prob = 1 / (1+exp(- _linp_));
_mu_ = n * prob ;
_variance_ = n * prob *(1-prob);
model y = block entry;
run;
The first GLIMMIX invocation models the proportion y=n as a binomial proportion with a logit
link. The DIST= and LINK= options are superfluous in this case, because the GLIMMIX procedure defaults to the binomial distribution in light of the events/trials syntax. The logit link is that
distribution’s default link. The second set of GLIMMIX statements models the count variable y and
takes the binomial sample size into account through assignments to the mean and variance function.
In contrast to the first set of GLIMMIX statements, the distribution of y is unknown. Only its mean
and variance are known. The model parameters are estimated by maximum likelihood in the first
case and by quasi-likelihood in the second case.
2210 F Chapter 38: The GLIMMIX Procedure
Details: GLIMMIX Procedure
Generalized Linear Models Theory
A generalized linear model consists of the following:
a linear predictor D x0 ˇ
a monotonic mapping between the mean of the data and the linear predictor
a response distribution in the exponential family of distributions
A density or mass function in this family can be written as
y b. /
C c.y; f .//
f .y/ D exp
for some functions b./ and c./. The parameter is called the natural (canonical) parameter. The
parameter is a scale parameter, and it is not present in all exponential family distributions. See
Table 38.15 for a list of distributions for which 1. In the case where observations are weighted,
the scale parameter is replaced with =w in the preceding density (or mass function), where w is
the weight associated with the observation y.
The mean and variance of the data are related to the components of the density, EŒY  D D b 0 . /,
VarŒY  D b 00 . /, where primes denote first and second derivatives. If you express as a function
of , the relationship is known as the natural link or the canonical link function. In other words,
modeling data with a canonical link assumes that D x0 ˇ; the effect contributions are additive on
the canonical scale. The second derivative of b./, expressed as a function of , is the variance function of the generalized linear model, a./ D b 00 ..//. Note that because of this relationship, the
distribution determines the variance function and the canonical link function. You cannot, however,
proceed in the opposite direction. If you provide a user-specified variance function, the GLIMMIX
procedure assumes that only the first two moments of the response distribution are known. The
full distribution of the data is then unknown and maximum likelihood estimation is not possible.
Instead, the GLIMMIX procedure then estimates parameters by quasi-likelihood.
Maximum Likelihood
The GLIMMIX procedure forms the log likelihoods of generalized linear models as
L.; I y/ D
n
X
fi l.i ; I yi ; wi /
i D1
where l.i ; I yi ; wi / is the log likelihood contribution of the i th observation with weight wi and
fi is the value of the frequency variable. For the determination of wi and fi , see the WEIGHT and
FREQ statements. The individual log likelihood contributions for the various distributions are as
follows.
Generalized Linear Models Theory F 2211
Beta
€.=wi /
l.i ; I yi ; wi / D log
€.=wi /€..1 /=wi /
C .=wi 1/ logfyi g
C ..1
VarŒY  D .1
/=wi
1/ logf1
yi g
/=.1 C /; > 0. See Ferrari and Cribari-Neto (2004).
Binary
l.i ; I yi ; wi / D wi .yi logfi g C .1
VarŒY  D .1
yi / logf1
i g/
/; 1.
Binomial
l.i ; I yi ; wi / D wi .yi logfi g C .ni
C wi .logf€.ni C 1/g
yi / logf1
i g/
logf€.yi C 1/g
logf€.ni
yi C 1/g/
where yi and ni are the events and trials in the events/trials syntax, and 0 < <
1. VarŒY =n D .1 /=n; 1.
Exponential
(
l.i ; I yi ; wi / D
logf
n i g oyi =i
wi yi
wi log wi yi i
i
wi D 1
logfyi €.wi /g wi 6D 1
VarŒY  D 2 ; 1.
Gamma
wi yi l.i ; I yi ; wi / D wi log
i
wi yi i
logfyi g
log f€.wi /g
VarŒY  D 2 ; > 0.
Geometric
i
i
l.i ; I yi ; wi / D yi log
.yi C wi / log 1 C
wi
wi
€.yi C wi /
C log
€.wi /€.yi C 1/
VarŒY  D C 2 ; 1.
Inverse Gaussian
l.i ; I yi ; wi / D
VarŒY  D 3 ; > 0.
1
2
"
(
)
#
yi3
wi .yi i /2
C log
C logf2g
wi
yi 2i
2212 F Chapter 38: The GLIMMIX Procedure
“Lognormal”
l.i ; I logfyi g; wi / D
1 wi .logfyi g
2
i /2
C log
wi
C logf2g
VarŒlogfY g D ; > 0.
If you specify DIST=LOGNORMAL with response variable Y, the GLIMMIX
procedure assumes that logfY g N.; 2 /. Note that the preceding density is
not the density of Y .
Multinomial
l.i ; I yi ; wi / D wi
J
X
yij logfij g
j D1
1.
Negative Binomial
ki
ki
l.i ; I yi ; wi / D yi log
.yi C wi =k/ log 1 C
wi
wi
€.yi C wi =k/
C log
€.wi =k/€.yi C 1/
VarŒY  D C k2 ; k > 0; 1.
For a given k, the negative binomial distribution is a member of the exponential
family. The parameter k is related to the scale of the data, because it is part
of the variance function. However, it cannot be factored from the variance, as
is the case with the parameter in many other distributions. The parameter k
is designated as “Scale” in the “Parameter Estimates” table of the GLIMMIX
procedure.
Normal (Gaussian)
l.i ; I yi ; wi / D
1 wi .yi i /2
C log
C logf2g
2
wi
VarŒY  D ; > 0.
Poisson
l.i ; I yi ; wi / D wi .yi logfi g
i
logf€.yi C 1/g/
VarŒY  D ; 1.
Shifted T
zi D
p
0:5 logf= wi g C log f€.0:5. C 1/g
0:5 log fg
wi .yi i /2
.=2 C 0:5/ log 1 C
C zi
log f€.0:5/g
l.i ; I yi ; wi / D
> 0; > 0; VarŒY  D =.
2/.
Generalized Linear Models Theory F 2213
Define the parameter vector for the generalized linear model as D ˇ, if 1, and as D Œˇ 0 ; 0
otherwise. ˇ denotes the fixed-effects parameters in the linear predictor. For the negative binomial
distribution, the relevant parameter vector is D Œˇ 0 ; k0 . The gradient and Hessian of the negative
log likelihood are then
gD
@L.I y/
@
HD
@2 L.I y/
@ @ 0
The GLIMMIX procedure computes the gradient vector and Hessian matrix analytically, unless
your programming statements involve functions whose derivatives are determined by finite differences. If the procedure is in scoring mode, H is replaced by its expected value. PROC GLIMMIX
is in scoring mode when the number n of SCORING=n iterations has not been exceeded and the
optimization technique uses second derivatives, or when the Hessian is computed at convergence
and the EXPHESSIAN option is in effect. Note that the objective function is the negative log likelihood when the GLIMMIX procedure fits a GLM model. The procedure performs a minimization
problem in this case.
In models for independent data with known distribution, parameter estimates are obtained by the
method of maximum likelihood. No parameters are profiled from the optimization. The default
optimization technique for GLMs is the Newton-Raphson algorithm, except for Gaussian models
with identity link, which do not require iterative model fitting. In the case of a Gaussian model, the
scale parameter is estimated by restricted maximum likelihood, because this estimate is unbiased.
The results from the GLIMMIX procedure agree with those from the GLM and REG procedure
for such models. You can obtain the maximum likelihood estimate of the scale parameter with the
NOREML option in the PROC GLIMMIX statement. To change the optimization algorithm, use
the TECHNIQUE= option in the NLOPTIONS statement.
Standard errors of the parameter estimates are obtained from the inverse of the (observed or expected) second derivative matrix H.
Scale and Dispersion Parameters
The parameter in the log-likelihood functions is a scale parameter. McCullagh and Nelder (1989,
p. 29) refer to it as the dispersion parameter. With the exception of the normal distribution, does
not correspond to the variance of an observation, the variance of an observation in a generalized
linear model is a function of and . In a generalized linear model (GLM mode), the GLIMMIX
procedure displays the estimate of is as “Scale” in the “Parameter Estimates” table. Note that
for some distributions this scale is different from that reported by the GENMOD procedure in its
“Parameter Estimates” table. The scale reported by PROC GENMOD is sometimes a transformation
of the dispersion parameter in the log-likelihood function. Table 38.15 displays the relationship
between the “Scale” entries reported by the two procedures in terms of the (or k) parameter in the
GLIMMIX log-likelihood functions.
2214 F Chapter 38: The GLIMMIX Procedure
Table 38.15
Scales in Parameter Estimates Table
Distribution
GLIMMIX Reports
GENMOD Reports
Beta
Gamma
b
b
Inverse gaussian
Negative binomial
b
b
k
Normal
b
c 
D VarŒY
N/A
b
q
b
b
k
q
b
Note that for normal linear models, PROC GLIMMIX by default estimates the parameters by restricted maximum likelihood, whereas PROC GENMOD estimates the parameters by maximum
likelihood. As a consequence, the scale parameter in the “Parameter Estimates” table of the GLIMMIX procedure coincides for these models with the mean-squared error estimate of the GLM or
REG procedures. To obtain maximum likelihood estimates in a normal linear model in the GLIMMIX procedure, specify the NOREML option in the PROC GLIMMIX statement.
Quasi-likelihood for Independent Data
Quasi-likelihood estimation uses only the first and second moment of the response. In the case
of independent data, this requires only a specification of the mean and variance of your data. The
GLIMMIX procedure estimates parameters by quasi-likelihood, if the following conditions are met:
The response distribution is unknown, because of a user-specified variance function.
There are no G-side random effects.
There are no R-side covariance structures or at most an overdispersion parameter.
Under some mild regularity conditions, the function
Z i
yi t
Q.i ; yi / D
dt
a.
i/
yi
known as the log quasi-likelihood of the i th observation, has some properties of a log-likelihood
function (McCullagh and Nelder 1989, p. 325). For example, the expected value of its derivative
is zero, and the variance of its derivative equals the negative of the expected value of the second
derivative. Consequently,
QL.; ; y/ D
n
X
i D1
fi wi
Yi i
a.i /
can serve as the score function for estimation. Quasi-likelihood estimation takes as the gradient and
“Hessian” matrix—with respect to the fixed-effects parameters ˇ—the quantities
@QL.; ; y/
gql D gql;j D
D D0 V 1 .Y /=
@ˇj
2
@ QL.; ; y/
Hql D hql;j k D
D D0 V 1 D=
@ˇj @ˇk
Generalized Linear Models Theory F 2215
In this expression, D is a matrix of derivatives of with respect to the elements in ˇ, and V is a
diagonal matrix containing variance functions, V D Œa.1 /; ; a.n /. Notice that Hql is not the
second derivative matrix of Q.; y/. Rather, it is the negative of the expected value of @ggl [email protected]ˇ.
Hql thus has the form of a “scoring Hessian.”
The GLIMMIX procedure fixes the scale parameter at 1.0 by default. To estimate the parameter,
add the statement
random _residual_;
The resulting estimator (McCullagh and Nelder 1989, p. 328) is
n
.yi b
1 X
i /2
b
D
fi wi
m
a.b
i /
i D1
where m D f rankfXg if the NOREML option is in effect, m D f otherwise, and f is the sum
of the frequencies.
See Example 38.4 for an application of quasi-likelihood estimation with PROC GLIMMIX.
Effects of Adding Overdispersion
You can add a multiplicative overdispersion parameter to a generalized linear model in the GLIMMIX procedure with the statement
random _residual_;
For models in which 1, this effectively lifts the constraint of the parameter. In models that already contain a or k scale parameter—such as the normal, gamma, or negative binomial model—
the statement adds a multiplicative scalar (the overdispersion parameter, o ) to the variance function.
The overdispersion parameter is estimated from Pearson’s statistic after all other parameters have
been determined by (restricted) maximum likelihood or quasi-likelihood. This estimate is
n
b
o D
1 X
.yi i /2
f
w
i i
pm
a.i /
iD1
where m D f rankfXg if the NOREML option is in effect, and m D f otherwise, and f is the
sum of the frequencies. The power p is 1 for the gamma distribution and 1 otherwise.
Adding an overdispersion parameter does not alter any of the other parameter estimates. It only
changes the variance-covariance matrix of the estimates by a certain factor. If overdispersion arises
from correlations among the observations, then you should investigate more complex randomeffects structures.
2216 F Chapter 38: The GLIMMIX Procedure
Generalized Linear Mixed Models Theory
Model or Integral Approximation
In a generalized linear model, the log likelihood is well defined, and an objective function for
estimation of the parameters is simple to construct based on the independence of the data. In a
GLMM, several problems must be overcome before an objective function can be computed.
The model might be vacuous in the sense that no valid joint distribution can be constructed
either in general or for a particular set of parameter values. For example, if Y is an equicorrelated .n 1/ vector of binary responses with the same success probability and a symmetric
distribution, then the lower bound on the correlation parameter depends on n and (Gilliland
and Schabenberger 2001). If further restrictions are placed on the joint distribution, as in Bahadur (1961), the correlation is also restricted from above.
The dependency between mean and variance for nonnormal data places constraints on the
possible correlation models that simultaneously yield valid joint distributions and a desired
conditional distributions. Thus, for example, aspiring for conditional Poisson variates that are
marginally correlated according to a spherical spatial process might not be possible.
Even if the joint distribution is feasible mathematically, it still can be out of reach computationally. When data are independent, conditional on the random effects, the marginal log
likelihood can in principle be constructed by integrating out the random effects from the joint
distribution. However, numerical integration is practical only when the number of random
effects is small and when the data have a clustered (subject) structure.
Because of these special features of generalized linear mixed models, many estimation methods
have been put forth in the literature. The two basic approaches are (1) to approximate the objective
function and (2) to approximate the model. Algorithms in the second category can be expressed
in terms of Taylor series (linearizations) and are hence also known as linearization methods. They
employ expansions to approximate the model by one based on pseudo-data with fewer nonlinear
components. The process of computing the linear approximation must be repeated several times
until some criterion indicates lack of further progress. Schabenberger and Gregoire (1996) list numerous algorithms based on Taylor series for the case of clustered data alone. The fitting methods
based on linearizations are usually doubly iterative. The generalized linear mixed model is approximated by a linear mixed model based on current values of the covariance parameter estimates. The
resulting linear mixed model is then fit, which is itself an iterative process. On convergence, the
new parameter estimates are used to update the linearization, which results in a new linear mixed
model. The process stops when parameter estimates between successive linear mixed model fits
change only within a specified tolerance.
Integral approximation methods approximate the log likelihood of the GLMM and submit the approximated function to numerical optimization. Various techniques are used to compute the approximation: Laplace methods, quadrature methods, Monte Carlo integration, and Markov chain
Monte Carlo methods. The advantage of integral approximation methods is to provide an actual objective function for optimization. This enables you to perform likelihood ratio tests among nested
models and to compute likelihood-based fit statistics. The estimation process is singly iterative.
Generalized Linear Mixed Models Theory F 2217
The disadvantage of integral approximation methods is the difficulty of accommodating crossed
random effects and multiple subject effects, and the inability to accommodate R-side covariance
structures, even only R-side overdispersion. The number of random effects should be small for
integral approximation methods to be practically feasible.
The advantages of linearization-based methods include a relatively simple form of the linearized
model that typically can be fit based on only the mean and variance in the linearized form. Models
for which the joint distribution is difficult—or impossible—to ascertain can be fit with linearizationbased approaches. Models with correlated errors, a large number of random effects, crossed random
effects, and multiple types of subjects are thus excellent candidates for linearization methods. The
disadvantages of this approach include the absence of a true objective function for the overall optimization process and potentially biased estimates, especially for binary data when the number of
observations per subject is small (see the section “Notes on Bias of Estimators” on page 2230 for
further comments and considerations about the bias of estimates in generalized linear mixed models). Because the objective function to be optimized after each linearization update depends on the
current pseudo-data, objective functions are not comparable across linearizations. The estimation
process can fail at both levels of the double iteration scheme.
By default the GLIMMIX procedure fits generalized linear mixed models based on linearizations.
The default estimation method in GLIMMIX for models containing random effects is a technique
known as restricted pseudo-likelihood (RPL) (Wolfinger and O’Connell 1993) estimation with an
expansion around the current estimate of the best linear unbiased predictors of the random effects
(METHOD=RSPL).
Two maximum likelihood estimation methods based on integral approximation are available in the
GLIMMIX procedure. If you choose METHOD=LAPLACE in a GLMM, the GLIMMIX procedure
performs maximum likelihood estimation based on a Laplace approximation of the marginal log
likelihood. See the section “Maximum Likelihood Estimation Based on Laplace Approximation”
on page 2222 for details about the Laplace approximation with PROC GLIMMIX. If you choose
METHOD=QUAD in the PROC GLIMMIX statement in a generalized linear mixed model, the
GLIMMIX procedure estimates the model parameters by adaptive Gauss-Hermite quadrature. See
the section “Maximum Likelihood Estimation Based on Adaptive Quadrature” on page 2225 for
details about the adaptive Gauss-Hermite quadrature approximation with PROC GLIMMIX.
The following subsections discuss the three estimation methods in turn. Keep in mind that your
modeling possibilities are increasingly restricted in the order of these subsections. For example,
in the class of generalized linear mixed models, the pseudo-likelihood estimation methods place
no restrictions on the covariance structure, and Laplace estimation adds restriction with respect to
the R-side covariance structure. Adaptive quadrature estimation further requires a clustered data
structure—that is, the data must be processed by subjects.
Table 38.16
Model Restrictions Depending on Estimation Method
Method
Restriction
RSPL, RMPL
None
MSPL, MMPL
None
LAPLACE
No R-side effects
QUAD
No R-side effects
Requires SUBJECT= effect
Requires processing by subjects
2218 F Chapter 38: The GLIMMIX Procedure
Pseudo-likelihood Estimation Based on Linearization
The Pseudo-model
Recall from the section “Notation for the Generalized Linear Mixed Model” on page 2081 that
EŒYj D g
1
.Xˇ C Z/ D g
1
./ D where N.0; G/ and VarŒYj D A1=2 RA1=2 . Following Wolfinger and O’Connell (1993), a
first-order Taylor series of about ě and e
yields
:
e
ě/ C Z.
e
e
g 1 ./ D g 1 .e
/ C X.ˇ
/
where
eD

1 ./ @g
@
ě;e
is a diagonal matrix of derivatives of the conditional mean evaluated at the expansion locus. Rearranging terms yields the expression
:
e 1 . g 1 .e
// C Xě C Ze
D Xˇ C Z

The left side is the expected value, conditional on , of
e

1
.Y
g
1
.e
// C Xě C Ze
P
and
e
VarŒPj D 
1
e
A1=2 RA1=2 
1
You can thus consider the model
P D Xˇ C Z C which is a linear mixed model with pseudo-response P, fixed effects ˇ, random effects , and
VarŒ D VarŒPj.
Objective Functions
Now define
e
V./ D ZGZ0 C 
1
e
A1=2 RA1=2 
1
as the marginal variance in the linear mixed pseudo-model, where is the .q 1/ parameter vector
containing all unknowns in G and R. Based on this linearized model, an objective function can be
defined, assuming that the distribution of P is known. The GLIMMIX procedure assumes that has a normal distribution. The maximum log pseudo-likelihood (MxPL) and restricted log pseudolikelihood (RxPL) for P are then
l.; p/ D
lR .; p/ D
1
log jV./j
2
1
log jV./j
2
1 0
r V./
2
1 0
r V./
2
1
r
1
r
f
logf2g
2
1
log jX0 V./
2
1
Xj
f
k
2
logf2g
Generalized Linear Mixed Models Theory F 2219
with r D p X.X0 V 1 X/ X0 V 1 p. f denotes the sum of the frequencies used in the analysis,
and k denotes the rank of X. The fixed-effects parameters ˇ are profiled from these expressions.
The parameters in are estimated by the optimization techniques specified in the NLOPTIONS
statement. The objective function for minimization is 2l.; p/ or 2lR .; p/. At convergence,
the profiled parameters are estimated and the random effects are predicted as
b̌ D .X0 V.b
/
1
X/ X0 V.b
/
1
p
b 0 V.b
b
D GZ
/ 1b
r
With these statistics, the pseudo-response and error weights of the linearized model are recomputed
and the objective function is minimized again. The predictors b
are the estimated BLUPs in the
approximated linear model. This process continues until the relative change between parameter
estimates at two successive (outer) iterations is sufficiently small. See the PCONV= option in the
PROC GLIMMIX statement for the computational details about how the GLIMMIX procedure
compares parameter estimates across optimizations.
If the conditional distribution contains a scale parameter 6D 1 (Table 38.15), the GLIMMIX
procedure profiles this parameter in GLMMs from the log pseudo-likelihoods as well. To this end
define
e
V. / D 
1
e
A1=2 R A1=2 
1
C ZG Z0
where is the covariance parameter vector with q 1 elements. The matrices G and R are
appropriately reparameterized versions of G and R. For example, if G has a variance component
structure and R D I, then contains ratios of the variance components and , and R D I. The
solution for b
is
b
/ 1b
r=m
Db
r0 V.b
where m D f for MxPL and m D f
the profiled log pseudo-likelihoods,
l. ; p/ D
lR . ; p/ D
k for RxPL. Substitution into the previous functions yields
˚
f
1
f
log jV. /j
log r0 V. / 1 r
.1 C logf2=f g/
2
2
2
˚
1
f k
log jV. /j
log r0 V. / 1 r
2
2
f k
1
log jX0 V. / 1 Xj
.1 C logf2=.f k/g/
2
2
Profiling of can be suppressed with the NOPROFILE option in the PROC GLIMMIX statement.
Where possible, the objective function, its gradient, and its Hessian employ the sweep-based Wtransformation (Hemmerle and Hartley 1973; Goodnight 1979; Goodnight and Hemmerle 1979).
Further details about the minimization process in the general linear mixed model can be found in
Wolfinger, Tobias, and Sall (1994).
Estimated Precision of Estimates
The GLIMMIX procedure produces estimates of the variability of b̌, b
, and estimates of the prediction variability for b
, VarŒb
. Denote as S the matrix
e
c
S VarŒPj
D
1
e
A1=2 RA1=2 
1
2220 F Chapter 38: The GLIMMIX Procedure
where all components on the right side are evaluated at the converged estimates. The mixed model
equations (Henderson 1984) in the linear mixed (pseudo-)model are then
0 1
0 1 b̌
XS X
X0 S 1 Z
XS p
D
0
1
0
1
1
b
Z0 S 1 p
b
Z S X Z S Z C G./
and
CD
"
D
X0 S
Z0 S
1X
X0 S 1 Z
1 X Z0 S 1 Z C G.b
/
b

0
b
G./Z V.b
/
1
b 0 V.b
X
/ 1 ZG.b
/
1 X
0
1
0
b M C G.b
b
b
/Z V./ XX V.b
/
#
1 ZG.b
/
is the approximate estimated variance-covariance matrix of Œb̌0 ; b
0
.X0 V.b
/ 1 X/ and M D .Z0 S 1 Z C G.b
/ 1 / 1 .
0 0 .
b D
Here, 
b are reported in the Standard Error column of the “PaThe square roots of the diagonal elements of 
rameter Estimates” table. This table is produced with the SOLUTION option in the MODEL statement. The prediction standard errors of the random-effects solutions are reported in the Std Err Pred
column of the “Solution for Random Effects” table. This table is produced with the SOLUTION
option in the RANDOM statement.
As a cautionary note, C tends to underestimate the true sampling variability of [b̌0 ; b
0 0 , because
no account is made for the uncertainty in estimating G and R. Although inflation factors have
been proposed (Kackar and Harville 1984; Kass and Steffey 1989; Prasad and Rao 1990), they
tend to be small for data sets that are fairly well balanced. PROC GLIMMIX does not compute
any inflation factors by default. The DDFM=KENWARDROGER option in the MODEL statement
prompts PROC GLIMMIX to compute a specific inflation factor (Kenward and Roger 1997), along
with Satterthwaite-based degrees of freedom.
If G.b
/ is singular, or if you use the CHOL option of the PROC GLIMMIX statement, the mixed
model equations are modified as follows. Let L denote the lower triangular matrix so that LL0 D
G.b
/. PROC GLIMMIX then solves the equations
b̌
X0 S 1 X
X0 S 1 ZL
X0 S 1 p
D
L0 Z0 S 1 p
L0 Z0 S 1 X L0 Z0 S 1 ZL C I
b
and transforms b
and a generalized inverse of the left-side coefficient matrix by using L.
The asymptotic covariance matrix of the covariance parameter estimator b
is computed based on
the observed or expected Hessian matrix of the optimization procedure. Consider first the case
where the scale parameter is not present or not profiled. Because ˇ is profiled from the pseudolikelihood, the objective function for minimization is f ./ D 2l.; p/ for METHOD=MSPL and
METHOD=MMPL and f ./ D 2lR .; p/ for METHOD=RSPL and METHOD=RMPL. Denote
the observed Hessian (second derivative) matrix as
HD
@2 f ./
@ @ 0
The GLIMMIX procedure computes the variance of b
by default as 2H 1 . If the Hessian is not positive definite, a sweep-based generalized inverse is used instead. When the EXPHESSIAN option of
Generalized Linear Mixed Models Theory F 2221
the PROC GLIMMIX statement is used, or when the procedure is in scoring mode at convergence
(see the SCORING option in the PROC GLIMMIX statement), the observed Hessian is replaced
with an approximated expected Hessian matrix in these calculations.
Following Wolfinger, Tobias, and Sall (1994), define the following components of the gradient and
Hessian in the optimization:
@ 0
r V./ 1 r
@
@2
H1 D
logfV./g
@ @ 0
@2
H2 D
r0 V./ 1 r
@ @ 0
@2
H3 D
logfjX0 V./
@ @ 0
g1 D
1
Xjg
Table 38.17 gives expressions for the Hessian matrix H depending on estimation method, profiling,
and scoring.
Table 38.17
Hessian Computation in GLIMMIX
Profiling
Scoring
MxPL
RxPL
No
No
H1 C H2
H1 C H2 C H3
No
Yes
H1
H1 C H3
No
Mod.
H1
H1
Yes
Yes
Yes
No
H1 C H2 =
g02 = 2
H1 C H2 = C H3
g2 = 2
0
2
g2 =
.f k/= 2
H1
0
g2 = 2
g2 = 2
f = 2
H1 C H3
g2 = 2
g02 = 2
.f k/= 2
H1
0
g2 = 2
g2 = 2
f = 2
H1 H3
g2 = 2
g02 = 2
.f k/= 2
Yes
Mod.
g2 = 2
f = 2
H3
The “Mod.” expressions for the Hessian under scoring in RxPL estimation refer to a modified scoring method. In some cases, the modification leads to faster convergence than the standard scoring
algorithm. The modification is requested with the SCOREMOD option in the PROC GLIMMIX
statement.
Finally, in the case of a profiled scale parameter , the Hessian for the . ; / parameterization is
converted into that for the parameterization as
H./ D BH. ; /B0
2222 F Chapter 38: The GLIMMIX Procedure
where
2
6
BD6
4
1=
0
0
1 =
0
1=
2 =
0
0
1=
q 1 =
3
0
0 7
7
0 5
1
Subject-Specific and Population-Averaged (Marginal) Expansions
There are two basic choices for the expansion locus of the linearization. A subject-specific (SS)
expansion uses
ě D b̌ e
Db
which are the current estimates of the fixed effects and estimated BLUPs. The population-averaged
(PA) expansion expands about the same fixed effects and the expected value of the random effects
ě D b̌ e
D0
To recompute the pseudo-response and weights in the SS expansion, the BLUPs must be computed
every time the objective function in the linear mixed model is maximized. The PA expansion does
not require any BLUPs. The four pseudo-likelihood methods implemented in the GLIMMIX procedure are the 2 2 factorial combination between two expansion loci and residual versus maximum
pseudo-likelihood estimation. The following table shows the combination and the corresponding
values of the METHOD= option (PROC GLIMMIX statement); METHOD=RSPL is the default.
Type of
PL
Expansion Locus
b
EŒ
residual
maximum
RSPL
MSPL
RMPL
MMPL
Maximum Likelihood Estimation Based on Laplace Approximation
Objective Function
Let ˇ denote the vector of fixed-effects parameters and the vector of covariance parameters. For
Laplace estimation in the GLIMMIX procedure, includes the G-side parameters and a possible scale parameter , provided that the conditional distribution of the data contains such a scale
parameter. is the vector of the G-side parameters.
The marginal distribution of the data in a mixed model can be expressed as
Z
p.y/ D p.yj; ˇ; / p.j / d Z
˚
D exp logfp.yj; ˇ; /g C logfp.j /g d Z
D exp fcl f .y; ˇ; I /g d Generalized Linear Mixed Models Theory F 2223
If the constant cl is large, the Laplace approximation of this integral is
L.ˇ; I b
; y/ D
2
cl
n =2
j
f 00 .y; ˇ; I b
/j
1=2 cl f .y;ˇ;Ib
/
e
where n is the number of elements in , f 00 is the second derivative matrix
f 00 .y; ˇ; I b
/ D
@2 f .y; ˇ; I /
jb
@@ 0
and b
satisfies the first-order condition
@f .y; ˇ; I /
D0
@
The objective function for Laplace parameter estimation in the GLIMMIX procedure is
2 logfL.ˇ; b
; y/g. The optimization process is singly iterative, but because b
depends on
b̌ and b
, the GLIMMIX procedure solves a suboptimization problem to determine for given values
of b̌ and b
the random-effects solution vector that maximizes f .y; ˇ; I /.
When you have longitudinal or clustered data with m independent subjects or clusters, the vector of
observations can be written as y D Œy01 ; ; y0m 0 , where yi is an ni 1 vector of observations for
subject (cluster) i (i D 1; ; m). In this case, assuming conditional independence such that
p.yi ji / D
ni
Y
p.yij ji /
j D1
the marginal distribution of the data can be expressed as
p.y/ D
m
Y
p.yi / D
i D1
D
m Z
Y
i D1
m Z
Y
p.yi ji /p.i / d i
exp fni f .yi ; ˇ; I i /g d i
i D1
where
ni f .yi ; ˇ; I i / D log fp.yi ji / p.i /g
D
ni
X
˚
log p.yij ji / C ni log fp.i /g
j D1
When the number of observations within a cluster, ni , is large, the Laplace approximation to the i th
individual’s marginal probability density function is
Z
p.yi jˇ; / D exp fni f .yi ; ˇ; I i /g d i
D
j
.2/n =2
ni f 00 .yi ; ˇ; I b
i /j
1=2
exp fni f .yi ; ˇ; I b
i /g
2224 F Chapter 38: The GLIMMIX Procedure
where ni is the common dimension of the random effects, i . In this case, provided that the
constant cl D minfni g is large, the Laplace approximation to the marginal log likelihood is
log fL.ˇ; I b
; y/g D
m n
X
n i
logf2g
2
00
ni f .ˇ; I b
i /j
ni f .y; ˇ; I b
i / C
i D1
1
log j
2
which serves as the objective function for the METHOD=LAPLACE estimator in PROC GLIMMIX.
The Laplace approximation implemented in the GLIMMIX procedure differs from that in Wolfinger (1993) and Pinheiro and Bates (1995) in important respects. Wolfinger (1993) assumed a flat
prior for ˇ and expanded the integrand around ˇ and , leaving only the covariance parameters
for the overall optimization. The “fixed” effects ˇ and the random effects are determined in a
suboptimization that takes the form of a linear mixed model step with pseudo-data. The GLIMMIX
procedure involves only the random effects vector in the suboptimization. Pinheiro and Bates
(1995) and Wolfinger (1993) consider a modified Laplace approximation that replaces the second
derivative f 00 .y; ˇ; I b
/ with an (approximate) expected value, akin to scoring. The GLIMMIX
procedure does not use an approximation to f 00 .y; ˇ; I b
/. The METHOD=RSPL estimates in
PROC GLIMMIX are equivalent to the estimates obtained with the modified Laplace approximation in Wolfinger (1993). The objective functions of METHOD=RSPL and Wolfinger (1993) differ
in a constant that depends on the number of parameters.
Asymptotic Properties and the Importance of Subjects
Suppose that the GLIMMIX procedure processes your data by subjects (see the section “Processing
by Subjects” on page 2245) and let ni denote the number of observations per subject, i D 1; : : : ; s.
Arguments in Vonesh (1996) show that the maximum likelihood estimator based on the Laplace
p
approximation is a consistent estimator to order Op fmaxf1= sg; 1= minfni gg. In other words, as
the number of subjects and the number of observations per subject grows, the small-sample bias
of the Laplace estimator disappears. Note that the term involving the number of subjects in this
maximum relates to standard asymptotic theory, and the term involving the number of observations
per subject relates to the accuracy of the Laplace approximation (Vonesh 1996). In the case where
random effects enter the model linearly, the Laplace approximation is exact and the requirement
that minfni g ! 1 can be dropped.
If your model is not processed by subjects but is equivalent to a subject model, the asymptotics
with respect to s still apply, because the Hessian matrix of the suboptimization for breaks into
s separate blocks. For example, the following two models are equivalent with respect to s and ni ,
although only for the first model does PROC GLIMMIX process the data explicitly by subjects:
proc glimmix method=laplace;
class sub A;
model y = A;
random intercept / subject=sub;
run;
Generalized Linear Mixed Models Theory F 2225
proc glimmix method=laplace;
class sub A;
model y = A;
random sub;
run;
The same holds, for example, for models with independent nested random effects. The following
two models are equivalent, and you can derive asymptotic properties related to s and minfni g from
the model in the first run:
proc glimmix method=laplace;
class A B block;
model y = A B A*B;
random intercept A / subject=block;
run;
proc glimmix method=laplace;
class A B block;
model y = A B A*B;
random block a*block;
run;
The Laplace approximation requires that the dimension of the integral does not increase with the
size of the sample. Otherwise the error of the likelihood approximation does not diminish with
ni . This is the case, for example, with exchangeable arrays (Shun and McCullagh 1995), crossed
random effects (Shun 1997), and correlated random effects of arbitrary dimension (Raudenbush,
Yang, and Yosef 2000). Results in Shun (1997), for example, show that even in this case the
standard Laplace approximation has smaller bias than pseudo-likelihood estimates.
Maximum Likelihood Estimation Based on Adaptive Quadrature
Quadrature methods, like the Laplace approximation, approximate integrals. If you choose
METHOD=QUAD for a generalized linear mixed model, the GLIMMIX procedure approximates
the marginal log likelihood with an adaptive Gauss-Hermite quadrature rule. Gaussian quadrature is
particularly well suited to numerically evaluate integrals against probability measures (Lange 1999,
Ch. 16). And Gauss-Hermite quadrature is appropriate when the density has kernel expf x 2 g
and integration extends over the real line, as is the case for the normal distribution. Suppose that
p.x/ is a probability density function and the function f .x/ is to be integrated against it. Then the
quadrature rule is
Z
1
1
f .x/p.x/ dx N
X
wi f .xi /
i D1
where N denotes the number of quadrature points, the wi are the quadrature weights, and the xi
are the abscissas. The Gaussian quadrature chooses abscissas in areas of high density, and if p.x/
is continuous, the quadrature rule is exact if f .x/ is a polynomial of up to degree 2N 1. In the
generalized linear mixed model the roles of f .x/ and p.x/ are played by the conditional distribution
of the data given the random effects, and the random-effects distribution, respectively. Quadrature
2226 F Chapter 38: The GLIMMIX Procedure
abscissas and weights are those of the standard Gauss-Hermite quadrature (Golub and Welsch 1969;
see also Table 25.10 of Abramowitz and Stegun 1972; Evans 1993).
A numerical integration rule is called adaptive when it uses a variable step size to control the error
of the approximation. For example, an adaptive trapezoidal rule uses serial splitting of intervals at
midpoints until a desired tolerance is achieved. The quadrature rule in the GLIMMIX procedure
is adaptive in the following sense: if you do not specify the number of quadrature points (nodes)
with the QPOINTS= suboption of the METHOD=QUAD option, then the number of quadrature
points is determined by evaluating the log likelihood at the starting values at a successively larger
number of nodes until a tolerance is met (for more details see the text under the heading “Starting
Values” in the next section). Furthermore, the GLIMMIX procedure centers and scales the quadrature points by using the empirical Bayes estimates (EBEs) of the random effects and the Hessian
(second derivative) matrix from the EBE suboptimization. This centering and scaling improves the
likelihood approximation by placing the abscissas according to the density function of the random
effects. It is not, however, adaptiveness in the previously stated sense.
Objective Function
Let ˇ denote the vector of fixed-effects parameters and the vector of covariance parameters.
For quadrature estimation in the GLIMMIX procedure, includes the G-side parameters and a
possible scale parameter , provided that the conditional distribution of the data contains such a
scale parameter. is the vector of the G-side parameters. The marginal distribution of the data for
subject i in a mixed model can be expressed as
Z
Z
p.yi / D
p.yi ji ; ˇ; / p.i j / d i
Suppose Nq denotes the number of quadrature points in each dimension (for each random effect)
and r denotes the number of random effects. For each subject, obtain the empirical Bayes estimates
of i as the vector b
i that minimizes
˚
log p.yi ji ; ˇ; /p.i j / D f .yi ; ˇ; I i /
If z D Œz1 ; ; zNq  are the standard abscissas for Gauss-Hermite quadrature, and zj D
Œzj1 ; ; zjr  is a point on the r-dimensional quadrature grid, then the centered and scaled abscissas
are
aj D b
i C 21=2 f 00 .yi ; ˇ; I b
i /
1=2 zj
As for the Laplace approximation, f 00 is the second derivative matrix with respect to the random
effects,
f 00 .yi ; ˇ; I b
i / D
@2 f .yi ; ˇ; I i /
jb
i
@i @i0
These centered and scaled abscissas, along with the Gauss-Hermite quadrature weights w D
Œw1 ; ; wNq , are used to construct the r-dimensional integral by a sequence of one-dimensional
Generalized Linear Mixed Models Theory F 2227
rules
Z
p.yi / D
Z
p.yi ji ; ˇ; / p.i j / d i
2r=2 jf 00 .yi ; ˇ; I b
i /j 1=2
#
Nq
Nq "
r
X
X
Y
2
p.yi jaj ; ˇ; /p.aj j /
wj k exp zjk
j1 D1
jr D1
kD1
The right-hand side of this expression, properly accumulated across subjects, is the objective function for adaptive quadrature estimation in the GLIMMIX procedure.
Quadrature or Laplace Approximation
If you select the quadrature rule with a single quadrature point, namely
proc glimmix method=quad(qpoints=1);
the results will be identical to METHOD=LAPLACE. Computationally, the two methods are not
identical, however. METHOD=LAPLACE can be applied to a considerably larger class of models. For example, crossed random effects, models without subjects, or models with non-nested
subjects can be handled with the Laplace approximation but not with quadrature. Furthermore,
METHOD=LAPLACE draws on a number of computational simplifications that can increase its
efficiency compared to a quadrature algorithm with a single node. For example, the Laplace approximation is possible with unbounded covariance parameter estimates (NOBOUND option in the
PROC GLIMMIX statement) and can permit certain types of negative definite or indefinite G matrices. The adaptive quadrature approximation with scaled abscissas typically breaks down when G
is not at least positive semidefinite.
As the number of random effects grows—for example, if you have nested random effects—
quadrature quickly becomes computationally infeasible, due to the high dimensionality of the integral. To this end it is worthwhile to clarify the issues of dimensionality and computational effort as
related to the number of quadrature nodes. Suppose that the A effect has 4 levels and consider the
following statements:
proc glimmix method=quad(qpoints=5);
class A id;
model y = / dist=negbin;
random A / subject=id;
run;
For each subject, computing the marginal log likelihood requires the numerical evaluation of a fourdimensional integral. As part of this evaluation 54 D 625 conditional log likelihoods need to be
computed for each observation on each pass through the data. As the number of quadrature points
or the number of random effects increases, this constitutes a sizable computational effort. Suppose,
for example, that an additional random effect with b D 2 levels is added as an interaction. The
following statements then require evaluation of 5.4C8/ D 244140625 conditional log likelihoods
for each observation one each pass through the data:
2228 F Chapter 38: The GLIMMIX Procedure
proc glimmix method=quad(qpoints=5);
class A B id;
model y = / dist=negbin;
random A A*B / subject=id;
run;
As the number of random effects increases, Laplace approximation presents a computationally more
expedient alternative.
If you wonder whether METHOD=LAPLACE would present a viable alternative to a model that you
can fit with METHOD=QUAD, the “Optimization Information” table can provide some insights.
The table contains as its last entry the number of quadrature points determined by PROC GLIMMIX
to yield a sufficiently accurate approximation of the log likelihood (at the starting values). In many
cases, a single quadrature node is sufficient, in which case the estimates are identical to those of
METHOD=LAPLACE.
Aspects Common to Adaptive Quadrature and Laplace Approximation
Estimated Precision of Estimates
Denote as H the second derivative matrix
HD
@2 logfL.ˇ; b
/g
@Œˇ; @Œˇ 0 ; 0 
evaluated at the converged solution of the optimization process. Partition its inverse as
C.ˇ; ˇ/ C.ˇ; /
H 1D
C.; ˇ/ C.; /
For METHOD=LAPLACE and METHOD=QUAD, the GLIMMIX procedure computes H by finite forward differences based on the analytic gradient of logfL.ˇ; b
/g. The partition C.; /
serves as the asymptotic covariance matrix of the covariance parameter estimates (ASYCOV option
in the PROC GLIMMIX statement). The standard errors reported in the “Covariance Parameter
Estimates” table are based on the diagonal entries of this partition.
If you request an empirical standard error matrix with the EMPIRICAL option in the PROC GLIMMIX statement, a likelihood-based sandwich estimator is computed based on the subject-specific
gradients of the Laplace or quadrature approximation. The sandwich estimator then replaces H 1
in calculations following convergence.
To compute the standard errors and prediction standard errors of linear combinations of ˇ and ,
PROC GLIMMIX forms an approximate prediction variance matrix for Œb̌; b
0 from
2
3
@b
H 1
H 1 @Œˇ;
5
PD4 @b
@b
H 1 € 1C
H 1 @b
@Œˇ 0 ; 0 
@Œˇ 0 ; 0 
@Œˇ;
where € is the second derivative matrix from the suboptimization that maximizes f .y; ˇ; I /
for given values of ˇ and . The prediction variance submatrix for the random effects is based
Generalized Linear Mixed Models Theory F 2229
on approximating the conditional mean squared error of prediction as in Booth and Hobert (1998).
Note that even in the normal linear mixed model, the approximate conditional prediction standard
errors are not identical to the prediction standard errors you obtain by inversion of the mixed model
equations.
Conditional Fit and Output Statistics
When you estimate the parameters of a mixed model by Laplace approximation or quadrature, the
GLIMMIX procedure displays fit statistics related to the marginal distribution as well as the conditional distribution p.yjb
; b̌; b
/. For ODS purposes, the name of the “Conditional Fit Statistics”
table is “CondFitStatistics.” Because the marginal likelihood is approximated numerically for these
methods, statistics based on the marginal distribution are not available. Instead of the generalized Pearson chi-square statistic in the “Fit Statistics” table, PROC GLIMMIX reports the Pearson
statistic of the conditional distribution in the “Conditional Fit Statistics” table.
The unavailability of the marginal distribution also affects the set of output statistics that can be produced with METHOD=LAPLACE and METHOD=QUAD. Output statistics and statistical graphics
that depend on the marginal variance of the data are not available with these estimation methods.
User-Defined Variance Function
If you provide your own variance function, PROC GLIMMIX generally assumes that the (conditional) distribution of the data is unknown. Laplace or quadrature estimation would then not be
possible. When you specify a variance function with METHOD=LAPLACE or METHOD=QUAD,
the procedure assumes that the conditional distribution is normal. For example, consider the following statements to fit a mixed model to count data:
proc glimmix method=laplace;
class sub;
_variance_ = _phi_*_mu_;
model count = x / s link=log;
random int / sub=sub;
run;
The variance function and the link suggest an overdispersed Poisson model. The Poisson distribution cannot accommodate the extra scale parameter _PHI_, however. In this situation, the GLIMMIX procedure fits a mixed model with random intercepts, log link function, and variance function
, assuming that the count variable is normally distributed, given the random effects.
Starting Values
Good starting values for the fixed effects and covariance parameters are important for Laplace and
quadrature methods because the process commences with a suboptimization in which the empirical
Bayes estimates of the random effects must be obtained before the optimization can get under way.
Furthermore, the starting values are important for the adaptive choice of the number of quadrature
points.
2230 F Chapter 38: The GLIMMIX Procedure
If you choose METHOD=LAPLACE or METHOD=QUAD and you do not provide starting values
for the covariance parameters through the PARMS statement, the GLIMMIX procedure determines
starting values in the following steps.
1. A GLM is fit initially to obtain starting values for the fixed-effects parameters. No output is
produced from this stage. The number of initial iterations of this GLM fit can be controlled
with the INITITER= option in the PROC GLIMMIX statement. You can suppress this step
with the NOINITGLM option in the PROC GLIMMIX statement.
2. Given the fixed-effects estimates, starting values for the covariance parameters are computed
by a MIVQUE0 step (Goodnight 1978b).
3. For METHOD=QUAD you can follow these steps with several pseudo-likelihood updates
to improve on the estimates and to obtain solutions for the random effects. The number of
pseudo-likelihood steps is controlled by the INITPL= suboption of METHOD=QUAD.
4. For METHOD=QUAD, if you do not specify the number of quadrature points with the suboptions of the METHOD option, the GLIMMIX procedure attempts to determine a sufficient
number of points adaptively as follows. Suppose that Nq denotes the number of nodes in each
dimension. If Nmi n and Nmax denote the values from the QMIN= and QMAX= suboptions,
respectively, the sequence for values less than 11 is constructed in increments of 2 starting
at Nmi n . Values greater than 11 are incremented in steps of r. The default value is r D 10.
The default sequence, without specifying the QMIN=, QMAX=, or QFAC= option, is thus
1; 3; 5; 7; 9; 11; 21; 31. If the relative difference of the log-likelihood approximation for two
values in the sequence is less than the QTOL=t value (default t D 0:0001), the GLIMMIX
procedure uses the lesser value for Nq in the subsequent optimization. If the relative difference does not fall below the tolerance t for any two subsequent values in the sequence, no
estimation takes place.
Notes on Bias of Estimators
Generalized linear mixed models are nonlinear models, and the estimation techniques rely on approximations to the log likelihood or approximations of the model. It is thus not surprising that the
estimates of the covariance parameters and the fixed effects are usually not unbiased. Whenever
estimates are biased, questions arise about the magnitude of the bias, its dependence on other model
quantities, and the order of the bias. The order is important because it determines how quickly the
bias vanishes while some aspect of the data increases. Typically, studies of asymptotic properties
in models for hierarchical data suppose that the number of subjects (clusters) tends to infinity while
the size of the clusters is held constant or grows at a particular rate. Note that asymptotic results so
established do not extend to designs with fully crossed random effects, for example.
The following paragraphs summarize some important findings from the literature regarding the bias
in covariance parameter and fixed-effects estimates with pseudo-likelihood, Laplace, and adaptive
quadrature methods. The remarks draw in particular on results in Breslow and Lin (1995), Lin
and Breslow (1996), and Pinheiro and Chao (2006). Breslow and Lin (1995) and Lin and Breslow
(1996) study the “worst case” scenario of binary responses in a matched-pairs design. Their models
have a variance component structure, comprising either a single variance component (a subjectspecific random intercept; Breslow and Lin 1995) or a diagonal G matrix (Lin and Breslow 1996).
GLM Mode or GLMM Mode F 2231
They study the bias in the estimates of the fixed-effects ˇ and the covariance parameters when
the variance components are near the origin and for a canonical link function.
The matched-pairs design gives rise to a generalized linear mixed model with a cluster (subject)
size of 2. Recall that the pseudo-likelihood methods rely on a linearization and a probabilistic
assumption that the pseudo-data so obtained follow a normal linear mixed model. Obviously, it is
difficult to imagine how the subject-specific (conditional) distribution would follow a normal linear
mixed models with binary data in a cluster size of 2. The bias in the pseudo-likelihood estimator of
ˇ is of order jjjj. The bias for the Laplace estimator of ˇ is of smaller magnitude; its asymptotic
bias has order jjjj2 .
The Laplace methods and the pseudo-likelihood method produce biased estimators of the variance
component for the model considered in Breslow and Lin (1995). The order of the asymptotic
bias for both estimation methods is , as approaches zero. Breslow and Lin (1995) comment on
the fact that even with matched pairs, the bias vanishes very quickly in the binomial setting. If the
conditional mean in the two groups is equal to 0:5, then the asymptotic bias factor of the pseudolikelihood estimator is 1 1=.2n/, where n is the binomial denominator. This term goes to 1 quickly
as n increases. This result underlines the importance of grouping binary observations into binomial
responses whenever possible.
The results of Breslow and Lin (1995) and Lin and Breslow (1996) are echoed in the simulation
study in Pinheiro and Chao (2006). These authors also consider adaptive quadrature in models
with nested, hierarchical, random effects and show that adaptive quadrature with a sufficient number of nodes leads to nearly unbiased—or least biased—estimates. Their results also show that
results for binary data cannot so easily be ported to other distributions. Even with a cluster size
of 2, the pseudo-likelihood estimates of fixed effects and covariance parameters are virtually unbiased in their simulation of a Poisson GLMM. Breslow and Lin (1995) and Lin and Breslow
(1996) “eschew” the residual PL version (METHOD=RSPL) over the maximum likelihood form
(METHOD=MSPL). Pinheiro and Chao (2006) consider both forms in their simulation study. As
expected, the residual form shows less bias than the MSPL form, for the same reasons REML estimation leads to less biased estimates compared to ML estimation in linear mixed models. The gain
is modest, however; see, for example, Table 1 in Pinheiro and Chao (2006). When the variance
components are small, there is a sufficient number of observations per cluster, and a reasonable
number of clusters, then pseudo-likelihood methods for binary data are very useful—they provide
a computational expedient alternative to numerical integration, and they allow the incorporation of
R-side covariance structure into the model. Because many group randomized trials involve many
observations per group and small random effects variances, Murray et al. (2004) term questioning
the use of conditional models for trials with binary outcome an “overreaction.”
GLM Mode or GLMM Mode
The GLIMMIX procedure knows two basic modes of parameter estimation, and it can be important
for you to understand the differences between the two modes.
In GLM mode, the data are never correlated and there can be no G-side random effects. Typical
examples are logistic regression and normal linear models. When you fit a model in GLM mode,
the METHOD= option in the PROC GLIMMIX statement has no effect. PROC GLIMMIX esti-
2232 F Chapter 38: The GLIMMIX Procedure
mates the parameters of the model by maximum likelihood, (restricted) maximum likelihood, or
quasi-likelihood, depending on the distributional properties of the model (see the section “Default
Estimation Techniques” on page 2271). The “Model Information” table tells you which estimation
method was applied. In GLM mode, the individual observations are considered the sampling units.
This has bearing, for example, on how sandwich estimators are computed (see the EMPIRICAL
option and the section “Empirical Covariance (“Sandwich”) Estimators” on page 2241).
In GLMM mode, the procedure assumes that the model contains random effects or possibly correlated errors, or that the data have a clustered structure. The parameters are then estimated by the
techniques specified with the METHOD= option in the PROC GLIMMIX statement.
In general, adding one overdispersion parameter to a generalized linear model does not trigger the
GLMM mode. For example, the model defined by the following statements is fit in GLM mode:
proc glimmix;
model y = x1 x2 / dist=poisson;
random _residual_;
run;
The parameters of the fixed effects are estimated by maximum likelihood, and the covariance matrix
of the fixed-effects parameters is adjusted by the overdispersion parameter.
In a model with uncorrelated data you can trigger the GLMM mode by specifying a SUBJECT= or
GROUP= effect in the RANDOM statement. For example, the following statements fit the model
by using the residual pseudo-likelihood algorithm:
proc glimmix;
class id;
model y = x1 x2 / dist=poisson;
random _residual_ / subject=id;
run;
If in doubt, you can determine whether a model was fit in GLM mode or GLMM mode. In GLM
mode the “Covariance Parameter Estimates” table is not produced. Scale and dispersion parameters
in the model appear in the “Parameter Estimates” table.
Statistical Inference for Covariance Parameters
The Likelihood Ratio Test
The likelihood ratio test (LRT) compares the likelihoods of two models where parameter estimates
are obtained in two parameter spaces, the space  and the restricted subspace 0 . In the GLIMMIX procedure, the full model defines  and the test-specification in the COVTEST statement
determines the null parameter space 0 . The likelihood ratio procedure consists of the following
steps (see, for example, Bickel and Doksum 1977, p. 210):
1. Find the estimate b
of 2 . Compute the likelihood L.b
/.
Statistical Inference for Covariance Parameters F 2233
2. Find the estimate b
0 of 2 0 . Compute the likelihood L.b
0 /.
3. Form the likelihood ratio
D
L.b
/
L.b
0/
4. Find a function f that has a known distribution. f ./ serves as the test statistic for the
likelihood ratio test.
Please note the following regarding the implementation of these steps in the COVTEST statement
of the GLIMMIX procedure.
The function f ./ in step 4 is always taken to be
n o
D 2 log which is twice the difference between the log likelihoods for the full model and the model
under the COVTEST restriction.
For METHOD=RSPL and METHOD=RMPL, the test statistic is based on the restricted likelihood.
For GLMMs involving pseudo-data, the test statistics are based on the pseudo-likelihood or
the restricted pseudo-likelihood and are based on the final pseudo-data.
The parameter space  for the full model is typically not an unrestricted space. The GLIMMIX procedure imposes boundary constraints for variance components and scale parameters,
for example. The specification of the subspace 0 must be consistent with these full-model
constraints; otherwise the test statistic does not have the needed distribution. You can
remove the boundary restrictions with the NOBOUND option in the PROC GLIMMIX statement or the NOBOUND option in the PARMS statement.
One- and Two-Sided Testing, Mixture Distributions
Consider testing the hypothesis H0 W i D 0. If  is the open interval .0; 1/, then only a one-sided
alternative hypothesis is meaningful,
H0 W i D 0
Ha W i > 0
This is the appropriate set of hypotheses, for example, when i is the variance of a G-side random
effect. The positivity constraint on  is required for valid conditional and marginal distributions of
the data. Verbeke and Molenberghs (2003) refer to this situation as the constrained case.
However, if one focuses on the validity of the marginal distribution alone, then negative values
for i might be permissible, provided that the marginal variance remains positive definite. In the
vernacular or Verbeke and Molenberghs (2003), this is the unconstrained case. The appropriate
alternative hypothesis is then two-sided,
H0 W i D 0
Ha W i 6D 0
2234 F Chapter 38: The GLIMMIX Procedure
Several important issues are connected to the choice of hypotheses. The GLIMMIX procedure
by default imposes constraints on some covariance parameters. For example, variances and scale
parameters have a lower bound of 0. This implies a contrained setting with one-sided alternatives.
If you specify the NOBOUND option in the PROC GLIMMIX statement, or the NOBOUND option
in the PARMS statement, the boundary restrictions are lifted from the covariance parameters and
the GLIMMIX procedure takes an unconstrained stance in the sense of Verbeke and Molenberghs
(2003). The alternative hypotheses for variance components are then two-sided.
When H0 W i D 0 and  D .0; 1/, the value of i under the null hypothesis is on the boundary
of the parameter space. The distribution of the likelihood ratio test statistic is then nonstandard.
In general, it is a mixture of distributions, and in certain special cases, it is a mixture of central
chi-square distributions. Important contributions to the understanding of the asymptotic behavior
of the likelihood ratio and score test statistic in this situation have been made by, for example, Self
and Liang (1987), Shapiro (1988), and Silvapulle and Silvapulle (1995). Stram and Lee (1994,
1995) applied the results of Self and Liang (1987) to likelihood ratio testing in the mixed model
with uncorrelated errors. Verbeke and Molenberghs (2003) compared the score and likelihood ratio
tests in random effects models with unstructured G matrix and provide further results on mixture
distributions.
The GLIMMIX procedure recognizes the following special cases in the computation of p-values (b
denotes the realized value of the test statistic). Notice that the probabilities of general chi-square
mixture distributions do not equal linear combination of central chi-square probabilities (Davis
1977; Johnson, Kotz, and Balakrishnan 1994, Section 18.8).
1. parameters are tested, and neither parameters specified under H0 nor nuisance parameters
are on the boundary of the parameters space (Case 4 in Self and Liang 1987). The p-value is
computed by the classical result:
p D Pr 2 b
2. One parameter is specified under H0 and it falls on the boundary. No other parameters are on
the boundary (Case 5 in Self and Liang 1987).
(
b
1
D0
pD
0:5 Pr 21 b
b
>0
Note that this implies a 50:50 mixture of a 20 and a 21 distribution. This is also Case 1 in
Verbeke and Molenberghs (2000, p. 69).
3. Two parameters are specified under H0 , and one falls on the boundary. No nuisance parameters are on the boundary (Case 6 in Self and Liang 1987).
p D 0:5 Pr 21 b
C 0:5 Pr 22 b
A special case of this scenario is the addition of a random effect to a model with a single
random effect and unstructured covariance matrix (Case 2 in Verbeke and Molenberghs 2000,
p. 70).
Statistical Inference for Covariance Parameters F 2235
4. Removing j random effects from j C k uncorrelated random effects (Verbeke and Molenberghs 2003).
pD2
j
j X
j
Pr 2i b
i
i D0
Note that this case includes the case of testing a single random effects variance against zero,
which leads to a 50:50 mixture of a 20 and a 21 as in 2.
5. Removing a random effect from an unstructured G matrix (Case 3 in Verbeke and Molenberghs 2000, p. 71).
p D 0:5 Pr 2k b
C 0:5 Pr 2k 1 b
where k is the number of random effects (columns of G) in the full model. Case 5 in Self and
Liang (1987) describes a special case.
When the GLIMMIX procedure determines that estimates of nuisance parameters (parameters not
specified under H0 ) fall on the boundary, no mixture results are computed.
You can request that the procedure not use mixtures with the CLASSICAL option in the COVTEST
statement. If mixtures are used, the Note column of the “Likelihood Ratio Tests of Covariance Parameters” table contains the “MI” entry. The “DF” entry is used when PROC GLIMMIX determines
that the standard computation of p-values is appropriate. The “–” entry is used when the classical
computation was used because the testing and model scenario does not match one of the special
cases described previously.
Handling the Degenerate Distribution
Likelihood ratio testing in mixed models invariably involves the chi-square distribution with zero
degrees of freedom. The 20 random variable is degenerate at 0, and it occurs in two important
circumstances. First, it is a component of mixtures, where typically the value of the test statistic
is not zero. In that case, the contribution of the 20 component of the mixture to the p-value is nil.
Second, a degenerate distribution of the test statistic occurs when the null model is identical to the
full model—for example, if you test a hypothesis that does not impose any (new) constraints on
the parameter space. The following statements test whether the R matrix in a variance component
model is diagonal:
proc glimmix;
class a b;
model y = a;
random b a*b;
covtest diagR;
run;
Because no R-side covariance structure is specified (all random effects are G-side effects), the R
matrix is diagonal in the full model and the COVTEST statement does not impose any further restriction on the parameter space. The likelihood ratio test statistic is zero. The GLIMMIX procedure
2236 F Chapter 38: The GLIMMIX Procedure
computes the p-value as the probability to observe a value at least as large as the test statistic under
the null hypothesis. Hence,
p D Pr.20 0/ D 1
Wald Versus Likelihood Ratio Tests
The Wald test and the likelihood ratio tests are asymptotic tests, meaning that the distribution from
which p-values are calculated for a finite number of samples draws on the distribution of the test
statistic as the sample size grows to infinity. The Wald test is a simple test that is easy to compute
based only on parameter estimates and their (asymptotic) standard errors. The likelihood ratio test,
on the other hand, requires the likelihoods of the full model and the model reduced under H0 . It is
computationally more demanding, but also provides the asymptotically more powerful and reliable
test. The likelihood ratio test is almost always preferable to the Wald test, unless computational
demands make it impractical to refit the model.
Confidence Bounds Based on Likelihoods
Families of statistical tests can be inverted to produce confidence limits for parameters. The confidence region for parameter is the set of values for which the corresponding test fails to reject
H W D 0 . When parameters are estimated by maximum likelihood or a likelihood-based technique, it is natural to consider the likelihood ratio test statistic for H in the test inversion. When
there are multiple parameters in the model, however, you need to supply values for these nuisance
parameters during the test inversion as well.
In the following, suppose that is the covariance parameter vector and that one of its elements, , is
the parameter of interest for which you want to construct a confidence interval. The other elements
of are collected in the nuisance parameter vector 2 . Suppose that b
is the estimate of from the
b
overall optimization and that L./ is the likelihood evaluated at that estimate. If estimation is based
on pseudo-data, then L.b
/ is the pseudo-likelihood based on the final pseudo-data. If estimation
uses a residual (restricted) likelihood, then L denotes the restricted maximum likelihood and b
is
the REML estimate.
Profile Likelihood Bounds
The likelihood ratio test statistic for testing H W D 0 is
n n
o
n
oo
2 log L.b
/
log L.0 ; b
2/
where b
2 is the likelihood estimate of 2 under the restriction that D 0 . To invert this test,
a function is defined that returns the maximum likelihood for a fixed value of by seeking the
maximum over the remaining parameters. This function is termed the profile likelihood (Pawitan
2001, Ch. 3.4),
p D L.2 je
/ D sup L.e
; 2 /
2
Statistical Inference for Covariance Parameters F 2237
In computing p , is fixed at e
and 2 is estimated. In mixed models, this step typically requires a
separate, iterative optimization to find the estimate of 2 while is held fixed. The .1 ˛/ 100%
profile likelihood confidence interval for is then defined as the set of values for e
that satisfy
n n
o
n
oo
2 log L.b
/
log L.2 je
/ 21;.1 ˛/
The GLIMMIX procedure seeks the values e
l and e
u that mark the endpoints of the set around b
e
e
that satisfy the inequality. The values . l and u / are then called the .1 ˛/ 100% confidence
bounds for . Note that the GLIMMIX procedure assumes that the confidence region is not disjoint
and relies on the convexity of L.b
/.
It is not always possible to find values e
l and e
u that satisfy the inequalities. For example, when
the parameter space is (0; 1/ and
n n
o
o
2 log L.b
/
log fL.2 j0/g > 21;.1 ˛/
a lower bound cannot be found at the desired confidence level. The GLIMMIX procedure reports
the right-tail probabilities that are achieved by the underlying likelihood ratio statistic separately for
lower and upper bounds.
Effect of Scale Parameter
When a scale parameter is eliminated from the optimization by profiling from the likelihood,
some parameters might be expressed as ratios with in the optimization. This is the case, for
example, in variance component models. The profile likelihood confidence bounds are reported
on the scale of the parameter in the overall optimization. In case parameters are expressed as
ratios with or functions of , the column RatioEstimate is added to the “Covariance Parameter
Estimates” table. If parameters are expressed as ratios with and you want confidence bounds for
the unscaled parameter, you can prevent profiling of from the optimization with the NOPROFILE
option in the PROC GLIMMIX statement, or choose estimated likelihood confidence bounds with
the TYPE=ELR suboption of the CL option in the COVTEST statement. Note that the NOPROFILE
option is automatically in effect with METHOD=LAPLACE and METHOD=QUAD.
Estimated Likelihood Bounds
Computing profile likelihood ratio confidence bounds can be computationally expensive, because
of the need to repeatedly estimate 2 in a constrained optimization. A computationally simpler
method to construct confidence bounds from likelihood-based quantities is to use the estimated
likelihood (Pawitan 2001, Ch. 10.7) instead of the profile likelihood. An estimated likelihood
technique replaces the nuisance parameters in the test inversion with some other estimate. If you
choose the TYPE=ELR suboption of the CL option in the COVTEST statement, the GLIMMIX
procedure holds the nuisance parameters fixed at the likelihood estimates. The estimated likelihood
statistic for inversion is then
e D L.e
;b
2/
where b
2 are the elements of b
that correspond to the nuisance parameters. As the values of e
are
varied, no reestimation of 2 takes place. Although computationally more economical, estimated
2238 F Chapter 38: The GLIMMIX Procedure
likelihood intervals do not take into account the variability associated with the nuisance parameters.
Their coverage can be satisfactory if the parameter of interest is not (or only weakly) correlated with
the nuisance parameters. Estimated likelihood ratio intervals can fall short of the nominal coverage
otherwise.
Figure 38.11 depicts profile and estimated likelihood ratio intervals for the parameter in a twoparameter compound-symmetric model, D Œ; 0 , in which the correlation between the covariance parameters is small. The elliptical shape traces the set of values for which the likelihood ratio
test rejects the hypothesis of equality with the solution. The interior of the ellipse is the “acceptance” region of the test. The solid and dashed lines depict the PLR and ELR confidence limits
for , respectively. Note that both confidence limits intersect the ellipse and that the ELR interval
passes through the REML estimate of . The PLR bounds are found as those points intersecting the
ellipse, where equals the constrained REML estimate.
Figure 38.11 PLR and ELR Intervals, Small Correlation between Parameters
The major axes of the ellipse in Figure 38.11 are nearly aligned with the major axes of the coordinate
system. As a consequence, the line connecting the PLR bounds passes close to the REML estimate
in the full model. As a result, ELR bounds will be similar to PLR bounds. Figure 38.12 displays a
different scenario, a two-parameter AR(1) covariance structure with a more substantial correlation
between the AR(1) parameter () and the residual variance ().
Satterthwaite Degrees of Freedom Approximation F 2239
Figure 38.12 PLR and ELR Intervals, Large Correlation between Parameters
The correlation between the parameters yields an acceptance region whose major axes are not
aligned with the axes of the coordinate system. The ELR bound for passes through the REML
estimate of from the full model and is much shorter than the PLR interval. The PLR interval
aligns with the major axis of the acceptance region; it is the preferred confidence interval.
Satterthwaite Degrees of Freedom Approximation
The DDFM=SATTERTHWAITE option in the MODEL statement requests denominator degrees of
freedom in t tests and F tests computed according to a general Satterthwaite approximation. The
DDFM=KENWARDROGER option also entails the computation of Satterthwaite-type degrees of
freedom.
The general Satterthwaite approximation computed in PROC GLIMMIX for the test
b̌
HWL
D0
b
2240 F Chapter 38: The GLIMMIX Procedure
is based on the F statistic
b̌ 0 0
L .LCL0 /
b
F D
r
1L
b̌
b
where r D rank.LCL0 /, and C is the approximate variance matrix of Œb̌0 ; b
0 0 0 ; see the section
“Estimated Precision of Estimates” on page 2219 and the section “Aspects Common to Adaptive
Quadrature and Laplace Approximation” on page 2228.
The approximation proceeds by first performing the spectral decomposition LCL0 D U0 DU, where
U is an orthogonal matrix of eigenvectors and D is a diagonal matrix of eigenvalues, both of dimension r r. Define bj to be the j th row of UL, and let
j D
2.Dj /2
g0j Agj
where Dj is the j th diagonal element of D and gj is the gradient of bj Cb0j with respect to ,
evaluated at b
. The matrix A is the asymptotic variance-covariance matrix of b
, obtained from the
second derivative matrix of the likelihood equations. You can display this matrix with the ASYCOV
option in the PROC GLIMMIX statement.
Finally, let
ED
r
X
j
I.j > 2/
j 2
j D1
where the indicator function eliminates terms for which j 2. The degrees of freedom for F are
then computed as
D
E
2E
rank.L/
provided E > r; otherwise is set to zero.
In the one-dimensional case, when PROC GLIMMIX computes a t test, the Satterthwaite degrees
of freedom for the t statistic
b̌
0
l
b
tD
l0 Cl
are computed as
D
2.l0 Cl/2
g0 Ag
where g is the gradient of l0 Cl with respect to , evaluated at b
.
Empirical Covariance (“Sandwich”) Estimators F 2241
Empirical Covariance (“Sandwich”) Estimators
Residual-Based Estimators
The GLIMMIX procedure can compute the classical sandwich estimator of the covariance matrix
of the fixed effects, as well as several bias-adjusted estimators. This requires that the model is either
an (overdispersed) GLM or a GLMM that can be processed by subjects (see the section “Processing
by Subjects” on page 2245).
Consider a statistical model of the form
Y D C ;
.0; †/
The general expression of a sandwich covariance estimator is then
!
m
X
b
b 1 F0 ei e0 Fi †
b
b 1b
c
Ai b
D0 †
Di Ai 
i
i
i
i
i
1 D/
.
i D1
where ei D yi
b
i ,  D .D0 †
For a GLMM estimated by one of the pseudo-likelihood techniques that involve linearization, you
can make the following substitutions: Y ! P, † ! V./, D ! X, b
! Xb̌. These matrices are
defined in the section “Pseudo-likelihood Estimation Based on Linearization” on page 2218.
The various estimators computed by the GLIMMIX procedure differ in the choice of the constant
c and the matrices Fi and Ai . You obtain the classical estimator, for example, with c D 1, and
Fi D Ai equal to the identity matrix.
The EMPIRICAL=ROOT estimator of Kauermann and Carroll (2001) is based on the approximation
Var ei e0i .I Hi /†i
where Hi D Di D0i †i 1 . The EMPIRICAL=FIRORES estimator is based on the approximation
Var ei e0i .I Hi /†i .I H0i /
of Mancl and DeRouen (2001). Finally, the EMPIRICAL=FIROEEQ estimator is based on approximating an unbiased estimating equation (Fay and Graubard 2001). For this estimator, Ai is a
diagonal matrix with entries
ŒAi jj D 1
minfr; ŒQjj g
1=2
b 1 Di .
b The optional number 0 r < 1 is chosen to provide an upper bound
where Q D D0i †
i
on the correction factor. For r D 0, the classical sandwich estimator results. PROC GLIMMIX
chooses as default value r D 3=4. The diagonal entries of Ai are then no greater than 2.
Table 38.18 summarizes the components of the computation for the GLMM based on linearization,
where m denotes the number of subjects and k is the rank of X.
2242 F Chapter 38: The GLIMMIX Procedure
Table 38.18
Empirical Covariance Estimators for a Linearized GLMM
c
EMPIRICAL=
CLASSICAL
DF
m
m k
1
ROOT
FIRORES
FIROEEQ(r)
1
m>k
otherwise
1
1
1
Diagf.1
Ai
Fi
I
I
I
I
I
I
minfr; ŒQjj g/
1=2 g
.I H0i / 1=2
.I H0i / 1
I
Computation of an empirical variance estimator requires that the data can be processed by independent sampling units. This is always the case in GLMs. In this case, m equals the sum of all
frequencies. In GLMMs, the empirical estimators require that the data consist of multiple subjects.
In that case, m equals the number of subjects as per the “Dimensions” table. The following section
discusses how the GLIMMIX procedure determines whether the data can be processed by subjects.
The section “GLM Mode or GLMM Mode” on page 2231 explains how PROC GLIMMIX determines whether a model is fit in GLM mode or in GLMM mode.
Design-Adjusted MBN Estimator
Morel (1989) and Morel, Bokossa, and Neerchal (2003) suggested a bias correction of the classical
sandwich estimator that rests on an additive correction of the residual crossproducts and a sample size correction. This estimator is available with the EMPIRICAL=MBN option in the PROC
GLIMMIX statement. In the notation of the previous section, the residual-based MBN estimator
can be written as
!
m
X
0
1
0
1
b
b
b
b b
b

D†
cei e C Bi †
Di 
i
i
i
i
i D1
where
c D .f
option
1/=.f
k/m=.m 1/ or c D 1 when you specify the EMPIRICAL=MBN(NODF)
f is the sum of the frequencies
k equals the rank of X
bi
B i D ım †
n
o
b
D max r; trace M
=k MD
Pm
b0 b 1 0 b 1 b
i D1 Di † i ei ei † i Di
b
k D k if m k, otherwise k equals the number of nonzero singular values of M
ım D k=.m
k/ if m > .d C 1/k and ım D 1=d otherwise
Empirical Covariance (“Sandwich”) Estimators F 2243
d 1 and 0 r 1 are parameters supplied with the mbn-options of the
EMPIRICAL=MBN(mbn-options) option. The default values are d D 2 and r D 1. When
the NODF option is in effect, the factor c is set to 1.
Rearranging terms, the MBN estimator can also be written as an additive adjustment to a samplesize corrected classical sandwich estimator
!
m
X
b
b C ım 
b
b
b 1 ei e0 †
b 1b
Di 
c
D0 †
i
i
i
i
i D1
Because ım is of order m 1 , the additive adjustment to the classical estimator vanishes as the
number of independent sampling units (subjects) increases. The parameter is a measure of the
design effect (Morel, Bokossa, and Neerchal 2003). Besides good statistical properties in terms
of Type I error rates in small-m situations, the MBN estimator also has the desirable property of
recovering rank when the number of sampling units is small. If m < k, the “meat” piece of the
classical sandwich estimator is essentially a sum of rank one matrices. A small number of subjects
relative to the rank of X can result in a loss of rank and subsequent loss of numerator degrees of
freedom in tests. The additive MBN adjustment counters the rank exhaustion. You can examine the
rank of an adjusted covariance matrix with the COVB(DETAILS) option in the MODEL statement.
When the principle of the MBN estimator is applied to the likelihood-based empirical estimator,
you obtain
!
m
X
˛/ 1
cgi .b
˛/gi .b
˛/0 C Bi H.b
H.b
˛/ 1
i D1
where Bi D ım Hi .b
˛/, and Hi .b
˛/ is the second derivative of the log likelihood for the i th
sampling unit (subject) evaluated at the vector of parameter estimates, b
˛. Also, gi .b
˛/ is the first
derivative of the log likelihood for the i th sampling unit. This estimator is computed if you request
EMPIRICAL=MBN with METHOD=LAPLACE or METHOD=QUAD.
In terms of adjusting the classical likelihood-based estimator (White 1982), the likelihood MBN
estimator can be written as
!
m
X
˛/ 1 ım H.b
˛/ 1
c H.b
˛/ 1
gi .b
˛/gi .b
˛/0 H.b
i D1
The parameter is determined as
˚
D max r; trace H.b
˛/
Pm
M D i D1 gi .b
˛/gi .b
˛/0
1M
=k k D k if m k, otherwise k equals the number of nonzero singular values of H.b
˛/
1M
2244 F Chapter 38: The GLIMMIX Procedure
Exploring and Comparing Covariance Matrices
If you use an empirical (sandwich) estimator with the EMPIRICAL= option in the PROC GLIMMIX statement, the procedure replaces the model-based estimator of the covariance of the fixed
effects with the sandwich estimator. This affects aspects of inference, such as prediction standard errors, tests of fixed effects, estimates, contrasts, and so forth. Similarly, if you choose the
DDFM=KENWARDROGER degrees-of-freedom method in the MODEL statement, PROC GLIMMIX adjusts the model-based covariance matrix of the fixed effects according to Kenward and
Roger (1997) or according to Kackar and Harville (1984) and Harville and Jeske (1992).
In this situation, the COVB(DETAILS) option in the MODEL statement has two effects. The GLIMMIX procedure displays the (adjusted) covariance matrix of the fixed effects and the model-based
covariance matrix (for ODS purposes, the name of the table with the model-based covariance matrix is “CovBModelBased”). The procedure also displays a table of statistics for the unadjusted and
adjusted covariance matrix and for their comparison. For ODS purposes, the name of this table is
“CovBDetails.”
If the model-based covariance matrix is not replaced with an adjusted estimator, the
COVB(DETAILS) option displays the model-based covariance matrix and provides diagnostic
measures for it in the “CovBDetails” table.
The table generated by the COVB(DETAILS) option consists of several sections. See Example 38.8
for an application.
The trace and log determinant of covariance matrices are general scalar summaries that are sometimes used in direct comparisons, or in formulating other statistics, such as the difference of log
determinants. The trace simply represents the sum of the variances of all fixed-effects parameters.
If a matrix is indefinite, the determinant is reported instead of the log determinant.
The model-based and adjusted covariance matrices should have the same general makeup of eigenvalues. There should not be any negative eigenvalues, and they should have the same numbers of
positive and zero eigenvalues. A reduction in rank due to the adjustment is troublesome for aspects
of inference. Negative eigenvalues are listed in the table only if they occur, because a covariance
matrix should be at least positive semi-definite. However, the GLIMMIX procedure examines the
model-based and adjusted covariance matrix for negative eigenvalues. The condition numbers reported by PROC GLIMMIX for positive (semi-)definite matrices are computed as the ratio of the
largest and smallest nonzero eigenvalue. A large condition number reflects poor conditioning of the
matrix.
Matrix norms are extensions of the concept of vector norms to measure the “length” of a matrix.
The Frobenius norm of an .n m/ matrix A is the direct equivalent of the Euclidean vector norm,
the square root of the sum of the squared elements,
v
uX
n
u n X
2
jjAjjF D t
aij
i D1 j D1
Processing by Subjects F 2245
The 1- and 1-norms of matrix A are the maximum absolute row and column sums, respectively:
8
9
m
<X
=
jjAjj1 D max
jaij j W i D 1; ; n
:
;
j D1
( n
)
X
jjAjj1 D max
jaij j W j D 1; ; m
i D1
These two norms are identical for symmetric matrices.
The “Comparison” section of the “CovBDetails” table provides several statistics that set the matrices in relationship. The concordance correlation reported by the GLIMMIX procedure is a standardized measure of the closeness of the model-based and adjusted covariance matrix. It is a slight
modification of the covariance concordance correlation in Vonesh, Chinchilli, and Pu (1996) and
Vonesh and Chinchilli (1997, Ch. 8.3). Denote as  the .p p/ model-based covariance matrix
and as a the adjusted matrix. Suppose that K is the matrix obtained from the identity matrix of
size p by replacing diagonal elements corresponding to singular rows in  with zeros. The lower
triangular portion of  1=2 a  1=2 is stored in vector ! and the lower triangular portion of K is
stored in vector k. The matrix  1=2 is constructed from an eigenanalysis of  and is symmetric.
The covariance concordance correlation is then
r.!/ D 1
jj! kjj2
jj!jj2 C jjkjj2
This measure is 1 if  = a . If ! is orthogonal to k, there is total disagreement between the
model-based and the adjusted covariance matrix and r.!/ is zero.
The discrepancy function reported by PROC GLIMMIX is computed as
d D logfjjg
logfja jg C tracefa  g
rankfg
In diagnosing departures between an assumed covariance structure and VarŒY—using an empirical
estimator—Vonesh, Chinchilli, and Pu (1996) find that the concordance correlation is useful in
detecting gross departures and propose D ns d to test the correctness of the assumed model,
where ns denotes the number of subjects.
Processing by Subjects
Some mixed models can be expressed in different but mathematically equivalent ways with PROC
GLIMMIX statements. While equivalent statements lead to equivalent statistical models, the data
processing and estimation phase can be quite different, depending on how you write the GLIMMIX
statements. For example, the particular use of the SUBJECT= option in the RANDOM statement affects data processing and estimation. Certain options are available only when the data are processed
by subject, such as the EMPIRICAL option in the PROC GLIMMIX statement.
Consider a GLIMMIX model where variables A and Rep are classification variables with a and r
levels, respectively. The following pairs of statements produce the same random-effects structure:
2246 F Chapter 38: The GLIMMIX Procedure
class Rep A;
random Rep*A;
class Rep A;
random intercept / subject=Rep*A;
class Rep A;
random Rep / subject=A;
class Rep A;
random A / subject=Rep;
In the first case, PROC GLIMMIX does not process the data by subjects because no SUBJECT=
option was given. The computation of empirical covariance estimators, for example, will not be
possible. The marginal variance-covariance matrix has the same block-diagonal structure as for
cases 2–4, where each block consists of the observations belonging to a unique combination of Rep
and A. More importantly, the dimension of the Z matrix of this model will be n ra, and Z will be
sparse. In the second case, the Zi matrix for each of the ra subjects is a vector of ones.
If the data can be processed by subjects, the procedure typically executes faster and requires less
memory. The differences can be substantial, especially if the number of subjects is large. Recall
that fitting of generalized linear mixed models might be doubly iterative. Small gains in efficiency
for any one optimization can produce large overall savings.
If you interpret the intercept as “1,” then a RANDOM statement with TYPE=VC (the default) and
no SUBJECT= option can be converted into a statement with subject by dividing the random effect
by the eventual subject effect. However, the presence of the SUBJECT= option does not imply
processing by subject. If a RANDOM statement does not have a SUBJECT= effect, processing
by subjects is not possible unless the random effect is a pure R-side overdispersion effect. In the
following example, the data will not be processed by subjects, because the first RANDOM statement
specifies a G-side component and does not use a SUBJECT= option:
proc glimmix;
class A B;
model y = B;
random A;
random B / subject=A;
run;
To allow processing by subjects, you can write the equivalent model with the following statements:
proc glimmix;
class A B;
model y = B;
random int / subject=A;
random B
/ subject=A;
run;
Processing by Subjects F 2247
If you denote a variance component effect X with subject effect S as X–(S), then the “calculus of
random effects” applied to the first RANDOM statement reads A = Int*A = Int–(A) = A–(Int). For the
second statement there are even more equivalent formulations: A*B = A*B*Int = A*B–(Int) = A–(B) =
B–(A) = Int–(A*B).
If there are multiple subject effects, processing by subjects is possible if the effects are equal or
contained in each other. Note that in the last example the A*B interaction is a random effect. The
following statements give an equivalent specification to the previous model:
proc glimmix;
class A B;
model y = B;
random int / subject=A;
random A
/ subject=B;
run;
Processing by subjects is not possible in this case, because the two subject effects are not syntactically equal or contained in each other. The following statements depict a case where subject effects
are syntactically contained:
proc glimmix;
class A B;
model y = B;
random int / subject=A;
random int / subject=A*B;
run;
The A main effect is contained in the A*B interaction. The GLIMMIX procedure chooses as the
subject effect for processing the effect that is contained in all other subject effects. In this case, the
subjects are defined by the levels of A.
You can examine the “Model Information” and “Dimensions” tables to see whether the GLIMMIX
procedure processes the data by subjects and which effect is used to define subjects. The “Model Information” table displays whether the marginal variance matrix is diagonal (GLM models), blocked,
or not blocked. The “Dimensions” table tells you how many subjects (=blocks) there are.
Finally, nesting and crossing of interaction effects in subject effects are equivalent. The following
two RANDOM statements are equivalent:
class Rep A;
random intercept / subject=Rep*A;
class Rep A;
random intercept / subject=Rep(A);
2248 F Chapter 38: The GLIMMIX Procedure
Radial Smoothing Based on Mixed Models
The radial smoother implemented with the TYPE=RSMOOTH option in the RANDOM statement
is an approximate low-rank thin-plate spline as described in Ruppert, Wand, and Carroll (2003,
Chapter 13.4–13.5). The following sections discuss in more detail the mathematical-statistical connection between mixed models and penalized splines and the determination of the number of spline
knots and their location as implemented in the GLIMMIX procedure.
From Penalized Splines to Mixed Models
The connection between splines and mixed models arises from the similarity of the penalized spline
fitting criterion to the minimization problem that yields the mixed model equations and solutions
for ˇ and . This connection is made explicit in the following paragraphs. An important distinction
between classical spline fitting and its mixed model smoothing variant, however, lies in the nature
of the spline coefficients. Although they address similar minimization criteria, the solutions for the
spline coefficients in the GLIMMIX procedure are the solutions of random effects, not fixed effects.
Standard errors of predicted values, for example, account for this source of variation.
Consider the linearized mixed pseudo-model from the section “The Pseudo-model” on page 2218,
P D Xˇ C Z C . One derivation of the mixed model equations, whose solutions are b̌ and b
,
is to maximize the joint density of f .; / with respect to ˇ and . This is not a true likelihood
problem, because is not a parameter, but a random vector.
In the special case with VarŒ D I and VarŒ D 2 I, the maximization of f .; / is equivalent
to the minimization of
Q.ˇ; / D 1
.p
Xˇ
Z/0 .p
Xˇ
Z/ C 2 0
Now consider a linear spline as in Ruppert, Wand, and Carroll (2003, p. 108),
pi D ˇ0 C ˇ1 xi C
K
X
j .xi
tj / C
j D1
where the j denote the spline coefficients at knots t1 ; ; tK . The truncated line function is defined
as
x t
x>t
.x t /C D
0
otherwise
If you collect the intercept and regressor x into the matrix X, and if you collect the truncated line
functions into the .n K/ matrix Z, then fitting the linear spline amounts to minimization of the
penalized spline criterion
Q .ˇ; / D .p
Xˇ
Z/0 .p
Xˇ
Z/ C 2 0 where is the smoothing parameter.
Because minimizing Q .ˇ; / with respect to ˇ and is equivalent to minimizing Q .ˇ; /=,
both problems lead to the same solution, and D = is the smoothing parameter. The mixed
Radial Smoothing Based on Mixed Models F 2249
model formulation of spline smoothing has the advantage that the smoothing parameter is selected
“automatically.” It is a function of the covariance parameter estimates, which, in turn, are estimated
according to the method you specify with the METHOD= option in the PROC GLIMMIX statement.
To accommodate nonnormal responses and general link functions, the GLIMMIX procedure uses
e 1 A
e 1 , where A is the matrix of variance functions and  is the diagonal matrix of
VarŒ D 
mean derivatives defined earlier. The correspondence between spline smoothing and mixed modeling is then one between a weighted linear mixed model and a weighted spline. In other words, the
minimization criterion that yields the estimates b̌ and solutions b
is then
Q.ˇ; / D 1
.p
Xˇ
e
Z/0 A
1e
.p
Xˇ
Z/0 C 2 0
If you choose the TYPE=RSMOOTH covariance structure, PROC GLIMMIX chooses radial basis
functions as the spline basis and transforms them to approximate a thin-plate spline as in Chapter
13.4 of Ruppert, Wand, and Carroll (2003). For computational expediency, the number of knots is
chosen to be less than the number of data points. Ruppert, Wand, and Carroll (2003) recommend one
knot per every four unique regressor values for one-dimensional smoothers. In the multivariate case,
general recommendations are more difficult, because the optimal number and placement of knots
depend on the spatial configuration of samples. Their recommendation for a bivariate smoother is
one knot per four samples, but at least 20 and no more than 150 knots (Ruppert, Wand, and Carroll
2003, p. 257).
The magnitude of the variance component 2 depends on the metric of the random effects. For
example, if you apply radial smoothing in time, the variance changes if you measure time in days
or minutes. If the solution for the variance component is near zero, then a rescaling of the random
effect data can help the optimization problem by moving the solution for the variance component
away from the boundary of the parameter space.
Knot Selection
The GLIMMIX procedure computes knots for low-rank smoothing based on the vertices or centroids of a k-d tree. The default is to use the vertices of the tree as the knot locations, if you use
the TYPE=RSMOOTH covariance structure. The construction of this tree amounts to a partitioning
of the random regressor space until all partitions contain at most b observations. The number b is
called the bucket size of the k-d tree. You can exercise control over the construction of the tree by
changing the bucket size with the BUCKET= suboption of the KNOTMETHOD=KDTREE option
in the RANDOM statement. A large bucket size leads to fewer knots, but it is not correct to assume
that K, the number of knots, is simply bn=bc. The number of vertices depends on the configuration
of the values in the regressor space. Also, coordinates of the bounding hypercube are vertices of the
tree. In the one-dimensional case, for example, the extreme values of the random effect are vertices.
To demonstrate how the k-d tree partitions the random-effects space based on observed data and the
influence of the bucket size, consider the following example from Chapter 50, “The LOESS Procedure.” The SAS data set Gas contains the results of an engine exhaust emission study (Brinkman
1981). The covariate in this analysis, E, is a measure of the air-fuel mixture richness. The response,
NOx, measures the nitric oxide concentration (in micrograms per joule, and normalized).
2250 F Chapter 38: The GLIMMIX Procedure
data Gas;
input NOx E;
format NOx E f5.3;
datalines;
4.818 0.831
2.849 1.045
3.275 1.021
4.691 0.97
4.255 0.825
5.064 0.891
2.118 0.71
4.602 0.801
2.286 1.074
0.97
1.148
3.965 1
5.344 0.928
3.834 0.767
1.99
0.701
5.199 0.807
5.283 0.902
3.752 0.997
0.537 1.224
1.64
1.089
5.055 0.973
4.937 0.98
1.561 0.665
;
There are 22 observations in the data set, and the values of the covariate are unique. If you want
to smooth these data with a low-rank radial smoother, you need to choose the number of knots, as
well as their placement within the support of the variable E. The k-d tree construction depends on
the observed values of the variable E; it is independent of the values of nitric oxide in the data. The
following statements construct a tree based on a bucket size of b D 11 and display information
about the tree and the selected knots:
ods select KDtree KnotInfo;
proc glimmix data=gas nofit;
model NOx = e;
random e / type=rsmooth
knotmethod=kdtree(bucket=11 treeinfo knotinfo);
run;
The NOFIT option prevents the GLIMMIX procedure from fitting the model. This option is useful
if you want to investigate the knot construction for various bucket sizes. The TREEINFO and
KNOTINFO suboptions of the KNOTMETHOD=KDTREE option request displays of the k-d tree
and the knot coordinates derived from it. Construction of the tree commences by splitting the data
in half. For b D 11, n D 22, neither of the two splits contains more than b observations and the
process stops. With a single split value, and the two extreme values, the tree has two terminal nodes
and leads to three knots (Figure 38.13). Note that for one-dimensional problems, vertices of the k-d
tree always coincide with data values.
Radial Smoothing Based on Mixed Models F 2251
Figure 38.13 K-d Tree and Knots for Bucket Size 11
The GLIMMIX Procedure
kd-Tree for RSmooth(E)
Node
Number
Left
Child
Right
Child
0
1
2
1
2
Split
Direction
E
TERMINAL
TERMINAL
Split
Value
0.9280
Radial Smoother
Knots for
RSmooth(E)
Knot
Number
E
1
2
3
0.6650
0.9280
1.2240
If the bucket size is reduced to b D 8, the following statements produce the tree and knots in
Figure 38.14:
ods select KDtree KnotInfo;
proc glimmix data=gas nofit;
model NOx = e;
random e / type=rsmooth
knotmethod=kdtree(bucket=8 treeinfo knotinfo);
run;
The initial split value of 0.9280 leads to two sets of 11 observations. In order to achieve a partition
into cells that contain at most eight observations, each initial partition is split at its median one more
time. Note that one split value is greater and one split value is less than 0.9280.
Figure 38.14 K-d Tree and Knots for Bucket Size 8
The GLIMMIX Procedure
kd-Tree for RSmooth(E)
Node
Number
Left
Child
Right
Child
0
1
2
3
4
5
6
1
3
5
2
4
6
Split
Direction
E
E
E
TERMINAL
TERMINAL
TERMINAL
TERMINAL
Split
Value
0.9280
0.8070
1.0210
2252 F Chapter 38: The GLIMMIX Procedure
Figure 38.14 continued
Radial Smoother
Knots for
RSmooth(E)
Knot
Number
E
1
2
3
4
5
0.6650
0.8070
0.9280
1.0210
1.2240
A further reduction in bucket size to b D 4 leads to the tree and knot information shown in
Figure 38.15.
Figure 38.15 K-d Tree and Knots for Bucket Size 4
The GLIMMIX Procedure
kd-Tree for RSmooth(E)
Node
Number
Left
Child
Right
Child
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
1
3
9
5
7
2
4
10
6
8
11
13
12
14
Split
Direction
Split
Value
E
E
E
E
E
TERMINAL
TERMINAL
TERMINAL
TERMINAL
E
E
TERMINAL
TERMINAL
TERMINAL
TERMINAL
0.9280
0.8070
1.0210
0.7100
0.8910
0.9800
1.0890
Radial Smoothing Based on Mixed Models F 2253
Figure 38.15 continued
Radial Smoother
Knots for
RSmooth(E)
Knot
Number
E
1
2
3
4
5
6
7
8
9
0.6650
0.7100
0.8070
0.8910
0.9280
0.9800
1.0210
1.0890
1.2240
The split value for b D 11 is also a split value for b D 8, the split values for b D 8 are a subset of
those for b D 4, and so forth. Figure 38.16 displays the data and the location of split values for the
three cases. For a one-dimensional problem (a univariate smoother), the vertices comprise the split
values and the values on the bounding interval.
You might want to move away from the boundary, in particular for an irregular data configuration
or for multivariate smoothing. The KNOTTYPE=CENTER suboption of the KNOTMETHOD=
option chooses centroids of the leaf node cells instead of vertices. This tends to move the outer knot
locations closer to the convex hull, but not necessarily to data locations. In the emission example,
choosing a bucket size of b D 11 and centroids as knot locations yields two knots at E=0.7956 and
E=1.076. If you choose the NEAREST suboption, then the nearest neighbor of a vertex or centroid
will serve as the knot location. In this case, the knot locations are a subset of the data locations,
regardless of the dimension of the smooth.
2254 F Chapter 38: The GLIMMIX Procedure
Figure 38.16 Vertices of k-d Trees for Various Bucket Sizes
Odds and Odds Ratio Estimation
In models with a logit, generalized logit, or cumulative logit link, you can obtain estimates of odds
ratios through the ODDSRATIO options in the PROC GLIMMIX, LSMEANS, and MODEL statements. This section provides details about the computation and interpretation of the computed quantities. Note that for these link functions the EXP option in the ESTIMATE and LSMESTIMATE
statements also produces odds or odds ratios.
Consider first a model with a dichotomous outcome variable, linear predictor D x0 ˇ C z0 , and
logit link function. Suppose that 0 represents the linear predictor for a condition of interest. For
example, in a simple logistic regression model with D ˛ C ˇx, 0 might correspond to the linear
predictor at a particular value of the covariate—say, 0 D ˛ C ˇx0 .
The modeled probability is D 1=.1 C expf g/, and the odds for D 0 are
0
1=.1 C expf 0 g/
D
D expf0 g
1 0
expf 0 g=.1 C expf 0 g/
Because 0 is a logit, it represents the log odds. The odds ratio
.1 ; 0 / is defined as the ratio of
Odds and Odds Ratio Estimation F 2255
odds for 1 and 0 ,
.1 ; 0 / D expf1
0 g
The odds ratio compares the odds of the outcome under the condition expressed by 1 to the odds
under the condition expressed by 0 . In the preceding simple logistic regression example, this ratio
equals expfˇ.x1 x0 /g. The exponentiation of the estimate of ˇ is thus an estimate of the odds ratio
comparing conditions for which x1 x0 D 1. If x and x C 1 represent standard and experimental
conditions, for example, expfˇg compares the odds of the outcome under the experimental condition
to the odds under the standard condition. For many other types of models, odds ratios can be
expressed as simple functions of parameter estimates. For example, suppose you are fitting a logistic
model with a single classification effect with three levels:
proc glimmix;
class A;
model y = A / dist=binary;
run;
The estimated linear predictor for level j of A is b
j D b
ˇ Cb
˛ j , j D 1; 2; 3. Because the X matrix
is singular in this model due to the presence of an overall intercept, the solution for the intercept
estimates ˇ C ˛3 , and the solution for the j th treatment effect estimates ˛j ˛3 . Exponentiating
the solutions for ˛1 and ˛2 thus produces odds ratios comparing the odds for these levels against
the third level of A.
Results designated as odds or odds ratios in the GLIMMIX procedure might reduce to simple exponentiations of solutions in the “Parameter Estimates” table, but they are computed by a different
mechanism if the model contains classification variables. The computations rely on general estimable functions; for the MODEL, LSMEANS, and LSMESTIMATE statements, these functions
are based on least squares means. This enables you to obtain odds ratio estimates in more complicated models that involve main effects and interactions, including interactions between continuous
and classification variables.
In all cases, the results represent the exponentiation of a linear function of the fixed-effects parameters, D l0 ˇ. If L and U are the confidence limits for on the logit scale, confidence limits for
the odds or the odds ratio are obtained as expfL g and expfU g.
The Odds Ratio Estimates Table
This table is produced by the ODDSRATIO option in the MODEL statement. It consists of estimates
of odds ratios and their confidence limits. Odds ratios are produced for the following:
classification main effects, if they appear in the MODEL statement
continuous variables in the MODEL statement, unless they appear in an interaction with a
classification effect
continuous variables in the MODEL statement at fixed levels of a classification effect, if the
MODEL statement contains an interaction of the two.
continuous variables in the MODEL statements if they interact with other continuous variables
2256 F Chapter 38: The GLIMMIX Procedure
The Default Table
Consider the following PROC GLIMMIX statements that fit a logistic model with one classification
effect, one continuous variable, and their interaction (the ODDSRATIO option in the MODEL
statement requests the “Odds Ratio Estimates” table).
proc glimmix;
class A;
model y = A x A*x / dist=binary oddsratio;
run;
By default, odds ratios are computed as follows:
The covariate is set to its average, x, and the least squares means for the A effect are obtained.
Suppose L.1/ denotes the matrix of coefficients defining the estimable functions that produce
.1/
the a least squares means Lb̌, and lj denotes the j th row of L.1/ . Differences of the least
squares means against the last level of the A factor are computed and exponentiated:
n
o
.1/
b̌
.A1 ; Aa / D exp l1
l.1/
a
n
o
.1/
b̌
.A2 ; Aa / D exp l2
l.1/
a
::
:
.Aa
1 ; Aa /
D exp
n
.1/
la
1
o
b̌
l.1/
a
The differences are checked for estimability. Notice that this set of odds ratios can also be
obtained with the following LSMESTIMATE statement (assuming A has five levels):
lsmestimate A 1
0
0
0
0
1
0
0
0
0
1
0
0
0
0
1
-1,
-1,
-1,
-1 / exp cl;
You can also obtain the odds ratios with this LSMEANS statement (assuming the last level of
A is coded as 5):
lsmeans A / diff=control(’5’) oddsratio cl;
The odds ratios for the covariate must take into account that x occurs in an interaction with
the A effect. A second set of least squares means are computed, where x is set to x C 1.
Denote the coefficients of the estimable functions for this set of least squares means as L.2/ .
Differences of the least squares means at a given level of factor A are then computed and
exponentiated:
n
o
.2/
.1/
.A.x C 1/1 ; A.x/1 / D exp l1
l1 b̌
n
o
.2/
.1/
.A.x C 1/2 ; A.x/2 / D exp l2
l2 b̌
::
:
.A.x C 1/a ; A.x/a / D exp
n
l.2/
a
o
b̌
l.1/
a
Odds and Odds Ratio Estimation F 2257
The differences are checked for estimability. If the continuous covariate does not appear in
an interaction with the A variable, only a single odds ratio estimate related to x would be
produced, relating the odds of a one-unit shift in the regressor from x.
Suppose you fit a model that contains interactions of continuous variables, as with the following
statements:
proc glimmix;
class A;
model y = A x x*z / dist=binary oddsratio;
run;
In the computation of the A least squares means, the continuous effects are set to their means—that
is, x and xz. In the computation of odds ratios for x, linear predictors are computed at x = x, x*z =
x z and at x = x C 1, x*z = .x C 1/z.
Modifying the Default Table, Customized Odds Ratios
Several suboptions of the ODDSRATIO option in the MODEL statement are available to obtain
customized odds ratio estimates. For customized odds ratios that cannot be obtained with these
suboptions, use the EXP option in the ESTIMATE or LSMESTIMATE statement.
The type of differences constructed when the levels of a classification factor are varied is controlled
by the DIFF= suboption. By default, differences against the last level are taken. DIFF=FIRST
computes differences from the first level, and DIFF=ALL computes odds ratios based on all pairwise
differences.
For continuous variables in the model, you can change both the reference value (with the AT suboption) and the units of change (with the UNIT suboption). By default, a one-unit change from
the mean of the covariate is assessed. For example, the following statements produce all pairwise
differences for the A factor:
proc glimmix;
class A;
model y = A x A*x / dist=binary
oddsratio(diff=all
at
x=4
unit x=3);
run;
The covariate x is set to the reference value x D 4 in the computation of the least squares means
for the A odds ratio estimates. The odds ratios computed for the covariate are based on differencing
this set of least squares means with a set of least squares means computed at x D 4 C 3.
Odds or Odds Ratio
The odds ratio is the exponentiation of a difference on the logit scale,
.1 ; 0 / D exp f.l1
l0 /ˇg
2258 F Chapter 38: The GLIMMIX Procedure
and expfl1 ˇg and expfl0 ˇg are the corresponding odds. If the ODDSRATIO option is specified
in a suitable model in the PROC GLIMMIX statement or the individual statements that support
the option, odds ratios are computed in the “Odds Ratio Estimates” table (MODEL statement), the
“Differences of Least Squares Means” table (LSMEANS / DIFF), and the “Simple Effect Comparisons of Least Squares Means” table (LSMEANS / SLICEDIFF=). Odds are computed in the “Least
Squares Means” table.
Odds Ratios in Multinomial Models
The GLIMMIX procedure fits two kinds of models to multinomial data. Models with cumulative
link functions apply to ordinal data, and generalized logit models are fit to nominal data. If you
model a multinomial response with LINK=CUMLOGIT or LINK=GLOGIT, odds ratio results are
available for these models.
In the generalized logit model, you model baseline category logits. By default, the GLIMMIX
procedure chooses the last category as the reference category. If your nominal response has J
categories, the baseline logit for category j is
˚
log j =J D j D x0 ˇj C z0 uj
and
expfj g
j D PJ
kD1 expfk g
J D 0
As before, suppose that the two conditions to be compared are identified with subscripts 1 and 0.
The log odds ratio of outcome j versus J for the two conditions is then
˚
expfj1 g
j1 =J1
log
j1 ; j 0 D log
D log
j 0 =J 0
expfj 0 g
D j1
j 0
Note that the log odds ratios are again differences on the scale of the linear predictor, but they
depend on the response category. The GLIMMIX procedure determines the estimable functions
whose differences represent log odds ratios as discussed previously but produces separate estimates
for each nonreference response category.
In models for ordinal data, PROC GLIMMIX models the logits of cumulative probabilities. Thus,
the estimates on the linear scale represent log cumulative odds. The cumulative logits are formed as
Pr.Y j /
log
D j D ˛j C x0 ˇ C z0 D ˛j C Q
Pr.Y > j /
so that the linear predictor depends on the response category only through the intercepts (cutoffs)
˛1 ; ; ˛J 1 . The odds ratio comparing two conditions represented by linear predictors j1 and
j 0 is then
˚
j1 ; j 0 D exp j1 j 0
D exp fQ 1
and is independent of category.
Q 0 g
Parameterization of Generalized Linear Mixed Models F 2259
Parameterization of Generalized Linear Mixed Models
PROC GLIMMIX constructs a generalized linear mixed model according to the specifications in
the CLASS, MODEL, and RANDOM statements. Each effect in the MODEL statement generates
one or more columns in the matrix X, and each G-side effect in the RANDOM statement generates
one or more columns in the matrix Z. R-side effects in the RANDOM statement do not generate
model matrices; they serve only to index observations within subjects. This section shows how the
GLIMMIX procedure builds X and Z. You can output the X and Z matrices to a SAS data set with
the OUTDESIGN= option in the PROC GLIMMIX statement.
The general rules and techniques for parameterization of a linear model are given in “GLM Parameterization of Classification Variables and Effects” on page 369 of Chapter 18, “Shared Concepts and
Topics.” The following paragraphs discuss how these rules differ in a mixed model, in particular,
how parameterization differs between the X and the Z matrix.
Intercept
By default, all models automatically include a column of 1s in X to estimate a fixed-effect intercept
parameter. You can use the NOINT option in the MODEL statement to suppress this intercept. The
NOINT option is useful when you are specifying a classification effect in the MODEL statement
and you want the parameter estimates to be in terms of the (linked) mean response for each level of
that effect, rather than in terms of a deviation from an overall mean.
By contrast, the intercept is not included by default in Z. To obtain a column of 1s in Z, you must
specify in the RANDOM statement either the INTERCEPT effect or some effect that has only one
level.
Interaction Effects
Often a model includes interaction (crossed) effects. With an interaction, PROC GLIMMIX first
reorders the terms to correspond to the order of the variables in the CLASS statement. Thus, B*A
becomes A*B if A precedes B in the CLASS statement. Then, PROC GLIMMIX generates columns
for all combinations of levels that occur in the data. The order of the columns is such that the
rightmost variables in the cross index faster than the leftmost variables. Empty columns (which
would contain all 0s) are not generated for X, but they are for Z.
See Table 18.5 in the section “GLM Parameterization of Classification Variables and Effects” on
page 369 of Chapter 18, “Shared Concepts and Topics,” for an example of an interaction parameterization.
Nested Effects
Nested effects are generated in the same manner as crossed effects. Hence, the design columns generated by the following two statements are the same (but the ordering of the columns is different):
2260 F Chapter 38: The GLIMMIX Procedure
Note that nested effects are often distinguished from interaction effects by the implied randomization structure of the design. That is, they usually indicate random effects within a fixed-effects
framework. The fact that random effects can be modeled directly in the RANDOM statement might
make the specification of nested effects in the MODEL statement unnecessary.
See Table 18.6 in the section “GLM Parameterization of Classification Variables and Effects” on
page 369 of Chapter 18, “Shared Concepts and Topics,” for an example of the parameterization of
a nested effect.
Implications of the Non-Full-Rank Parameterization
For models with fixed effects involving classification variables, there are more design columns in
X constructed than there are degrees of freedom for the effect. Thus, there are linear dependencies
among the columns of X. In this event, all of the parameters are not estimable; there is an infinite
number of solutions to the mixed model equations. The GLIMMIX procedure uses a generalized
inverse (a g2 -inverse, Pringle and Rayner 1971), to obtain values for the estimates (Searle 1971).
The solution values are not displayed unless you specify the SOLUTION option in the MODEL
statement. The solution has the characteristic that estimates are 0 whenever the design column for
that parameter is a linear combination of previous columns. With this parameterization, hypothesis
tests are constructed to test linear functions of the parameters that are estimable.
Some procedures (such as the CATMOD and LOGISTIC procedures) reparameterize models to full
rank by using restrictions on the parameters. PROC GLM, PROC MIXED, and PROC GLIMMIX
do not reparameterize, making the hypotheses that are commonly tested more understandable. See
Goodnight (1978a) for additional reasons for not reparameterizing.
Missing Level Combinations
PROC GLIMMIX handles missing level combinations of classification variables in the same manner as PROC GLM and PROC MIXED. These procedures delete fixed-effects parameters corresponding to missing levels in order to preserve estimability. However, PROC GLIMMIX does not
delete missing level combinations for random-effects parameters because linear combinations of
the random-effects parameters are always predictable. These conventions can affect the way you
specify your CONTRAST and ESTIMATE coefficients.
Notes on the EFFECT Statement
Some restrictions and limitations for models that contain constructed effects are in place with the
GLIMMIX procedure. Also, you should be aware of some special defaults and handling that apply
only when the model contains constructed fixed and/or random effects.
Constructed effects can be used in the MODEL and RANDOM statements but not to specify
SUBJECT= or GROUP= effects.
Parameterization of Generalized Linear Mixed Models F 2261
Computed variables are not supported in the specification of a constructed effect. All variables needed to form the collection of columns for a constructed effect must be in the data
set.
You cannot use constructed effects that comprise continuous variables or interactions with
other constructed effects as the LSMEANS or LSMESTIMATE effect.
The calculation of quantities that depend on least squares means, such as odds ratios in the
“Odds Ratio Estimates” table, is not possible if the model contains fixed effects that consist
of more than one constructed effects, unless all constructed effects are of spline type. For
example, least squares means computations are not possible in the following model because
the MM_AB*cvars effect contains two constructed effects:
proc glimmix;
class A B C;
effect MM_AB = MM(A B);
effect cvars = COLLECTION(x1 x2 x3);
model y = C MM_AB*cvars;
run;
If the MODEL or RANDOM statement contains constructed effects, the default degreesof-freedom method for mixed models is DDFM=KENWARDROGER. The containment
degrees-of-freedom method (DDFM=CONTAIN) is not available in these models.
If the model contains fixed spline effects, least squares means are computed at the average
spline coefficients across the usable data, possibly further averaged over levels of class variables that interact with the spline effects in the model. You can use the AT option in the
LSMEANS and LSMESTIMATE statements to construct the splines for particular values of
the covariates involved. Consider, for example, the following statements:
proc glimmix;
class A;
effect spl = spline(x);
model y = A spl;
lsmeans A;
lsmeans A / at means;
lsmeans A / at x=0.4;
run;
Suppose that the spl effect contributes seven columns Œs1 ; ; s7  to the X matrix. The least
squares means coefficients for the spl effect in the first LSMEANS statement are Œs 1 ; ; s 7 
with the averages taken across the observations used in the analysis. The second LSMEANS
statement computes the spline coefficient at the average value of x: Œs.x/1 ; ; s.x/7 . The
final LSMEANS statement uses Œs.0:4/1 ; ; s.0:4/7 . Using the AT option for least squares
means calculations with spline effects can resolve inestimability issues.
Using a spline effect with B-spline basis in the RANDOM statement is not the same as
using a penalized B-spline (P-spline) through the TYPE=PSPLINE option in the RANDOM
statement. The following statement constructs a penalized B-spline by using mixed model
methodology:
random x / type=pspline;
2262 F Chapter 38: The GLIMMIX Procedure
The next set of statements defines a set of B-spline columns in the Z matrix with uncorrelated
random effects and homogeneous variance:
effect bspline = spline(x);
random bspline / type=vc;
This does not lead to a properly penalized fit. See the documentation on TYPE=PSPLINE
about the construction of penalties for B-splines through the covariance matrix of random
effects.
Positional and Nonpositional Syntax for Contrast Coefficients
When you define custom linear hypotheses with the CONTRAST or ESTIMATE statement, the
GLIMMIX procedure sets up an L vector or matrix that conforms to the fixed-effects solutions or the
fixed- and random-effects solutions. With the LSMESTIMATE statement, you specify coefficients
of the matrix K that is then converted into a coefficient matrix that conforms to the fixed-effects
solutions.
There are two methods for specifying the entries in a coefficient matrix (hereafter simply referred to
as the L matrix), termed the positional and nonpositional methods. In the positional form, and this
is the traditional method, you provide a list of values that occupy the elements of the L matrix associated with the effect in question in the order in which the values are listed. For traditional model
effects comprising continuous and classification variables, the positional syntax is simpler in some
cases (main effects) and more cumbersome in others (interactions). When you work with effects
constructed through the experimental EFFECT statement, the nonpositional syntax is essential.
Consider, for example, the following two-way model with interactions where factors A and B have
three and two levels, respectively:
proc glimmix;
class a b block;
model y = a b a*b / ddfm=kr;
random block a*block;
run;
To test the difference of the B levels at the second level of A with a CONTRAST statement (a slice),
you need to assign coefficients 1 and 1 to the levels of B and to the levels of the interaction where
A is at the second level. Two examples of equivalent CONTRAST statements by using positional
and nonpositional syntax are as follows:
contrast ’B at A2’ b 1 -1 a*b 0 0 1 -1
;
contrast ’B at A2’ b 1 -1 a*b [1 2 1] [-1 2 2];
Because A precedes B in the CLASS statement, the levels of the interaction are formed as
˛1 ˇ1 ; ˛1 ˇ2 ; ˛2 ˇ1 ; ˛2 ˇ2 ; . If B precedes A in the CLASS statement, you need to modify the
coefficients accordingly:
Parameterization of Generalized Linear Mixed Models F 2263
proc glimmix;
class b a block;
model y = a b a*b / ddfm=kr;
random block a*block;
contrast ’B at A2’ b 1 -1 a*b 0 1 0 0 -1
;
contrast ’B at A2’ b 1 -1 a*b [1 1 2] [-1 2 2];
contrast ’B at A2’ b 1 -1 a*b [1, 1 2] [-1, 2 2];
run;
You can optionally separate the L value entry from the level indicators with a comma, as in the last
CONTRAST statement.
The general syntax for defining coefficients with the nonpositional syntax is as follows:
effect-name [multiplier < , > level-values] . . . < [multiplier < , > level-values] >
The first entry in square brackets is the multiplier that is applied to the elements of L for the effect
after the level-values have been resolved and any necessary action forming L has been taken.
The level-values are organized in a specific form:
The number of entries should equal the number of terms needed to construct the effect. For
effects that do not contain any constructed effects, this number is simply the number of terms
in the name of the effect.
Values of continuous variables needed for the construction of the L matrix precede the level
indicators of CLASS variables.
If the effect involves constructed effects, then you need to provide as many continuous and
classification variables as are needed for the effect formation. For example, if a grouping
effect is defined as
class c;
effect v = vars(x1 x2 c);
then a proper nonpositional syntax would be, for example,
v [0.5,
0.2 0.3 3]
If an effect contains both regular terms (old-style effects) and constructed effects, then the
order of the coefficients is as follows: continuous values for old-style effects, class levels for
CLASS variables in old-style effects, continuous values for constructed effects, and finally
class levels needed for constructed effects.
Assume that C has four levels so that effect v contributes six elements to the L matrix. When
PROC GLIMMIX resolves this syntax, the values 0.2 and 0.3 are assigned to the positions for
x1 and x2 and a 1 is associated with the third level of C. The resulting vector is then multiplied
by 0.5 to produce
Œ0:1
0:15
0
0
0:5
0
2264 F Chapter 38: The GLIMMIX Procedure
Note that you enter the levels of the classification variables in the square brackets, not their formatted values. The ordering of the levels of CLASS variables can be gleaned from the “Class Level
Information” table.
To specify values for continuous variables, simply give their value as one of the terms in the effect.
The nonpositional syntax in the following ESTIMATE statement is read as “1-time the value 0.4 in
the column associated with level 2 of A”
proc glimmix;
class a;
model y = a a*x / s;
lsmeans a / e at x=0.4;
estimate ’A2 at x=0.4’ intercept 1 a 0 1 a*x [1,0.4 2] / e;
run;
Because the value before the comma serves as a multiplier, the same estimable function could also
be constructed with the following statements:
estimate ’A2 at x=0.4’ intercept 1 a 0 1 a*x [ 4, 0.1 2];
estimate ’A2 at x=0.4’ intercept 1 a 0 1 a*x [ 2, 0.2 2];
estimate ’A2 at x=0.4’ intercept 1 a 0 1 a*x [-1, -0.4 2];
Note that continuous variables needed to construct an effect are always listed before any CLASS
variables.
When you work with constructed effects, the nonpositional syntax works in the same way. For example, the following model contains a classification effect and a B-spline. The first two ESTIMATE
statements produce predicted values for level one of C when the continuous variable x takes on the
values 20 and 10, respectively.
proc glimmix;
class c;
effect spl = spline(x /
model y = c spl;
estimate ’C = 1 @ x=20’
’C = 1 @ x=10’
estimate ’Difference’
run;
knotmethod=equal(5));
intercept 1 c 1 spl [1,20],
intercept 1 c 1 spl [1,10];
spl [1,20] [-1,10];
The GLIMMIX procedure computes the spline coefficients for the first ESTIMATE statement based
on x D 20, and similarly in the second statement for x D 10. The third ESTIMATE statement
computes the difference of the predicted values. Because the spline effect does not interact with
the classification variable, this difference does not depend on the level of C. If such an interaction
is present, you can estimate the difference in predicted values for a given level of C by using the
nonpositional syntax. Because the effect C*spl contains both old-style terms (C) and a constructed
effect, you specify the values for the old-style terms before assigning values to constructed effects:
Response-Level Ordering and Referencing F 2265
proc glimmix;
class c;
effect spl = spline(x / knotmethod=equal(5));
model y = spl*c;
estimate ’C2 = 1, x=20’ intercept 1 c*spl [1,1 20];
estimate ’C2 = 2, x=20’ intercept 1 c*spl [1,2 20];
estimate ’C diff at x=20’ c*spl [1,1 20] [-1,2 20];
run;
It is recommended to add the E option to the CONTRAST, ESTIMATE, or LSMESTIMATE statement to verify that the L matrix is formed according to your expectations.
In any row of an ESTIMATE or CONTRAST statement you can choose positional and nonpositional
syntax separately for each effect. You cannot mix the two forms of syntax for coefficients of a single
effect, however. For example, the following statement is not proper because both forms of syntax
are used for the interaction effect:
estimate ’A1B1 - A1B2’ b 1 -1
a*b 0 1
[-1, 1 2];
Response-Level Ordering and Referencing
In models for binary and multinomial data, the response-level ordering is important because it
reflects the following:
which probability is modeled with binary data
how categories are ordered for ordinal data
which category serves as the reference category in nominal generalized logit models (models
for nominal data)
You should view the “Response Profile” table to ensure that the categories are properly arranged
and that the desired outcome is modeled. In this table, response levels are arranged by Ordered
Value. The lowest response level is assigned Ordered Value 1, the next lowest is assigned Ordered
Value 2, and so forth. In binary models, the probability modeled is the probability of the response
level with the lowest Ordered Value.
You can change which probability is modeled and the Ordered Value in the “Response Profile”
table with the DESCENDING, EVENT=, ORDER=, and REF= response variable options in the
MODEL statement. See the section “Response Level Ordering” on page 3316 in Chapter 51, “The
LOGISTIC Procedure,” for examples about how to use these options to affect the probability being
modeled for binary data.
For multinomial models, the response-level ordering affects two important aspects. In cumulative
link models the categories are assumed ordered according to their Ordered Value in the “Response
Profile” table. If the response variable is a character variable or has a format, you should check this
table carefully as to whether the Ordered Values reflect the correct ordinal scale.
2266 F Chapter 38: The GLIMMIX Procedure
In generalized logit models (for multinomial data with unordered categories), one response category
is chosen as the reference category in the formulation of the generalized logits. By default, the linear
predictor in the reference category is set to 0, and the reference category corresponds to the entry
in the “Response Profile” table with the highest Ordered Value. You can affect the assignment of
Ordered Values with the DESCENDING and ORDER= options in the MODEL statement. You can
choose a different reference category with the REF= option. The choice of the reference category
for generalized logit models affects the results. It is sometimes recommended that you choose the
category with the highest frequency as the reference (see, for example, Brown and Prescott 1999, p.
160). You can achieve this with the GLIMMIX procedure by combining the ORDER= and REF=
options, as in the following statements:
proc glimmix;
class preference;
model preference(order=freq ref=first) = feature price /
dist=multinomial
link=glogit;
random intercept / subject=store group=preference;
run;
The ORDER=FREQ option arranges the categories by descending frequency. The REF=FIRST
option then selects the response category with the lowest Ordered Value—the most frequent
category—as the reference.
Comparing the GLIMMIX and MIXED Procedures
The MIXED procedure is subsumed by the GLIMMIX procedure in the following sense:
Linear mixed models are a special case in the family of generalized linear mixed models; a
linear mixed model is a generalized linear mixed model where the conditional distribution is
normal and the link function is the identity function.
Most models that can be fit with the MIXED procedure can also be fit with the GLIMMIX
procedure.
Despite this overlap in functionality, there are also some important differences between the two
procedures. Awareness of these differences enables you to select the most appropriate tool in situations where you have a choice between procedures and to identify situations where a choice
does not exist. Furthermore, the %GLIMMIX macro, which fits generalized linear mixed models
by linearization methods, essentially calls the MIXED procedure repeatedly. If you are aware of
the syntax differences between the procedures, you can easily convert your %GLIMMIX macro
statements.
Important functional differences between PROC GLIMMIX and PROC MIXED for linear models
and linear mixed models include the following:
The MIXED procedure models R-side effects through the REPEATED statement and G-side
effects through the RANDOM statement. The GLIMMIX procedure models all random components of the model through the RANDOM statement. You use the _RESIDUAL_ keyword
Comparing the GLIMMIX and MIXED Procedures F 2267
or the RESIDUAL option in the RANDOM statement to model R-side covariance structure
in the GLIMMIX procedure. For example, the PROC MIXED statement
repeated / subject=id type=ar(1);
is equivalent to the following RANDOM statement in the GLIMMIX procedure:
random _residual_ / subject=id type=ar(1);
If you need to specify an effect for levelization—for example, because the construction of
the R matrix is order-dependent or because you need to account for missing values—the
RESIDUAL option in the RANDOM statement of the GLIMMIX procedure is used to indicate that you are modeling an R-side covariance nature. For example, the PROC MIXED
statements
class time id;
repeated time / subject=id type=ar(1);
are equivalent to the following PROC GLIMMIX statements:
class time id;
random time / subject=id type=ar(1) residual;
There is generally considerable overlap in the covariance structures available through the
TYPE= option in the RANDOM statement in PROC GLIMMIX and through the TYPE=
options in the RANDOM and REPEATED statements in PROC MIXED. However, the
Kronecker-type structures, the geometrically anisotropic spatial structures, and the GDATA=
option in the RANDOM statement of the MIXED procedure are currently not supported
in the GLIMMIX procedure. The MIXED procedure, on the other hand, does not support
TYPE=RSMOOTH and TYPE=PSPLINE.
For normal linear mixed models, the (default) METHOD=RSPL in PROC GLIMMIX is
identical to the default METHOD=REML in PROC MIXED. Similarly, METHOD=MSPL
in PROC GLIMMIX is identical for these models to METHOD=ML in PROC MIXED. The
GLIMMIX procedure does not support Type I through Type III (ANOVA) estimation methods
for variance component models. Also, the procedure does not have a METHOD=MIVQUE0
option, but you can produce these estimates through the NOITER option in the PARMS statement.
The MIXED procedure solves the iterative optimization problem by means of a ridgestabilized Newton-Raphson algorithm. With the GLIMMIX procedure, you can choose from
a variety of optimization methods via the NLOPTIONS statement. The default method for
most GLMMs is a quasi-Newton algorithm. A ridge-stabilized Newton-Raphson algorithm,
akin to the optimization method in the MIXED procedure, is available in the GLIMMIX procedure through the TECHNIQUE=NRRIDG option in the NLOPTIONS statement. Because
of differences in the line-search methods, update methods, and the convergence criteria, you
might get slightly different estimates with the two procedures in some instances. The GLIMMIX procedure, for example, monitors several convergence criteria simultaneously.
You can produce predicted values, residuals, and confidence limits for predicted values with
both procedures. The mechanics are slightly different, however. With the MIXED procedure
you use the OUTPM= and OUTP= options in the MODEL statement to write statistics to
data sets. With the GLIMMIX procedure you use the OUTPUT statement and indicate with
keywords which “flavor” of a statistic to compute.
2268 F Chapter 38: The GLIMMIX Procedure
The following GLIMMIX statements are not available in the MIXED procedure: COVTEST,
EFFECT, FREQ, LSMESTIMATE, OUTPUT, and programming statements.
A sampling-based Bayesian analysis as through the PRIOR statement in the MIXED procedure is not available in the GLIMMIX procedure.
In the GLIMMIX procedure, several RANDOM statement options apply to the RANDOM
statement in which they are specified. For example, the following statements in the GLIMMIX procedure request that the solution vector be printed for the A and A*B*C random effects
and that the G matrix corresponding to the A*B interaction random effect be displayed:
random a
/ s;
random a*b
/ G;
random a*b*c / alpha=0.04;
Confidence intervals with a 0.96 coverage probability are produced for the solutions of the
A*B*C effect. In the MIXED procedure, the S option, for example, when specified in one
RANDOM statement, applies to all RANDOM statements.
If you select nonmissing values in the value-list of the DDF= option in the MODEL statement,
PROC GLIMMIX uses these values to override degrees of freedom for this effect that might
be determined otherwise. For example, the following statements request that the denominator
degrees of freedom for tests and confidence intervals involving the A effect be set to 4:
proc glimmix;
class block a b;
model y = a b a*b / s ddf=4,.,. ddfm=satterthwaite;
random block a*block / s;
lsmeans a b a*b / diff;
run;
In the example, this applies to the “Type III Tests of Fixed Effects,” “Least Squares Means,”
and “Differences of Least Squares Means” tables. In the MIXED procedure, the Satterthwaite
approximation overrides the DDF= specification.
The DDFM=BETWITHIN degrees-of-freedom method in the GLIMMIX procedure requires
that the data be processed by subjects; see the section “Processing by Subjects” on page 2245.
When you add the response variable to the CLASS statement, PROC GLIMMIX defaults to
the multinomial distribution. Adding the response variable to the CLASS statement in PROC
MIXED has no effect on the fitted model.
For ODS purposes, the name of the table for the solution of fixed effects is “SolutionF” in
the MIXED procedure. In PROC GLIMMIX, the name of the table that contains fixed-effects
solutions is “ParameterEstimates.” In generalized linear models, this table also contains scale
parameters and overdispersion parameters. The MIXED procedure always produces a “Covariance Parameter Estimates” table. The GLIMMIX procedure produces this table only in
mixed models or models with nontrivial R-side covariance structure.
Singly or Doubly Iterative Fitting F 2269
If you compute predicted values in the GLIMMIX procedure in a model with only R-side
random components and missing values for the dependent variable, the predicted values will
not be kriging predictions as is the case with the MIXED procedure.
Singly or Doubly Iterative Fitting
Depending on the structure of your model, the GLIMMIX procedure determines the appropriate
approach for estimating the parameters of the model. The elementary algorithms fall into three
categories:
1. Noniterative algorithms
A closed form solution exists for all model parameters. Standard linear models with homoscedastic, uncorrelated errors can be fit with noniterative algorithms.
2. Singly iterative algorithms
A single optimization, consisting of one or more iterations, is performed to obtain solutions
for the parameter estimates by numerical techniques. Linear mixed models for normal data
can be fit with singly iterative algorithms. Laplace and quadrature estimation for generalized linear mixed models uses a singly iterative algorithm with a separate suboptimization to
compute the random-effects solutions as modes of the log-posterior distribution.
3. Doubly iterative algorithms
A model of simpler structure is derived from the target model. The parameters of the simpler
model are estimated by noniterative or singly iterative methods. Based on these new estimates, the model of simpler structure is rederived and another estimation step follows. The
process continues until changes in the parameter estimates are sufficiently small between two
recomputations of the simpler model or until some other criterion is met. The rederivation of
the model can often be cast as a change of the response to some pseudo-data along with an
update of implicit model weights.
Obviously, noniterative algorithms are preferable to singly iterative ones, which in turn are preferable to doubly iterative algorithms. Two drawbacks of doubly iterative algorithms based on linearization are that likelihood-based measures apply to the pseudo-data, not the original data, and
that at the outer level the progress of the algorithm is tied to monitoring the parameter estimates.
The advantage of doubly iterative algorithms, however, is to offer—at convergence—the statistical
inference tools that apply to the simpler models.
The output and log messages contain information about which algorithm is employed. For a noniterative algorithm, PROC GLIMMIX produces a message that no optimization was performed.
Noniterative algorithms are employed automatically for normal data with identity link.
You can determine whether a singly or doubly iterative algorithm was used, based on the “Iteration
History” table and the “Convergence Status” table (Figure 38.17).
2270 F Chapter 38: The GLIMMIX Procedure
Figure 38.17 Iteration History and Convergence Status in Singly Iterative Fit
The GLIMMIX Procedure
Iteration History
Iteration
Restarts
Evaluations
Objective
Function
Change
Max
Gradient
0
1
2
3
0
0
0
0
4
3
3
3
83.039723731
82.189661988
82.189255211
82.189255211
.
0.85006174
0.00040678
0.00000000
13.63536
0.281308
0.000174
1.05E-10
Convergence criterion (GCONV=1E-8) satisfied.
The “Iteration History” table contains the Evaluations column that shows how many function evaluations were performed in a particular iteration. The convergence status message informs you which
convergence criterion was met when the estimation process concluded. In a singly iterative fit, the
criterion is one that applies to the optimization. In other words, it is one of the criteria that can be
controlled with the NLOPTIONS statement: see the ABSCONV=, ABSFCONV=, ABSGCONV=,
ABSXCONV=, FCONV=, or GCONV= option.
In a doubly iterative fit, the “Iteration History” table does not contain an Evaluations column. Instead
it displays the number of iterations within an optimization (Subiterations column in Figure 38.18).
Figure 38.18 Iteration History and Convergence Status in Doubly Iterative Fit
Iteration History
Iteration
Restarts
Subiterations
Objective
Function
Change
Max
Gradient
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
5
3
2
1
1
1
1
1
1
1
1
1
1
1
1
0
79.688580269
81.294622554
81.438701534
81.444083567
81.444265216
81.444277364
81.444266322
81.44427636
81.444267235
81.44427553
81.44426799
81.444274844
81.444268614
81.444274277
81.444269129
81.444273808
0.11807224
0.02558021
0.00166079
0.00006263
0.00000421
0.00000383
0.00000348
0.00000316
0.00000287
0.00000261
0.00000237
0.00000216
0.00000196
0.00000178
0.00000162
0.00000000
7.851E-7
8.209E-7
4.061E-8
2.292E-8
0.000025
0.000023
0.000021
0.000019
0.000017
0.000016
0.000014
0.000013
0.000012
0.000011
9.772E-6
9.102E-6
Convergence criterion (PCONV=1.11022E-8) satisfied.
Default Estimation Techniques F 2271
The Iteration column then counts the number of optimizations. The “Convergence Status” table
indicates that the estimation process concludes when a criterion is met that monitors the parameter
estimates across optimization, namely the PCONV= or ABSPCONV= criterion.
You can control the optimization process with the GLIMMIX procedure through the NLOPTIONS
statement. Its options affect the individual optimizations. In a doubly iterative scheme, these apply
to all optimizations.
The default optimization techniques are TECHNIQUE=NONE for noniterative estimation, TECHNIQUE=NEWRAP for singly iterative methods in GLMs, TECHNIQUE=NRRIDG for pseudolikelihood estimation with binary data, and TECHNIQUE=QUANEW for other mixed models.
Default Estimation Techniques
Based on the structure of the model, the GLIMMIX procedure selects the estimation technique for
estimating the model parameters. If you fit a generalized linear mixed model, you can change the
estimation technique with the METHOD= option in the PROC GLIMMIX statement. The defaults
are determined as follows:
generalized linear model
– normal distribution: restricted maximum likelihood
– all other distributions: maximum likelihood
generalized linear model with overdispersion
Parameters (ˇ; , if present) are estimated by (restricted) maximum likelihood as for generalized linear models. The overdispersion parameter is estimated from the Pearson statistic
after all other parameters have been estimated.
generalized linear mixed models
The default technique is METHOD=RSPL, corresponding to maximizing the residual log
pseudo-likelihood with an expansion about the current solutions of the best linear unbiased predictors of the random effects. In models for normal data with identity link,
METHOD=RSPL and METHOD=RMPL are equivalent to restricted maximum likelihood
estimation, and METHOD=MSPL and METHOD=MMPL are equivalent to maximum likelihood estimation. This is reflected in the labeling of statistics in the “Fit Statistics” table.
Default Output
The following sections describe the output that PROC GLIMMIX produces by default. The output
is organized into various tables, which are discussed in the order of appearance. Note that the
contents of a table can change with the estimation method or the model being fit.
2272 F Chapter 38: The GLIMMIX Procedure
Model Information
The “Model Information” table displays basic information about the fitted model, such as the link
and variance functions, the distribution of the response, and the data set. If important model
quantities—for example, the response, weights, link, or variance function—are user-defined, the
“Model Information” table displays the final assignment to the respective variable, as determined
from your programming statements. If the table indicates that the variance matrix is blocked by an
effect, then PROC GLIMMIX processes the data by subjects. The “Dimensions” table displays the
number of subjects. For more information about processing by subjects, see the section “Processing
by Subjects” on page 2245. For ODS purposes, the name of the “Model Information” table is
“ModelInfo.”
Class Level Information
The “Class Level Information” table lists the levels of every variable specified in the CLASS statement. You should check this information to make sure that the data are correct. You can adjust the
order of the CLASS variable levels with the ORDER= option in the PROC GLIMMIX statement.
For ODS purposes, the name of the “Class Level Information” table is “ClassLevels.”
Number of Observations
The “Number of Observations” table displays the number of observations read from the input data
set and the number of observations used in the analysis. If you specify a FREQ statement, the
table also displays the sum of frequencies read and used. If the events/trials syntax is used for the
response, the table also displays the number of events and trials used in the analysis. For ODS
purposes, the name of the “Number of Observations” table is “NObs.”
Response Profile
For binary and multinomial data, the “Response Profile” table displays the Ordered Value from
which the GLIMMIX procedure determines the following:
the probability being modeled for binary data
the ordering of categories for ordinal data
the reference category for generalized logit models
For each response category level, the frequency used in the analysis is reported. The section “Response-Level Ordering and Referencing” on page 2265 explains how you can use the
DESCENDING, EVENT=, ORDER=, and REF= options to affect the assignment of Ordered Values to the response categories. For ODS purposes, the name of the “Response Profile” table is
“ResponseProfile.”
Default Output F 2273
Dimensions
The “Dimensions” table displays information from which you can determine the size of relevant
matrices in the model. This table is useful in determining CPU time and memory requirements. For
ODS purposes, the name of the “Dimensions” table is “Dimensions.”
Optimization Information
The “Optimization Information” table displays important details about the optimization process.
The optimization technique that is displayed in the table is the technique that applies to any single
optimization. For singly iterative methods that is the optimization method.
The number of parameters that are updated in the optimization equals the number of parameters
in this table minus the number of equality constraints. The number of constraints is displayed if
you fix covariance parameters with the HOLD= option in the PARMS statement. The GLIMMIX
procedure also lists the number of upper and lower boundary constraints. Note that the procedure
might impose boundary constraints for certain parameters, such as variance components and correlation parameters. Covariance parameters for which a HOLD= was issued have an upper and lower
boundary equal to the parameter value.
If a residual scale parameter is profiled from the optimization, it is also shown in the “Optimization
Information” table.
In a GLMM for which the parameters are estimated by one of the linearization methods, you need to
initiate the process of computing the pseudo-response. This can be done based on existing estimates
of the fixed effects, or by using the data themselves—possibly after some suitable adjustment—as
an estimate of the initial mean. The default in PROC GLIMMIX is to use the data themselves to
derive initial estimates of the mean function and to construct the pseudo-data. The “Optimization
Information” table shows how the pseudo-data are determined initially. Note that this issue is separate from the determination of starting values for the covariance parameters. These are computed
as minimum variance quadratic unbiased estimates (with 0 priors, MIVQUE0; Goodnight 1978b)
or obtained from the value-list in the PARMS statement.
For ODS purposes, the name of the table is “OptInfo.”
Iteration History
The “Iteration History” table describes the progress of the estimation process. In singly iterative
methods, the table displays the following:
the iteration count, Iteration
the number of restarts, Restarts
the number of function evaluations, Evaluations
the objective function, Objective
2274 F Chapter 38: The GLIMMIX Procedure
the change in the objective function, Change
the absolute value of the largest (projected) gradient, MaxGradient
Note that the change in the objective function is not the convergence criterion monitored by the
GLIMMIX procedure. PROC GLIMMIX tracks several convergence criteria simultaneously; see
the ABSCONV=, ABSFCONV=, ABSGCONV=, ABSXCONV=, FCONV=, or GCONV= option
in the NLOPTIONS statement.
For doubly iterative estimation methods, the “Iteration History” table does not display the progress
of the individual optimizations; instead, it reports on the progress of the outer iterations. Every
row of the table then corresponds to an update of the linearization, the computation of a new set of
pseudo-data, and a new optimization. In the listing, PROC GLIMMIX displays the following:
the optimization count, Iteration
the number of restarts, Restarts
the number of iterations per optimization, Subiterations
the change in the parameter estimates, Change
the absolute value of the largest (projected) gradient at the end of the optimization, MaxGradient
By default, the change in the parameter estimates is expressed in terms of the relative PCONV
criterion. If you request an absolute criterion with the ABSPCONV option of the PROC GLIMMIX
statement, the change reflects the largest absolute difference since the last optimization.
If you specify the ITDETAILS option in the PROC GLIMMIX statement, parameter estimates and
their gradients are added to the “Iteration History” table. For ODS purposes, the name of the
“Iteration History” table is “IterHistory.”
Convergence Status
The “Convergence Status” table contains a status message describing the reason for termination
of the optimization. The message is also written to the log. For ODS purposes, the name of the
“Convergence Status” table is “ConvergenceStatus,” and you can query the nonprinting numeric
variable Status to check for a successful optimization. This is useful in batch processing, or when
processing BY groups, such as in simulations. Successful optimizations are indicated by the value
0 of the Status variable.
Fit Statistics
The “Fit Statistics” table provides statistics about the estimated model. The first entry of the table
corresponds to the negative of twice the (possibly restricted) log likelihood, log pseudo-likelihood,
or log quasi-likelihood. If the estimation method permits the true log likelihood or residual log likelihood, the description of the first entry reads accordingly. Otherwise, the fit statistics are preceded
by the words Pseudo- or Quasi-, for Pseudo- and Quasi-Likelihood estimation, respectively.
Default Output F 2275
Note that the (residual) log pseudo-likelihood in a GLMM is the (residual) log likelihood of a
linearized model. You should not compare these values across different statistical models, even if
the models are nested with respect to fixed and/or G-side random effects. It is possible that between
two nested models the larger model has a smaller pseudo-likelihood. For this reason, IC=NONE is
the default for GLMMs fit by pseudo-likelihood methods.
See the IC= option of the PROC GLIMMIX statement and Table 38.2 for the definition and computation of the information criteria reported in the “Fit Statistics” table.
For generalized linear models, the GLIMMIX procedure reports Pearson’s chi-square statistic
X2 D
b
i /2
a.b
i /
X wi .yi
i
where a.b
i / is the variance function evaluated at the estimated mean.
For GLMMs, the procedure typically reports a generalized chi-square statistic,
Xg2 D b
r0 V.b
/ 1b
r
so that the ratio of X 2 or Xg2 and the degrees of freedom produces the usual residual dispersion
estimate.
If the R-side scale parameter is not extracted from V, the GLIMMIX procedure computes
Xg2 D b
r0 V.b
/ 1b
r
as the generalized chi-square statistic. This is the case, for example, if R-side covariance structures
are varied by a GROUP= effect or if the scale parameter is not profiled for an R-side TYPE=CS,
TYPE=SP, TYPE=AR, TYPE=TOEP, or TYPE=ARMA covariance structure.
For METHOD=LAPLACE, the generalized chi-square statistic is not reported. Instead, the Pearson
statistic for the conditional distribution appears in the “Conditional Fit Statistics” table.
If your model contains smooth components (such as TYPE=RSMOOTH), then the “Fit Statistics”
table also displays the residual degrees of freedom of the smoother. These degrees of freedom are
computed as
dfsmooth;res D f
trace.S/
where S is the “smoother” matrix—that is, the matrix that produces the predicted values on the
linked scale.
For ODS purposes, the name of the “Fit Statistics” table is “FitStatistics.”
Covariance Parameter Estimates
In a GLMM, the “Covariance Parameter Estimates” table displays the estimates of the covariance
parameters and their asymptotic standard errors. This table is produced only for generalized linear
mixed models. In generalized linear models with scale parameter, or when an overdispersion parameter is present, the estimates of parameters related to the dispersion are displayed in the “Parameter
Estimates” table.
2276 F Chapter 38: The GLIMMIX Procedure
The standard error of the covariance parameters is determined from the diagonal entries of the
asymptotic variance matrix of the covariance parameter estimates. You can display this matrix with
the ASYCOV option in the PROC GLIMMIX statement.
For ODS purposes, the name of the “Covariance Parameter Estimates” table is “CovParms.”
Type III Tests of Fixed Effects
The “Type III Tests of Fixed Effects” table contains hypothesis tests for the significance of each of
the fixed effects specified in the MODEL statement. By default, PROC GLIMMIX computes these
tests by first constructing a Type III L matrix for each effect; see Chapter 15, “The Four Types of
Estimable Functions.” The L matrix is then used to construct the test statistic
F D
b̌0 L0 .LQL0 /
1 Lb̌
rank.LQL0 /
where the matrix Q depends on the estimation method and options. For example, in a GLMM,
the default is Q D .X0 V.b
/ 1 X/ , where V./ is the marginal variance of the pseudo-response.
If you specify the DDFM=KENWARDROGER option, Q is the estimated variance matrix of the
fixed effects, adjusted by the method of Kenward and Roger (1997). If the EMPIRICAL= option is
in effect, Q corresponds to the selected sandwich estimator.
You can use the HTYPE= option in the MODEL statement to obtain tables of Type I (sequential)
tests and Type II (adjusted) tests in addition to or instead of the table of Type III (partial) tests.
For ODS purposes, the names of the “Type I Tests of Fixed Effects” through the “Type III Tests of
Fixed Effects” tables are “Tests1” through “Tests3,” respectively.
Notes on Output Statistics
Table 38.11 lists the statistics computed with the OUTPUT statement of the GLIMMIX procedure
and their default names. This section provides further details about these statistics.
The distinction between prediction and confidence limits in Table 38.11 stems from the involvement
of the predictors of the random effects. If the random-effect solutions (BLUPs, EBES) are involved,
then the associated standard error used in computing the limits are standard errors of prediction
rather than standard errors of estimation. The prediction limits are not limits for the prediction of a
new observation.
The Pearson residuals in Table 38.11 are “Pearson-type” residuals, because the residuals are standardized by the square root of the marginal or conditional variance of an observation. Traditionally,
Pearson residuals in generalized linear models are divided by the square root of the variance function. The GLIMMIX procedure divides by the square root of the variance so that marginal and
conditional residuals have similar expressions. In other words, scale and overdispersion parameters
are included.
When residuals or predicted values involve only the fixed effects part of the linear predictor (that
is, b
m D x0b̌), then all model quantities are computed based on this predictor. For example, if
Notes on Output Statistics F 2277
the variance by which to standardize a marginal residual involves the variance function, then the
variance function is also evaluated at the marginal mean, g 1 .b
m /. Thus the residuals p b
and
pm b
m can also be expressed as .y /[email protected] and .y m /[email protected] , respectively, where @ is the
derivative with respect to the linear predictor. To construct the residual p b
m in a GLMM, you
can add the value of _ZGAMMA_ to the conditional residual p b
. If the predictor involves the
BLUPs, then all relevant expressions and evaluations involve the conditional mean g 1 .b
/.
The naming convention to add “PA” to quantities not involving the BLUPs is chosen to suggest the
concept of a population average. When the link function is nonlinear, these are not truly populationaveraged quantities, because g 1 .x0 ˇ/ does not equal EŒY  in the presence of random effects. For
example, if
i D g
1
.x0i ˇ C z0i i /
is the conditional mean for subject i , then
g
1
.x0i b̌/
does not estimate the average response in the population of subjects but the response of the average
subject (the subject for which i D 0). For models with identity link, the average response and the
response of the average subject are identical.
The GLIMMIX procedure obtains standard errors on the scale of the mean by the delta method. If
the link is a nonlinear function of the linear predictor, these standard errors are only approximate.
For example,
2
: @g 1 .t /
1
VarŒg .b
m / D
VarŒb
m 
@t jb
m
Confidence limits on the scale of the data are usually computed by applying the inverse link function
to the confidence limits on the linked scale. The resulting limits on the data scale have the same
coverage probability as the limits on the linked scale, but they are possibly asymmetric.
In generalized logit models, confidence limits on the mean scale are based on symmetric limits
about the predicted mean in a category. Suppose that the multinomial response in such a model has
J categories. The probability of a response in category i is computed as
exp fb
i g
b
i D PJ
i g
j D1 exp fb
The variance of b
i is then approximated as
:
1 b
2 b
J i
VarŒb
i  D D 0i Var b
where i is a J 1 vector with kth element
b
i .1 b
i / i D k
b
i b
k
iD
6 k
The confidence limits in the generalized logit model are then obtained as
p
b
i ˙ t;˛=2 where t;˛=2 is the 100 .1 ˛=2/ percentile from a t distribution with degrees of freedom.
Confidence limits are truncated if they fall outside the Œ0; 1 interval.
2278 F Chapter 38: The GLIMMIX Procedure
ODS Table Names
Each table created by PROC GLIMMIX has a name associated with it, and you must use this name
to reference the table when you use ODS statements. These names are listed in Table 38.19.
Table 38.19
ODS Tables Produced by PROC GLIMMIX
Table Name
Description
Required Statement / Option
AsyCorr
asymptotic correlation matrix of covariance parameters
asymptotic covariance matrix of covariance parameters
Cholesky root of the estimated G
matrix
Cholesky root of blocks of the estimated V matrix
level information from the CLASS
statement
L matrix coefficients
PROC GLIMMIX ASYCORR
AsyCov
CholG
CholV
ClassLevels
Coef
ColumnNames
CondFitStatistics
Contrasts
ConvergenceStatus
CorrB
CovB
CovBDetails
CovBI
CovBModelBased
CovParms
CovTests
name association for OUTDESIGN
data set
conditional fit statistics
results from the CONTRAST statements
status of optimization at conclusion
approximate correlation matrix of
fixed-effects parameter estimates
approximate covariance matrix of
fixed-effects parameter estimates
details about model-based and/or
adjusted covariance matrix of fixed
effects
inverse of approximate covariance
matrix of fixed-effects parameter estimates
model-based (unadjusted) covariance matrix of fixed effects if
DDFM=KR or EMPIRICAL option
is used
estimated covariance parameters in
GLMMs
results from COVTEST statements
(except for confidence bounds)
PROC GLIMMIX ASYCOV
RANDOM / GC
RANDOM / VC
default output
E option in MODEL, CONTRAST,
ESTIMATE,
LSMESTIMATE,
or LSMEANS; ELSM option in
LSMESTIMATE
PROC GLIMMIX
OUTDESIGN(NAMES)
PROC GLIMMIX
METHOD=LAPLACE
CONTRAST
default output
MODEL / CORRB
MODEL / COVB
MODEL / COVB(DETAILS)
MODEL / COVBI
MODEL / COVB(DETAILS)
default output (in GLMMs)
COVTEST
ODS Table Names F 2279
Table 38.19
continued
Table Name
Description
Required Statement / Option
Diffs
Dimensions
Estimates
FitStatistics
G
GCorr
differences of LS-means
dimensions of the model
results from ESTIMATE statements
fit statistics
estimated G matrix
correlation matrix from the estimated G matrix
Hessian matrix (observed or expected)
inverse Cholesky root of the estimated G matrix
inverse Cholesky root of the blocks
of the estimated V matrix
inverse of the estimated G matrix
inverse of blocks of the estimated V
matrix
iteration history
k-d tree information
LSMEANS / DIFF (or PDIFF)
default output
ESTIMATE
default
RANDOM / G
RANDOM / GCORR
Hessian
InvCholG
InvCholV
InvG
InvV
IterHistory
kdTree
KnotInfo
LSMeans
LSMEstimates
LSMFtest
LSMLines
ModelInfo
NObs
OddsRatios
OptInfo
ParameterEstimates
ParmSearch
QuadCheck
ResponseProfile
Slices
SliceDiffs
SolutionR
StandardizedCoefficients
Tests1
knot coordinates of low-rank spline
smoother
LS-means
estimates among LS-means
F test for LSMESTIMATEs
lines display for LS-means
model information
number of observations read and
used, number of trials and events
odds ratios of parameter estimates
optimization information
fixed-effects solution; overdispersion and scale parameter in GLMs
parameter search values
adaptive recalculation of quadrature
approximation at solution
response categories and category
modeled
tests of LS-means slices
differences of simple LS-means effects
random-effects solution vector
fixed-effects solutions from centered
and/or scaled model
Type I tests of fixed effects
PROC GLIMMIX HESSIAN
RANDOM / GCI
RANDOM / VCI
RANDOM / GI
RANDOM / VI
default output
RANDOM / TYPE=RSMOOTH
KNOTMETHOD=
KDTREE(TREEINFO)
RANDOM / TYPE=RSMOOTH
KNOTINFO
LSMEANS
LSMESTIMATE
LSMESTIMATE / FTEST
LSMEANS / LINES
default output
default output
MODEL / ODDSRATIO
default output
MODEL / S
PARMS
METHOD=QUAD(QCHECK)
default output in models with binary
or nominal response
LSMEANS / SLICE=
LSMEANS / SLICEDIFF=
RANDOM / S
MODEL / STDCOEF
MODEL / HTYPE=1
2280 F Chapter 38: The GLIMMIX Procedure
Table 38.19
continued
Table Name
Description
Required Statement / Option
Tests2
Tests3
V
VCorr
Type II tests of fixed effects
Type III tests of fixed effects
blocks of the estimated V matrix
correlation matrix from the blocks of
the estimated V matrix
MODEL / HTYPE=2
default output
RANDOM / V
RANDOM / VCORR
ODS Graphics
The following subsections provide information about the basic ODS statistical graphics produced
by the GLIMMIX procedure. The graphics fall roughly into two categories: diagnostic plots and
graphics for least squares means.
Diagnostic Plots
Residual Panels
There are three types of residual panels in the GLIMMIX procedure. Their makeup of four component plots is the same; the difference lies in the type of residual from which the panel is computed. Raw residuals are displayed with the PLOTS=RESIDUALPANEL option. Studentized
residuals are displayed with the PLOTS=STUDENTPANEL option, and Pearson residuals with
the PLOTS==PEARSONPANEL option. By default, conditional residuals are used in the construction of the panels if the model contains G-side random effects. For example, consider the following
statements:
proc glimmix plots=residualpanel;
class A;
model y = x1 x2 / dist=Poisson;
random int / sub=A;
run;
The parameters are estimated by a pseudo-likelihood method, and at the final stage pseudo-data are
related to a linear mixed model with random intercepts. The residual panel is constructed from
r Dp
x0b̌ C z0b
where p is the pseudo-data.
The following hypothetical data set contains yields of an industrial process. Material was available
from five randomly selected vendors to produce a chemical reaction whose yield depends on two
factors (pressure and temperature at 3 and 2 levels, respectively).
ODS Graphics F 2281
data Yields;
input Vendor Pressure Temp Yield @@;
datalines;
1 1 1 10.20
1 1 2
9.48
1 2
1 2 2
8.92
1 3 1 11.79
1 3
2 1 1 10.43
2 1 2 10.59
2 2
2 2 2 10.15
2 3 1 11.12
2 3
3 1 1
6.46
3 1 2
7.34
3 2
3 2 2
8.11
3 3 1
9.38
3 3
4 1 1
7.36
4 1 2
9.92
4 2
4 2 2 10.34
4 3 1 10.24
4 3
5 1 1 11.72
5 1 2 10.60
5 2
5 2 2
9.03
5 3 1 14.09
5 3
1
2
1
2
1
2
1
2
1
2
9.74
8.85
10.29
9.30
9.44
8.37
10.99
9.96
11.28
8.92
;
We consider here a linear mixed model with a two-way factorial fixed-effects structure for pressure and temperature effects and independent, homoscedastic random effects for the vendors. The
following statements fit this model and request panels of marginal and conditional residuals:
ods graphics on;
proc glimmix data=Yields
plots=residualpanel(conditional marginal);
class Vendor Pressure Temp;
model Yield = Pressure Temp Pressure*Temp;
random vendor;
run;
ods graphics off;
The suboptions of the RESIDUALPANEL request produce two panels. The panel of conditional
residuals is constructed from y x0b̌ z0b
(Figure 38.19). The panel of marginal residuals is
0
b̌
constructed from y x (Figure 38.20). Note that these residuals are deviations from the observed
data, because the model is a normal linear mixed model, and hence it does not involve pseudodata. Whenever the random-effects solutions b
are involved in constructing residuals, the title of
the residual graphics identifies them as conditional residuals (Figure 38.19).
2282 F Chapter 38: The GLIMMIX Procedure
Figure 38.19 Conditional Residuals
ODS Graphics F 2283
Figure 38.20 Marginal Residuals
The predictor takes on only six values for the marginal residuals, corresponding to the combinations
of three temperature and two pressure levels. The assumption of a zero mean for the vendor random
effect seems justified; the marginal residuals in the upper-left plot of Figure 38.20 do not exhibit
any trend. The conditional residuals in Figure 38.19 are smaller and somewhat closer to normality
compared to the marginal residuals.
Box Plots
You can produce box plots of observed data, pseudo-data, and various residuals for effects in your
model that consist of classification variables. Because you might not want to produce box plots for
all such effects, you can request subsets with the suboptions of the BOXPLOT option in the PLOTS
option. The BOXPLOT request in the following PROC GLIMMIX statement produces box plots
for the random effects—in this case, the vendor effect. By default, PROC GLIMMIX constructs
box plots from conditional residuals. The MARGINAL, CONDITIONAL, and OBSERVED suboptions instruct the procedure to construct three box plots for each random effect: box plots of the
observed data (Figure 38.21), the marginal residuals (Figure 38.22), and the conditional residuals
(Figure 38.23).
2284 F Chapter 38: The GLIMMIX Procedure
ods graphics on;
proc glimmix data=Yields
plots=boxplot(random marginal conditional observed);
class Vendor Pressure Temp;
model Yield = Pressure Temp Pressure*Temp;
random vendor;
run;
ods graphics off;
The observed vendor means in Figure 38.21 are different; in particular, vendors 3 and 5 appear to
differ from the other vendors and from each other. There is also heterogeneity of variance in the
five groups. The marginal residuals in Figure 38.22 continue to show the differences in means by
vendor, because vendor enters the model as a random effect. The marginal means are adjusted for
vendor effects only in the sense that the vendor variance component affects the marginal variance
that is involved in the generalized least squares solution for the pressure and temperature effects.
Figure 38.21 Box Plots of Observed Values
ODS Graphics F 2285
Figure 38.22 Box Plots of Marginal Residuals
The conditional residuals account for the vendor effects through the empirical BLUPs. The
means and medians have stabilized near zero, but some heterogeneity in these residuals remains
(Figure 38.23).
2286 F Chapter 38: The GLIMMIX Procedure
Figure 38.23 Box Plots of Conditional Residuals
Graphics for LS-Mean Comparisons
The following subsections provide information about the ODS statistical graphics for least squares
means produced by the GLIMMIX procedure. Mean plots display marginal or interaction means.
The diffogram, control plot, and ANOM plot display least squares mean comparisons.
Mean Plots
The following SAS statements request a plot of the PressureTemp means in which the pressure
trends are plotted for each temperature.
ods graphics on;
ods select CovParms Tests3 MeanPlot;
proc glimmix data=Yields;
class Vendor Pressure Temp;
model Yield = Pressure Temp Pressure*Temp;
random Vendor;
lsmeans Pressure*Temp / plot=mean(sliceby=Temp join);
run;
ods graphics off;
ODS Graphics F 2287
There is a significant effect of temperature and an interaction between pressure and temperature
(Figure 38.24). Notice that the pressure main effect might be masked by the interaction. Because
of the interaction, temperature comparisons depend on the pressure and vice versa. The mean plot
option requests a display of the Pressure Temp least squares means with separate trends for each
temperature (Figure 38.25).
Figure 38.24 Tests for Fixed Effects
The GLIMMIX Procedure
Covariance Parameter Estimates
Cov Parm
Estimate
Standard
Error
Vendor
Residual
0.8602
1.1039
0.7406
0.3491
Type III Tests of Fixed Effects
Effect
Pressure
Temp
Pressure*Temp
Num
DF
Den
DF
F Value
Pr > F
2
1
2
20
20
20
1.42
6.48
3.82
0.2646
0.0193
0.0393
The interaction between the two effects is evident in the lack of parallelism in Figure 38.25. The
masking of the pressure main effect can be explained by slopes of different sign for the two trends.
Based on these results, inferences about the pressure effects are conducted for a specific temperature. For example, Figure 38.26 is produced by adding the following statement:
lsmeans pressure*temp / slicediff=temp slice=temp;
2288 F Chapter 38: The GLIMMIX Procedure
Figure 38.25 Interaction Plot for Pressure x Temperature
Figure 38.26 Pressure Comparisons at a Given Temperature
The GLIMMIX Procedure
Tests of Effect Slices for Pressure*Temp
Sliced By Temp
Temp
1
2
Num
DF
Den
DF
F Value
Pr > F
2
2
20
20
4.95
0.29
0.0179
0.7508
ODS Graphics F 2289
Figure 38.26 continued
Simple Effect Comparisons of Pressure*Temp Least Squares Means By Temp
Simple
Effect
Level
Pressure
_Pressure
Temp
Temp
Temp
Temp
Temp
Temp
1
1
2
1
1
2
2
3
3
2
3
3
1
1
1
2
2
2
Estimate
Standard
Error
DF
t Value
Pr > |t|
-1.1140
-2.0900
-0.9760
0.2760
0.5060
0.2300
0.6645
0.6645
0.6645
0.6645
0.6645
0.6645
20
20
20
20
20
20
-1.68
-3.15
-1.47
0.42
0.76
0.35
0.1092
0.0051
0.1575
0.6823
0.4553
0.7329
The slope differences are evident by the change in sign for comparisons within temperature 1 and
within temperature 2. There is a significant effect of pressure at temperature 1 (p = 0.0179), but not
at temperature 2 (p = 0.7508).
Pairwise Difference Plot (Diffogram)
Graphical displays of LS-means-related analyses consist of plots of all pairwise differences (DiffPlot), plots of differences against a control level (ControlPlot), and plots of differences against an
overall average (AnomPlot). The following data set is from an experiment to investigate how snapdragons grow in various soils (Stenstrom 1940). To eliminate the effect of local fertility variations,
the experiment is run in blocks, with each soil type sampled in each block. See the “Examples”
section of Chapter 39, “The GLM Procedure,” for an in-depth analysis of these data.
data plants;
input Type $ @;
do Block = 1 to 3;
input StemLength
output;
end;
datalines;
Clarion
32.7 32.3
Clinton
32.1 29.7
Knox
35.7 35.9
ONeill
36.0 34.2
Compost
31.8 28.0
Wabash
38.2 37.8
Webster
32.5 31.1
;
@;
31.5
29.1
33.1
31.2
29.2
31.9
29.7
The following statements perform the analysis of the experiment with the GLIMMIX procedure:
2290 F Chapter 38: The GLIMMIX Procedure
ods graphics on;
ods select LSMeans DiffPlot;
proc glimmix data=plants order=data plots=Diffogram;
class Block Type;
model StemLength = Block Type;
lsmeans Type;
run;
ods graphics off;
The PLOTS= option in the PROC GLIMMIX statement requests that plots of pairwise least squares
means differences are produced for effects that are listed in corresponding LSMEANS statements.
This is the Type effect.
The Type LS-means are shown in Figure 38.27. Note that the order in which the levels appear
corresponds to the order in which they were read from the data set. This was accomplished with the
ORDER=DATA option in the PROC GLIMMIX statement.
Figure 38.27 Least Squares Means for Type Effect
The GLIMMIX Procedure
Type Least Squares Means
Type
Clarion
Clinton
Knox
ONeill
Compost
Wabash
Webster
Estimate
Standard
Error
DF
t Value
Pr > |t|
32.1667
30.3000
34.9000
33.8000
29.6667
35.9667
31.1000
0.7405
0.7405
0.7405
0.7405
0.7405
0.7405
0.7405
12
12
12
12
12
12
12
43.44
40.92
47.13
45.64
40.06
48.57
42.00
<.0001
<.0001
<.0001
<.0001
<.0001
<.0001
<.0001
Because there are seven levels of Type in this analysis, there are 7.6 1/=2 D 21 pairwise comparisons among the least squares means. The comparisons are performed in the following fashion:
the first level of Type is compared against levels 2 through 7; the second level of Type is compared
against levels 3 through 7; and so forth.
The default difference plot for these data is shown in Figure 38.28. The display is also known as
a “mean-mean scatter plot” (Hsu 1996; Hsu and Peruggia 1994). It contains 21 lines rotated by
45 degrees counterclockwise, and a reference line (dashed 45-degree line). The .x; y/ coordinate
for the center of each line corresponds to the two least squares means being compared. Suppose
that b
:i and b
:j denote the i th and j th least squares mean, respectively, for the effect in question,
where i < j according to the ordering of the effect levels. If the ABS option is in effect, which
is the default, the line segment is centered at .minfb
i: ;b
j: g; maxfb
i: ;b
j: g/. Take, for example, the
comparison of “Clarion” and “Compost” types. The respective estimates of their LS-means are
b
:1 D 32:1667 and b
:5 D 29:6667. The center of the line segment for H0 W :1 D :5 is placed at
.29:6667; 32:1667/.
ODS Graphics F 2291
The length of the line segment for the comparison between means i and j corresponds to the
width of the confidence interval for the difference :i :j . This length is adjusted for the rotation
in the plot. As a consequence, comparisons whose confidence interval covers zero cross the 45degree reference line. These are the nonsignificant comparisons. Lines associated with significant
comparisons do not touch or cross the reference line. Because these data are balanced, the estimated
standard errors of all pairwise comparisons are identical, and the widths of the line segments are the
same.
Figure 38.28 LS-Means Plot of Pairwise Differences
The background grid of the difference plot is drawn at the values of the least squares means for the
seven type levels. These grid lines are used to find a particular comparison by intersection. Also,
the labels of the grid lines indicate the ordering of the least squares means.
In the next set of statements, the NOABS and CENTER suboptions of the PLOTS=DIFFOGRAM
option in the LSMEANS statement modify the appearance of the diffogram:
ods graphics on;
proc glimmix data=plants order=data;
class Block Type;
model StemLength = Block Type;
lsmeans Type / plots=diffogram(noabs center);
run;
ods graphics off;
2292 F Chapter 38: The GLIMMIX Procedure
The NOABS suboption of the difference plot changes the way in which the GLIMMIX procedure
places the line segments (Figure 38.29). If the NOABS suboption is in effect, the line segment is
centered at the point .b
:i ;b
:j /, i < j . For example, the center of the line segment for a comparison of “Clarion” and “Compost” types is centered at .b
:1 ;b
:5 / D .32:1667; 29:6667/. Whether
a line segment appears above or below the reference line depends on the magnitude of the least
squares means and the order of their appearance in the “Least Squares Means” table. The CENTER
suboption places a marker at the intersection of the least squares means.
Because the ABS option places lines on the same side of the 45-degree reference, it can help to
visually discover groups of significant and nonsignificant differences. On the other hand, when the
number of levels in the effect is large, the display can get crowded. The NOABS option can then
provide a more accessible resolution.
Figure 38.29 Diffogram with NOABS and CENTER Options
ODS Graphics F 2293
Least Squares Mean Control Plot
The following SAS statements create the same data set as before, except that one observation for
Type=“Knox” has been removed for illustrative purposes:
data plants;
input Type $ @;
do Block = 1 to 3;
input StemLength
output;
end;
datalines;
Clarion
32.7 32.3
Clinton
32.1 29.7
Knox
35.7 35.9
ONeill
36.0 34.2
Compost
31.8 28.0
Wabash
38.2 37.8
Webster
32.5 31.1
;
@;
31.5
29.1
.
31.2
29.2
31.9
29.7
The following statements request control plots for effects in LSMEANS statements with compatible
option:
ods graphics on;
ods select Diffs ControlPlot;
proc glimmix data=plants order=data plots=ControlPlot;
class Block Type;
model StemLength = Block Type;
lsmeans Type / diff=control(’Clarion’) adjust=dunnett;
run;
ods graphics off;
The LSMEANS statement for the Type effect is compatible; it requests comparisons of Type levels
against “Clarion,” adjusted for multiplicity with Dunnett’s method. Because “Clarion” is the first
level of the effect, the LSMEANS statement is equivalent to
lsmeans type / diff=control adjust=dunnett;
The “Differences of Type Least Squares Means” table in Figure 38.30 shows the six comparisons
between Type levels and the control level.
2294 F Chapter 38: The GLIMMIX Procedure
Figure 38.30 Least Squares Means Differences
The GLIMMIX Procedure
Differences of Type Least Squares Means
Adjustment for Multiple Comparisons: Dunnett
Type
_Type
Clinton
Knox
ONeill
Compost
Wabash
Webster
Clarion
Clarion
Clarion
Clarion
Clarion
Clarion
Estimate
Standard
Error
DF
t Value
Pr > |t|
Adj P
-1.8667
2.7667
1.6333
-2.5000
3.8000
-1.0667
1.0937
1.2430
1.0937
1.0937
1.0937
1.0937
11
11
11
11
11
11
-1.71
2.23
1.49
-2.29
3.47
-0.98
0.1159
0.0479
0.1635
0.0431
0.0052
0.3504
0.3936
0.1854
0.5144
0.1688
0.0236
0.8359
The two rightmost columns of the table give the unadjusted and multiplicity-adjusted p-values. At
the 5% significance level, both “Knox” and “Wabash” differ significantly from “Clarion” according
to the unadjusted tests. After adjusting for multiplicity, only “Wabash” has a least squares mean
significantly different from the control mean. Note that the standard error for the comparison involving “Knox” is larger than that for other comparisons because of the reduced sample size for that
soil type.
In the plot of control differences a horizontal line is drawn at the value of the “Clarion” least squares
mean. Vertical lines emanating from this reference line terminate in the least squares means for the
other levels (Figure 38.31).
The dashed upper and lower horizontal reference lines are the upper and lower decision limits
for tests against the control level. If a vertical line crosses the upper or lower decision limit, the
corresponding least squares mean is significantly different from the LS-mean in the control group.
If the data had been balanced, the UDL and LDL would be straight lines, because all estimates
b
:i b
:j would have had the same standard error. The limits for the comparison between “Knox”
and “Clarion” are wider than for other comparisons, because of the reduced sample size for the
“Knox” soil type.
ODS Graphics F 2295
Figure 38.31 LS-Means Plot of Differences against a Control
The significance level of the decision limits is determined from the ALPHA= level in the
LSMEANS statement. The default are 95% limits. If you choose one-sided comparisons with
DIFF=CONTROLL or DIFF=CONTROLU in the LSMEANS statement, only one of the decision
limits is drawn.
Analysis of Means (ANOM) Plot
The analysis of means in PROC GLIMMIX compares least squares means not by contrasting them
against each other as with all pairwise differences or control differences. Instead, the least squares
means are compared against an average value. Consequently, there are k comparisons for a factor
with k levels. The following statements request ANOM differences for the Type least squares means
(Figure 38.32) and plots the differences (Figure 38.33):
ods graphics on;
ods select Diffs AnomPlot;
proc glimmix data=plants order=data plots=AnomPlot;
class Block Type;
model StemLength = Block Type;
lsmeans Type / diff=anom;
run;
ods graphics off;
2296 F Chapter 38: The GLIMMIX Procedure
Figure 38.32 ANOM LS-Mean Differences
The GLIMMIX Procedure
Differences of Type Least Squares Means
Type
_Type
Clarion
Clinton
Knox
ONeill
Compost
Wabash
Webster
Avg
Avg
Avg
Avg
Avg
Avg
Avg
Estimate
Standard
Error
DF
t Value
Pr > |t|
-0.2635
-2.1302
2.5032
1.3698
-2.7635
3.5365
-1.3302
0.7127
0.7127
0.9256
0.7127
0.7127
0.7127
0.7127
11
11
11
11
11
11
11
-0.37
-2.99
2.70
1.92
-3.88
4.96
-1.87
0.7186
0.0123
0.0205
0.0809
0.0026
0.0004
0.0888
At the 5% level, the “Clarion,” “O’Neill,” and “Webster” soil types are not significantly different
from the average. Note that the artificial lack of balance introduced previously reduces the precision
of the ANOM comparison for the “Knox” soil type.
Figure 38.33 LS-Means Analysis of Means (ANOM) Plot
ODS Graphics F 2297
The reference line in the ANOM plot is drawn at the average. Vertical lines extend from this reference line upward or downward, depending on the magnitude of the least squares means compared
to the reference value. This enables you to quickly see which levels perform above and below the
average. The horizontal reference lines are 95% upper and lower decision limits. If a vertical line
crosses the limits, you conclude that the least squares mean is significantly different (at the 5%
significance level) from the average. You can adjust the comparisons for multiplicity by adding the
ADJUST=NELSON option in the LSMEANS statement.
ODS Graph Names
To request graphics with PROC GLIMMIX, you must first enable ODS Graphics by specifying
the ODS GRAPHICS ON statement. See Chapter 21, “Statistical Graphics Using ODS,” for more
information. The GLIMMIX procedure does not produce graphs by default. You can reference
every graph produced through ODS Graphics with a name. The names of the graphs that PROC
GLIMMIX generates are listed in Table 38.20, along with the required statements and options.
Table 38.20
ODS Graphics Produced by PROC GLIMMIX
ODS Graph Name
Plot Description
Option
AnomPlot
Plot of LS-mean differences
against the average LSmean
Box plots of residuals
and/or observed values for
model effects
Plot of LS-mean differences
against a control level
PLOTS=ANOMPLOT
LSMEANS / PLOTS=ANOMPLOT
PLOTS=CONTROLPLOT
LSMEANS / PLOTS=CONTROLPLOT
DiffPlot
Plot of LS-mean pairwise
differences
PLOTS=DIFFPLOT
LSMEANS / PLOTS=DIFFPLOT
MeanPlot
Plot of least squares means
PLOTS=MEANPLOT
LSMEANS / PLOTS=MEANPLOT
ORPlot
Plot of odds ratios
PLOTS=ODDSRATIO
PearsonBoxplot
Box plot of Pearson residuals
Pearson residuals vs. mean
PLOTS=PEARSONPANEL(UNPACK)
Histogram of Pearson residuals
Panel of Pearson residuals
PLOTS=PEARSONPANEL(UNPACK)
Q-Q plot of Pearson residuals
Box plot of (raw) residuals
PLOTS=PEARSONPANEL(UNPACK)
Residuals vs. mean or linear
predictor
Histogram of (raw) residuals
PLOTS=RESIDUALPANEL(UNPACK)
Boxplot
ControlPlot
PearsonByPredicted
PearsonHistogram
PearsonPanel
PearsonQQplot
ResidualBoxplot
ResidualByPredicted
ResidualHistogram
PLOTS=BOXPLOT
PLOTS=PEARSONPANEL(UNPACK)
PLOTS=PEARSONPANEL
PLOTS=RESIDUALPANEL(UNPACK)
PLOTS=RESIDUALPANEL(UNPACK)
2298 F Chapter 38: The GLIMMIX Procedure
Table 38.20
continued
ODS Graph Name
Plot Description
Option
ResidualPanel
Panel of (raw) residuals
PLOTS=RESIDUALPANEL
ResidualQQplot
Q-Q plot of (raw) residuals
PLOTS=RESIDUALPANEL(UNPACK)
StudentBoxplot
Box plot of studentized
residuals
Studentized residuals vs.
mean or linear predictor
Histogram of studentized
residuals
Panel of studentized residuals
Q-Q plot of studentized
residuals
PLOTS=STUDENTPANEL(UNPACK)
StudentByPredicted
StudentHistogram
StudentPanel
StudentQQplot
PLOTS=STUDENTPANEL(UNPACK)
PLOTS=STUDENTPANEL(UNPACK)
PLOTS=STUDENTPANEL
PLOTS=STUDENTPANEL(UNPACK)
Examples: GLIMMIX Procedure
Example 38.1: Binomial Counts in Randomized Blocks
In the context of spatial prediction in generalized linear models, Gotway and Stroup (1997) analyze
data from an agronomic field trial. Researchers studied 16 varieties (entries) of wheat for their
resistance to infestation by the Hessian fly. They arranged the varieties in a randomized complete
block design on an 8 8 grid. Each 4 4 quadrant of that arrangement constitutes a block.
The outcome of interest was the number of damaged plants (Yij ) out of the total number of plants
growing on the unit (nij ). The two subscripts identify the block (i D 1; ; 4) and the entry
(j D 1; ; 16). The following SAS statements create the data set. The variables lat and lng denote
the coordinate of an experimental unit on the 8 8 grid.
data HessianFly;
label Y = ’No. of damaged plants’
n = ’No. of plants’;
input block entry lat lng n Y @@;
datalines;
1 14 1 1 8 2
1 16 1 2 9 1
1 7 1 3 13 9
1 6 1 4 9 9
1 13 2 1 9 2
1 15 2 2 14 7
1 8 2 3 8 6
1 5 2 4 11 8
1 11 3 1 12 7
1 12 3 2 11 8
1 2 3 3 10 8
1 3 3 4 12 5
1 10 4 1 9 7
1 9 4 2 15 8
1 4 4 3 19 6
1 1 4 4 8 7
Example 38.1: Binomial Counts in Randomized Blocks F 2299
2
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0
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1
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2
8
9
6
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7
;
Analysis as a GLM
If infestations are independent among experimental units, and all plants within a unit have the same
propensity for infestation, then the Yij are binomial random variables. The first model considered
is a standard generalized linear model for independent binomial counts:
proc glimmix data=HessianFly;
class block entry;
model y/n = block entry / solution;
run;
The PROC GLIMMIX statement invokes the procedure. The CLASS statement instructs the GLIMMIX procedure to treat both block and entry as classification variables. The MODEL statement specifies the response variable and the fixed effects in the model. PROC GLIMMIX constructs the X
matrix of the model from the terms on the right side of the MODEL statement. The GLIMMIX procedure supports two kinds of syntax for the response variable. This example uses the events/trials
syntax. The variable y represents the number of successes (events) out of n Bernoulli trials. When
the events/trials syntax is used, the GLIMMIX procedure automatically selects the binomial distribution as the response distribution. Once the distribution is determined, the procedure selects the
link function for the model. The default link for binomial data is the logit link. The preceding
statements are thus equivalent to the following statements:
proc glimmix data=HessianFly;
class block entry;
model y/n = block entry / dist=binomial link=logit solution;
run;
2300 F Chapter 38: The GLIMMIX Procedure
The SOLUTION option in the MODEL statement requests that solutions for the fixed effects (parameter estimates) be displayed.
The “Model Information” table describes the model and methods used in fitting the statistical model
(Output 38.1.1).
The GLIMMIX procedure recognizes that this is a model for uncorrelated data (variance matrix
is diagonal) and that parameters can be estimated by maximum likelihood. The default degreesof-freedom method to denominator degrees of freedom for F tests and t tests is the RESIDUAL
method. This corresponds to choosing f rank.X/ as the degrees of freedom, where f is the sum
of the frequencies used in the analysis. You can change the degrees of freedom method with the
DDFM= option in the MODEL statement.
Output 38.1.1 Model Information in GLM Analysis
The GLIMMIX Procedure
Model Information
Data Set
Response Variable (Events)
Response Variable (Trials)
Response Distribution
Link Function
Variance Function
Variance Matrix
Estimation Technique
Degrees of Freedom Method
WORK.HESSIANFLY
Y
n
Binomial
Logit
Default
Diagonal
Maximum Likelihood
Residual
The “Class Level Information” table lists the levels of the variables specified in the CLASS statement and the ordering of the levels (Output 38.1.2). The “Number of Observations” table displays
the number of observations read and used in the analysis.
Output 38.1.2 Class Level Information and Number of Observations
Class Level Information
Class
Levels
block
entry
4
16
Number
Number
Number
Number
Values
1 2 3 4
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
of
of
of
of
Observations Read
Observations Used
Events
Trials
64
64
396
736
The “Dimensions” table lists the size of relevant matrices (Output 38.1.3).
Example 38.1: Binomial Counts in Randomized Blocks F 2301
Output 38.1.3 Model Dimensions Information in GLM Analysis
Dimensions
Columns in X
Columns in Z
Subjects (Blocks in V)
Max Obs per Subject
21
0
1
64
Because of the absence of G-side random effects in this model, there are no columns in the Z
matrix. The 21 columns in the X matrix comprise the intercept, 4 columns for the block effect
and 16 columns for the entry effect. Because no RANDOM statement with a SUBJECT= option
was specified, the GLIMMIX procedure does not process the data by subjects (see the section
“Processing by Subjects” on page 2245 for details about subject processing).
The “Optimization Information” table provides information about the methods and size of the optimization problem (Output 38.1.4).
Output 38.1.4 Optimization Information in GLM Analysis
Optimization Information
Optimization Technique
Parameters in Optimization
Lower Boundaries
Upper Boundaries
Fixed Effects
Newton-Raphson
19
0
0
Not Profiled
With few exceptions, models fit with the GLIMMIX procedure require numerical methods for
parameter estimation. The default optimization method for (overdispersed) GLM models is the
Newton-Raphson algorithm. In this example, the optimization involves 19 parameters, corresponding to the number of linearly independent columns of the X0 X matrix.
The “Iteration History” table shows that the procedure converged after 3 iterations and 13 function
evaluations (Output 38.1.5). The Change column measures the change in the objective function
between iterations; however, this is not the monitored convergence criterion. The GLIMMIX procedure monitors several features simultaneously to determine whether to stop an optimization.
Output 38.1.5 Iteration History in GLM Analysis
Iteration History
Iteration
Restarts
Evaluations
Objective
Function
Change
Max
Gradient
0
1
2
3
0
0
0
0
4
3
3
3
134.13393738
132.85058236
132.84724263
132.84724254
.
1.28335502
0.00333973
0.00000009
4.899609
0.206204
0.000698
3.029E-8
2302 F Chapter 38: The GLIMMIX Procedure
Output 38.1.5 continued
Convergence criterion (GCONV=1E-8) satisfied.
The “Fit Statistics” table lists information about the fitted model (Output 38.1.6). The 2 Log
Likelihood values are useful for comparing nested models, and the information criteria AIC, AICC,
BIC, CAIC, and HQIC are useful for comparing nonnested models. On average, the ratio between
the Pearson statistic and its degrees of freedom should equal one in GLMs. Values larger than one
indicate overdispersion. With a ratio of 2.37, these data appear to exhibit more dispersion than
expected under a binomial model with block and varietal effects.
Output 38.1.6 Fit Statistics in GLM Analysis
Fit Statistics
-2 Log Likelihood
AIC (smaller is better)
AICC (smaller is better)
BIC (smaller is better)
CAIC (smaller is better)
HQIC (smaller is better)
Pearson Chi-Square
Pearson Chi-Square / DF
265.69
303.69
320.97
344.71
363.71
319.85
106.74
2.37
The “Parameter Estimates” table displays the maximum likelihood estimates (Estimate), standard
errors, and t tests for the hypothesis that the estimate is zero (Output 38.1.7).
Example 38.1: Binomial Counts in Randomized Blocks F 2303
Output 38.1.7 Parameter Estimates in GLM Analysis
Parameter Estimates
Effect
Intercept
block
block
block
block
entry
entry
entry
entry
entry
entry
entry
entry
entry
entry
entry
entry
entry
entry
entry
entry
block
entry
Estimate
Standard
Error
DF
t Value
Pr > |t|
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
-1.2936
-0.05776
-0.1838
-0.4420
0
2.9509
2.8098
2.4608
1.5404
2.7784
2.0403
2.3253
1.3006
1.5605
2.3058
1.4957
1.5068
-0.6296
0.4460
0.8342
0
0.3908
0.2332
0.2303
0.2328
.
0.5397
0.5158
0.4956
0.4564
0.5293
0.4889
0.4966
0.4754
0.4569
0.5203
0.4710
0.4767
0.6488
0.5126
0.4698
.
45
45
45
45
.
45
45
45
45
45
45
45
45
45
45
45
45
45
45
45
.
-3.31
-0.25
-0.80
-1.90
.
5.47
5.45
4.97
3.38
5.25
4.17
4.68
2.74
3.42
4.43
3.18
3.16
-0.97
0.87
1.78
.
0.0018
0.8055
0.4289
0.0640
.
<.0001
<.0001
<.0001
0.0015
<.0001
0.0001
<.0001
0.0089
0.0014
<.0001
0.0027
0.0028
0.3370
0.3889
0.0826
.
1
2
3
4
The “Type III Tests of Fixed Effect” table displays significance tests for the two fixed effects in the
model (Output 38.1.8).
Output 38.1.8 Type III Tests of Block and Entry Effects in GLM Analysis
Type III Tests of Fixed Effects
Effect
Num
DF
Den
DF
F Value
Pr > F
3
15
45
45
1.42
6.96
0.2503
<.0001
block
entry
These tests are Wald-type tests, not likelihood ratio tests. The entry effect is clearly significant
in this model with a p-value of <0.0001, indicating that the 16 wheat varieties are not equally
susceptible to infestation by the Hessian fly.
Analysis with Random Block Effects
There are several possible reasons for the overdispersion noted in Output 38.1.6 (Pearson ratio =
2:37). The data might not follow a binomial distribution, one or more important effects might not
have been accounted for in the model, or the data might be positively correlated. If important fixed
2304 F Chapter 38: The GLIMMIX Procedure
effects have been omitted, then you might need to consider adding them to the model. Because
this is a designed experiment, it is reasonable not to expect further effects apart from the block and
entry effects that represent the treatment and error control design structure. The reasons for the
overdispersion must lie elsewhere.
If overdispersion stems from correlations among the observations, then the model should be appropriately adjusted. The correlation can have multiple sources. First, it might not be the case that
the plants within an experimental unit responded independently. If the probability of infestation
of a particular plant is altered by the infestation of a neighboring plant within the same unit, the
infestation counts are not binomial and a different probability model should be used. A second
possible source of correlations is the lack of independence of experimental units. Even if treatments
were assigned to units at random, they might not respond independently. Shared spatial soil effects,
for example, can be the underlying factor. The following analyses take these spatial effects into
account.
First, assume that the environmental effects operate at the scale of the blocks. By making the
block effects random, the marginal responses will be correlated due to the fact that observations
within a block share the same random effects. Observations from different blocks will remain
uncorrelated, in the spirit of separate randomizations among the blocks. The next set of statements
fits a generalized linear mixed model (GLMM) with random block effects:
proc glimmix data=HessianFly;
class block entry;
model y/n = entry / solution;
random block;
run;
Because the conditional distribution—conditional on the block effects—is binomial, the marginal
distribution will be overdispersed relative to the binomial distribution. In contrast to adding a multiplicative scale parameter to the variance function, treating the block effects as random changes the
estimates compared to a model with fixed block effects.
In the presence of random effects and a conditional binomial distribution, PROC GLIMMIX does
not use maximum likelihood for estimation. Instead, the GLIMMIX procedure applies a restricted
(residual) pseudo-likelihood algorithm (Output 38.1.9). The “restricted” attribute derives from the
same rationale by which restricted (residual) maximum likelihood methods for linear mixed models
attain their name; the likelihood equations are adjusted for the presence of fixed effects in the model
to reduce bias in covariance parameter estimates.
Example 38.1: Binomial Counts in Randomized Blocks F 2305
Output 38.1.9 Model Information in GLMM Analysis
The GLIMMIX Procedure
Model Information
Data Set
Response Variable (Events)
Response Variable (Trials)
Response Distribution
Link Function
Variance Function
Variance Matrix
Estimation Technique
Degrees of Freedom Method
WORK.HESSIANFLY
Y
n
Binomial
Logit
Default
Not blocked
Residual PL
Containment
The “Class Level Information” and “Number of Observations” tables are as before (Output 38.1.10).
Output 38.1.10 Class Level Information and Number of Observations
Class Level Information
Class
Levels
block
entry
4
16
Number
Number
Number
Number
Values
1 2 3 4
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
of
of
of
of
Observations Read
Observations Used
Events
Trials
64
64
396
736
The “Dimensions” table indicates that there is a single G-side parameter, the variance of the random block effect (Output 38.1.11). The “Dimensions” table has changed from the previous model
(compare Output 38.1.11 to Output 38.1.3). Note that although the block effect has four levels, only
a single variance component is estimated. The Z matrix has four columns, however, corresponding to the four levels of the block effect. Because no SUBJECT= option is used in the RANDOM
statement, the GLIMMIX procedure treats these data as having arisen from a single subject with 64
observations.
Output 38.1.11 Model Dimensions Information in GLMM Analysis
Dimensions
G-side Cov. Parameters
Columns in X
Columns in Z
Subjects (Blocks in V)
Max Obs per Subject
1
17
4
1
64
2306 F Chapter 38: The GLIMMIX Procedure
The “Optimization Information” table indicates that a quasi-Newton method is used to solve the
optimization problem. This is the default optimization method for GLMM models (Output 38.1.12).
Output 38.1.12 Optimization Information in GLMM Analysis
Optimization Information
Optimization Technique
Parameters in Optimization
Lower Boundaries
Upper Boundaries
Fixed Effects
Starting From
Dual Quasi-Newton
1
1
0
Profiled
Data
In contrast to the Newton-Raphson method, the quasi-Newton method does not require second
derivatives. Because the covariance parameters are not unbounded in this example, the procedure
enforces a lower boundary constraint (zero) for the variance of the block effect, and the optimization
method is changed to a dual quasi-Newton method. The fixed effects are profiled from the likelihood equations in this model. The resulting optimization problem involves only the covariance
parameters.
The “Iteration History” table appears to indicate that the procedure converged after four iterations
(Output 38.1.13). Notice, however, that this table has changed slightly from the previous analysis (see Output 38.1.5). The Evaluations column has been replaced by the Subiterations column,
because the GLIMMIX procedure applied a doubly iterative fitting algorithm. The entire process
consisted of five optimizations, each of which was iterative. The initial optimization required four
iterations, the next one required three iterations, and so on.
Output 38.1.13 Iteration History in GLMM Analysis
Iteration History
Iteration
Restarts
Subiterations
Objective
Function
Change
Max
Gradient
0
1
2
3
4
0
0
0
0
0
4
3
2
1
0
173.28473428
181.66726674
182.20789493
182.21315596
182.21317662
0.81019251
0.17550228
0.00614874
0.00004386
0.00000000
0.000197
0.000739
7.018E-6
1.213E-8
3.349E-6
Convergence criterion (PCONV=1.11022E-8) satisfied.
The “Fit Statistics” table shows information about the fit of the GLMM (Output 38.1.14). The log
likelihood reported in the table is not the residual log likelihood of the data. It is the residual log
likelihood for an approximated model. The generalized chi-square statistic measures the residual
sum of squares in the final model, and the ratio with its degrees of freedom is a measure of variability
of the observation about the mean model.
Example 38.1: Binomial Counts in Randomized Blocks F 2307
Output 38.1.14 Fit Statistics in GLMM Analysis
Fit Statistics
-2 Res Log Pseudo-Likelihood
Generalized Chi-Square
Gener. Chi-Square / DF
182.21
107.96
2.25
The variance of the random block effects is rather small (Output 38.1.15).
Output 38.1.15 Estimated Covariance Parameters and Approximate Standard Errors
Covariance Parameter
Estimates
Cov
Parm
block
Estimate
Standard
Error
0.01116
0.03116
If the environmental effects operate on a spatial scale smaller than the block size, the random block
model does not provide a suitable adjustment. From the coarse layout of the experimental area, it
is not surprising that random block effects alone do not account for the overdispersion in the data.
Adding a random component to a generalized linear model is different from adding a multiplicative
overdispersion component, for example, via the PSCALE option in PROC GENMOD or a
random _residual_;
statement in PROC GLIMMIX. Such overdispersion components do not affect the parameter estimates, only their standard errors. A genuine random effect, on the other hand, affects both the
parameter estimates and their standard errors (compare Output 38.1.16 to Output 38.1.7).
2308 F Chapter 38: The GLIMMIX Procedure
Output 38.1.16 Parameter Estimates for Fixed Effects in GLMM Analysis
Solutions for Fixed Effects
Effect
entry
Intercept
entry
entry
entry
entry
entry
entry
entry
entry
entry
entry
entry
entry
entry
entry
entry
entry
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
Estimate
Standard
Error
DF
t Value
Pr > |t|
-1.4637
2.9609
2.7807
2.4339
1.5347
2.7653
2.0014
2.3518
1.2927
1.5663
2.2896
1.5018
1.5075
-0.5955
0.4573
0.8683
0
0.3738
0.5384
0.5138
0.4934
0.4542
0.5276
0.4865
0.4952
0.4739
0.4554
0.5179
0.4682
0.4752
0.6475
0.5111
0.4682
.
3
45
45
45
45
45
45
45
45
45
45
45
45
45
45
45
.
-3.92
5.50
5.41
4.93
3.38
5.24
4.11
4.75
2.73
3.44
4.42
3.21
3.17
-0.92
0.89
1.85
.
0.0296
<.0001
<.0001
<.0001
0.0015
<.0001
0.0002
<.0001
0.0091
0.0013
<.0001
0.0025
0.0027
0.3626
0.3758
0.0702
.
Output 38.1.17 Type III Test of Entry in GLMM Analysis
Type III Tests of Fixed Effects
Effect
Num
DF
Den
DF
F Value
Pr > F
entry
15
45
6.90
<.0001
Because the block variance component is small, the Type III test for the variety effect in
Output 38.1.17 is affected only very little compared to the GLM (Output 38.1.8).
Analysis with Smooth Spatial Trends
You can also consider these data in an observational sense, where the covariation of the observations
is subject to modeling. Rather than deriving model components from the experimental design alone,
environmental effects can be modeled by adjusting the mean and/or correlation structure. Gotway
and Stroup (1997) and Schabenberger and Pierce (2002) supplant the coarse block effects with
smooth-scale spatial components.
The model considered by Gotway and Stroup (1997) is a marginal model in that the correlation
structure is modeled through residual-side (R-side) random components. This exponential covariance model is fit with the following statements:
Example 38.1: Binomial Counts in Randomized Blocks F 2309
proc glimmix data=HessianFly;
class entry;
model y/n = entry / solution ddfm=contain;
random _residual_ / subject=intercept type=sp(exp)(lng lat);
run;
Note that the block effects have been removed from the statements. The keyword _RESIDUAL_
in the RANDOM statement instructs the GLIMMIX procedure to model the R matrix. Here, R is
to be modeled as an exponential covariance structure matrix. The SUBJECT=INTERCEPT option
means that all observations are considered correlated. Because the random effects are residual-type
(R-side) effects, there are no columns in the Z matrix for this model (Output 38.1.18).
Output 38.1.18 Model Dimension Information in Marginal Spatial Analysis
The GLIMMIX Procedure
Dimensions
R-side Cov. Parameters
Columns in X
Columns in Z per Subject
Subjects (Blocks in V)
Max Obs per Subject
2
17
0
1
64
In addition to the fixed effects, the GLIMMIX procedure now profiles one of the covariance parameters, the variance of the exponential covariance model (Output 38.1.19). This reduces the size of the
optimization problem. Only a single parameter is part of the optimization, the “range” (SP(EXP))
of the spatial process.
Output 38.1.19 Optimization Information in Spatial Analysis
Optimization Information
Optimization Technique
Parameters in Optimization
Lower Boundaries
Upper Boundaries
Fixed Effects
Residual Variance
Starting From
Dual Quasi-Newton
1
1
0
Profiled
Profiled
Data
The practical range of a spatial process is that distance at which the correlation between data points
has decreased to at most 0.05. The parameter reported by the GLIMMIX procedure as SP(EXP) in
Output 38.1.20 corresponds to one-third of the practical range. The practical range in this process
is 3 0:9052 D 2:7156. Correlations extend beyond a single experimental unit, but they do not
appear to exist on the scale of the block size.
2310 F Chapter 38: The GLIMMIX Procedure
Output 38.1.20 Estimates of Covariance Parameters
Covariance Parameter Estimates
Cov Parm
Subject
SP(EXP)
Residual
Intercept
Estimate
Standard
Error
0.9052
2.5315
0.4404
0.6974
The sill of the spatial process, the variance of the underlying residual effect, is estimated as 2.5315.
Output 38.1.21 Type III Test of Entry Effect in Spatial Analysis
Type III Tests of Fixed Effects
Effect
Num
DF
Den
DF
F Value
Pr > F
entry
15
48
3.60
0.0004
The F value for the entry effect has been sharply reduced compared to the previous analyses. The
smooth spatial variation accounts for some of the variation among the varieties (Output 38.1.21).
In this example three models were considered for the analysis of a randomized block design with
binomial outcomes. If data are correlated, a standard generalized linear model often will indicate
overdispersion relative to the binomial distribution. Two courses of action are considered in this
example to address this overdispersion. First, the inclusion of G-side random effects models the
correlation indirectly; it is induced through the sharing of random effects among responses from
the same block. Second, the R-side spatial covariance structure models covariation directly. In
generalized linear (mixed) models these two modeling approaches can lead to different inferences,
because the models have different interpretation. The random block effects are modeled on the
linked (logit) scale, and the spatial effects were modeled on the mean scale. Only in a linear mixed
model are the two scales identical.
Example 38.2: Mating Experiment with Crossed Random Effects
McCullagh and Nelder (1989, Ch. 14.5) describe a mating experiment—conducted by S. Arnold
and P. Verell at the University of Chicago, Department of Ecology and Evolution—involving two
geographically isolated populations of mountain dusky salamanders. One goal of the experiment
was to determine whether barriers to interbreeding have evolved in light of the geographical isolation of the populations. In this case, matings within a population should be more successful than
matings between the populations. The experiment conducted in the summer of 1986 involved 40
animals, 20 rough butt (R) and 20 whiteside (W) salamanders, with equal numbers of males and
females. The animals were grouped into two sets of R males, two sets of R females, two sets of
W males, and two sets of W females, so that each set comprised five salamanders. Each set was
Example 38.2: Mating Experiment with Crossed Random Effects F 2311
mated against one rough butt and one whiteside set, creating eight crossings. Within the pairings
of sets, each female was paired to three male animals. The salamander mating data have been used
by a number of authors; see, for example, McCullagh and Nelder (1989), Schall (1991), Karim and
Zeger (1992), Breslow and Clayton (1993), Wolfinger and O’Connell (1993), and Shun (1997).
The following DATA step creates the data set for the analysis.
data salamander;
input day fpop$
datalines;
4 rb 1 rb 1 1
4 rb 3 rb 2 1
4 rb 5 rb 3 1
4 rb 7 ws 8 0
4 rb 9 ws 10 0
4 ws 1 rb 9 0
4 ws 3 rb 8 0
4 ws 5 rb 6 0
4 ws 7 ws 4 1
4 ws 9 ws 3 1
8 rb 1 ws 4 1
8 rb 3 ws 1 0
8 rb 5 ws 3 1
8 rb 7 rb 8 0
8 rb 9 rb 7 0
8 ws 1 ws 9 1
8 ws 3 ws 7 0
8 ws 5 ws 8 1
8 ws 7 rb 1 1
8 ws 9 rb 3 1
12 rb 1 rb 5 1
12 rb 3 rb 1 1
12 rb 5 rb 4 1
12 rb 7 ws 9 0
12 rb 9 ws 8 1
12 ws 1 rb 7 1
12 ws 3 rb 6 0
12 ws 5 rb 10 0
12 ws 7 ws 5 1
12 ws 9 ws 1 1
16 rb 1 ws 1 0
16 rb 3 ws 4 1
16 rb 5 ws 2 1
16 rb 7 rb 9 1
16 rb 9 rb 6 1
16 ws 1 ws 10 1
16 ws 3 ws 9 0
16 ws 5 ws 6 0
16 ws 7 rb 2 0
16 ws 9 rb 1 1
20 rb 1 rb 4 1
20 rb 3 rb 3 1
20 rb 5 rb 2 1
fnum mpop$ mnum mating @@;
4
4
4
4
4
4
4
4
4
4
8
8
8
8
8
8
8
8
8
8
12
12
12
12
12
12
12
12
12
12
16
16
16
16
16
16
16
16
16
16
20
20
20
rb
rb
rb
rb
rb
ws
ws
ws
ws
ws
rb
rb
rb
rb
rb
ws
ws
ws
ws
ws
rb
rb
rb
rb
rb
ws
ws
ws
ws
ws
rb
rb
rb
rb
rb
ws
ws
ws
ws
ws
rb
rb
rb
2
4
6
8
10
2
4
6
8
10
2
4
6
8
10
2
4
6
8
10
2
4
6
8
10
2
4
6
8
10
2
4
6
8
10
2
4
6
8
10
2
4
6
rb
rb
ws
ws
ws
rb
rb
ws
ws
ws
ws
ws
rb
rb
rb
ws
ws
rb
rb
rb
rb
rb
ws
ws
ws
rb
rb
ws
ws
ws
ws
ws
rb
rb
rb
ws
ws
rb
rb
rb
rb
rb
ws
5
4
9
6
7
7
10
5
1
2
5
2
9
6
10
6
10
2
4
5
3
2
10
7
6
9
8
3
2
4
3
5
7
10
8
7
8
4
5
3
1
5
6
1
1
1
0
0
0
0
0
1
1
1
1
1
1
0
0
1
0
0
0
1
1
1
0
1
0
1
1
1
0
1
0
0
0
0
1
1
0
0
1
1
1
1
2312 F Chapter 38: The GLIMMIX Procedure
20
20
20
20
20
20
20
24
24
24
24
24
24
24
24
24
24
;
rb
rb
ws
ws
ws
ws
ws
rb
rb
rb
rb
rb
ws
ws
ws
ws
ws
7
9
1
3
5
7
9
1
3
5
7
9
1
3
5
7
9
ws 7 0
ws 9 1
rb 10 0
rb 7 0
rb 8 0
ws 1 1
ws 4 1
ws 5 1
ws 3 1
ws 1 1
rb 6 0
rb 10 1
ws 8 1
ws 6 1
ws 7 0
rb 5 1
rb 4 0
20
20
20
20
20
20
20
24
24
24
24
24
24
24
24
24
24
rb
rb
ws
ws
ws
ws
ws
rb
rb
rb
rb
rb
ws
ws
ws
ws
ws
8
10
2
4
6
8
10
2
4
6
8
10
2
4
6
8
10
ws 10 1
ws 8 1
rb 6 0
rb 9 0
ws 2 0
ws 5 1
ws 3 1
ws 2 1
ws 4 1
rb 8 1
rb 9 1
rb 7 0
ws 10 0
ws 9 1
rb 1 0
rb 3 0
rb 2 0
The first observation, for example, indicates that rough butt female 1 was paired in the laboratory
on day 4 of the experiment with rough butt male 1, and the pair mated. On the same day rough butt
female 7 was paired with whiteside male 8, but the pairing did not result in mating of the animals.
The model adopted by many authors for these data comprises fixed effects for gender and population, their interaction, and male and female random effects. Specifically, let RR , RW , WR , and
W W denote the mating probabilities between the populations, where the first subscript identifies
the female partner of the pair. Then, we model
kl
D kl C f C m k; l 2 fR; W g
log
1 kl
where f and m are independent random variables representing female and male random effects
(20 each), and kl denotes the average logit of mating between females of population k and males
of population l.
The following statements fit this model by pseudo-likelihood:
proc glimmix data=salamander;
class fpop fnum mpop mnum;
model mating(event=’1’) = fpop|mpop / dist=binary;
random fpop*fnum mpop*mnum;
lsmeans fpop*mpop / ilink;
run;
The response variable is the two-level variable mating. Because it is coded as zeros and ones,
and because PROC GLIMMIX models by default the probability of the first level according to the
response-level ordering, the EVENT=’1’ option instructs PROC GLIMMIX to model the probability of a successful mating. The distribution of the mating variable, conditional on the random
effects, is binary.
The fpop*fnum effect in the RANDOM statement creates a random intercept for each female animal.
Because fpop and fnum are CLASS variables, the effect has 20 levels (10 rb and 10 ws females).
Similarly, the mpop*mnum effect creates the random intercepts for the male animals. Because no
Example 38.2: Mating Experiment with Crossed Random Effects F 2313
TYPE= is specified in the RANDOM statement, the covariance structure defaults to TYPE=VC. The
random effects and their levels are independent, and each effect has its own variance component.
Because the conditional distribution of the data, conditioned on the random effects, is binary, no
extra scale parameter () is added.
The LSMEANS statement requests least squares means for the four levels of the fpop*mpop effect,
which are estimates of the cell means in the 2 2 classification of female and male populations. The
ILINK option in the LSMEANS statement requests that the estimated means and standard errors are
also reported on the scale of the data. This yields estimates of the four mating probabilities, RR ,
RW , WR , and W W .
The “Model Information” table displays general information about the model being fit
(Output 38.2.1).
Output 38.2.1 Analysis of Mating Experiment with Crossed Random Effects
The GLIMMIX Procedure
Model Information
Data Set
Response Variable
Response Distribution
Link Function
Variance Function
Variance Matrix
Estimation Technique
Degrees of Freedom Method
WORK.SALAMANDER
mating
Binary
Logit
Default
Not blocked
Residual PL
Containment
The response variable mating follows a binary distribution (conditional on the random effects).
Hence, the mean of the data is an event probability, , and the logit of this probability is linearly related to the linear predictor of the model. The variance function is the default function that
is implied by the distribution, a./ D .1 /. The variance matrix is not blocked, because
the GLIMMIX procedure does not process the data by subjects (see the section “Processing by
Subjects” on page 2245 for details). The estimation technique is the default method for GLMMs,
residual pseudo-likelihood (METHOD=RSPL), and degrees of freedom for tests and confidence
intervals are determined by the containment method.
The “Class Level Information” table in Output 38.2.2 lists the levels of the variables listed in the
CLASS statement, as well as the order of the levels.
Output 38.2.2 Class Level Information and Number of Observations
Class Level Information
Class
fpop
fnum
mpop
mnum
Levels
2
10
2
10
Values
rb ws
1 2 3 4 5 6 7 8 9 10
rb ws
1 2 3 4 5 6 7 8 9 10
2314 F Chapter 38: The GLIMMIX Procedure
Output 38.2.2 continued
Number of Observations Read
Number of Observations Used
120
120
Note that there are two female populations and two male populations; also, the variables fnum and
mnum have 10 levels each. As a consequence, the effects fpop*fnum and mpop*mnum identify the 20
females and males, respectively. The effect fpop*mpop identifies the four mating types.
The “Response Profile Table,” which is displayed for binary or multinomial data, lists the levels of
the response variable and their order (Output 38.2.3). With binary data, the table also provides information about which level of the response variable defines the event. Because of the EVENT=’1’
response variable option in the MODEL statement, the probability being modeled is that of the
higher-ordered value.
Output 38.2.3 Response Profiles
Response Profile
Ordered
Value
1
2
mating
0
1
Total
Frequency
50
70
The GLIMMIX procedure is modeling the probability that mating=’1’.
There are two covariance parameters in this model, the variance of the fpop*fnum effect and the variance of the mpop*mnum effect (Output 38.2.4). Both parameters are modeled as G-side parameters.
The nine columns in the X matrix comprise the intercept, two columns each for the levels of the
fpop and mpop effects, and four columns for their interaction. The Z matrix has 40 columns, one for
each animal. Because the data are not processed by subjects, PROC GLIMMIX assumes the data
consist of a single subject (a single block in V).
Output 38.2.4 Model Dimensions Information
Dimensions
G-side Cov. Parameters
Columns in X
Columns in Z
Subjects (Blocks in V)
Max Obs per Subject
2
9
40
1
120
The “Optimization Information” table displays basic information about the optimization
(Output 38.2.5). The default technique for GLMMs is the quasi-Newton method. There are
two parameters in the optimization, which correspond to the two variance components. The 17
fixed effects parameters are not part of the optimization. The initial optimization computes pseudo-
Example 38.2: Mating Experiment with Crossed Random Effects F 2315
data based on the response values in the data set rather than from estimates of a generalized linear
model fit.
Output 38.2.5 Optimization Information
Optimization Information
Optimization Technique
Parameters in Optimization
Lower Boundaries
Upper Boundaries
Fixed Effects
Starting From
Newton-Raphson with Ridging
2
2
0
Profiled
Data
The GLIMMIX procedure performs eight optimizations after the initial optimization
(Output 38.2.6). That is, following the initial pseudo-data creation, the pseudo-data were updated eight more times and a total of nine linear mixed models were estimated.
Output 38.2.6 Iteration History and Convergence Status
Iteration History
Iteration
Restarts
Subiterations
Objective
Function
Change
Max
Gradient
0
1
2
3
4
5
6
7
0
0
0
0
0
0
0
0
4
3
2
2
1
1
1
0
537.09173501
544.12516903
545.89139118
546.10489538
546.13075146
546.13374731
546.13409761
546.13413861
2.00000000
0.66319780
0.13539318
0.01742065
0.00212475
0.00025072
0.00002931
0.00000000
1.719E-8
1.14E-8
1.609E-6
5.89E-10
9.654E-7
1.346E-8
1.84E-10
4.285E-6
Convergence criterion (PCONV=1.11022E-8) satisfied.
The “Covariance Parameter Estimates” table lists the estimates for the two variance components
and their estimated standard errors (Output 38.2.7). The heterogeneity (in the logit of the mating
probabilities) among the females is considerably larger than the heterogeneity among the males.
Output 38.2.7 Estimated Covariance Parameters and Approximate Standard Errors
Covariance Parameter Estimates
Cov Parm
Estimate
Standard
Error
fpop*fnum
mpop*mnum
1.4099
0.08963
0.8871
0.4102
2316 F Chapter 38: The GLIMMIX Procedure
The “Type III Tests of Fixed Effects” table indicates a significant interaction between the male and
female populations (Output 38.2.8). A comparison in the logits of mating success in pairs with R
females and W females depends on whether the male partner in the pair is the same species. The
“fpop*mpop Least Squares Means” table shows this effect more clearly (Output 38.2.9).
Output 38.2.8 Tests of Main Effects and Interaction
Type III Tests of Fixed Effects
Effect
fpop
mpop
fpop*mpop
Num
DF
Den
DF
F Value
Pr > F
1
1
1
18
17
81
2.86
4.71
9.61
0.1081
0.0444
0.0027
Output 38.2.9 Interaction Least Squares Means
fpop*mpop Least Squares Means
fpop
mpop
rb
rb
ws
ws
rb
ws
rb
ws
Estimate
Standard
Error
DF
t Value
Pr > |t|
Mean
Standard
Error
Mean
1.1629
0.7839
-1.4119
1.0151
0.5961
0.5729
0.6143
0.5871
81
81
81
81
1.95
1.37
-2.30
1.73
0.0545
0.1750
0.0241
0.0876
0.7619
0.6865
0.1959
0.7340
0.1081
0.1233
0.09678
0.1146
In a pairing with a male rough butt salamander, the logit drops sharply from 1:1629 to 1:4119
when the male is paired with a whiteside female instead of a female from its own population. The
corresponding estimated probabilities of mating success are b
RR D 0:7619 and b
WR D 0:1959.
If the same comparisons are made in pairs with whiteside males, then you also notice a drop in
the logit if the female comes from a different population, 1:0151 versus 0:7839. The change is
considerably less, though, corresponding to mating probabilities of b
W W D 0:7340 and b
RW D
0:6865. Whiteside females appear to be successful with their own population. Whiteside males
appear to succeed equally well with female partners of the two populations.
This insight into the factor-level comparisons can be amplified by graphing the least squares mean
comparisons and by subsetting the differences of least squares means. This is accomplished with
the following statements:
ods graphics on;
ods select DiffPlot SliceDiffs;
proc glimmix data=salamander;
class fpop fnum mpop mnum;
model mating(event=’1’) = fpop|mpop / dist=binary;
random fpop*fnum mpop*mnum;
lsmeans fpop*mpop / plots=diffplot;
lsmeans fpop*mpop / slicediff=(mpop fpop);
run;
ods graphics off;
Example 38.2: Mating Experiment with Crossed Random Effects F 2317
The PLOTS=DIFFPLOT option in the first LSMEANS statement requests a comparison plot that
displays the result of all pairwise comparisons (Output 38.2.10). The SLICEDIFF=(mpop fpop)
option requests differences of simple effects.
The comparison plot in Output 38.2.10 is also known as a mean-mean scatter plot (Hsu 1996). Each
solid line in the plot corresponds to one of the possible 4 3=2 D 6 unique pairwise comparisons.
The line is centered at the intersection of two least squares means, and the length of the line segments
corresponds to the width of a 95% confidence interval for the difference between the two least
squares means. The length of the segment is adjusted for the rotation. If a line segment crosses the
dashed 45-degree line, the comparison between the two factor levels is not significant; otherwise, it
is significant. The horizontal and vertical axes of the plot are drawn in least squares means units,
and the grid lines are placed at the values of the least squares means.
The six pairs of least squares means comparisons separate into two sets of three pairs. Comparisons
in the first set are significant; comparisons in the second set are not significant. For the significant set, the female partner in one of the pairs is a whiteside salamander. For the nonsignificant
comparisons, the male partner in one of the pairs is a whiteside salamander.
Output 38.2.10 LS-Means Diffogram
The “Simple Effect Comparisons” tables show the results of the SLICEDIFF= option in the second
LSMEANS statement (Output 38.2.11).
2318 F Chapter 38: The GLIMMIX Procedure
Output 38.2.11 Simple Effect Comparisons
Simple Effect Comparisons of fpop*mpop Least Squares Means By mpop
Simple
Effect
Level
fpop
_fpop
mpop rb
mpop ws
rb
rb
ws
ws
Estimate
Standard
Error
DF
t Value
Pr > |t|
2.5748
-0.2312
0.8458
0.8092
81
81
3.04
-0.29
0.0031
0.7758
Simple Effect Comparisons of fpop*mpop Least Squares Means By fpop
Simple
Effect
Level
mpop
_mpop
fpop rb
fpop ws
rb
rb
ws
ws
Estimate
Standard
Error
DF
t Value
Pr > |t|
0.3790
-2.4270
0.6268
0.6793
81
81
0.60
-3.57
0.5471
0.0006
The first table of simple effect comparisons holds fixed the level of the mpop factor and compares the
levels of the fpop factor. Because there is only one possible comparison for each male population,
there are two entries in the table. The first entry compares the logits of mating probabilities when
the male partner is a rough butt, and the second entry applies when the male partner is from the
whiteside population. The second table of simple effects comparisons applies the same logic, but
holds fixed the level of the female partner in the pair. Note that these four comparisons are a subset
of all six possible comparisons, eliminating those where both factors are varied at the same time.
The simple effect comparisons show that there is no difference in mating probabilities if the male
partner is a whiteside salamander, or if the female partner is a rough butt. Rough butt females also
appear to mate indiscriminately.
Example 38.3: Smoothing Disease Rates; Standardized Mortality Ratios
Clayton and Kaldor (1987, Table 1) present data on observed and expected cases of lip cancer in the
56 counties of Scotland between 1975 and 1980. The expected number of cases was determined by a
separate multiplicative model that accounted for the age distribution in the counties. The goal of the
analysis is to estimate the county-specific log-relative risks, also known as standardized mortality
ratios (SMR).
If Yi is the number of incident cases in county i and Ei is the expected number of incident cases,
then the ratio of observed to expected counts, Yi =Ei , is the standardized mortality ratio. Clayton
and Kaldor (1987) assume there exists a relative risk i that is specific to each county and is a
random variable. Conditional on i , the observed counts are independent Poisson variables with
mean Ei i .
An elementary mixed model for i specifies only a random intercept for each county, in addition to a
fixed intercept. Breslow and Clayton (1993), in their analysis of these data, also provide a covariate
that measures the percentage of employees in agriculture, fishing, and forestry. The expanded model
Example 38.3: Smoothing Disease Rates; Standardized Mortality Ratios F 2319
for the region-specific relative risk in Breslow and Clayton (1993) is
i D exp fˇ0 C ˇ1 xi =10 C i g ;
i D 1; ; 56
where ˇ0 and ˇ1 are fixed effects, and the i are county random effects.
The following DATA step creates the data set lipcancer. The expected number of cases is based
on the observed standardized mortality ratio for counties with lip cancer cases, and based on the
expected counts reported by Clayton and Kaldor (1987, Table 1) for the counties without cases.
The sum of the expected counts then equals the sum of the observed counts.
data lipcancer;
input county observed expected employment SMR;
if (observed > 0) then expCount = 100*observed/SMR;
else expCount = expected;
datalines;
1 9 1.4 16 652.2
2 39 8.7 16 450.3
3 11 3.0 10 361.8
4 9 2.5 24 355.7
5 15 4.3 10 352.1
6 8 2.4 24 333.3
7 26 8.1 10 320.6
8 7 2.3 7 304.3
9 6 2.0 7 303.0
10 20 6.6 16 301.7
11 13 4.4 7 295.5
12 5 1.8 16 279.3
13 3 1.1 10 277.8
14 8 3.3 24 241.7
15 17 7.8 7 216.8
16 9 4.6 16 197.8
17 2 1.1 10 186.9
18 7 4.2 7 167.5
19 9 5.5 7 162.7
20 7 4.4 10 157.7
21 16 10.5 7 153.0
22 31 22.7 16 136.7
23 11 8.8 10 125.4
24 7 5.6 7 124.6
25 19 15.5 1 122.8
26 15 12.5 1 120.1
27 7 6.0 7 115.9
28 10 9.0 7 111.6
29 16 14.4 10 111.3
30 11 10.2 10 107.8
31 5 4.8 7 105.3
32 3 2.9 24 104.2
33 7 7.0 10 99.6
34 8 8.5 7 93.8
35 11 12.3 7 89.3
36 9 10.1 0 89.1
37 11 12.7 10 86.8
38 8 9.4 1 85.6
39 6 7.2 16 83.3
2320 F Chapter 38: The GLIMMIX Procedure
40 4 5.3 0
41 10 18.8 1
42 8 15.8 16
43 2 4.3 16
44 6 14.6 0
45 19 50.7 1
46 3 8.2 7
47 2 5.6 1
48 3 9.3 1
49 28 88.7 0
50 6 19.6 1
51 1 3.4 1
52 1 3.6 0
53 1 5.7 1
54 1 7.0 1
55 0 4.2 16
56 0 1.8 10
;
75.9
53.3
50.7
46.3
41.0
37.5
36.6
35.8
32.1
31.6
30.6
29.1
27.6
17.4
14.2
0.0
0.0
Because the mean of the Poisson variates, conditional on the random effects, is i D Ei i , applying
a log link yields
logfi g D logfEi g C ˇ0 C ˇ1 xi =10 C i
The term logfEi g is an offset, a regressor variable whose coefficient is known to be one. Note that
it is assumed that the Ei are known; they are not treated as random variables.
The following statements fit this model by residual pseudo-likelihood:
proc glimmix data=lipcancer;
class county;
x
= employment / 10;
logn = log(expCount);
model observed = x / dist=poisson offset=logn
solution ddfm=none;
random county;
SMR_pred = 100*exp(_zgamma_ + _xbeta_);
id employment SMR SMR_pred;
output out=glimmixout;
run;
The offset is created with the assignment statement
logn = log(expCount);
and is associated with the linear predictor through the OFFSET= option in the MODEL statement.
The statement
x = employment / 10;
transforms the covariate measuring percentage of employment in agriculture, fisheries, and forestry
to agree with the analysis of Breslow and Clayton (1993). The DDFM=NONE option in the
MODEL statement requests chi-square tests and z tests instead of the default F tests and t tests
by setting the denominator degrees of freedom in tests of fixed effects to 1.
Example 38.3: Smoothing Disease Rates; Standardized Mortality Ratios F 2321
The statement
SMR_pred = 100*exp(_zgamma_ + _xbeta_);
calculates the fitted standardized mortality rate. Note that the offset variable does not contribute to
the exponentiated term.
The OUTPUT statement saves results of the calculations to the output data set glimmixout. The ID
statement specifies that only the listed variables are written to the output data set.
Output 38.3.1 Model Information in Poisson GLMM
The GLIMMIX Procedure
Model Information
Data Set
Response Variable
Response Distribution
Link Function
Variance Function
Offset Variable
Variance Matrix
Estimation Technique
Degrees of Freedom Method
WORK.LIPCANCER
observed
Poisson
Log
Default
logn = log(expCount);
Not blocked
Residual PL
None
Class Level Information
Class
county
Levels
56
Values
1 2 3
19 20
34 35
49 50
4 5 6
21 22
36 37
51 52
7 8 9
23 24
38 39
53 54
10
25
40
55
11 12 13 14 15 16 17 18
26 27 28 29 30 31 32 33
41 42 43 44 45 46 47 48
56
Number of Observations Read
Number of Observations Used
56
56
Dimensions
G-side Cov. Parameters
Columns in X
Columns in Z
Subjects (Blocks in V)
Max Obs per Subject
1
2
56
1
56
The GLIMMIX procedure displays in the “Model Information” table that the offset variable was
computed with programming statements and the final assignment statement from your GLIMMIX
statements (Output 38.3.1). There are two columns in the X matrix, corresponding to the intercept
and the regressor x=10. There are 56 columns in the Z matrix, however, one for each observation
in the data set (Output 38.3.1).
The optimization involves only a single covariance parameter, the variance of the county effect
(Output 38.3.2). Because this parameter is a variance, the GLIMMIX procedure imposes a lower
2322 F Chapter 38: The GLIMMIX Procedure
boundary constraint; the solution for the variance is bounded by zero from below.
Output 38.3.2 Optimization Information in Poisson GLMM
Optimization Information
Optimization Technique
Parameters in Optimization
Lower Boundaries
Upper Boundaries
Fixed Effects
Starting From
Dual Quasi-Newton
1
1
0
Profiled
Data
Following the initial creation of pseudo-data and the fit of a linear mixed model, the procedure
goes through five more updates of the pseudo-data, each associated with a separate optimization
(Output 38.3.3). Although the objective function in each optimization is the negative of twice the
restricted maximum likelihood for that pseudo-data, there is no guarantee that across the outer
iterations the objective function decreases in subsequent optimizations. In this example, minus
twice the residual maximum likelihood at convergence takes on its smallest value at the initial
optimization and increases in subsequent optimizations.
Output 38.3.3 Iteration History in Poisson GLMM
Iteration History
Iteration
Restarts
Subiterations
Objective
Function
Change
Max
Gradient
0
1
2
3
4
5
0
0
0
0
0
0
4
3
2
1
1
0
123.64113992
127.05866018
127.48839749
127.50502469
127.50528068
127.50528481
0.20997891
0.03393332
0.00223427
0.00006946
0.00000118
0.00000000
3.848E-8
0.000048
5.753E-6
1.938E-7
1.09E-7
1.299E-6
Convergence criterion (PCONV=1.11022E-8) satisfied.
The “Covariance Parameter Estimates” table in Output 38.3.4 shows the estimate of the variance of
the region-specific log-relative risks. There is significant county-to-county heterogeneity in risks. If
the covariate were removed from the analysis, as in Clayton and Kaldor (1987), the heterogeneity
in county-specific risks would increase. (The fitted SMRs in Table 6 of Breslow and Clayton (1993)
were obtained without the covariate x in the model.)
Example 38.3: Smoothing Disease Rates; Standardized Mortality Ratios F 2323
Output 38.3.4 Estimated Covariance Parameters in Poisson GLMM
Covariance Parameter Estimates
Cov
Parm
county
Estimate
Standard
Error
0.3567
0.09869
The “Solutions for Fixed Effects” table displays the estimates of ˇ0 and ˇ1 along with their standard
errors and test statistics (Output 38.3.5). Because of the DDFM=NONE option in the MODEL
statement, PROC GLIMMIX assumes that the degrees of freedom for the t tests of H0 W ˇj D 0
are infinite. The p-values correspond to probabilities under a standard normal distribution. The
covariate measuring employment percentages in agriculture, fisheries, and forestry is significant.
This covariate might be a surrogate for the exposure to sunlight, an important risk factor for lip
cancer.
Output 38.3.5 Fixed-Effects Parameter Estimates in Poisson GLMM
Solutions for Fixed Effects
Effect
Intercept
x
Estimate
Standard
Error
DF
t Value
Pr > |t|
-0.4406
0.6799
0.1572
0.1409
Infty
Infty
-2.80
4.82
0.0051
<.0001
You can examine the quality of the fit of this model with various residual plots. A panel of studentized residuals is requested with the following statements:
ods graphics on;
ods select StudentPanel;
proc glimmix data=lipcancer plots=studentpanel;
class county;
x
= employment / 10;
logn = log(expCount);
model observed = x / dist=poisson offset=logn s ddfm=none;
random county;
run;
ods graphics off;
The graph in the upper-left corner of the panel displays studentized residuals plotted against
the linear predictor (Output 38.3.6). The default of the GLIMMIX procedure is to use
the estimated BLUPs in the construction of the residuals and to present them on the linear
scale, which in this case is the logarithmic scale. You can change the type of the computed residual with the TYPE= suboptions of each paneled display. For example, the option
PLOTS=STUDENTPANEL(TYPE=NOBLUP) would request a paneled display of the marginal
residuals on the linear scale.
2324 F Chapter 38: The GLIMMIX Procedure
Output 38.3.6 Panel of Studentized Residuals
The graph in the upper-right corner of the panel shows a histogram with overlaid normal density. A
Q-Q plot and a box plot are shown in the lower cells of the panel.
The following statements produce a graph of the observed and predicted standardized mortality
ratios (Output 38.3.7):
proc template;
define statgraph scatter;
BeginGraph;
layout overlayequated / yaxisopts=(label=’Predicted SMR’)
xaxisopts=(label=’Observed SMR’)
equatetype=square;
lineparm y=0 slope=1 x=0 /
lineattrs = GraphFit(pattern=dash)
extend
= true;
scatterplot y=SMR_pred x=SMR /
markercharacter = employment;
endlayout;
EndGraph;
end;
run;
proc sgrender data=glimmixout template=scatter;
run;
Example 38.3: Smoothing Disease Rates; Standardized Mortality Ratios F 2325
In Output 38.3.7, fitted SMRs tend to be larger than the observed SMRs for counties with small
observed SMR and smaller than the observed SMRs for counties with high observed SMR.
Output 38.3.7 Observed and Predicted SMRs; Data Labels Indicate Covariate Values
To demonstrate the impact of the random effects adjustment to the log-relative risks, the following
statements fit a Poisson regression model (a GLM) by maximum likelihood:
proc glimmix data=lipcancer;
x
= employment / 10;
logn = log(expCount);
model observed = x / dist=poisson offset=logn
solution ddfm=none;
SMR_pred = 100*exp(_zgamma_ + _xbeta_);
id employment SMR SMR_pred;
output out=glimmixout;
run;
The GLIMMIX procedure defaults to maximum likelihood estimation because these statements fit
a generalized linear model with nonnormal distribution. As a consequence, the SMRs are county
specific only to the extent that the risks vary with the value of the covariate. But risks are no longer
adjusted based on county-to-county heterogeneity in the observed incidence count.
2326 F Chapter 38: The GLIMMIX Procedure
Because of the absence of random effects, the GLIMMIX procedure recognizes the model as a
generalized linear model and fits it by maximum likelihood (Output 38.3.8). The variance matrix is
diagonal because the observations are uncorrelated.
Output 38.3.8 Model Information in Poisson GLM
The GLIMMIX Procedure
Model Information
Data Set
Response Variable
Response Distribution
Link Function
Variance Function
Offset Variable
Variance Matrix
Estimation Technique
Degrees of Freedom Method
WORK.LIPCANCER
observed
Poisson
Log
Default
logn = log(expCount);
Diagonal
Maximum Likelihood
None
The “Dimensions” table shows that there are no G-side random effects in this model and no R-side
scale parameter either (Output 38.3.9).
Output 38.3.9 Model Dimensions Information in Poisson GLM
Dimensions
Columns in X
Columns in Z
Subjects (Blocks in V)
Max Obs per Subject
2
0
1
56
Because this is a GLM, the GLIMMIX procedure defaults to the Newton-Raphson algorithm, and
the fixed effects (intercept and slope) comprise the parameters in the optimization (Output 38.3.10).
(The default optimization technique for a GLM is the Newton-Raphson method.)
Output 38.3.10 Optimization Information in Poisson GLM
Optimization Information
Optimization Technique
Parameters in Optimization
Lower Boundaries
Upper Boundaries
Fixed Effects
Newton-Raphson
2
0
0
Not Profiled
The estimates of ˇ0 and ˇ1 have changed from the previous analysis. In the GLMM, the estimates
were b
ˇ 0 D 0:4406 and b
ˇ 1 D 0:6799 (Output 38.3.11).
Example 38.3: Smoothing Disease Rates; Standardized Mortality Ratios F 2327
Output 38.3.11 Parameter Estimates in Poisson GLM
Parameter Estimates
Effect
Intercept
x
Estimate
Standard
Error
DF
t Value
Pr > |t|
-0.5419
0.7374
0.06951
0.05954
Infty
Infty
-7.80
12.38
<.0001
<.0001
More importantly, without the county-specific adjustments through the best linear unbiased predictors of the random effects, the predicted SMRs are the same for all counties with the same
percentage of employees in agriculture, fisheries, and forestry (Output 38.3.12).
Output 38.3.12 Observed and Predicted SMRs in Poisson GLM
2328 F Chapter 38: The GLIMMIX Procedure
Example 38.4: Quasi-likelihood Estimation for Proportions with
Unknown Distribution
Wedderburn (1974) analyzes data on the incidence of leaf blotch (Rhynchosporium secalis) on barley. The data represent the percentage of leaf area affected in a two-way layout with 10 barley
varieties at nine sites. The following DATA step converts these data to proportions, as analyzed
in McCullagh and Nelder (1989, Ch. 9.2.4). The purpose of the analysis is to make comparisons
among the varieties, adjusted for site effects.
data blotch;
array p{9} pct1-pct9;
input variety pct1-pct9;
do site = 1 to 9;
prop = p{site}/100;
output;
end;
drop pct1-pct9;
datalines;
1 0.05 0.00 1.25 2.50 5.50 1.00 5.00 5.00 17.50
2 0.00 0.05 1.25 0.50 1.00 5.00 0.10 10.00 25.00
3 0.00 0.05 2.50 0.01 6.00 5.00 5.00 5.00 42.50
4 0.10 0.30 16.60 3.00 1.10 5.00 5.00 5.00 50.00
5 0.25 0.75 2.50 2.50 2.50 5.00 50.00 25.00 37.50
6 0.05 0.30 2.50 0.01 8.00 5.00 10.00 75.00 95.00
7 0.50 3.00 0.00 25.00 16.50 10.00 50.00 50.00 62.50
8 1.30 7.50 20.00 55.00 29.50 5.00 25.00 75.00 95.00
9 1.50 1.00 37.50 5.00 20.00 50.00 50.00 75.00 95.00
10 1.50 12.70 26.25 40.00 43.50 75.00 75.00 75.00 95.00
;
Little is known about the distribution of the leaf area proportions. The outcomes are not binomial
proportions, because they do not represent the ratio of a count over a total number of Bernoulli trials.
However, because the mean proportion ij for variety j on site i must lie in the interval Œ0; 1, you
can commence the analysis with a model that treats Prop as a “pseudo-binomial” variable:
EŒPropij  D ij
ij D 1=.1 C expf ij g/
ij D ˇ0 C ˛i C j
VarŒPropij  D ij .1
ij /
Here, ij is the linear predictor for variety j on site i , ˛i denotes the i th site effect, and j denotes
the j th barley variety effect. The logit of the expected leaf area proportions is linearly related to
these effects. The variance funcion of the model is that of a binomial(n,ij ) variable, and is an
overdispersion parameter. The moniker “pseudo-binomial” derives not from the pseudo-likelihood
methods used to estimate the parameters in the model, but from treating the response variable as if
it had first and second moment properties akin to a binomial random variable.
The model is fit in the GLIMMIX procedure with the following statements:
Example 38.4: Quasi-likelihood Estimation for Proportions with Unknown Distribution F 2329
proc glimmix data=blotch;
class site variety;
model prop = site variety / link=logit dist=binomial;
random _residual_;
lsmeans variety / diff=control(’1’);
run;
The MODEL statement specifies the distribution as binomial and the logit link. Because the variance function of the binomial distribution is a./ D .1 /, you use the statement
random _residual_;
to specify the scale parameter . The LSMEANS statement requests estimates of the least squares
means for the barley variety. The DIFF=CONTROL(’1’) option requests tests of least squares
means differences against the first variety.
The “Model Information” table in Output 38.4.1 describes the model and methods used in fitting
the statistical model. It is assumed here that the data are binomial proportions.
Output 38.4.1 Model Information in Pseudo-binomial Analysis
The GLIMMIX Procedure
Model Information
Data Set
Response Variable
Response Distribution
Link Function
Variance Function
Variance Matrix
Estimation Technique
Degrees of Freedom Method
WORK.BLOTCH
prop
Binomial
Logit
Default
Diagonal
Maximum Likelihood
Residual
The “Class Level Information” table in Output 38.4.2 lists the number of levels of the Site and
Variety effects and their values. All 90 observations read from the data are used in the analysis.
Output 38.4.2 Class Levels and Number of Observations
Class Level Information
Class
site
variety
Levels
9
10
Values
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9 10
Number of Observations Read
Number of Observations Used
90
90
2330 F Chapter 38: The GLIMMIX Procedure
In Output 38.4.3, the “Dimensions” table shows that the model does not contain G-side random effects. There is a single covariance parameter, which corresponds to . The “Optimization Information” table shows that the optimization comprises 18 parameters (Output 38.4.3). These correspond
to the 18 nonsingular columns of the X0 X matrix.
Output 38.4.3 Model Fit in Pseudo-binomial Analysis
Dimensions
Covariance Parameters
Columns in X
Columns in Z
Subjects (Blocks in V)
Max Obs per Subject
1
20
0
1
90
Optimization Information
Optimization Technique
Parameters in Optimization
Lower Boundaries
Upper Boundaries
Fixed Effects
Newton-Raphson
18
0
0
Not Profiled
Fit Statistics
-2 Log Likelihood
AIC (smaller is better)
AICC (smaller is better)
BIC (smaller is better)
CAIC (smaller is better)
HQIC (smaller is better)
Pearson Chi-Square
Pearson Chi-Square / DF
57.15
93.15
102.79
138.15
156.15
111.30
6.39
0.09
There are significant site and variety effects in this model based on the approximate Type III F tests
(Output 38.4.4).
Output 38.4.4 Tests of Site and Variety Effects in Pseudo-binomial Analysis
Type III Tests of Fixed Effects
Effect
site
variety
Num
DF
Den
DF
F Value
Pr > F
8
9
72
72
18.25
13.85
<.0001
<.0001
Output 38.4.5 displays the Variety least squares means for this analysis. These are obtained by
averaging
logit.b
ij / D b
ij
Example 38.4: Quasi-likelihood Estimation for Proportions with Unknown Distribution F 2331
across the sites. In other words, LS-means are computed on the linked scale where the model effects
are additive. Note that the least squares means are ordered by variety. The estimate of the expected
proportion of infected leaf area for the first variety is
b
:;1 D
1
D 0:0124
1 C expf4:38g
and that for the last variety is
b
:;10 D
1
D 0:468
1 C expf0:127g
Output 38.4.5 Variety Least Squares Means in Pseudo-binomial Analysis
variety Least Squares Means
variety
1
2
3
4
5
6
7
8
9
10
Estimate
Standard
Error
DF
t Value
Pr > |t|
-4.3800
-4.2300
-3.6906
-3.3319
-2.7653
-2.0089
-1.8095
-1.0380
-0.8800
-0.1270
0.5643
0.5383
0.4623
0.4239
0.3768
0.3320
0.3228
0.2960
0.2921
0.2808
72
72
72
72
72
72
72
72
72
72
-7.76
-7.86
-7.98
-7.86
-7.34
-6.05
-5.61
-3.51
-3.01
-0.45
<.0001
<.0001
<.0001
<.0001
<.0001
<.0001
<.0001
0.0008
0.0036
0.6523
Because of the ordering of the least squares means, the differences against the first variety are also
ordered from smallest to largest (Output 38.4.6).
Output 38.4.6 Variety Differences against the First Variety
Differences of variety Least Squares Means
variety
2
3
4
5
6
7
8
9
10
_variety
1
1
1
1
1
1
1
1
1
Estimate
Standard
Error
DF
t Value
Pr > |t|
0.1501
0.6895
1.0482
1.6147
2.3712
2.5705
3.3420
3.5000
4.2530
0.7237
0.6724
0.6494
0.6257
0.6090
0.6065
0.6015
0.6013
0.6042
72
72
72
72
72
72
72
72
72
0.21
1.03
1.61
2.58
3.89
4.24
5.56
5.82
7.04
0.8363
0.3086
0.1109
0.0119
0.0002
<.0001
<.0001
<.0001
<.0001
2332 F Chapter 38: The GLIMMIX Procedure
This analysis depends on your choice for the variance function that was implied by the binomial
distribution. You can diagnose the distributional assumption by examining various graphical diagnotics measures. The following statements request a panel display of the Pearson-type residuals:
ods graphics on;
ods select PearsonPanel;
proc glimmix data=blotch plots=pearsonpanel;
class site variety;
model prop = site variety / link=logit dist=binomial;
random _residual_;
run;
ods graphics off;
Output 38.4.7 clearly indicates that the chosen variance function is not appropriate for these data.
As approaches zero or one, the variability in the residuals is less than that implied by the binomial
variance function.
Output 38.4.7 Panel of Pearson-Type Residuals in Pseudo-binomial Analysis
To remedy this situation, McCullagh and Nelder (1989) consider instead the variance function
VarŒPropij  D 2ij .1
ij /2
Example 38.4: Quasi-likelihood Estimation for Proportions with Unknown Distribution F 2333
Imagine two varieties with :i D 0:1 and :k D 0:5. Under the binomial variance function, the
variance of the proportion for variety k is 2.77 times larger than that for variety i . Under the revised
model this ratio increases to 2:772 D 7:67.
The analysis of the revised model is obtained with the next set of GLIMMIX statements. Because
you need to model a variance function that does not correspond to any of the built-in distributions,
you need to supply a function with an assignment to the automatic variable _VARIANCE_. The
GLIMMIX procedure then considers the distribution of the data as unknown. The corresponding
estimation technique is quasi-likelihood. Because this model does not include an extra scale parameter, you can drop the RANDOM _RESIDUAL_ statement from the analysis.
ods graphics on;
ods select ModelInfo FitStatistics LSMeans Diffs PearsonPanel;
proc glimmix data=blotch plots=pearsonpanel;
class site variety;
_variance_ = _mu_**2 * (1-_mu_)**2;
model prop = site variety / link=logit;
lsmeans variety / diff=control(’1’);
run;
ods graphics off;
The “Model Information” table in Output 38.4.8 now displays the distribution as “Unknown,” because of the assignment made in the GLIMMIX statements to _VARIANCE_. The table also shows
the expression evaluated as the variance function.
Output 38.4.8 Model Information in Quasi-likelihood Analysis
The GLIMMIX Procedure
Model Information
Data Set
Response Variable
Response Distribution
Link Function
Variance Function
Variance Matrix
Estimation Technique
Degrees of Freedom Method
WORK.BLOTCH
prop
Unknown
Logit
_mu_**2 * (1-_mu_)**2
Diagonal
Quasi-Likelihood
Residual
The fit statistics of the model are now expressed in terms of the log quasi-likelihood. It is computed
as
9 X
10 Z
X
ij
i D1 j D1 yij
yij
2
t .1
t
dt
t /2
Twice the negative of this sum equals
(Output 38.4.9).
85:74, which is displayed in the “Fit Statistics” table
The scaled Pearson statistic is now 0.99. Inclusion of an extra scale parameter would have little
or no effect on the results.
2334 F Chapter 38: The GLIMMIX Procedure
Output 38.4.9 Fit Statistics in Quasi-likelihood Analysis
Fit Statistics
-2 Log Quasi-Likelihood
Quasi-AIC (smaller is better)
Quasi-AICC (smaller is better)
Quasi-BIC (smaller is better)
Quasi-CAIC (smaller is better)
Quasi-HQIC (smaller is better)
Pearson Chi-Square
Pearson Chi-Square / DF
-85.74
-49.74
-40.11
-4.75
13.25
-31.60
71.17
0.99
The panel of Pearson-type residuals now shows a much more adequate distribution for the residuals
and a reduction in the number of outlying residuals (Output 38.4.10).
Output 38.4.10 Panel of Pearson-Type Residuals (Quasi-likelihood)
The least squares means are no longer ordered in size by variety (Output 38.4.11). For example,
logit.b
:1 / > logit.b
:2 /. Under the revised model, the second variety has a greater percentage of its
leaf area covered by blotch, compared to the first variety. Varieties 5 and 6 and varieties 8 and 9
show similar reversal in ranking.
Example 38.4: Quasi-likelihood Estimation for Proportions with Unknown Distribution F 2335
Output 38.4.11 Variety Least Squares Means in Quasi-likelihood Analysis
variety Least Squares Means
variety
1
2
3
4
5
6
7
8
9
10
Estimate
Standard
Error
DF
t Value
Pr > |t|
-4.0453
-4.5126
-3.9664
-3.0912
-2.6927
-2.7167
-1.7052
-0.7827
-0.9098
-0.1580
0.3333
0.3333
0.3333
0.3333
0.3333
0.3333
0.3333
0.3333
0.3333
0.3333
72
72
72
72
72
72
72
72
72
72
-12.14
-13.54
-11.90
-9.27
-8.08
-8.15
-5.12
-2.35
-2.73
-0.47
<.0001
<.0001
<.0001
<.0001
<.0001
<.0001
<.0001
0.0216
0.0080
0.6369
Interestingly, the standard errors are constant among the LS-means (Output 38.4.11) and among the
LS-means differences (Output 38.4.12). This is due to the fact that for the logit link
@
D .1
@
/
which cancels with the square root of the variance function in the estimating equations. The analysis
is thus orthogonal.
Output 38.4.12 Variety Differences in Quasi-likelihood Analysis
Differences of variety Least Squares Means
variety
2
3
4
5
6
7
8
9
10
_variety
1
1
1
1
1
1
1
1
1
Estimate
Standard
Error
DF
t Value
Pr > |t|
-0.4673
0.07885
0.9541
1.3526
1.3286
2.3401
3.2626
3.1355
3.8873
0.4714
0.4714
0.4714
0.4714
0.4714
0.4714
0.4714
0.4714
0.4714
72
72
72
72
72
72
72
72
72
-0.99
0.17
2.02
2.87
2.82
4.96
6.92
6.65
8.25
0.3249
0.8676
0.0467
0.0054
0.0062
<.0001
<.0001
<.0001
<.0001
2336 F Chapter 38: The GLIMMIX Procedure
Example 38.5: Joint Modeling of Binary and Count Data
Clustered data arise when multiple observations are collected on the same sampling or experimental
unit. Often, these multiple observations refer to the same attribute measured at different points in
time or space. This leads to repeated measures, longitudinal, and spatial data, which are special
forms of multivariate data. A different class of multivariate data arises when the multiple observations refer to different attributes.
The data set hernio, created in the following DATA step, provides an example of a bivariate outcome
variable. It reflects the condition and length of hospital stay for 32 herniorrhaphy patients. These
data are based on data given by Mosteller and Tukey (1977) and reproduced in Hand et al. (1994,
pp. 390, 391). The data set that follows does not contain all the covariates given in these sources.
The response variables are leave and los; these denote the condition of the patient upon leaving the
operating room and the length of hospital stay after the operation (in days). The variable leave takes
on the value one if a patient experiences a routine recovery, and the value zero if postoperative
intensive care was required. The binary variable OKstatus distinguishes patients based on their
postoperative physical status (“1” implies better status).
data hernio;
input patient age gender$ OKstatus leave los;
datalines;
1
78 m
1
0
9
2
60 m
1
0
4
3
68 m
1
1
7
4
62 m
0
1 35
5
76 m
0
0
9
6
76 m
1
1
7
7
64 m
1
1
5
8
74 f
1
1 16
9
68 m
0
1
7
10
79 f
1
0 11
11
80 f
0
1
4
12
48 m
1
1
9
13
35 f
1
1
2
14
58 m
1
1
4
15
40 m
1
1
3
16
19 m
1
1
4
17
79 m
0
0
3
18
51 m
1
1
5
19
57 m
1
1
8
20
51 m
0
1
8
21
48 m
1
1
3
22
48 m
1
1
5
23
66 m
1
1
8
24
71 m
1
0
2
25
75 f
0
0
7
26
2 f
1
1
0
27
65 f
1
0 16
28
42 f
1
0
3
29
54 m
1
0
2
Example 38.5: Joint Modeling of Binary and Count Data F 2337
30
31
32
;
43
4
52
m
m
m
1
1
1
1
1
1
3
3
8
While the response variable los is a Poisson count variable, the response variable leave is a binary
variable. You can perform separate analysis for the two outcomes, for example, by fitting a logistic model for the operating room exit condition and a Poisson regression model for the length of
hospital stay. This, however, would ignore the correlation between the two outcomes. Intuitively,
you would expect that the length of postoperative hospital stay is longer for those patients who had
more tenuous exit conditions.
The following DATA step converts the data set hernio from the multivariate form to the univariate
form. In the multivariate form the responses are stored in separate variables. The GLIMMIX
procedure requires the univariate data structure.
data hernio_uv;
length dist $7;
set hernio;
response = (leave=1);
dist
= "Binary";
output;
response = los;
dist
= "Poisson";
output;
keep patient age OKstatus response dist;
run;
This DATA step expands the 32 observations in the data set hernio into 64 observations, stacking
two observations per patient. The character variable dist identifies the distribution that is assumed
for the respective observations within a patient. The first observation for each patient corresponds
to the binary response.
The following GLIMMIX statements fit a logistic regression model with two regressors (age and
OKStatus) to the binary observations:
proc glimmix data=hernio_uv(where=(dist="Binary"));
model response(event=’1’) = age OKStatus / s dist=binary;
run;
The EVENT=(’1’) response option requests that PROC GLIMMIX model the probability
Pr.leave D 1/—that is, the probability of routine recovery. The fit statistics and parameter estimates for this univariate analysis are shown in Output 38.5.1. The coefficient for the age effect
is negative ( 0:07725) and marginally significant at the 5% level (p D 0:0491). The negative
sign indicates that the probability of routine recovery decreases with age. The coefficient for the
OKStatus variable is also negative. Its large standard error and the p-value of 0:7341 indicate,
however, that this regressor is not significant.
2338 F Chapter 38: The GLIMMIX Procedure
Output 38.5.1 Univariate Logistic Regression
The GLIMMIX Procedure
Fit Statistics
-2 Log Likelihood
AIC (smaller is better)
AICC (smaller is better)
BIC (smaller is better)
CAIC (smaller is better)
HQIC (smaller is better)
Pearson Chi-Square
Pearson Chi-Square / DF
32.77
38.77
39.63
43.17
46.17
40.23
30.37
1.05
Parameter Estimates
Effect
Estimate
Standard
Error
DF
t Value
Pr > |t|
Intercept
age
OKstatus
5.7694
-0.07725
-0.3516
2.8245
0.03761
1.0253
29
29
29
2.04
-2.05
-0.34
0.0503
0.0491
0.7341
Based on the univariate logistic regression analysis, you would probably want to revisit the model,
examine other regressor variables, test for gender effects and interactions, and so forth. For this
example, we are content with the two-regressor model. It will be illustrative to trace the relative
importance of the two regressors through various types of models.
The next statements fit the same regressors to the count data:
proc glimmix data=hernio_uv(where=(dist="Poisson"));
model response = age OKStatus / s dist=Poisson;
run;
For this response, both regressors appear to make significant contributions at the 5% significance
level (Output 38.5.2). The sign of the coefficient seems appropriate; the length of hospital stay
should increase with patient age and be shorter for patients with better preoperative health. The magnitude of the scaled Pearson statistic (4:48) indicates, however, that there is considerable overdispersion in this model. This could be due to omitted variables or an improper distributional assumption.
The importance of preoperative health status, for example, can change with a patient’s age, which
could call for an interaction term.
Example 38.5: Joint Modeling of Binary and Count Data F 2339
Output 38.5.2 Univariate Poisson Regression
The GLIMMIX Procedure
Fit Statistics
-2 Log Likelihood
AIC (smaller is better)
AICC (smaller is better)
BIC (smaller is better)
CAIC (smaller is better)
HQIC (smaller is better)
Pearson Chi-Square
Pearson Chi-Square / DF
215.52
221.52
222.38
225.92
228.92
222.98
129.98
4.48
Parameter Estimates
Effect
Intercept
age
OKstatus
Estimate
Standard
Error
DF
t Value
Pr > |t|
1.2640
0.01525
-0.3301
0.3393
0.004454
0.1562
29
29
29
3.72
3.42
-2.11
0.0008
0.0019
0.0433
You can also model both responses jointly. The following statements request a multivariate analysis:
proc glimmix data=hernio_uv;
class dist;
model response(event=’1’) = dist dist*age dist*OKstatus /
noint s dist=byobs(dist);
run;
The DIST=BYOBS option in the MODEL statement instructs the GLIMMIX procedure to examine
the variable dist in order to identify the distribution of an observation. The variable can be character
or numeric. See the DIST= option of the MODEL statement for a list of the numeric codes for the
various distributions that are compatible with the DIST=BYOBS formulation. Because no LINK=
option is specified, the link functions are chosen as the default links that correspond to the respective
distributions. In this case, the logit link is applied to the binary observations and the log link is
applied to the Poisson outcomes. The dist variable is also listed in the CLASS statement, which
enables you to use interaction terms in the MODEL statement to vary the regression coefficients by
response distribution. The NOINT option is used here so that the parameter estimates of the joint
model are directly comparable to those in Output 38.5.1 and Output 38.5.2.
The “Fit Statistics” and “Parameter Estimates” tables of this bivariate estimation process are shown
in Output 38.5.3.
2340 F Chapter 38: The GLIMMIX Procedure
Output 38.5.3 Bivariate Analysis – Independence
The GLIMMIX Procedure
Fit Statistics
Description
-2 Log Likelihood
AIC (smaller is better)
AICC (smaller is better)
BIC (smaller is better)
CAIC (smaller is better)
HQIC (smaller is better)
Pearson Chi-Square
Pearson Chi-Square / DF
Binary
Poisson
Total
32.77
44.77
48.13
53.56
59.56
47.68
30.37
1.05
215.52
227.52
230.88
236.32
242.32
230.44
129.98
4.48
248.29
260.29
261.77
273.25
279.25
265.40
160.35
2.76
Parameter Estimates
Effect
dist
Estimate
Standard
Error
DF
t Value
Pr > |t|
dist
dist
age*dist
age*dist
OKstatus*dist
OKstatus*dist
Binary
Poisson
Binary
Poisson
Binary
Poisson
5.7694
1.2640
-0.07725
0.01525
-0.3516
-0.3301
2.8245
0.3393
0.03761
0.004454
1.0253
0.1562
58
58
58
58
58
58
2.04
3.72
-2.05
3.42
-0.34
-2.11
0.0456
0.0004
0.0445
0.0011
0.7329
0.0389
The “Fit Statistics” table now contains a separate column for each response distribution, as well as
an overall contribution. Because the model does not specify any random effects or R-side correlations, the log likelihoods are additive. The parameter estimates and their standard errors in this joint
model are identical to those in Output 38.5.1 and Output 38.5.2. The p-values reflect the larger
“sample size” in the joint analysis. Note that the coefficients would be different from the separate
analyses if the dist variable had not been used to form interactions with the model effects.
There are two ways in which the correlations between the two responses for the same patient can
be incorporated. You can induce them through shared random effects or model the dependency
directly. The following statements fit a model that induces correlation:
proc glimmix data=hernio_uv;
class patient dist;
model response(event=’1’) = dist dist*age dist*OKstatus /
noint s dist=byobs(dist);
random int / subject=patient;
run;
Notice that the patient variable has been added to the CLASS statement and as the SUBJECT= effect
in the RANDOM statement.
The “Fit Statistics” table in Output 38.5.4 no longer has separate columns for each response distribution, because the data are not independent. The log (pseudo-)likelihood does not factor into
Example 38.5: Joint Modeling of Binary and Count Data F 2341
additive component that correspond to distributions. Instead, it factors into components associated
with subjects.
Output 38.5.4 Bivariate Analysis – Mixed Model
The GLIMMIX Procedure
Fit Statistics
-2 Res Log Pseudo-Likelihood
Generalized Chi-Square
Gener. Chi-Square / DF
226.71
52.25
0.90
Covariance Parameter Estimates
Cov Parm
Subject
Estimate
Standard
Error
Intercept
patient
0.2990
0.1116
Solutions for Fixed Effects
Effect
dist
Estimate
Standard
Error
DF
t Value
Pr > |t|
dist
dist
age*dist
age*dist
OKstatus*dist
OKstatus*dist
Binary
Poisson
Binary
Poisson
Binary
Poisson
5.7783
0.8410
-0.07572
0.01875
-0.4697
-0.1856
2.9048
0.5696
0.03791
0.007383
1.1251
0.3020
29
29
29
29
29
29
1.99
1.48
-2.00
2.54
-0.42
-0.61
0.0562
0.1506
0.0552
0.0167
0.6794
0.5435
The estimate of the variance of the random patient intercept is 0:2990, and the estimated standard
error of this variance component estimate is 0:1116. There appears to be significant patient-topatient variation in the intercepts. The estimates of the fixed effects as well as their estimated
standard errors have changed from the bivariate-independence analysis (see Output 38.5.3). When
the length of hospital stay and the postoperative condition are modeled jointly, the preoperative
health status (variable OKStatus) no longer appears significant. Compare this result to Output 38.5.3;
in the separate analyses the initial health status was a significant predictor of the length of hospital
stay. A further joint analysis of these data would probably remove this predictor from the model
entirely.
A joint model of the second kind, where correlations are modeled directly, is fit with the following
GLIMMIX statements:
proc glimmix data=hernio_uv;
class patient dist;
model response(event=’1’) = dist dist*age dist*OKstatus /
noint s dist=byobs(dist);
random _residual_ / subject=patient type=chol;
run;
2342 F Chapter 38: The GLIMMIX Procedure
Instead of a shared G-side random effect, an R-side covariance structure is used to model the correlations. It is important to note that this is a marginal model that models covariation on the scale of
the data. The previous model involves the Z random components inside the linear predictor.
The _RESIDUAL_ keyword instructs PROC GLIMMIX to model the R-side correlations. Because
of the SUBJECT=PATIENT option, data from different patients are independent, and data from
a single patient follow the covariance model specified with the TYPE= option. In this case, a
generally unstructured 2 2 covariance matrix is modeled, but in its Cholesky parameterization.
This ensures that the resulting covariance matrix is at least positive semidefinite and stabilizes the
numerical optimizations.
Output 38.5.5 Bivariate Analysis – Marginal Correlated Error Model
The GLIMMIX Procedure
Fit Statistics
-2 Res Log Pseudo-Likelihood
Generalized Chi-Square
Gener. Chi-Square / DF
240.98
58.00
1.00
Covariance Parameter Estimates
Cov Parm
Subject
Estimate
Standard
Error
CHOL(1,1)
CHOL(2,1)
CHOL(2,2)
patient
patient
patient
1.0162
0.3942
2.0819
0.1334
0.3893
0.2734
Solutions for Fixed Effects
Effect
dist
Estimate
Standard
Error
DF
t Value
Pr > |t|
dist
dist
age*dist
age*dist
OKstatus*dist
OKstatus*dist
Binary
Poisson
Binary
Poisson
Binary
Poisson
5.6514
1.2463
-0.07568
0.01548
-0.3421
-0.3253
2.8283
0.7189
0.03765
0.009432
1.0384
0.3310
26
26
26
26
26
26
2.00
1.73
-2.01
1.64
-0.33
-0.98
0.0563
0.0948
0.0549
0.1128
0.7445
0.3349
The “Covariance Parameter Estimates” table in Output 38.5.5 contains three entries for this model,
corresponding to a .2 2/ covariance matrix for each patient. The Cholesky root of the R matrix is
1:0162
0
LD
0:3942 2:0819
so that the covariance matrix can be obtained as
1:0162
0
1:0162 0:3942
1:0326 0:4005
0
LL D
D
0:3942 2:0819
0
2:0819
0:4005 4:4897
Example 38.6: Radial Smoothing of Repeated Measures Data F 2343
This is not the covariance matrix of the data, however, because the variance functions need to be
accounted for.
The p-values in the “Solutions for Fixed Effects” table indicate the same pattern of significance and
nonsignificance as in the conditional model with random patient intercepts.
Example 38.6: Radial Smoothing of Repeated Measures Data
This example of a repeated measures study is taken from Diggle, Liang, and Zeger (1994, p. 100).
The data consist of body weights of 27 cows, measured at 23 unequally spaced time points over
a period of approximately 22 months. Following Diggle, Liang, and Zeger (1994), one animal is
removed from the analysis, one observation is removed according to their Figure 5.7, and the time
is shifted to start at 0 and is measured in 10-day increments. The design is a 2 2 factorial, and the
factors are the infection of an animal with M. paratuberculosis and whether the animal is receiving
iron dosing.
The following DATA steps create the data and arrange them in univariate format.
data times;
input time1-time23;
datalines;
122 150 166 179 219
478 508 536 569 599
;
247
627
276
655
296
668
324
723
data cows;
if _n_ = 1 then merge times;
array t{23} time1
- time23;
array w{23} weight1 - weight23;
input cow iron infection weight1-weight23
do i=1 to 23;
weight = w{i};
tpoint = (t{i}-t{1})/10;
output;
end;
keep cow iron infection tpoint weight;
datalines;
1 0 0 4.7
4.905 5.011 5.075 5.136
5.416 5.438 5.541 5.652 5.687
5.784 5.844 5.886 5.914 5.979
2 0 0 4.868 5.075 5.193 5.22
5.298
5.617 5.635 5.687 5.768 5.799
5.914 5.966 5.991 6.016 6.087
3 0 0 4.868 5.011 5.136 5.193 5.273
5.521 5.58
5.617 5.687 5.72
5.784 5.814 5.829 5.872 5.927
4 0 0 4.828 5.011 5.136 5.193 5.273
5.541 5.598 5.67
.
5.737
5.886 5.927 5.94
5.979 6.052
5 1 0 4.787 4.977 5.043 5.136 5.106
354
751
380
781
445
@@;
5.165
5.737
5.927
5.416
5.872
6.098
5.323
5.753
5.9
5.347
5.844
6.028
5.298
5.298
5.814
5.94
5.481
5.886
6.153
5.416
5.784
5.991
5.438
5.858
6.12
5.298
5.323
5.799
5.521
5.872
5.46
5.784
5.561
5.872
5.371
2344 F Chapter 38: The GLIMMIX Procedure
6 1 0
7 1 0
8 0 1
9 0 1
10 0 1
11 0 1
12 0 1
13 0 1
14 0 1
15 0 1
16 0 1
17 1 1
18 1 1
19 1 1
20 1 1
21 1 1
22 1 1
23 1 1
5.438
5.784
4.745
5.416
5.737
4.745
5.416
5.768
4.942
5.561
5.94
4.605
5.22
5.635
4.7
5.22
5.541
4.828
5.46
5.704
4.7
5.298
5.687
4.828
5.438
5.799
4.828
5.323
5.704
4.745
5.394
5.753
4.7
5.347
5.784
4.605
5.273
5.501
4.828
5.416
5.799
4.7
5.247
5.501
4.745
5.416
5.652
4.787
5.394
5.687
4.605
5.247
5.521
4.7
5.501
5.784
4.868
5.501
5.753
4.905
5.521
5.814
5.106
5.58
5.991
4.745
5.247
5.67
4.868
5.22
5.598
5.011
5.501
5.72
4.828
5.323
5.72
5.011
5.416
5.858
4.942
5.298
5.753
4.905
5.394
5.768
4.868
5.371
5.768
4.787
5.247
5.635
4.977
5.416
5.858
4.905
5.22
5.561
4.905
5.394
5.687
4.942
5.371
5.72
4.828
5.22
5.561
4.905
5.561
5.829
5.043
5.561
5.768
5.011
5.541
5.829
5.136
5.635
6.016
4.868
5.298
5.72
4.905
5.273
5.58
5.075
5.541
5.737
4.905
5.416
5.72
5.075
5.521
5.872
5.011
5.394
5.768
4.977
5.438
5.814
5.011
5.438
5.814
4.828
5.347
5.652
5.011
5.438
5.886
4.942
5.323
5.541
4.977
5.521
5.652
4.977
5.438
5.737
4.828
5.298
5.617
5.011
5.652
5.858
5.106
5.58
5.784
5.106
5.635
5.858
5.193
5.704
6.064
4.905
5.416
5.753
4.977
5.384
5.635
5.165
5.609
5.768
5.011
5.505
5.737
5.136
5.628
5.914
5.075
5.489
5.814
5.075
5.583
5.844
5.043
5.455
5.844
4.942
5.366
5.598
5.136
5.557
5.914
5.011
5.338
5.58
5.043
5.617
5.617
5.106
5.521
5.737
4.977
5.375
5.635
5.075
5.67
5.914
5.22
5.687
5.844
5.165
5.687
5.94
5.298
5.784
6.052
4.977
5.501
5.799
5.011
5.438
5.687
5.247
5.687
5.858
5.075
5.561
5.784
5.22
5.67
5.94
5.075
5.541
5.872
5.193
5.617
5.886
5.106
5.617
5.886
5.011
5.416
5.635
5.273
5.617
5.979
5.043
5.371
5.652
5.136
5.617
5.687
5.165
5.521
5.768
5.043
5.371
5.72
5.106
5.737
5.9
5.298
5.72
5.844
5.273
5.704
5.94
5.347
5.823
6.016
5.22
5.521
5.829
5.106
5.438
5.72
5.323
5.704
5.9
5.165
5.58
5.814
5.273
5.687
5.991
5.22
5.58
5.927
5.22
5.652
5.886
5.165
5.635
5.94
5.136
5.46
5.635
5.298
5.67
6.004
5.136
5.394
5.67
5.273
5.617
5.768
5.247
5.561
5.768
5.165
5.416
5.737
5.22
5.784
5.94
5.347
5.737
5.9
5.371
5.784
6.004
5.46
5.858
5.979
5.165
5.58
5.858
5.165
5.501
5.704
5.394
5.72
5.94
5.247
5.561
5.799
5.347
5.72
6.016
5.273
5.617
5.927
5.298
5.687
5.886
5.247
5.704
5.927
5.22
5.541
5.598
5.371
5.72
6.028
5.193
5.438
5.704
5.347
5.67
5.814
5.323
5.635
5.704
5.22
5.501
5.768
5.22
5.768
5.347
5.72
5.416
5.768
5.521
5.9
5.22
5.58
5.22
5.501
5.46
5.704
5.298
5.635
5.416
5.72
5.298
5.67
5.323
5.72
5.298
5.737
5.247
5.481
5.46
5.72
5.193
5.416
5.394
5.635
5.416
5.617
5.273
5.501
5.298
Example 38.6: Radial Smoothing of Repeated Measures Data F 2345
24 1 1
25 1 1
26 1 1
5.323
5.598
4.745
5.347
5.635
4.654
5.165
5.46
4.828
5.371
5.72
5.347
5.652
4.942
5.371
5.687
4.828
5.165
5.58
4.977
5.394
5.784
5.416
5.67
5.011
5.416
5.704
4.828
5.193
5.635
5.011
5.46
5.784
5.472
5.704
5.075
5.481
5.72
4.977
5.204
5.67
5.106
5.576
5.784
5.501
5.737
5.106
5.501
5.829
4.977
5.22
5.753
5.165
5.652
5.829
5.541
5.768
5.247
5.541
5.844
5.043
5.273
5.799
5.22
5.617
5.814
5.598
5.784
5.273
5.598
5.9
5.136
5.371
5.844
5.273
5.687
5.844
5.598
5.323
5.598
5.165
5.347
5.323
5.67
;
The mean response profiles of the cows are not of particular interest; what matters are inferences
about the Iron effect, the Infection effect, and their interaction. Nevertheless, the body weight of
the cows changes over the 22-month period, and you need to account for these changes in the
analysis. A reasonable approach is to apply the approximate low-rank smoother to capture the trends
over time. This approach frees you from having to stipulate a parametric model for the response
trajectories over time. In addition, you can test hypotheses about the smoothing parameter; for
example, whether it should be varied by treatment.
The following statements fit a model with a 2 2 factorial treatment structure and smooth trends
over time, choosing the Newton-Raphson algorithm with ridging for the optimization:
proc glimmix data=cows;
t2 = tpoint / 100;
class cow iron infection;
model weight = iron infection iron*infection tpoint;
random t2 / type=rsmooth subject=cow
knotmethod=kdtree(bucket=100 knotinfo);
output out=gmxout pred(blup)=pred;
nloptions tech=newrap;
run;
The continuous time effect appears in both the MODEL statement (tpoint) and the RANDOM statement (t2). Because the variance of the radial smoothing component depends on the temporal metric,
the time scale was rescaled for the RANDOM effect to move the parameter estimate away from the
boundary. The knots of the radial smoother are selected as the vertices of a k-d tree. Specifying
BUCKET=100 sets the bucket size of the tree to b D 100. Because measurements at each time point
are available for 26 (or 25) cows, this groups approximately four time points in a single bucket. The
KNOTINFO keyword of the KNOTMETHOD= option requests a printout of the knot locations for
the radial smoother. The OUTPUT statement saves the predictions of the mean of each observations to the data set gmxout. Finally, the TECH=NEWRAP option in the NLOPTIONS statement
specifies the Newton-Raphson algorithm for the optimization technique.
The “Class Level Information” table lists the number of levels of the Cow, Iron, and Infection effects
(Output 38.6.1).
2346 F Chapter 38: The GLIMMIX Procedure
Output 38.6.1 Model Information and Class Levels in Repeated Measures Analysis
The GLIMMIX Procedure
Model Information
Data Set
Response Variable
Response Distribution
Link Function
Variance Function
Variance Matrix Blocked By
Estimation Technique
Degrees of Freedom Method
WORK.COWS
weight
Gaussian
Identity
Default
cow
Restricted Maximum Likelihood
Containment
Class Level Information
Class
cow
iron
infection
Levels
26
2
2
Values
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
19 20 21 22 23 24 25 26
0 1
0 1
The “Radial Smoother Knots for RSmooth(t2)” table displays the knots computed from the vertices
of the t2 k-d tree (Output 38.6.2). Notice that knots are spaced unequally and that the extreme
time points are among the knot locations. The “Number of Observations” table shows that one
observation was not used in the analysis. The 12th observation for cow 4 has a missing value.
Output 38.6.2 Knot Information and Number of Observations
Radial Smoother
Knots for
RSmooth(t2)
Knot
Number
t2
1
2
3
4
5
6
7
8
9
0
0.04400
0.1250
0.2020
0.3230
0.4140
0.5050
0.6010
0.6590
Number of Observations Read
Number of Observations Used
598
597
The “Dimensions” table shows that the model contains only two covariance parameters, the G-side
variance of the spline coefficients ( 2 ) and the R-side scale parameter (, Output 38.6.3). For
Example 38.6: Radial Smoothing of Repeated Measures Data F 2347
each subject (cow), there are nine columns in the Z matrix, one per knot location. The GLIMMIX
procedure processes these data by subjects (cows).
Output 38.6.3 Dimensions Information in Repeated Measures Analysis
Dimensions
G-side Cov. Parameters
R-side Cov. Parameters
Columns in X
Columns in Z per Subject
Subjects (Blocks in V)
Max Obs per Subject
1
1
10
9
26
23
The “Optimization Information” table displays information about the optimization process. Because fixed effects and the residual scale parameter can be profiled from the optimization, the iterative algorithm involves only a single covariance parameter, the variance of the spline coefficients
(Output 38.6.4).
Output 38.6.4 Optimization Information in Repeated Measures Analysis
Optimization Information
Optimization Technique
Parameters in Optimization
Lower Boundaries
Upper Boundaries
Fixed Effects
Residual Variance
Starting From
Newton-Raphson
1
1
0
Profiled
Profiled
Data
After 11 iterations, the optimization process terminates (Output 38.6.5). In this case, the absolute
gradient convergence criterion was met.
2348 F Chapter 38: The GLIMMIX Procedure
Output 38.6.5 Iteration History and Convergence Status
Iteration History
Iteration
Restarts
Evaluations
Objective
Function
Change
Max
Gradient
0
1
2
3
4
5
6
7
8
9
10
11
0
0
0
0
0
0
0
0
0
0
0
0
4
3
3
3
3
3
3
3
3
3
3
3
-1302.549272
-1451.587367
-1585.640946
-1694.516203
-1775.290458
-1829.966584
-1862.878184
-1879.329133
-1885.175082
-1886.238032
-1886.288519
-1886.288673
.
149.03809501
134.05357887
108.87525722
80.77425512
54.67612585
32.91160012
16.45094875
5.84594887
1.06295071
0.05048659
0.00015425
20.33682
9.940495
4.71531
2.176741
0.978577
0.425724
0.175992
0.066061
0.020137
0.00372
0.000198
6.364E-7
Convergence criterion (ABSGCONV=0.00001) satisfied.
The generalized chi-square statistic in the “Fit Statistics” table is small for this model
(Output 38.6.6). There is very little residual variation. The radial smoother is associated with
433.55 residual degrees of freedom, computed as 597 minus the trace of the smoother matrix.
Output 38.6.6 Fit Statistics in Repeated Measures Analysis
Fit Statistics
-2 Res Log Likelihood
AIC (smaller is better)
AICC (smaller is better)
BIC (smaller is better)
CAIC (smaller is better)
HQIC (smaller is better)
Generalized Chi-Square
Gener. Chi-Square / DF
Radial Smoother df(res)
-1886.29
-1882.29
-1882.27
-1879.77
-1877.77
-1881.56
0.47
0.00
433.55
The “Covariance Parameter Estimates” table in Output 38.6.7 displays the estimates of the covariance parameters. The variance of the random spline coefficients is estimated as b
2 D 0:5961, and
b
the scale parameter (=residual variance) estimate is = 0.0008.
Example 38.6: Radial Smoothing of Repeated Measures Data F 2349
Output 38.6.7 Estimated Covariance Parameters
Covariance Parameter Estimates
Cov Parm
Subject
Estimate
Standard
Error
Var[RSmooth(t2)]
Residual
cow
0.5961
0.000800
0.08144
0.000059
The “Type III Tests of Fixed Effects” table displays F tests for the fixed effects in the MODEL
statement (Output 38.6.8). There is a strong infection effect as well as the absence of an interaction
between infection with M. paratuberculosis and iron dosing. It is important to note, however, that
the interpretation of these tests rests on the assumption that the random effects in the mixed model
have zero mean; in this case, the radial smoother coefficients.
Output 38.6.8 Tests of Fixed Effects
Type III Tests of Fixed Effects
Effect
iron
infection
iron*infection
tpoint
Num
DF
Den
DF
F Value
Pr > F
1
1
1
1
358
358
358
358
3.59
21.16
0.09
53.88
0.0588
<.0001
0.7637
<.0001
A graph of the observed data and fitted profiles in the four groups is produced with the following
statements (Output 38.6.9):
data plot; set gmxout;
length group $26;
if (iron=0) and (infection=0) then group=’Control Group (n=4)’;
else if (iron=1) and (infection=0) then group=’Iron - No Infection (n=3)’;
else if (iron=0) and (infection=1) then group=’No Iron - Infection (n=9)’;
else group = ’Iron - Infection (n=10)’;
run;
proc sort data=plot; by group cow;
run;
proc sgpanel data=plot noautolegend;
title ’Radial Smoothing With Cow-Specific Trends’;
label tpoint=’Time’ weight=’log(Weight)’;
panelby group / columns=2 rows=2;
scatter x=tpoint y=weight;
series x=tpoint y=pred / group=cow lineattrs=GraphFit;
run;
2350 F Chapter 38: The GLIMMIX Procedure
Output 38.6.9 Observed and Predicted Profiles
The trends are quite smooth, and you can see how the radial smoother adapts to the cow-specific
profile. This is the reason for the small scale parameter estimate, b
D 0:008. Comparing the panels
at the top to the panels at the bottom of Output 38.6.9 reveals the effect of Infection. A comparison
of the panels on the left to those on the right indicates the weak Iron effect.
The smoothing parameter in this analysis is related to the covariance parameter estimates. Because
there is only one radial smoothing variance component, the amount of smoothing is the same in
all four treatment groups. To test whether the smoothing parameter should be varied by group,
you can refine the analysis of the previous model. The following statements fit the same general
model, but they vary the covariance parameters by the levels of the Iron*Infection interaction. This is
accomplished with the GROUP= option in the RANDOM statement.
Example 38.6: Radial Smoothing of Repeated Measures Data F 2351
ods select OptInfo FitStatistics CovParms;
proc glimmix data=cows;
t2 = tpoint / 100;
class cow iron infection;
model weight = iron infection iron*infection tpoint;
random t2 / type=rsmooth
subject=cow
group=iron*infection
knotmethod=kdtree(bucket=100);
nloptions tech=newrap;
run;
All observations that have the same value combination of the Iron and Infection effects share the same
covariance parameter. As a consequence, you obtain different smoothing parameters result in the
four groups.
In Output 38.6.10, the “Optimization Information” table shows that there are now four covariance
parameters in the optimization, one spline coefficient variance for each group.
Output 38.6.10 Analysis with Group-Specific Smoothing Parameter
The GLIMMIX Procedure
Optimization Information
Optimization Technique
Parameters in Optimization
Lower Boundaries
Upper Boundaries
Fixed Effects
Residual Variance
Starting From
Newton-Raphson
4
4
0
Profiled
Profiled
Data
Fit Statistics
-2 Res Log Likelihood
AIC (smaller is better)
AICC (smaller is better)
BIC (smaller is better)
CAIC (smaller is better)
HQIC (smaller is better)
Generalized Chi-Square
Gener. Chi-Square / DF
Radial Smoother df(res)
-1887.95
-1877.95
-1877.85
-1871.66
-1866.66
-1876.14
0.48
0.00
434.72
Covariance Parameter Estimates
Cov Parm
Subject
Group
Var[RSmooth(t2)]
Var[RSmooth(t2)]
Var[RSmooth(t2)]
Var[RSmooth(t2)]
Residual
cow
cow
cow
cow
iron*infection
iron*infection
iron*infection
iron*infection
0
0
1
1
0
1
0
1
Estimate
Standard
Error
0.4788
0.5152
0.4904
0.7105
0.000807
0.1922
0.1182
0.2195
0.1409
0.000060
2352 F Chapter 38: The GLIMMIX Procedure
Varying this variance component by groups has changed the 2 Res Log Likelihood from 1886:29
to 1887:95 (Output 38.6.10). The difference, 1:66, can be viewed (asymptotically) as the realization of a chi-square random variable with three degrees of freedom. The difference is not significant
(p D 0:64586). The “Covariance Parameter Estimates” table confirms that the estimates of the
spline coefficient variance are quite similar in the four groups, ranging from 0:4788 to 0:7105.
Finally, you can apply a different technique for varying the temporal trends among the cows. From
Output 38.6.9 it appears that an assumption of parallel trends within groups might be reasonable.
In other words, you can fit a model in which the “overall” trend over time in each group is modeled
nonparametrically, and this trend is shifted up or down to capture the behavior of the individual
cow. You can accomplish this with the following statements:
ods select FitStatistics CovParms;
proc glimmix data=cows;
t2 = tpoint / 100;
class cow iron infection;
model weight = iron infection iron*infection tpoint;
random t2 / type=rsmooth
subject=iron*infection
knotmethod=kdtree(bucket=100);
random intercept / subject=cow;
output out=gmxout pred(blup)=pred;
nloptions tech=newrap;
run;
There are now two subject effects in this analysis. The first RANDOM statement applies the radial
smoothing and identifies the experimental conditions as the subject. For each condition, a separate
realization of the random spline coefficients is obtained. The second RANDOM statement adds a
random intercept to the trend for each cow. This random intercept results in the parallel shift of the
trends over time.
Results from this analysis are shown in Output 38.6.11.
Output 38.6.11 Analysis with Parallel Shifts
The GLIMMIX Procedure
Fit Statistics
-2 Res Log Likelihood
AIC (smaller is better)
AICC (smaller is better)
BIC (smaller is better)
CAIC (smaller is better)
HQIC (smaller is better)
Generalized Chi-Square
Gener. Chi-Square / DF
Radial Smoother df(res)
-1788.52
-1782.52
-1782.48
-1788.52
-1785.52
-1788.52
1.17
0.00
547.21
Example 38.6: Radial Smoothing of Repeated Measures Data F 2353
Output 38.6.11 continued
Covariance Parameter Estimates
Cov Parm
Subject
Estimate
Standard
Error
Var[RSmooth(t2)]
Intercept
Residual
iron*infection
cow
0.5398
0.007122
0.001976
0.1940
0.002173
0.000121
Because the parallel shift model is not nested within either one of the previous models, the models
cannot be compared with a likelihood ratio test. However, you can draw on the other fit statistics.
All statistics indicate that this model does not fit the data as well as the initial model that varies the
spline coefficients by cow. The Pearson chi-square statistic is more than twice as large as in the
previous model, indicating much more residual variation in the fit. On the other hand, this model
generates only four sets of spline coefficients, one for each treatment group, and thus retains more
residual degrees of freedom.
The “Covariance Parameter Estimates” table in Output 38.6.11 displays the solutions for the covariance parameters. The estimate of the variance of the spline coefficients is not that different from
the estimate obtained in the first model (0:5961). The residual variance, however, has more than
doubled.
Using similar SAS statements as previously, you can produce a plot of the observed and predicted
profiles (Output 38.6.12).
The parallel shifts of the nonparametric smooths are clearly visible in Output 38.6.12. In the groups
receiving only iron or only an infection, the parallel lines assumption holds quite well. In the control
group and the group receiving iron and the infection, the parallel shift assumption does not hold as
well. Two of the profiles in the iron-only group are nearly indistinguishable.
2354 F Chapter 38: The GLIMMIX Procedure
Output 38.6.12 Observed and Predicted Profiles
This example demonstrates that mixed model smoothing techniques can be applied not only to
achieve scatter plot smoothing, but also to longitudinal or repeated measures data. You can then use
the SUBJECT= option in the RANDOM statement to obtain independent sets of spline coefficients
for different subjects, and the GROUP= option in the RANDOM statement to vary the degree of
smoothing across groups. Also, radial smoothers can be combined with other random effects. For
the data considered here, the appropriate model is one with a single smoothing parameter for all
treatment group and cow-specific spline coefficients.
Example 38.7: Isotonic Contrasts for Ordered Alternatives F 2355
Example 38.7: Isotonic Contrasts for Ordered Alternatives
Dose response studies often focus on testing for monotone increasing or decreasing behavior in the
mean values of the dependent variable. Hirotsu and Srivastava (2000) demonstrate one approach by
using data that originally appeared in Moriguchi (1976). The data, which follow, consist of ferrite
cores subjected to four increasing temperatures. The response variable is the magnetic force of each
core.
data FerriteCores;
do Temp = 1 to 4;
do rep = 1 to 5; drop rep;
input MagneticForce @@;
output;
end;
end;
datalines;
10.8 9.9 10.7 10.4 9.7
10.7 10.6 11.0 10.8 10.9
11.9 11.2 11.0 11.1 11.3
11.4 10.7 10.9 11.3 11.7
;
It is of interest to test whether the magnetic force of the cores rises monotonically with temperature.
The approach of Hirotsu and Srivastava (2000) depends on the lower confidence limits of the isotonic contrasts of the force means at each temperature, adjusted for multiplicity. The corresponding
isotonic contrast compares the average of a particular group and the preceding groups with the average of the succeeding groups. You can compute adjusted confidence intervals for isotonic contrasts
by using the LSMESTIMATE statement.
The following statements request an analysis of the FerriteCores data as a one-way design and
multiplicity-adjusted lower confidence limits for the isotonic contrasts. For the multiplicity adjustment, the LSMESTIMATE statement employs simulation, which provides adjusted p-values
and lower confidence limits that are exact up to Monte Carlo error.
proc glimmix data=FerriteCores;
class Temp;
model MagneticForce = Temp;
lsmestimate Temp
’avg(1:1)<avg(2:4)’ -3 1 1 1 divisor=3,
’avg(1:2)<avg(3:4)’ -1 -1 1 1 divisor=2,
’avg(1:3)<avg(4:4)’ -1 -1 -1 3 divisor=3
/ adjust=simulate(seed=1) cl upper;
ods select LSMestimates;
run;
The results are shown in Output 38.7.1.
2356 F Chapter 38: The GLIMMIX Procedure
Output 38.7.1 Analysis of LS-Means with Isotonic Contrasts
The GLIMMIX Procedure
Least Squares Means Estimates
Adjustment for Multiplicity: Simulated
Effect
Label
Temp
Temp
Temp
avg(1:1)<avg(2:4)
avg(1:2)<avg(3:4)
avg(1:3)<avg(4:4)
Estimate
Standard
Error
DF
t Value
Pr > t
Adj P
0.8000
0.7000
0.4000
0.1906
0.1651
0.1906
16
16
16
4.20
4.24
2.10
0.0003
0.0003
0.0260
0.0010
0.0009
0.0625
Least Squares Means Estimates
Adjustment for Multiplicity: Simulated
Effect
Label
Temp
Temp
Temp
avg(1:1)<avg(2:4)
avg(1:2)<avg(3:4)
avg(1:3)<avg(4:4)
Alpha
Lower
0.05
0.05
0.05
0.4672
0.4118
0.06721
Upper
Infty
Infty
Infty
Adj
Lower
0.3771
0.3337
-0.02291
Adj
Upper
Infty
Infty
Infty
With an adjusted p-value of 0.001, the magnetic force at the first temperature is significantly less
than the average of the other temperatures. Likewise, the average of the first two temperatures is
significantly less than the average of the last two (p D 0:0009). However, the magnetic force at
the last temperature is not significantly greater than the average magnetic force of the others (p D
0:0625). These results indicate a significant monotone increase over the first three temperatures, but
not across all four temperatures.
Example 38.8: Adjusted Covariance Matrices of Fixed Effects
The following data are from Pothoff and Roy (1964) and consist of growth measurements for 11
girls and 16 boys at ages 8, 10, 12, and 14. Some of the observations are suspect (for example, the
third observation for person 20); however, all of the data are used here for comparison purposes.
data pr;
input child gender$ y1 y2 y3 y4;
array yy y1-y4;
do time=1 to 4;
age = time*2 + 6;
y
= yy{time};
output;
end;
drop y1-y4;
datalines;
1
F
21.0
20.0
21.5
23.0
2
F
21.0
21.5
24.0
25.5
Example 38.8: Adjusted Covariance Matrices of Fixed Effects F 2357
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
;
F
F
F
F
F
F
F
F
F
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
20.5
23.5
21.5
20.0
21.5
23.0
20.0
16.5
24.5
26.0
21.5
23.0
25.5
20.0
24.5
22.0
24.0
23.0
27.5
23.0
21.5
17.0
22.5
23.0
22.0
24.0
24.5
23.0
21.0
22.5
23.0
21.0
19.0
25.0
25.0
22.5
22.5
27.5
23.5
25.5
22.0
21.5
20.5
28.0
23.0
23.5
24.5
25.5
24.5
21.5
24.5
25.0
22.5
21.0
23.0
23.5
22.0
19.0
28.0
29.0
23.0
24.0
26.5
22.5
27.0
24.5
24.5
31.0
31.0
23.5
24.0
26.0
25.5
26.0
23.5
26.0
26.5
23.5
22.5
25.0
24.0
21.5
19.5
28.0
31.0
26.5
27.5
27.0
26.0
28.5
26.5
25.5
26.0
31.5
25.0
28.0
29.5
26.0
30.0
25.0
Jennrich and Schluchter (1986) analyze these data with various models for the fixed effects and
the covariance structure. The strategy here is to fit a growth curve model for the boys and girls
and to account for subject-to-subject variation through G-side random effects. In addition, serial
correlation among the observations within each child is accounted for by a time series process. The
data are assumed to be Gaussian, and their 2 restricted log likelihood is minimized to estimate the
model parameters.
The following statements fit a mixed model in which a separate growth curve is assumed for each
gender:
proc glimmix data=pr;
class child gender time;
model y = gender age gender*age / covb(details) ddfm=kr;
random intercept age / type=chol sub=child;
random time / subject=child type=ar(1) residual;
ods select ModelInfo CovB CovBModelBased CovBDetails;
run;
The growth curve for an individual child differs from the gender-specific trend because of a random
intercept and a random slope. The two G-side random effects are assumed to be correlated. Their
unstructured covariance matrix is parameterized in terms of the Cholesky root to guarantee a positive (semi-)definite estimate. An AR(1) covariance structure is modeled for the observations over
time for each child. Notice the RESIDUAL option in the second RANDOM statement. It identifies
this as an R-side random effect.
2358 F Chapter 38: The GLIMMIX Procedure
The DDFM=KR option requests that the covariance matrix of the fixed-effect parameter estimates
and denominator degrees of freedom for t and F tests are determined according to Kenward and
Roger (1997). This is reflected in the “Model Information” table (Output 38.8.1).
Output 38.8.1 Model Information with DDFM=KR
The GLIMMIX Procedure
Model Information
Data Set
Response Variable
Response Distribution
Link Function
Variance Function
Variance Matrix Blocked By
Estimation Technique
Degrees of Freedom Method
Fixed Effects SE Adjustment
WORK.PR
y
Gaussian
Identity
Default
child
Restricted Maximum Likelihood
Kenward-Roger
Kenward-Roger
The COVB option in the MODEL statement requests that the covariance matrix used for inference about fixed effects in this model is displayed; this is the Kenward-Roger-adjusted covariance
matrix. The DETAILS suboption requests that the unadjusted covariance matrix is also displayed
(Output 38.8.2). In addition, a table of diagnostic measures for the covariance matrices is produced.
Output 38.8.2 Model-Based and Adjusted Covariance Matrix
Model Based Covariance Matrix for Fixed Effects (Unadjusted)
Effect
Intercept
gender
gender
age
age*gender
age*gender
gender
F
M
F
M
Row
Col1
Col2
1
2
3
4
5
6
0.9969
-0.9969
-0.07620
0.07620
Col3
Col4
Col5
-0.9969
2.4470
-0.07620
0.07620
0.07620
-0.1870
0.07620
-0.1870
0.007581
-0.00758
-0.00758
0.01861
Col6
Covariance Matrix for Fixed Effects
Effect
Intercept
gender
gender
age
age*gender
age*gender
gender
F
M
F
M
Row
Col1
Col2
1
2
3
4
5
6
0.9724
-0.9724
-0.07412
0.07412
Col3
Col4
Col5
-0.9724
2.3868
-0.07412
0.07412
0.07412
-0.1819
0.07412
-0.1819
0.007256
-0.00726
-0.00726
0.01781
Col6
Example 38.8: Adjusted Covariance Matrices of Fixed Effects F 2359
Output 38.8.2 continued
Diagnostics for Covariance Matrices of Fixed Effects
ModelBased
Adjusted
6
16
6
16
Dimensions
Rows
Non-zero entries
Summaries
Trace
Log determinant
3.4701
-11.95
3.3843
-12.17
Eigenvalues
> 0
= 0
max abs
min abs non-zero
Condition number
4
2
2.972
0.0009
3467.8
4
2
2.8988
0.0008
3698.2
Norms
Frobenius
Infinity
3.0124
3.7072
2.9382
3.6153
Comparisons
Concordance correlation
Discrepancy function
Frobenius norm of difference
Trace(Adjusted Inv(MBased))
0.9979
0.0084
0.0742
3.7801
Determinant and inversion results apply to the nonsingular
partitions of the covariance matrices.
The “Diagnostics for Covariance Matrices” table in Output 38.8.2 consists of several sections. The
trace and log determinant of covariance matrices are general scalar summaries that are sometimes
used in direct comparisons, or in formulating further statistics, such as the difference of log determinants. The trace simply represents the sum of the variances of all fixed-effects parameters.
The two matrices have the same number of positive and zero eigenvalues; hence they are of the
same rank. There are no negative eigenvalues; hence the matrices are positive semi-definite.
The “Comparisons” section of the table provides several statistics that set the matrices in relationship. The statistics enable you to assess the extent to which the adjustment affected the model-based
matrix. If the two matrices are identical, the concordance correlation equals 1, the discrepancy function and the Frobenius norm of the differences equal 0, and the trace of the adjusted and the (generalized) inverse of the model-based matrix equals the rank. See the section “Exploring and Comparing
Covariance Matrices” on page 2244 for computational details regarding these statistics. With increasing discrepancy between the matrices, the difference norm and discrepancy function increase,
the concordance correlation falls below 1, and the trace deviates from the rank. In this particular example, there is strong agreement between the two matrices; the adjustment to the covariance matrix
associated with DDFM=KR is only slight. It is noteworthy, however, that the trace of the adjusted
covariance matrix falls short of the trace of the unadjusted one. Indeed, from Output 38.8.2 you can
see that the diagonal elements of the adjusted covariance matrices are uniformly smaller than those
of the model-based covariance matrix.
2360 F Chapter 38: The GLIMMIX Procedure
Standard error “shrinkage” for the Kenward-Roger covariance adjustment is due to the term
0:25Rij in equation (3) of Kenward and Roger (1997), which is nonzero for covariance structures with second derivatives, such as the TYPE=ANTE(1), TYPE=AR(1), TYPE=ARH(1),
TYPE=ARMA(1,1), TYPE=CHOL, TYPE=CSH, TYPE=FA0(q), TYPE=TOEPH, and
TYPE=UNR structures and all TYPE=SP() structures.
For covariance structures that are linear in the parameters, Rij D 0. You can add the FIRSTORDER
suboption to the DDFM=KR option to request that second derivative matrices Rij are excluded from
computing the covariance matrix adjustment. The resulting covariance adjustment is that of Kackar
and Harville (1984) and Harville and Jeske (1992). This estimator is denoted as m
[email protected] in Harville
and Jeske (1992) and is referred to there as the Prasad-Rao estimator after related work by Prasad
and Rao (1990). This standard error adjustment is guaranteed to be positive (semi-)definite. The
following statements fit the model with the Kackar-Harville-Jeske estimator and compare modelbased and adjusted covariance matrices:
proc glimmix data=pr;
class child gender time;
model y = gender age gender*age / covb(details)
ddfm=kr(firstorder);
random intercept age / type=chol sub=child;
random time / subject=child type=ar(1) residual;
ods select ModelInfo CovB CovBDetails;
run;
The standard error adjustment is reflected in the “Model Information” table (Output 38.8.3).
Output 38.8.3 Model Information with DDFM=KR(FIRSTORDER)
The GLIMMIX Procedure
Model Information
Data Set
Response Variable
Response Distribution
Link Function
Variance Function
Variance Matrix Blocked By
Estimation Technique
Degrees of Freedom Method
Fixed Effects SE Adjustment
WORK.PR
y
Gaussian
Identity
Default
child
Restricted Maximum Likelihood
Kenward-Roger
Prasad-Rao-Kackar-Harville-Jeske
Output 38.8.4 displays the adjusted covariance matrix. Notice that the elements of this matrix, in
particular the diagonal elements, are larger in absolute value than those of the model-based estimator
(Output 38.8.2).
Example 38.8: Adjusted Covariance Matrices of Fixed Effects F 2361
Output 38.8.4 Adjusted Covariance Matrix and Comparison to Model-Based Estimator
Covariance Matrix for Fixed Effects
Effect
Intercept
gender
gender
age
age*gender
age*gender
gender
Row
Col1
Col2
1
2
3
4
5
6
1.0122
-1.0122
-0.07758
0.07758
F
M
F
M
Col3
Col4
Col5
-1.0122
2.4845
-0.07758
0.07758
0.07758
-0.1904
0.07758
-0.1904
0.007706
-0.00771
-0.00771
0.01891
Col6
Diagnostics for Covariance Matrices of Fixed Effects
ModelBased
Adjusted
6
16
6
16
Dimensions
Rows
Non-zero entries
Summaries
Trace
Log determinant
3.4701
-11.95
3.5234
-11.91
Eigenvalues
> 0
= 0
max abs
min abs non-zero
Condition number
4
2
2.972
0.0009
3467.8
4
2
3.0176
0.0009
3513.4
Norms
Frobenius
Infinity
3.0124
3.7072
3.0587
3.7647
Comparisons
Concordance correlation
Discrepancy function
Frobenius norm of difference
Trace(Adjusted Inv(MBased))
0.9999
0.0003
0.0463
4.0352
Determinant and inversion results apply to the nonsingular
partitions of the covariance matrices.
The “Comparisons” statistics show that the model-based and adjusted covariance matrix of the
fixed-effects parameter estimates are very similar. The concordance correlation is near 1, the discrepancy is near zero, and the trace is very close to the number of positive eigenvalues. This is due
to the balanced nature of these repeated measures data. Shrinkage of standard errors, however, can
not occur with the Kackar-Harville-Jeske estimator.
2362 F Chapter 38: The GLIMMIX Procedure
Example 38.9: Testing Equality of Covariance and Correlation Matrices
Fisher’s iris data are widely used in multivariate statistics. They comprise measurements in millimeters of four flower attributes, the length and width of sepals and petals for 50 specimens from
each of three species, Iris setosa, I. versicolor, and I. virginica (Fisher 1936).
When modeling multiple attributes from the same specimen, correlations among measurements
from the same flower must be taken into account. Unstructured covariance matrices are common
in this multivariate setting. Species comparisons can focus on comparisons of mean response, but
comparisons of the variation and covariation are also of interest. In this example, the equivalence
of covariance and correlation matrices among the species are examined.
The following DATA step creates the iris data in multivariate format—that is, each observation
contains multiple response variables. The subsequent DATA step creates a data set in univariate
form, where each observation corresponds to a single response variable. This is the form needed by
the GLIMMIX procedure.
proc format;
value specname
1=’Setosa
’
2=’Versicolor’
3=’Virginica ’;
value variable
1=’Sepal Length’
2=’Sepal Width ’
3=’Petal Length’
4=’Petal Width’;
run;
data iris;
input Y1 Y2 Y3 Y4 Species @@;
format Species specname.;
label Y1 = ’Sepal Length (mm)’
Y2 = ’Sepal Width (mm)’
Y3 = ’Petal Length (mm)’
Y4 = ’Petal Width (mm)’;
datalines;
50 33 14 02 1 64 28 56 22 3 65 28
63 28 51 15 3 46 34 14 03 1 69 31
59 32 48 18 2 46 36 10 02 1 61 30
65 30 52 20 3 56 25 39 11 2 65 30
68 32 59 23 3 51 33 17 05 1 57 28
77 38 67 22 3 63 33 47 16 2 67 33
49 25 45 17 3 55 35 13 02 1 67 30
64 32 45 15 2 61 28 40 13 2 48 31
55 24 38 11 2 63 25 50 19 3 64 32
49 36 14 01 1 54 30 45 15 2 79 38
67 33 57 21 3 50 35 16 06 1 58 26
77 28 67 20 3 63 27 49 18 3 47 32
50 23 33 10 2 72 32 60 18 3 48 30
61 30 49 18 3 48 34 19 02 1 50 30
46
51
46
55
45
57
52
16
53
64
40
16
14
16
15
23
14
18
13
25
23
02
23
20
12
02
03
02
2
3
2
3
2
3
3
1
3
3
2
1
1
1
67
62
60
58
62
76
70
59
52
44
44
55
51
50
31
22
27
27
34
30
32
30
34
32
30
26
38
32
56
45
51
51
54
66
47
51
14
13
13
44
16
12
24
15
16
19
23
21
14
18
02
02
02
12
02
02
3
2
2
3
3
3
2
3
1
1
1
2
1
1
Example 38.9: Testing Equality of Covariance and Correlation Matrices F 2363
61
51
51
46
50
57
71
49
49
66
44
47
74
56
49
56
51
54
61
68
45
55
51
63
;
26
38
35
32
36
29
30
24
31
29
29
32
28
28
31
30
25
39
29
30
23
23
37
33
56
19
14
14
14
42
59
33
15
46
14
13
61
49
15
41
30
13
47
55
13
40
15
60
14
04
02
02
02
13
21
10
02
13
02
02
19
20
01
13
11
04
14
21
03
13
04
25
3
1
1
1
1
2
3
2
1
2
1
1
3
3
1
2
2
1
2
3
1
2
1
3
64
67
56
60
77
72
64
56
77
52
50
46
59
60
67
63
57
51
56
55
57
66
52
53
28
31
30
29
30
30
31
27
26
27
20
31
30
22
31
25
28
35
29
25
25
30
35
37
56
44
45
45
61
58
55
42
69
39
35
15
42
40
47
49
41
14
36
40
50
44
15
15
21
14
15
15
23
16
18
13
23
14
10
02
15
10
15
15
13
03
13
13
20
14
02
02
3
2
2
2
3
3
3
2
3
2
2
1
2
2
2
2
2
1
2
2
3
2
1
1
43
62
58
57
63
54
60
57
60
60
55
69
51
73
63
61
65
72
69
48
57
68
58
30
28
27
26
34
34
30
30
22
34
24
32
34
29
23
28
30
36
31
34
38
28
28
11
48
41
35
56
15
48
42
50
45
37
57
15
63
44
47
58
61
49
16
17
48
51
01
18
10
10
24
04
18
12
15
16
10
23
02
18
13
12
22
25
15
02
03
14
24
1
3
2
2
3
1
3
2
3
2
2
3
1
3
2
2
3
3
2
1
1
2
3
58
49
50
57
58
52
63
55
54
50
58
62
50
67
54
64
69
65
64
48
51
54
67
40
30
34
44
27
41
29
42
39
34
27
29
35
25
37
29
31
32
27
30
38
34
30
12
14
16
15
51
15
56
14
17
15
39
43
13
58
15
43
54
51
53
14
15
17
50
02
02
04
04
19
01
18
02
04
02
12
13
03
18
02
13
21
20
19
01
03
02
17
1
1
1
1
3
1
3
1
1
1
2
2
1
3
1
2
3
3
3
1
1
1
2
data iris_univ;
set iris;
retain id 0;
array y{4};
format var variable.;
id+1;
do var=1 to 4;
response = y{var};
output;
end;
drop y:;
run;
The following GLIMMIX statements fit a model with separate unstructured covariance matrices for
each species:
ods select FitStatistics CovParms CovTests;
proc glimmix data=iris_univ;
class species var id;
model response = species*var;
random _residual_ / type=un group=species subject=id;
covtest homogeneity;
run;
2364 F Chapter 38: The GLIMMIX Procedure
The mean function is modeled as a cell-means model that allows for different means for each species
and outcome variable. The covariances are modeled directly (R-side) rather than through random
effects. The ID variable identifies the individual plant, so that responses from different plants are
independent. The GROUP=SPECIES option varies the parameters of the unstructured covariance
matrix by species. Hence, this model has 30 covariance parameters: 10 unique parameters for a
.4 4/ covariance matrix for each of three species.
The COVTEST statement requests a test of homogeneity—that is, it tests whether varying the covariance parameters by the group effect provides a significantly better fit compared to a model in
which different groups share the same parameter.
Output 38.9.1 Fit Statistics for Analysis of Fisher’s Iris Data
The GLIMMIX Procedure
Fit Statistics
-2 Res Log Likelihood
AIC (smaller is better)
AICC (smaller is better)
BIC (smaller is better)
CAIC (smaller is better)
HQIC (smaller is better)
Generalized Chi-Square
Gener. Chi-Square / DF
2812.89
2872.89
2876.23
2963.21
2993.21
2909.58
588.00
1.00
The “Fit Statistics” table shows the 2 restricted (residual) log likelihood in the full model and
other fit statistics (Output 38.9.1). The “-2 Res Log Likelihood” sets the benchmark against which
a model with homogeneity constraint is compared. Output 38.9.2 displays the 30 covariance parameters in this model.
There appear to be substantial differences among the covariance parameters from different groups.
For example, the residual variability of the petal length of the three species is 12.4249, 26.6433,
and 40.4343, respectively. The homogeneity hypothesis restricts these variances to be equal and
similarly for the other covariance parameters. The results from the COVTEST statement are shown
in Output 38.9.3.
Example 38.9: Testing Equality of Covariance and Correlation Matrices F 2365
Output 38.9.2 Covariance Parameters Varied by Species (TYPE=UN)
Covariance Parameter Estimates
Cov
Parm
Subject
Group
UN(1,1)
UN(2,1)
UN(2,2)
UN(3,1)
UN(3,2)
UN(3,3)
UN(4,1)
UN(4,2)
UN(4,3)
UN(4,4)
UN(1,1)
UN(2,1)
UN(2,2)
UN(3,1)
UN(3,2)
UN(3,3)
UN(4,1)
UN(4,2)
UN(4,3)
UN(4,4)
UN(1,1)
UN(2,1)
UN(2,2)
UN(3,1)
UN(3,2)
UN(3,3)
UN(4,1)
UN(4,2)
UN(4,3)
UN(4,4)
id
id
id
id
id
id
id
id
id
id
id
id
id
id
id
id
id
id
id
id
id
id
id
id
id
id
id
id
id
id
Species
Species
Species
Species
Species
Species
Species
Species
Species
Species
Species
Species
Species
Species
Species
Species
Species
Species
Species
Species
Species
Species
Species
Species
Species
Species
Species
Species
Species
Species
Setosa
Setosa
Setosa
Setosa
Setosa
Setosa
Setosa
Setosa
Setosa
Setosa
Versicolor
Versicolor
Versicolor
Versicolor
Versicolor
Versicolor
Versicolor
Versicolor
Versicolor
Versicolor
Virginica
Virginica
Virginica
Virginica
Virginica
Virginica
Virginica
Virginica
Virginica
Virginica
Estimate
Standard
Error
12.4249
9.9216
14.3690
1.6355
1.1698
3.0159
1.0331
0.9298
0.6069
1.1106
26.6433
8.5184
9.8469
18.2898
8.2653
22.0816
5.5780
4.1204
7.3102
3.9106
40.4343
9.3763
10.4004
30.3290
7.1380
30.4588
4.9094
4.7629
4.8824
7.5433
2.5102
2.3775
2.9030
0.9052
0.9552
0.6093
0.5508
0.5859
0.2755
0.2244
5.3828
2.6144
1.9894
4.3398
2.4149
4.4612
1.6617
1.0641
1.6891
0.7901
8.1690
3.2213
2.1012
6.6262
2.7395
6.1536
2.5916
1.4367
2.2750
1.5240
Output 38.9.3 Likelihood Ratio Test of Homogeneity
Tests of Covariance Parameters
Based on the Restricted Likelihood
Label
DF
-2 Res Log Like
ChiSq
Pr > ChiSq
Homogeneity
20
2959.55
146.66
<.0001
DF: P-value based on a chi-square with DF degrees of freedom.
Note
DF
2366 F Chapter 38: The GLIMMIX Procedure
Denote as Rk the covariance matrix for species k D 1; 2; 3 with elements ij k . In processing the
COVTEST hypothesis H0 W R1 D R2 D R3 , the GLIMMIX procedure fits a model that satistfies
the constraints
111 D 112 D 113
211 D 212 D 213
231 D 232 D 233
::
:
441 D 442 D 443
where ij k is the covariance between the i th and j th variable for the kth species. The 2 restricted
log likelihood of this restricted model is 2959.55 (Output 38.9.3). The change of 146.66 compared
to the full model is highly significant. There is sufficient evidence to reject the notion of equal
covariance matrices among the three iris species.
Equality of covariance matrices implies equality of correlation matrices, but the reverse is not true.
Fewer constraints are needed to equate correlations because the diagonal entries of the covariance
matrices are free to vary. In order to test the equality of the correlation matrices among the three
species, you can parameterize the unstructured covariance matrix in terms of the correlations and
use a COVTEST statement with general contrasts, as shown in the following statements:
ods select FitStatistics CovParms CovTests;
proc glimmix data=iris_univ;
class species var id;
model response = species*var;
random _residual_ / type=unr group=species subject=id;
covtest ’Equal Covariance Matrices’ homogeneity;
covtest ’Equal Correlation Matrices’ general
0 0 0 0 1 0 0 0 0 0
0 0 0 0 -1 0 0 0 0 0,
0 0 0 0 1 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 -1 0 0 0 0 0,
0 0 0 0 0 1 0 0 0 0
0 0 0 0 0 -1 0 0 0 0,
0 0 0 0 0 1 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 -1 0 0 0 0,
0 0 0 0 0 0 1 0 0 0
0 0 0 0 0 0 -1 0 0 0,
0 0 0 0 0 0 1 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 -1 0 0 0,
0 0 0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 -1 0 0,
0 0 0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 -1 0 0,
0 0 0 0 0 0 0 0 1 0
0 0 0 0 0 0 0 0 -1 0,
0 0 0 0 0 0 0 0 1 0
Example 38.9: Testing Equality of Covariance and Correlation Matrices F 2367
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0 0
-1 0,
0 1
0 -1,
0 1
0 0
0 -1 / estimates;
run;
The TYPE=UNR structure is a reparameterization of TYPE=UN. The models provide the same
fit, as seen by comparison of the “Fit Statistics” tables in Output 38.9.1 and Output 38.9.4. The
covariance parameters are ordered differently, however. In each group, the four variances precede
the six correlations (Output 38.9.4). The first COVTEST statement tests the homogeneity hypothesis in terms of the UNR parameterization, and the result is identical to the test in Output 38.9.3.
The second COVTEST statement restricts the correlations to be equal across groups. If ij k is the
correlation between the i th and j th variable for the kth species, the 12 restrictions are
211 D 212 D 213
311 D 312 D 313
321 D 322 D 323
411 D 412 D 413
421 D 422 D 423
431 D 432 D 433
The ESTIMATES option in the COVTEST statement requests that the GLIMMIX procedure display
the covariance parameter estimates in the restricted model (Output 38.9.4).
Output 38.9.4 Fit Statistics, Covariance Parameters (TYPE=UNR), and Likelihood Ratio Tests
for Equality of Covariance and Correlation Matrices
The GLIMMIX Procedure
Fit Statistics
-2 Res Log Likelihood
AIC (smaller is better)
AICC (smaller is better)
BIC (smaller is better)
CAIC (smaller is better)
HQIC (smaller is better)
Generalized Chi-Square
Gener. Chi-Square / DF
2812.89
2872.89
2876.23
2963.21
2993.21
2909.58
588.00
1.00
2368 F Chapter 38: The GLIMMIX Procedure
Output 38.9.4 continued
Covariance Parameter Estimates
Cov Parm
Subject
Group
Var(1)
Var(2)
Var(3)
Var(4)
Corr(2,1)
Corr(3,1)
Corr(3,2)
Corr(4,1)
Corr(4,2)
Corr(4,3)
Var(1)
Var(2)
Var(3)
Var(4)
Corr(2,1)
Corr(3,1)
Corr(3,2)
Corr(4,1)
Corr(4,2)
Corr(4,3)
Var(1)
Var(2)
Var(3)
Var(4)
Corr(2,1)
Corr(3,1)
Corr(3,2)
Corr(4,1)
Corr(4,2)
Corr(4,3)
id
id
id
id
id
id
id
id
id
id
id
id
id
id
id
id
id
id
id
id
id
id
id
id
id
id
id
id
id
id
Species
Species
Species
Species
Species
Species
Species
Species
Species
Species
Species
Species
Species
Species
Species
Species
Species
Species
Species
Species
Species
Species
Species
Species
Species
Species
Species
Species
Species
Species
Setosa
Setosa
Setosa
Setosa
Setosa
Setosa
Setosa
Setosa
Setosa
Setosa
Versicolor
Versicolor
Versicolor
Versicolor
Versicolor
Versicolor
Versicolor
Versicolor
Versicolor
Versicolor
Virginica
Virginica
Virginica
Virginica
Virginica
Virginica
Virginica
Virginica
Virginica
Virginica
Estimate
Standard
Error
12.4249
14.3690
3.0159
1.1106
0.7425
0.2672
0.1777
0.2781
0.2328
0.3316
26.6433
9.8469
22.0816
3.9106
0.5259
0.7540
0.5605
0.5465
0.6640
0.7867
40.4343
10.4004
30.4588
7.5433
0.4572
0.8642
0.4010
0.2811
0.5377
0.3221
2.5102
2.9030
0.6093
0.2244
0.06409
0.1327
0.1383
0.1318
0.1351
0.1271
5.3828
1.9894
4.4612
0.7901
0.1033
0.06163
0.09797
0.1002
0.07987
0.05445
8.1690
2.1012
6.1536
1.5240
0.1130
0.03616
0.1199
0.1316
0.1015
0.1280
Example 38.9: Testing Equality of Covariance and Correlation Matrices F 2369
Output 38.9.4 continued
Tests of Covariance Parameters
Based on the Restricted Likelihood
Label
DF
-2 Res Log Like
ChiSq
Pr > ChiSq
--Estim
ates
H0-Est1
Equal Covariance Matrices
Equal Correlation Matrices
20
12
2959.55
2876.38
146.66
63.49
<.0001
<.0001
26.5
16.5
Tests of Covariance Parameters
Based on the Restricted Likelihood
--------------------Estimates H0-------------------Est2
Est3
Est4
Est5
Est6
Est7
Est8
Label
Equal Covariance Matrices
Equal Correlation Matrices
11.5
14.9
18.52
4.84
4.19
1.44
0.530
0.561
0.756
0.683
0.378
0.402
0.365
0.384
Tests of Covariance Parameters
Based on the Restricted Likelihood
-------------------Estimates H0-------------------Est9
Est10
Est11 Est12
Est13 Est14 Est15
Label
Equal Covariance Matrices
Equal Correlation Matrices
0.471
0.498
0.484
0.522
26.5
24.4
11.54
9.16
18.5
17.4
4.19
3.00
0.530
0.561
Tests of Covariance Parameters
Based on the Restricted Likelihood
--------------------Estimates H0-------------------Est16
Est17
Est18
Est19
Est20
Est21 Est22
Label
Equal Covariance Matrices
Equal Correlation Matrices
0.756
0.683
0.378
0.402
0.365
0.384
0.471
0.498
0.484
0.522
26.5
35.1
11.5
10.8
Tests of Covariance Parameters
Based on the Restricted Likelihood
--------------------Estimates H0-------------------Est23 Est24 Est25
Est26
Est27
Est28
Est29
Label
Equal Covariance Matrices
Equal Correlation Matrices
18.5
27.4
4.19
8.14
0.530
0.561
0.756
0.683
0.378
0.402
Tests of Covariance Parameters
Based on the Restricted Likelihood
Label
Equal Covariance Matrices
Equal Correlation Matrices
--Estima
tes
H0--Est30
0.484
0.522
Note
DF
DF
DF: P-value based on a chi-square with DF degrees of freedom.
0.365
0.384
0.471
0.498
2370 F Chapter 38: The GLIMMIX Procedure
The result of the homogeneity test is identical to that in Output 38.9.3. The hypothesis of equality of
the correlation matrices is also rejected with a chi-square value of 63:49 and a p-value of < 0:0001.
Notice, however, that the chi-square statistic is smaller than in the test of homogeneity due to the
smaller number of restrictions imposed on the full model. The estimate of the common correlation
matrix in the restricted model is
2
3
1
0:561 0:683 0:384
6 0:561 1
0:402 0:498 7
6
7
4 0:683 0:402 1
0:522 5
0:384 0:498 0:522 1
Example 38.10: Multiple Trends Correspond to Multiple Extrema in
Profile Likelihoods
The following data set contains observations for a period of 168 months for the “Southern Oscillation Index,” measurements of monthly averaged atmospheric pressure differences between Easter
Island and Darwin, Australia (Kahaner, Moler, and Nash 1989, Ch. 11.9; National Institute of Standards and Technology 1998). These data are also used as an example in Chapter 50, “The LOESS
Procedure,” in the SAS/STAT User’s Guide.
data ENSO;
input Pressure @@;
Year = _n_ / 12;
datalines;
12.9 11.3 10.6 11.2
12.9 14.3 10.9 13.7
5.7
5.5
7.6
8.6
12.7 12.9 13.0 10.9
13.6 10.5
9.2 12.4
4.8
3.0
2.5
6.3
10.5 13.3 10.4
8.1
10.9 11.7 11.4 13.7
9.6 11.7
5.0 10.8
15.7 12.6 14.8
7.8
5.2 12.0 10.2 12.7
5.7
6.7
3.9
8.5
12.5 12.5
9.8
7.2
11.9 13.6 16.3 17.6
14.3 14.5
8.5 12.0
10.4 11.5 13.4
7.5
4.6
8.2
9.9
9.2
7.6
9.5
8.4 10.7
16.8 17.1 15.4
9.5
11.2 16.6 15.6 12.0
8.6
8.6
8.7 12.8
;
10.9
17.1
7.3
10.4
12.7
9.7
3.7
14.1
12.7
7.1
10.2
8.3
4.1
15.5
12.7
0.6
12.5
13.6
6.1
11.5
13.2
7.5
14.0
7.6
10.2
13.3
11.6
10.7
14.0
10.8
11.2
14.7
10.8
10.6
16.0
11.3
0.3
10.9
13.7
10.1
8.6
14.0
7.7
15.3
12.7
8.0
10.1
8.6
5.1
12.5
11.8
8.1
12.2
16.7
10.1
15.2
14.5
5.5
9.9
13.7
9.3
13.8
13.4
11.7
8.5
11.0
10.9
7.8
12.4
10.4
6.3
12.6
6.4
7.1
12.6
10.1
11.2
15.1
5.0
8.9
16.5
5.3
8.7
14.8
Differences in atmospheric pressure create wind, and the differences recorded in the data set ENSO
drive the trade winds in the southern hemisphere. Such time series often do not consist of a single
trend or cycle. In this particular case, there are at least two known cycles that reflect the annual
Example 38.10: Multiple Trends Correspond to Multiple Extrema in Profile Likelihoods F 2371
weather pattern and a longer cycle that represents the periodic warming of the Pacific Ocean (El
Niño).
To estimate the trend in these data by using mixed model technology, you can apply a mixed model
smoothing technique such as TYPE=RSMOOTH or TYPE=PSPLINE. The following statements
fit a radial smoother to the ENSO data and obtain profile likelihoods for a series of values for the
variance of the random spline coefficients:
data tdata;
do covp1=0,0.0005,0.05,0.1,0.2,0.5,
1,2,3,4,5,6,8,10,15,20,50,
75,100,125,140,150,160,175,
200,225,250,275,300,350;
output;
end;
run;
ods select FitStatistics CovParms CovTests;
proc glimmix data=enso noprofile;
model pressure = year;
random year / type=rsmooth knotmethod=equal(50);
parms (2) (10);
covtest tdata=tdata / parms;
ods output covtests=ct;
run;
The tdata data set contains value for the variance of the radial smoother variance for which the
profile likelihood of the model is to be computed. The profile likelihood is obtained by setting the
radial smoother variance at the specified value and estimating all other parameters subject to that
constraint.
Because the model contains a residual variance and you need to specify nonzero values for the
first covariance parameter, the NOPROFILE is added to the PROC GLIMMIX statements. If the
residual variance is profiled from the estimation, you cannot fix covariance parameters at a given
value, because they would be reexpressed during model fitting in terms of ratios with the profiled
(and changing) variance.
The PARMS statement determines starting values for the covariance parameters for fitting the (full)
model. The PARMS option in the COVTEST statement requests that the input parameters be added
to the output and the output data set. This is useful for subsequent plotting of the profile likelihood
function.
2372 F Chapter 38: The GLIMMIX Procedure
The “Fit Statistics” table displays the 2 restricted log likelihood of the model (897:76,
Output 38.10.1). The estimate of the variance of the radial smoother coefficients is 3:5719.
The “Test of Covariance Parameters” table displays the 2 restricted log likelihood for each observation in the tdata set. Because the tdata data set specifies values for only the first covariance
parameter, the second covariance parameter is free to vary and the values for 2 Res Log Like are
profile likelihoods. Notice that for a number of values of CovP1 the chi-square statistic is missing in
this table. For these values the 2 Res Log Like is smaller than that of the full model. The model
did not converge to a global minimum of the negative restricted log likelihood.
Output 38.10.1 REML and Profile Likelihood Analysis
The GLIMMIX Procedure
Fit Statistics
-2 Res Log Likelihood
AIC (smaller is better)
AICC (smaller is better)
BIC (smaller is better)
CAIC (smaller is better)
HQIC (smaller is better)
Generalized Chi-Square
Gener. Chi-Square / DF
Radial Smoother df(res)
897.76
901.76
901.83
897.76
899.76
897.76
1554.38
9.36
153.52
Covariance Parameter Estimates
Cov Parm
Var[RSmooth(Year)]
Residual
Estimate
Standard
Error
3.5719
9.3638
3.7672
1.3014
Example 38.10: Multiple Trends Correspond to Multiple Extrema in Profile Likelihoods F 2373
Output 38.10.1 continued
Tests of Covariance Parameters
Based on the Restricted Likelihood
Label
WORK.TDATA
WORK.TDATA
WORK.TDATA
WORK.TDATA
WORK.TDATA
WORK.TDATA
WORK.TDATA
WORK.TDATA
WORK.TDATA
WORK.TDATA
WORK.TDATA
WORK.TDATA
WORK.TDATA
WORK.TDATA
WORK.TDATA
WORK.TDATA
WORK.TDATA
WORK.TDATA
WORK.TDATA
WORK.TDATA
WORK.TDATA
WORK.TDATA
WORK.TDATA
WORK.TDATA
WORK.TDATA
WORK.TDATA
WORK.TDATA
WORK.TDATA
WORK.TDATA
WORK.TDATA
DF
-2 Res Log Like
ChiSq
Pr > ChiSq
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
893.01
892.76
897.34
898.53
899.38
899.49
898.83
898.04
897.79
897.77
897.86
897.99
898.27
898.49
898.70
898.45
892.63
887.44
883.79
881.55
880.72
.
880.07
879.85
.
880.21
880.80
881.56
882.44
884.41
.
.
.
0.77
1.62
1.73
1.07
0.28
0.03
0.01
0.10
0.23
0.51
0.73
0.94
0.69
.
.
.
.
.
.
.
.
.
.
.
.
.
.
1.0000
1.0000
1.0000
0.3816
0.2038
0.1888
0.3016
0.5967
0.8693
0.9145
0.7517
0.6311
0.4761
0.3919
0.3318
0.4068
1.0000
1.0000
1.0000
1.0000
1.0000
.
1.0000
1.0000
.
1.0000
1.0000
1.0000
1.0000
1.0000
------Input
Parameters---CovP1
CovP2
0.0000
0.0005
0.0500
0.1000
0.2000
0.5000
1.0000
2.0000
3.0000
4.0000
5.0000
6.0000
8.0000
10.0000
15.0000
20.0000
50.0000
75.0000
100.0000
125.0000
140.0000
150.0000
160.0000
175.0000
200.0000
225.0000
250.0000
275.0000
300.0000
350.0000
9.36
9.36
9.36
9.36
9.36
9.36
9.36
9.36
9.36
9.36
9.36
9.36
9.36
9.36
9.36
9.36
9.36
9.36
9.36
9.36
9.36
9.36
9.36
9.36
9.36
9.36
9.36
9.36
9.36
9.36
DF: P-value based on a chi-square with DF degrees of freedom.
MI: P-value based on a mixture of chi-squares.
The following statements plot the 2 restricted profile log likelihood (Output 38.10.2):
proc sgplot data=ct;
series y=objective x=covp1;
run;
Note
MI
DF
DF
DF
DF
DF
DF
DF
DF
DF
DF
DF
DF
DF
DF
DF
DF
DF
DF
DF
DF
DF
DF
DF
DF
DF
DF
DF
2374 F Chapter 38: The GLIMMIX Procedure
Output 38.10.2
2 Restricted Profile Log Likelihood for Smoothing Variance
The local minimum at which the optimization stopped is clearly visible, as are a second local minimum near zero and the global minimum near 180.
The observed and predicted pressure differences that correspond to the three minima are shown in
Output 38.10.3. These results were produced with the following statements:
proc glimmix data=enso;
model pressure = year;
random year / type=rsmooth knotmethod=equal(50);
parms (0) (10);
output out=gmxout1 pred=pred1;
run;
proc glimmix data=enso;
model pressure = year;
random year / type=rsmooth knotmethod=equal(50);
output out=gmxout2 pred=pred2;
parms (2) (10);
run;
proc glimmix data=enso;
model pressure = year;
random year / type=rsmooth knotmethod=equal(50);
Example 38.10: Multiple Trends Correspond to Multiple Extrema in Profile Likelihoods F 2375
output out=gmxout3 pred=pred3;
parms (200) (10);
run;
data plotthis; merge gmxout1 gmxout2 gmxout3;
run;
proc sgplot data=plotthis;
scatter x=year y=Pressure;
series x=year y=pred1 /
lineattrs
= (pattern=solid thickness=2)
legendlabel = "Var[RSmooth] = 0.0005"
name
= "pred1";
series x=year y=pred2 /
lineattrs
= (pattern=dot thickness=2)
legendlabel = "Var[RSmooth] = 3.5719"
name
= "pred2";
series x=year y=pred3 /
lineattrs
= (pattern=dash thickness=2)
legendlabel = "Var[RSmooth] = 186.71"
name
= "pred3";
keylegend "pred1" "pred2" "pred3" / across=2;
run;
Output 38.10.3 Observed and Predicted Pressure Differences
2376 F Chapter 38: The GLIMMIX Procedure
The one-year cycle (b
2r D 186:71) and the El Niño cycle (b
2r D 3:5719) are clearly visible. Notice
that a larger smoother variance results in larger BLUPs and hence larger adjustments to the fixedeffects model. A large smoother variance thus results in a more wiggly fit. The third local minimum
at b
2r D 0:0005 applies only very small adjustments to the linear regression between pressure and
time, creating slight curvature.
Example 38.11: Maximum Likelihood in Proportional Odds Model with
Random Effects
The data for this example are taken from Gilmour, Anderson, and Rae (1987) and concern the
foot shape of 2,513 lambs that represent 34 sires. The foot shape of the animals was scored in
three ordered categories. The following DATA step lists the data in multivariate form, where each
observation corresponds to a sire and contains the outcomes for the three response categories in the
variables k1, k2, and k3. For example, for the first sire the first foot shape category was observed for
52 of its offspring, foot shape category 2 was observed for 25 lambs, and none of its offspring was
rated in foot shape category 3. The variables yr, b1, b2, and b3 represent contrasts of fixed effects.
data foot_mv;
input yr b1
sire = _n_;
datalines;
1 1 0 0
1 1 0 0
1 1 0 0
1 1 0 0
1 1 0 0
1 1 0 0
1 1 0 0
1 -1 1 0
1 -1 1 0
1 -1 1 0
1 -1 1 0
1 -1 1 0
1 -1 1 0
1 -1 -1 0
1 -1 -1 0
1 -1 -1 0
-1 0 0 1
-1 0 0 1
-1 0 0 1
-1 0 0 1
-1 0 0 1
-1 0 0 1
-1 0 0 -1
-1 0 0 -1
-1 0 0 -1
-1 0 0 -1
-1 0 0 -1
b2 b3 k1 k2 k3;
52
49
50
42
74
54
96
57
55
70
70
82
75
17
13
21
37
47
46
79
50
63
30
31
28
42
35
25
17
13
9
15
8
12
52
27
36
37
21
19
12
23
17
41
24
25
32
23
18
20
33
18
27
22
0
1
1
0
0
0
0
9
5
4
3
1
0
10
3
3
23
12
9
11
5
8
9
3
4
4
2
Example 38.11: Maximum Likelihood in Proportional Odds Model with Random Effects F 2377
-1
-1
-1
-1
-1
-1
-1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
-1
-1
-1
-1
-1
-1
-1
33
35
26
37
36
63
41
18
17
13
15
14
20
8
3
4
2
2
1
3
1
;
In order to analyze these data as multinomial data with PROC GLIMMIX, the data need to be
arranged in univariate form. The following DATA step creates three observations from each record
in data set foot_mv and stores the category counts in the variable count:
data footshape; set foot_mv;
array k{3};
do Shape = 1 to 3;
count = k{Shape};
output;
end;
drop k:;
run;
Because the sires were selected at random, we consider here a model for the three-category response
with fixed regression effects for yr, b1–b3, and with random sire effects. Because the response
categories are ordered, a proportional odds model is chosen (McCullagh 1980). Gilmour, Anderson,
and Rae (1987) consider various analyses for these data. The following GLIMMIX statements fit a
model with probit link for the cumulative probabilities by maximum likelihood where the marginal
log likelihood is approximated by adaptive quadrature:
proc glimmix data=footshape method=quad;
class sire;
model Shape = yr b1 b2 b3 / s link=cumprobit dist=multinomial;
random int / sub=sire s cl;
ods output Solutionr=solr;
freq count;
run;
The number of observations that share a particular response and covariate pattern (variable count) is
used in the FREQ statement. The S and CL options request solutions for the sire effects. These are
output to the data set solr for plotting.
The “Model Information” table shows that the parameters are estimated by maximum likelihood
and that the marginal likelihood is approximated by Gauss-Hermite quadrature (Output 38.11.1).
2378 F Chapter 38: The GLIMMIX Procedure
Output 38.11.1 Model and Data Information
The GLIMMIX Procedure
Model Information
Data Set
Response Variable
Response Distribution
Link Function
Variance Function
Frequency Variable
Variance Matrix Blocked By
Estimation Technique
Likelihood Approximation
Degrees of Freedom Method
Number
Number
Sum of
Sum of
WORK.FOOTSHAPE
Shape
Multinomial (ordered)
Cumulative Probit
Default
count
sire
Maximum Likelihood
Gauss-Hermite Quadrature
Containment
of Observations Read
of Observations Used
Frequencies Read
Frequencies Used
102
96
2513
2513
Response Profile
Ordered
Value
1
2
3
Shape
1
2
3
Total
Frequency
1636
731
146
The GLIMMIX procedure is modeling the probabilities of levels of
Shape having lower Ordered Values in the Response Profile table.
The distribution of the data is multinomial with ordered categories. The ordering is implied by the
choice of a link function for the cumulative probabilities. Because a frequency variable is specified,
the number of observations as well as the number of frequencies is displayed. Observations with
zero frequency—that is, foot shape categories that were not observed for a particular sire are not
used in the analysis. The “Response Profile Table” shows the ordering of the response variable and
gives a breakdown of the frequencies by category.
Output 38.11.2 Information about the Size of the Optimization Problem
Dimensions
G-side Cov. Parameters
Columns in X
Columns in Z per Subject
Subjects (Blocks in V)
Max Obs per Subject
1
6
1
34
3
Example 38.11: Maximum Likelihood in Proportional Odds Model with Random Effects F 2379
Output 38.11.2 continued
Optimization Information
Optimization Technique
Parameters in Optimization
Lower Boundaries
Upper Boundaries
Fixed Effects
Starting From
Quadrature Points
Dual Quasi-Newton
7
1
0
Not Profiled
GLM estimates
1
With METHOD=QUAD, the “Dimensions” and “Optimization Information” tables are particularly
important, because for this estimation methods both fixed effects and covariance parameters participate in the optimization (Output 38.11.2). For GLM models the optimization involves the fixed
effects and possibly a single scale parameter. For mixed models the fixed effects are typically
profiled from the optimization. Laplace and quadrature estimations are exceptions to these rules.
Consequently, there are seven parameters in this optimization, corresponding to six fixed effects and
one variance component. The variance component has a lower bound of 0. Also, because the fixed
effects are part of the optimizations, PROC GLIMMIX initially performs a few GLM iterations to
obtain starting values for the fixed effects. You can control the number of initial iterations with the
INITITER= option in the PROC GLIMMIX statement.
The last entry in the “Optimization Information” table shows that—at the starting values—PROC
GLIMMIX determined that a single quadrature point is sufficient to approximate the marginal log
likelihood with the required accuracy. This approximation is thus identical to the Laplace method
that is available with METHOD=LAPLACE.
For METHOD=LAPLACE and METHOD=QUAD, the GLIMMIX procedure produces fit statistics based on the conditional and marginal distribution (Output 38.11.3). Within the limits of the
numeric likelihood approximation, the information criteria shown in the “Fit Statistics” table can be
used to compare models, and the 2 log likelihood can be used to compare among nested models
(nested with respect to fixed effects and/or the covariance parameters).
Output 38.11.3 Marginal and Conditional Fit Statistics
Fit Statistics
-2 Log Likelihood
AIC (smaller is better)
AICC (smaller is better)
BIC (smaller is better)
CAIC (smaller is better)
HQIC (smaller is better)
3870.12
3884.12
3884.17
3894.81
3901.81
3887.76
Fit Statistics for Conditional
Distribution
-2 log L(Shape | r. effects)
3807.62
2380 F Chapter 38: The GLIMMIX Procedure
The variance of the sire effect is estimated as 0:04849 with estimated asymptotic standard error of
0:01673 (Output 38.11.4). Based on the magnitude of the estimate relative to the standard error,
one might conclude that there is significant sire-to-sire variability. Because parameter estimation
is based on maximum likelihood, a formal test of the hypothesis of no sire variability is possible.
The category cutoffs for the cumulative probabilities are 0:3781 and 1:6435. Except for b3, all fixed
effects contrasts are significant.
Output 38.11.4 Parameter Estimates
Covariance Parameter Estimates
Cov Parm
Subject
Intercept
sire
Estimate
Standard
Error
0.04849
0.01673
Solutions for Fixed Effects
Effect
Shape
Estimate
Standard
Error
DF
t Value
Pr > |t|
Intercept
Intercept
yr
b1
b2
b3
1
2
0.3781
1.6435
0.1422
0.3781
0.3157
-0.09887
0.04907
0.05930
0.04834
0.07154
0.09709
0.06508
29
29
2478
2478
2478
2478
7.71
27.72
2.94
5.28
3.25
-1.52
<.0001
<.0001
0.0033
<.0001
0.0012
0.1289
A likelihood ratio test for the sire variability can be carried out by adding a COVTEST statement to
the PROC GLIMMIX statements (Output 38.11.5):
ods select FitStatistics CovParms Covtests;
proc glimmix data=footshape method=quad;
class sire;
model Shape = yr b1 b2 b3 / link=cumprobit dist=multinomial;
random int / sub=sire;
covtest GLM;
freq count;
run;
The statement
covtest GLM;
compares the fitted model to a generalized linear model for independent data by removing the sire
variance component from the model. Equivalently, you can specify
covtest 0;
which compares the fitted model against one where the sire variance is fixed at zero.
Example 38.11: Maximum Likelihood in Proportional Odds Model with Random Effects F 2381
Output 38.11.5 Likelihood Ratio Test for Sire Variance
The GLIMMIX Procedure
Fit Statistics
-2 Log Likelihood
AIC (smaller is better)
AICC (smaller is better)
BIC (smaller is better)
CAIC (smaller is better)
HQIC (smaller is better)
3870.12
3884.12
3884.17
3894.81
3901.81
3887.76
Covariance Parameter Estimates
Cov Parm
Subject
Intercept
sire
Estimate
Standard
Error
0.04849
0.01673
Tests of Covariance Parameters
Based on the Likelihood
Label
Independence
DF
-2 Log Like
ChiSq
Pr > ChiSq
1
3915.29
45.17
<.0001
Note
MI
MI: P-value based on a mixture of chi-squares.
The 2 Log Likelihood in the reduced model without the sire effect is 3915:29. Compared to the
corresponding marginal fit statistic in the full model (3870:12), this results in a chi-square statistic
of 45.17. Because the variance component for the sire effect has a natural lower bound of zero,
PROC GLIMMIX performs the likelihood ratio test as a one-sided test. As indicated by the note,
the p-value for this test is computed from a mixture of chi-square distributions, applying the results
of Self and Liang (1987). There is significant evidence that the model without sire random effects
does not fit the data as well.
In studies of heritability, one is often interested to rank individuals according to some measure of
“breeding value.” The following statements display the empirical Bayes estimates of the sire effects
from ML estimation by quadrature along with prediction standard error bars (Output 38.11.6):
proc sort data=solr; by Estimate;
data solr; set solr;
length sire $2;
obs = _n_;
sire = left(substr(Subject,6,2));
run;
proc sgplot data=solr;
scatter x=obs y=estimate /
markerchar = sire
yerrorupper = upper
yerrorlower = lower;
xaxis grid label=’Sire Rank’ values=(1 5 10 15 20 25 30);
yaxis grid label=’Predicted Sire Effect’;
run;
2382 F Chapter 38: The GLIMMIX Procedure
Output 38.11.6 Ranked Predicted Sire Effects and Prediction Standard Errors
Example 38.12: Fitting a Marginal (GEE-Type) Model
A marginal GEE-type model for clustered data is a model for correlated data that is specified through
a mean function, a variance function, and a “working” covariance structure. Because the assumed
covariance structure can be wrong, the covariance matrix of the parameter estimates is not based on
the model alone. Rather, one of the empirical (“sandwich”) estimators is used to make inferences robust against the choice of working covariance structure. PROC GLIMMIX can fit marginal models
by using R-side random effects and drawing on the distributional specification in the MODEL statement to derive the link and variance functions. The EMPIRICAL= option in the PROC GLIMMIX
statement enables you to choose one of a number of empirical covariance estimators.
The data for this example are from Thall and Vail (1990) and reflect the number of seizures of
patients suffering fom epileptic episodes. After an eight-week period without treatment, patients
were observed four times in two-week intervals during which they received a placebo or the drug
Progabide in addition to other therapy. These data are also analyzed in Example 37.7 of Chapter 37,
“The GENMOD Procedure.” The following DATA step creates the data set seizures. The variable
id identifies the subjects in the study, and the variable trt identifies whether a subject received the
placebo (trt = 0) or the drug Progabide (trt = 1). The variable x1 takes on value 0 for the baseline
Example 38.12: Fitting a Marginal (GEE-Type) Model F 2383
measurement and 1 otherwise.
data seizures;
array c{5};
input id trt c1-c5;
do i=1 to 5;
x1
= (i > 1);
ltime = (i=1)*log(8) + (i ne 1)*log(2);
cnt
= c{i};
output;
end;
keep id cnt x1 trt ltime;
datalines;
101 1 76 11 14 9 8
102 1 38 8 7 9 4
103 1 19 0 4 3 0
104 0 11 5 3 3 3
106 0 11 3 5 3 3
107 0
6 2 4 0 5
108 1 10 3 6 1 3
110 1 19 2 6 7 4
111 1 24 4 3 1 3
112 1 31 22 17 19 16
113 1 14 5 4 7 4
114 0
8 4 4 1 4
116 0 66 7 18 9 21
117 1 11 2 4 0 4
118 0 27 5 2 8 7
121 1 67 3 7 7 7
122 1 41 4 18 2 5
123 0 12 6 4 0 2
124 1
7 2 1 1 0
126 0 52 40 20 23 12
128 1 22 0 2 4 0
129 1 13 5 4 0 3
130 0 23 5 6 6 5
135 0 10 14 13 6 0
137 1 46 11 14 25 15
139 1 36 10 5 3 8
141 0 52 26 12 6 22
143 1 38 19 7 6 7
145 0 33 12 6 8 4
147 1
7 1 1 2 3
201 0 18 4 4 6 2
202 0 42 7 9 12 14
203 1 36 6 10 8 8
204 1 11 2 1 0 0
205 0 87 16 24 10 9
206 0 50 11 0 0 5
208 1 22 4 3 2 4
209 1 41 8 6 5 7
210 0 18 0 0 3 3
211 1 32 1 3 1 5
213 0 111 37 29 28 29
2384 F Chapter 38: The GLIMMIX Procedure
214
215
217
218
219
220
221
222
225
226
227
228
230
232
234
236
238
;
1
0
0
1
0
0
1
0
1
0
0
1
0
1
0
1
0
56 18 11 28 13
18 3 5 2 5
20 3 0 6 7
24 6 3 4 0
12 3 4 3 4
9 3 4 3 4
16 3 5 4 3
17 2 3 3 5
22 1 23 19 8
28 8 12 2 8
55 18 24 76 25
25 2 3 0 1
9 2 1 2 1
13 0 0 0 0
10 3 1 4 2
12 1 4 3 2
47 13 15 13 12
The model fit initially with the following PROC GLIMMIX statements is a Poisson generalized
linear model with effects for an intercept, the baseline measurement, the treatment, and their interaction:
proc glimmix data=seizures;
model cnt = x1 trt x1*trt / dist=poisson offset=ltime
ddfm=none s;
run;
The DDFM=NONE option is chosen in the MODEL statement to produce chi-square and z tests
instead of F and t tests.
Because the initial pretreatment time period is four times as long as the subsequent measurement
intervals, an offset variable is used to standardize the counts. If Yij denotes the number of seizures
of subject i in time interval j of length tj , then Yij =tj is the number of seizures per time unit.
Modeling the average number per time unit with a log link leads to logfEŒYij =tj g D x0 ˇ or
logfEŒYij g D x0 ˇ C logftj g. The logarithm of time (variable ltime) thus serves as an offset. Suppose that ˇ0 denotes the intercept, ˇ1 the effect of x1, and ˇ2 the effect of trt. Then expfˇ0 g is the
expected number of seizures per week in the placebo group at baseline. The corresponding numbers
in the treatment group are expfˇ0 C ˇ2 g at baseline and expfˇ0 C ˇ1 C ˇ2 g for postbaseline visits.
The “Model Information” table shows that the parameters in this Poisson model are estimated by
maximum likelihood (Output 38.12.1). In addition to the default link and variance function, the
variable ltime is used as an offset.
Example 38.12: Fitting a Marginal (GEE-Type) Model F 2385
Output 38.12.1 Model Information in Poisson GLM
The GLIMMIX Procedure
Model Information
Data Set
Response Variable
Response Distribution
Link Function
Variance Function
Offset Variable
Variance Matrix
Estimation Technique
Degrees of Freedom Method
WORK.SEIZURES
cnt
Poisson
Log
Default
ltime
Diagonal
Maximum Likelihood
None
Fit statistics and parameter estimates are shown in Output 38.12.2.
Output 38.12.2 Results from Fitting Poisson GLM
Fit Statistics
-2 Log Likelihood
AIC (smaller is better)
AICC (smaller is better)
BIC (smaller is better)
CAIC (smaller is better)
HQIC (smaller is better)
Pearson Chi-Square
Pearson Chi-Square / DF
3442.66
3450.66
3450.80
3465.34
3469.34
3456.54
3015.16
10.54
Parameter Estimates
Effect
Intercept
x1
trt
x1*trt
Estimate
Standard
Error
DF
t Value
Pr > |t|
1.3476
0.1108
-0.1080
-0.3016
0.03406
0.04689
0.04865
0.06975
Infty
Infty
Infty
Infty
39.57
2.36
-2.22
-4.32
<.0001
0.0181
0.0264
<.0001
Because this is a generalized linear model, the large value for the ratio of the Pearson chi-square
statistic and its degrees of freedom is indicative of a model shortcoming. The data are considerably more dispersed than is expected under a Poisson model. There could be many reasons for
this overdispersion—for example, a misspecified mean model, data that might not be Poisson distributed, an incorrect variance function, and correlations among the observations. Because these
data are repeated measurements, the presence of correlations among the observations from the same
subject is a likely contributor to the overdispersion.
The following PROC GLIMMIX statements fit a marginal model with correlations. The model is
a marginal one, because no G-side random effects are specified on which the distribution could be
conditioned. The choice of the id variable as the SUBJECT effect indicates that observations from
2386 F Chapter 38: The GLIMMIX Procedure
different IDs are uncorrelated. Observations from the same ID are assumed to follow a compound
symmetry (equicorrelation) model. The EMPIRICAL option in the PROC GLIMMIX statement
requests the classical sandwich estimator as the covariance estimator for the fixed effects:
proc glimmix data=seizures empirical;
class id;
model cnt = x1 trt x1*trt / dist=poisson offset=ltime
ddfm=none covb s;
random _residual_ / subject=id type=cs vcorr;
run;
The “Model Information” table shows that the parameters are now estimated by residual pseudolikelihood (compare Output 38.12.3 and Output 38.12.1). And in this fact lies the main difference
between fitting marginal models with PROC GLIMMIX and with GEE methods as per Liang and
Zeger (1986), where parameters of the working correlation matrix are estimated by the method of
moments.
Output 38.12.3 Model Information in Marginal Model
The GLIMMIX Procedure
Model Information
Data Set
Response Variable
Response Distribution
Link Function
Variance Function
Offset Variable
Variance Matrix Blocked By
Estimation Technique
Degrees of Freedom Method
Fixed Effects SE Adjustment
WORK.SEIZURES
cnt
Poisson
Log
Default
ltime
id
Residual PL
None
Sandwich - Classical
According to the compound symmetry model, there is substantial correlation among the observations from the same subject (Output 38.12.4).
Output 38.12.4 Covariance Parameter Estimates and Correlation Matrix
Estimated V Correlation Matrix for id 101
Row
Col1
Col2
Col3
Col4
Col5
1
2
3
4
5
1.0000
0.6055
0.6055
0.6055
0.6055
0.6055
1.0000
0.6055
0.6055
0.6055
0.6055
0.6055
1.0000
0.6055
0.6055
0.6055
0.6055
0.6055
1.0000
0.6055
0.6055
0.6055
0.6055
0.6055
1.0000
Example 38.12: Fitting a Marginal (GEE-Type) Model F 2387
Output 38.12.4 continued
Covariance Parameter Estimates
Cov Parm
Subject
CS
Residual
id
Estimate
Standard
Error
6.4653
4.2128
1.3833
0.3928
The parameter estimates in Output 38.12.5 are the same as in the Poisson generalized linear model
(Output 38.12.2), because of the balance in these data. The standard errors have increased substantially, however, by taking into account the correlations among the observations.
Output 38.12.5 GEE-Type Inference for Fixed Effects
Solutions for Fixed Effects
Effect
Intercept
x1
trt
x1*trt
Estimate
Standard
Error
DF
t Value
Pr > |t|
1.3476
0.1108
-0.1080
-0.3016
0.1574
0.1161
0.1937
0.1712
Infty
Infty
Infty
Infty
8.56
0.95
-0.56
-1.76
<.0001
0.3399
0.5770
0.0781
Empirical Covariance Matrix for Fixed Effects
Effect
Intercept
x1
trt
x1*trt
Row
Col1
Col2
Col3
Col4
1
2
3
4
0.02476
-0.00115
-0.02476
0.001152
-0.00115
0.01348
0.001152
-0.01348
-0.02476
0.001152
0.03751
-0.00300
0.001152
-0.01348
-0.00300
0.02931
2388 F Chapter 38: The GLIMMIX Procedure
Example 38.13: Response Surface Comparisons with Multiplicity
Adjustments
Koch et al. (1990) present data for a multicenter clinical trial testing the efficacy of a respiratory
drug in patients with respiratory disease. Within each of two centers, patients were randomly assigned to a placebo (P) or an active (A) treatment. Prior to treatment and at four follow-up visits,
patient status was recorded in one of five ordered categories (0=terrible, 1=poor, . . . , 4=excellent).
The following DATA step creates the SAS data set clinical for this study.
data Clinical;
do Center =
1, 2;
do Gender = ’F’,’M’;
do Drug
= ’A’,’P’;
input nPatient @@;
do iPatient = 1 to nPatient;
input ID Age (t0-t4) (1.) @@;
output;
end;
end; end; end;
datalines;
2 53 32 12242 18 47 22344
5
5 13 44444 19 31 21022 25 35 10000
36 45 22221
25 54 11 44442 12 14 23332 51 15 02333
16 22 12223 50 22 21344
3 23 33443
56 25 23323 35 26 12232 26 26 22222
8 28 12212 30 28 00121 33 30 33442
42 31 12311
9 31 33444 37 31 02321
6 34 11211 22 46 43434 24 48 23202
48 57 33434
24 43 13 34444 41 14 22123 34 15 22332
15 20 44444 13 23 33111 27 23 44244
17 25 11222 45 26 24243 40 26 12122
49 27 33433 39 23 21111
2 28 20000
31 37 10000 10 37 32332
7 43 23244
4 44 34342
1 46 22222 46 49 22222
4 30 37 13444 52 39 23444 23 60 44334
12 28 31 34444
5 32 32234 21 36 33213
1 39 12112 48 39 32300
7 44 34444
8 48 22100 11 48 22222
4 51 34244
23 12 13 44444 10 14 14444 27 19 33233
16 20 21100 29 21 33444 20 24 44444
15 25 34433
2 25 22444
9 26 23444
55 31 44444 43 34 24424 26 35 44444
36 41 34434 51 43 33442 37 52 12122
32 55 22331
3 58 44444 53 68 23334
16 39 11 34444 40 14 21232 24 15 32233
33 19 42233 34 20 32444 13 20 14444
22 36 24334 18 38 43000 35 42 32222
6 45 34212 46 48 44000 31 52 23434
;
28 36 23322
20
32
21
11
23
38
20
23
26
30
32
50
33231
23444
24142
34443
34433
22222
29
55
44
14
52
47
54
50
38
17
47
25
49
14
19
19
24
27
30
43
63
63
38
47
58
20
25
28
37
55
23300
34443
12212
10000
11132
22222
44444
12000
23323
14220
24443
34331
23221
43224
44444
41
45
44
42
15
33
43
66
43334
33323
21000
33344
Example 38.13: Response Surface Comparisons with Multiplicity Adjustments F 2389
Westfall and Tobias (2007) define as the measure of efficacy the average of the ratings at the final
two visits and model this average as a function of drug, baseline assessment score, and age. Hence,
in their model, the expected efficacy for drug d 2 A; P can be written as
E ŒYd  D ˇ0d C ˇ1d t C ˇ2d a
where t is the baseline (pretreatment) assessment score and a is the patient’s age at baseline. The
age range for these data extends from 11 to 68 years. Suppose that the scientific question of interest
is the comparison of the two response surfaces at a set of values St Sa D f0; 1; 2; 3; 4g Sa . In
other words, we would like to know for which values of the covariates the average response differs
significantly between the treatment group and the placebo group. If the set of ages of interest is
f10; 13; 16; ; 70g, then this involves 5 21 D 105 comparisons, a massive multiple testing problem. The large number of comparisons and the fact that the set Sa is chosen somewhat arbitrarily
require the application of multiplicity corrections in order to protect the familywise Type I error
across the comparisons.
When testing hypotheses that have logical restrictions, the power of multiplicity corrected tests can
be increased by taking the restrictions into account. Logical restrictions exist, for example, when not
all hypotheses in a set can be simultaneously true. Westfall and Tobias (2007) extend the truncated
closed testing procedure (TCTP) of Royen (1989) for pairwise comparisons in ANOVA to general
contrasts. Their work is also an extension of the S2 method of Shaffer (1986); see also Westfall
(1997). These methods are all monotonic in the (unadjusted) p-values of the individual tests, in the
sense that if pj < pi then the multiple test will never retain Hj while rejecting Hi . In terms of
multiplicity-adjusted p-values pQj , monotonicity means that if pj < pi , then pQj < pQi .
Analysis as Normal Data with Averaged Endpoints
In order to apply the extended TCTP procedure of Westfall and Tobias (2007) to the problem of
comparing response surfaces in the clinical trial, the following convenience macro is helpful to
generate the comparisons for the ESTIMATE statement in PROC GLIMMIX:
%macro Contrast(from,to,byA,byT);
%let nCmp = 0;
%do age = &from %to &to %by &byA;
%do t0 = 0 %to 4 %by &byT;
%let nCmp = %eval(&nCmp+1);
%end;
%end;
%let iCmp = 0;
%do age = &from %to &to %by &byA;
%do t0 = 0 %to 4 %by &byT;
%let iCmp = %eval(&iCmp+1);
"%trim(%left(&age)) %trim(%left(&t0))"
drug
1
-1
drug*age &age -&age
drug*t0 &t0 -&t0
%if (&icmp < &nCmp) %then %do; , %end;
%end;
%end;
%mend;
2390 F Chapter 38: The GLIMMIX Procedure
The following GLIMMIX statements fit the model to the data and compute the 105 contrasts that
compare the placebo to the active response at 105 points in the two-dimensional regressor space:
proc glimmix data=clinical;
t = (t3+t4)/2;
class drug;
model t = drug t0 age drug*age drug*t0;
estimate %contrast(10,70,3,1)
/ adjust=simulate(seed=1)
stepdown(type=logical);
ods output Estimates=EstStepDown;
run;
Note that only a single ESTIMATE statement is used. Each of the 105 comparisons is one
comparison in the multirow statement. The ADJUST option in the ESTIMATE statement requests multiplicity-adjusted p-values. The extended TCTP method is applied by specifying the
STEPDOWN(TYPE=LOGICAL) option to compute step-down-adjusted p-values where logical
constraints among the hypotheses are taken into account. The results from the ESTIMATE statement are saved to a data set for subsequent processing. Note also that the response, the average of
the ratings at the final two visits, is computed with programming statements in PROC GLIMMIX.
The following statements print the 20 most significant estimated differences (Output 38.13.1):
proc sort data=EstStepDown;
by Probt;
proc print data=EstStepDown(obs=20);
var Label Estimate StdErr Probt AdjP;
run;
Output 38.13.1 The First 20 Observations of the Estimates Data Set
Obs
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Label
37
40
34
43
46
31
49
52
40
37
43
28
55
46
34
43
46
49
40
58
2
2
2
2
2
2
2
2
3
3
3
2
2
3
3
1
1
3
1
2
Estimate
StdErr
Probt
Adjp
0.8310
0.8813
0.7806
0.9316
0.9819
0.7303
1.0322
1.0825
0.7755
0.7252
0.8258
0.6800
1.1329
0.8761
0.6749
1.0374
1.0877
0.9264
0.9871
1.1832
0.2387
0.2553
0.2312
0.2794
0.3093
0.2338
0.3434
0.3807
0.2756
0.2602
0.2982
0.2461
0.4202
0.3265
0.2532
0.3991
0.4205
0.3591
0.3827
0.4615
0.0007
0.0008
0.0010
0.0012
0.0020
0.0023
0.0033
0.0054
0.0059
0.0063
0.0066
0.0068
0.0082
0.0085
0.0089
0.0107
0.0111
0.0113
0.0113
0.0118
0.0071
0.0071
0.0071
0.0071
0.0071
0.0081
0.0107
0.0167
0.0200
0.0201
0.0201
0.0215
0.0239
0.0239
0.0257
0.0329
0.0329
0.0329
0.0329
0.0329
Example 38.13: Response Surface Comparisons with Multiplicity Adjustments F 2391
Notice that the adjusted p-values (Adjp) are larger than the unadjusted p-values, as expected. Also
notice that several comparisons share the same adjusted p-values. This is a result of the monotonicity of the extended TCTP method.
In order to compare the step-down-adjusted p-values to adjusted p-values that do not use step-down
methods, replace the ESTIMATE statement in the previous statements with the following:
estimate %contrast2(10,70,3,1) / adjust=simulate(seed=1);
ods output Estimates=EstAdjust;
The following GLIMMIX invocations create output data sets named EstAdjust and EstUnAdjust
that contain (non-step-down-) adjusted and unadjusted p-values:
proc glimmix data=clinical;
t = (t3+t4)/2;
class drug;
model t = drug t0 age drug*age drug*t0;
estimate %contrast(10,70,3,1)
/ adjust=simulate(seed=1);
ods output Estimates=EstAdjust;
run;
proc glimmix data=clinical;
t = (t3+t4)/2;
class drug;
model t = drug t0 age drug*age drug*t0;
estimate %contrast(10,70,3,1);
ods output Estimates=EstUnAdjust;
run;
Output 38.13.2 shows a comparison of the significant comparisons (p < 0.05) based on unadjusted,
adjusted, and step-down (TCTP) adjusted p-values. Clearly, the unadjusted results indicate the most
significant results, but without protecting the Type I error rate for the group of tests. The adjusted
p-values (filled circles) lead to a much smaller region in which the response surfaces between treatment and placebo are significantly different. The increased power of the TCTP procedure (open
circles) over the standard multiplicity adjustment—without sacrificing Type I error protection—can
be seen in the considerably larger region covered by the open circles.
2392 F Chapter 38: The GLIMMIX Procedure
Output 38.13.2 Comparison of Significance Regions
Ordinal Repeated Measure Analysis
The outcome variable in this clinical trial is an ordinal rating of patients in categories 0=terrible,
1=poor, 2=fair, 3=good, and 4=excellent. Furthermore, the observations from repeat visits for the
same patients are likely correlated. The previous analysis removes the repeated measures aspect
by defining efficacy as the average score at the final two visits. These averages are not normally
distributed, however. The response surfaces for the two study arms can also be compared based on
a model for ordinal data that takes correlation into account through random effects. Keeping with
the theme of the previous analysis, the focus here for illustrative purposes is on the final two visits,
and the pretreatment assessment score serves as a covariate in the model.
The following DATA step rearranges the data from the third and fourth on-treatment visits in univariate form with separate observations for the visits by patient:
data clinical_uv;
set clinical;
array time{2} t3-t4;
do i=1 to 2; rating = time{i}; output; end;
run;
Example 38.13: Response Surface Comparisons with Multiplicity Adjustments F 2393
The basic model for the analysis is a proportional odds model with cumulative logit link (McCullagh
1980) and J D 5 categories. In this model, separate intercepts (cutoffs) are modeled for the first
J 1 D 4 cumulative categories and the intercepts are monotonically increasing. This guarantees
ordering of the cumulative probabilities and nonnegative category probabilities. Using the same
covariate structure as in the previous analysis, the probability to observe a rating in at most category
k 4 is
1
1 C expf kd g
D ˛k C ˇ0d C ˇ1d t C ˇ2d a
Pr.Yd k/ D
kd
Because only the intercepts are dependent on the category, contrasts comparing regression coefficients can be formulated in standard fashion. To accommodate the random and covariance structure
of the repeated measures model, a random intercept i is applied to the observations for each patient:
1
1 C expf ikd g
D ˛k C ˇ0d C ˇ1d t C ˇ2d a C i
Pr.Yid k/ D
ikd
i iid N.0; 2 /
The shared random effect of the two observations creates a marginal correlation. Note that the
random effects do not depend on category.
The following GLIMMIX statements fit this ordinal repeated measures model by maximum likelihood via the Laplace approximation and compute TCTP-adjusted p-values for the 105 estimates:
proc glimmix data=clinical_uv method=laplace;
class center id drug;
model rating = drug t0 age drug*age drug*t0 /
dist=multinomial link=cumlogit;
random intercept / subject=id(center);
covtest 0;
estimate %contrast(10,70,3,1)
/ adjust=simulate(seed=1)
stepdown(type=logical);
ods output Estimates=EstStepDownMulti;
run;
The combination of DIST=MULTINOMIAL and LINK=CUMLOGIT requests the proportional
odds model. The SUBJECT= effect nests patient IDs within centers, because patient IDs in the
data set clinical are not unique within centers. (Specifying SUBJECT=ID*CENTER would have the
same effect.) The COVTEST statement requests a likelihood ratio test for the significance of the
random patient effect.
The estimate of the variance component for the random patient effect is substantial (Output 38.13.3),
but so is its standard error.
2394 F Chapter 38: The GLIMMIX Procedure
Output 38.13.3 Model and Covariance Parameter Information
The GLIMMIX Procedure
Model Information
Data Set
Response Variable
Response Distribution
Link Function
Variance Function
Variance Matrix Blocked By
Estimation Technique
Likelihood Approximation
Degrees of Freedom Method
WORK.CLINICAL_UV
rating
Multinomial (ordered)
Cumulative Logit
Default
ID(Center)
Maximum Likelihood
Laplace
Containment
Covariance Parameter Estimates
Cov Parm
Subject
Intercept
ID(Center)
Estimate
Standard
Error
10.3483
3.2599
Tests of Covariance Parameters
Based on the Likelihood
Label
Parameter list
DF
-2 Log Like
ChiSq
Pr > ChiSq
1
604.70
57.64
<.0001
Note
MI
MI: P-value based on a mixture of chi-squares.
The likelihood ratio test provides a better picture of the significance of the variance component. The
difference in the 2 log likelihoods is 57.6, highly significant even if one does not apply the Self
and Liang (1987) correction that halves the p-value in this instance.
The results for the 20 most significant estimates are requested with the following statements and
shown in Output 38.13.4:
proc sort data=EstStepDownMulti;
by Probt;
proc print data=EstStepDownMulti(obs=20);
var Label Estimate StdErr Probt AdjP;
run;
The p-values again show the “repeat” pattern corresponding to the monotonicity of the step-down
procedure.
Example 38.13: Response Surface Comparisons with Multiplicity Adjustments F 2395
Output 38.13.4 The First 20 Estimates in the Ordinal Analysis
Obs
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Label
37
40
34
43
46
31
49
52
40
37
28
43
46
55
34
49
43
58
40
46
2
2
2
2
2
2
2
2
3
3
2
3
3
2
3
3
1
2
1
1
Estimate
StdErr
Probt
Adjp
-2.7224
-2.8857
-2.5590
-3.0491
-3.2124
-2.3957
-3.3758
-3.5391
-2.6263
-2.4630
-2.2323
-2.7897
-2.9530
-3.7025
-2.2996
-3.1164
-3.3085
-3.8658
-3.1451
-3.4718
0.8263
0.8842
0.7976
0.9659
1.0660
0.8010
1.1798
1.3037
0.9718
0.9213
0.8362
1.0451
1.1368
1.4351
0.8974
1.2428
1.3438
1.5722
1.2851
1.4187
0.0013
0.0015
0.0018
0.0021
0.0032
0.0034
0.0051
0.0077
0.0080
0.0087
0.0088
0.0088
0.0107
0.0112
0.0118
0.0136
0.0154
0.0155
0.0160
0.0160
0.0133
0.0133
0.0133
0.0133
0.0133
0.0133
0.0164
0.0236
0.0267
0.0267
0.0278
0.0278
0.0291
0.0324
0.0337
0.0344
0.0448
0.0448
0.0448
0.0448
As previously, the comparisons were also performed with standard p-value adjustment via simulation. Output 38.13.5 displays the components of the regressor space in which the response surfaces
differ significantly (p < 0:05) between the two treatment arms. As before, the most significant
differences occur with unadjusted p-values at the cost of protecting only the individual Type I error
rate. The standard multiplicity adjustment has considerably less power than the TCTP adjustment.
2396 F Chapter 38: The GLIMMIX Procedure
Output 38.13.5 Comparison of Significance Regions, Ordinal Analysis
Example 38.14: Generalized Poisson Mixed Model for Overdispersed
Count Data
Overdispersion is the condition by which data appear more dispersed than is expected under a reference model. For count data, the reference models are typically based on the binomial or Poisson
distributions. Among the many reasons for overdispersion are an incorrect model, an incorrect distributional specification, incorrect variance functions, positive correlation among the observations,
and so forth. In short, correcting an overdispersion problem, if it exists, requires the appropriate remedy. Adding an R-side scale parameter to multiply the variance function is not necessarily
the adequate correction. For example, Poisson-distributed data appear overdispersed relative to a
Poisson model with regressors when an important regressor is omitted.
If the reference model for count data is Poisson, a number of alternative model formulations are
available to increase the dispersion. For example, zero-inflated models add a proportion of zeros
(usually from a bernoulli process) to the zeros of a Poisson process. Hurdle models are two-part
models where zeros and nonzeros are generated by different stochastic processes. Zero-inflated
and hurdle models are described in detail by Cameron and Trivedi (1998) and cannot be fit with the
GLIMMIX procedure. See Section 15.5 in Littell et al. (2006) for examples of using the NLMIXED
Example 38.14: Generalized Poisson Mixed Model for Overdispersed Count Data F 2397
procedure to fit zero-inflated and hurdle models.
An alternative approach is to derive from the reference distribution a probability distribution that
exhibits increased dispersion. By mixing a Poisson process with a gamma distribution for the Poisson parameter, for example, the negative binomial distribution results, which is thus overdispersed
relative to the Poisson.
Joe and Zhu (2005) show that the generalized Poisson distribution can also be motivated as a Poisson
mixture and hence provides an alternative to the negative binomial (NB) distribution. Like the
NB, the generalized Poisson distribution has a scale parameter. It is heavier in the tails than the
NB distribution and easily reduces to the standard Poisson. Joe and Zhu (2005) discuss further
comparisons between these distributions.
The probability mass function of the generalized Poisson is given by
p.y/ D
˛
.˛ C y/y
yŠ
1
exp f ˛
yg
where y D 0; 1; 2; , ˛ > 0, and 0 < 1 (Joe and Zhu 2005). Notice that for D 0 the mass
function of the standard Poisson distribution with mean ˛ results. The mean and variance of Y in
terms of the parameters ˛ and are given by
EŒY  D
˛
D
˛
D
VarŒY  D
3
.1 /
.1 /2
1
The log likelihood of the generalized Poisson can thus be written in terms of the mean and scale
parameter as
l.; I y/ D log f.1
.
.
/g C .y
y//
1/ log f
.
y/g
log f€.y C 1/g
The data in the following DATA step are simulated counts. For each of i D 1; ; 30 subjects
a randomly varying number ni of observations were drawn from a count regression model with a
single covariate and excess zeros (compared to a Poisson distribution).
data counts;
input ni @@;
sub = _n_;
do i=1 to ni;
input x y @@;
output;
end;
datalines;
1 29 0
6 2 0 82 5 33 0 15
19 81 0 18 0 85 0 99
3 0 60 0 87 2 80
9 18 0 64 0 80 0 0
15 91 0 2 1 14 0 5
98 0 94 0 23 1
2
0
0
0
2
35
20
75
58
27
0 79 0
0 26 2 29 0 91 2 37 0 39 0
0 3 0 63 1
0 7 0 81 0 22 3 50 0
1 8 1 95 0 76 0 62 0 26 2
9 1 33 0
9 0 72 1
2398 F Chapter 38: The GLIMMIX Procedure
2 34 0
18 48 1
63 0
13 28 1
41 0
9 42 0
3 64 0
4 5 0
2 0 0
20 21 0
34 0
2 66 1
5 83 7
17 29 5
50 4
17 47 0
6 0
7 91 0
14 60 0
93 0
16 68 0
82 0
19 48 3
66 0
8 34 1
13 11 0
64 0
9 3 0
7 2 0
18 73 1
51 2
17 96 0
11 0
13 59 0
88 0
15 66 0
38 1
12 84 6
95
5
27
31
0
0 47 0 44 0 27 0 88 0 27 0 68 0 84 0 86 0 44 0 90 0
0 47 0 25 0 72 0 62 1
0 63 0 14 0 74 0 44 0 75 0 65 0 74 1 84 0 57 0 29 0
8
64
73
41
58
7
13
98
79
97
57
17
25
87
3
26
34
48
86
44
27
0
1
2
0
0
0
0
1
0
0
0
0
1
0
0
1
0
1
2
0
2
11
39
37
33
30
51
94
1
2
2
0
0
4
0
28
47
51
47
70
20
29
0
2
0
0
1
0
0
18
80
45
2
99
61
41
19
30
43
63
62
16
0
0
2
0
0
1
60
81
54
10
21
12
1
0
0
0
0
3
93 3 65 0 16 0 79 0 14 0
45
92
7
52
9 53 0 14 0 92 5 21 1 20 0 73 0 99 0
14 44 1 74 0
0 17 0 0 2 49 0
1 55 0 2 6 89 5 31 5 28 3 51 5 54 13
36
80
66
47
40
29
96
0
0
0
0
0
0
0
57
41
10
60
34
70
47
0
1
0
1
0
2
1
77
20
42
55
59
87
64
0
0
0
4
0
0
0
41
2
22
83
12
47
18
0
0
0
3
1
0
0
91 0 20 0 23 0 22 0 96 0 83 0 56 0
15 0
50 1 13 0
5 0 61 1 28 0 71 0 75 1 94 16 51 4 51 2 74 0
11 0 60 3 31 0 75 0 62 0 54 1
0
1
0
0
0
1
0
45 1 96 1 17 0 91 0
33 0 52 0
60 1 33 1 92 0 38 0
19 0 37 0 78 1 26 0 72 1
1 1
6 0 50 3
83 0 74 0 93 0 36 0 53 0 26 0 86 0
34 0 33 2
78 0 50 0 37 0 15 0 39 0 22 0 82 0
39
27
59
38
47
0
0
9
0
0
3 1 90 0 28 3
55 0 57 0 88 1
40 0
68 0 34 1 96 0 30 0 13 0 35 0
93 0 50 0 39 0 97 0 19 0 54 0
30 0 37 0 36 1 69 0 78 1 47 1 86 0
4 0 22 0
5 2 47 0 38 0 80 0
7 1
6 0 43 3 13 2 18 0 51 0 50 4 68 0
;
The following PROC GLIMMIX statements fit a standard Poisson regression model with random
intercepts by maximum likelihood. The marginal likelihood of the data is approximated by adaptive
quadrature (METHOD=QUAD).
proc glimmix data=counts method=quad;
class sub;
model y = x / link=log s dist=poisson;
random int / subject=sub;
run;
Output 38.14.1 displays various informational items about the model and the estimation process.
Example 38.14: Generalized Poisson Mixed Model for Overdispersed Count Data F 2399
Output 38.14.1 Poisson: Model and Optimization Information
The GLIMMIX Procedure
Model Information
Data Set
Response Variable
Response Distribution
Link Function
Variance Function
Variance Matrix Blocked By
Estimation Technique
Likelihood Approximation
Degrees of Freedom Method
WORK.COUNTS
y
Poisson
Log
Default
sub
Maximum Likelihood
Gauss-Hermite Quadrature
Containment
Optimization Information
Optimization Technique
Parameters in Optimization
Lower Boundaries
Upper Boundaries
Fixed Effects
Starting From
Quadrature Points
Dual Quasi-Newton
3
1
0
Not Profiled
GLM estimates
5
Iteration History
Iteration
Restarts
Evaluations
Objective
Function
Change
Max
Gradient
0
1
2
3
4
5
6
0
0
0
0
0
0
0
4
5
6
2
4
4
3
862.57645728
862.43893582
854.49131023
854.47983504
854.47396189
854.47006558
854.47006484
.
0.13752147
7.94762559
0.01147519
0.00587315
0.00389631
0.00000074
366.7105
22.36158
28.70814
6.036114
4.238363
0.332454
0.003104
The “Model Information” table shows that the parameters are estimated by ML with quadrature.
Using the starting values for fixed effects and covariance parameters that the GLIMMIX procedure
generates by default, the procedure determined that five quadrature nodes provide a sufficiently
accurate approximation of the marginal log likelihood (“Optimization Information” table). The
iterative estimation process converges after nine iterations.
The table of conditional fit statistics displays the sum of the independent contributions to the
conditional 2 log likelihood (854:47) and the Pearson statistics for the conditional distribution
(Output 38.14.2).
2400 F Chapter 38: The GLIMMIX Procedure
Output 38.14.2 Poisson: Fit Statistics and Estimates
Fit Statistics
-2 Log Likelihood
AIC (smaller is better)
AICC (smaller is better)
BIC (smaller is better)
CAIC (smaller is better)
HQIC (smaller is better)
854.47
860.47
860.54
864.67
867.67
861.81
Fit Statistics for Conditional
Distribution
-2 log L(y | r. effects)
Pearson Chi-Square
Pearson Chi-Square / DF
777.90
649.58
1.97
Covariance Parameter Estimates
Cov Parm
Subject
Intercept
sub
Estimate
Standard
Error
1.1959
0.4334
Solutions for Fixed Effects
Effect
Intercept
x
Estimate
Standard
Error
DF
t Value
Pr > |t|
-1.4947
0.01207
0.2745
0.002387
29
299
-5.45
5.06
<.0001
<.0001
The departure of the scaled Pearson statistic from 1.0 is fairly pronounced in this case (1:97). If
one deems it to far from 1.0, however, the conclusion has to be that the conditional variation is not
properly specified. This could be due to an incorrect variance function, for example. The “Solutions
for Fixed Effects” table shows the estimates of the slope and intercept in this model along with their
standard errors and tests of significance. Note that the slope in this model is highly significant. The
variance of the random subject-specific intercepts is estimated as 1:1959.
To fit the generalized Poisson distribution to these data we cannot draw on the built-in distributions.
Instead, the variance function and the log likelihood are computed directly with PROC GLIMMIX
programming statements. The CLASS, MODEL, and RANDOM statements in the following PROC
GLIMMIX program are as before, except for the omission of the DIST= option in the MODEL
statement:
proc glimmix data=counts method=quad;
class sub;
model y = x / link=log s;
random int / subject=sub;
xi = (1 - 1/exp(_phi_));
_variance_ = _mu_ / (1-xi)/(1-xi);
if (_mu_=.) or (_linp_ = .) then _logl_ = .;
Example 38.14: Generalized Poisson Mixed Model for Overdispersed Count Data F 2401
else do;
mustar = _mu_ - xi*(_mu_ - y);
if (mustar < 1E-12) or (_mu_*(1-xi) < 1e-12) then
_logl_ = -1E20;
else do;
_logl_ = log(_mu_*(1-xi)) + (y-1)*log(mustar) mustar - lgamma(y+1);
end;
end;
run;
The assignments to the variables xi and the reserved symbols _VARIANCE_ and _LOGL_ define
the variance function and the log likelihood. Because the scale parameter of the generalized Poisson
distribution has the range 0 < < 1, and the scale parameter _PHI_ in the GLIMMIX procedure
is bounded only from below (by 0), a reparameterization is applied so that D 0 , D 0 and approaches 1 as increases. The statements preceding the calculation of the actual log likelihood
are intended to prevent floating-point exceptions and to trap missing values.
Output 38.14.3 displays information about the model and estimation process. The “Model Information” table shows that the distribution is not a built-in distribution and echoes the expression for
the user-specified variance function. As in the case of the Poisson model, the GLIMMIX procedure determines that five quadrature points are sufficient for accurate estimation of the marginal log
likelihood at the starting values. The estimation process converges after 11 iterations.
Output 38.14.3 Generalized Poisson: Model, Optimization, and Iteration Information
The GLIMMIX Procedure
Model Information
Data Set
Response Variable
Response Distribution
Link Function
Variance Function
Variance Matrix Blocked By
Estimation Technique
Likelihood Approximation
Degrees of Freedom Method
WORK.COUNTS
y
User specified
Log
_mu_ / (1-xi)/(1-xi)
sub
Maximum Likelihood
Gauss-Hermite Quadrature
Containment
Optimization Information
Optimization Technique
Parameters in Optimization
Lower Boundaries
Upper Boundaries
Fixed Effects
Starting From
Quadrature Points
Dual Quasi-Newton
4
2
0
Not Profiled
GLM estimates
5
2402 F Chapter 38: The GLIMMIX Procedure
Output 38.14.3 continued
Iteration History
Iteration
Restarts
Evaluations
Objective
Function
Change
Max
Gradient
0
1
2
3
4
5
6
7
8
9
10
0
0
0
0
0
0
0
0
0
0
0
4
5
4
2
2
2
3
3
3
3
3
716.12976769
716.07585953
714.27148068
711.02643265
710.26952196
709.96824991
709.8419071
709.83122731
709.83047646
709.83046461
709.83046436
.
0.05390816
1.80437884
3.24504804
0.75691069
0.30127205
0.12634280
0.01067980
0.00075085
0.00001185
0.00000025
161.1184
11.88788
36.09657
108.4615
216.9822
96.2775
19.07487
0.649164
2.127665
0.383319
0.010279
The achieved 2 log likelihood is lower than in the Poisson model (compare “Fit Statistics” tables
in Output 38.14.4 and Output 38.14.1). The scaled Pearson statistic is now less than 1.0. The fixed
slope estimate remains significant at the 5% level, but the test statistics are not as large as in the
Poisson model, partly because the generalized Poisson model permits more variation.
Output 38.14.4 Generalized Poisson: Fit Statistics and Estimates
Fit Statistics
-2 Log Likelihood
AIC (smaller is better)
AICC (smaller is better)
BIC (smaller is better)
CAIC (smaller is better)
HQIC (smaller is better)
709.83
717.83
717.95
723.44
727.44
719.62
Fit Statistics for Conditional
Distribution
-2 log L(y | r. effects)
Pearson Chi-Square
Pearson Chi-Square / DF
665.56
241.42
0.73
Covariance Parameter Estimates
Cov Parm
Subject
Intercept
Scale
sub
Estimate
Standard
Error
0.5135
0.6401
0.2400
0.09718
Example 38.14: Generalized Poisson Mixed Model for Overdispersed Count Data F 2403
Output 38.14.4 continued
Solutions for Fixed Effects
Effect
Estimate
Standard
Error
DF
t Value
Pr > |t|
Intercept
x
-0.7264
0.003742
0.2749
0.003537
29
299
-2.64
1.06
0.0131
0.2910
Based on the large difference in the 2 log likelihoods between the Poisson and generalized Poisson
models, we conclude that a mixed model based on the latter provides a better fit to these data. From
the “Covariance Parameter Estimates” table in Output 38.14.4 you can see that the estimate of the
scale parameter is b
D 0:6401 and is considerably larger than 0, taking into account its standard
error. The hypothesis H W D 0, which articulates that a Poisson model fits the data as well as the
generalized Poisson model, can be formally tested with a likelihood ratio test. Adding the statement
covtest ’H: phi = 0’ . 0 / est;
to the previous PROC GLIMMIX run compares the model to one in which the variance of the
random intercepts (the first covariance parameter) is not constrained and the scale parameter is
fixed at zero. This COVTEST statement produces Output 38.14.5.
Output 38.14.5 Likelihood Ratio Test for Poisson Assumption
Tests of Covariance Parameters
Based on the Likelihood
Label
H:phi = 0
DF
-2 Log Like
ChiSq
Pr > ChiSq
1
854.47
144.64
<.0001
--Estimates H0-Est1
Est2
1.2
1.11E-12
Note
MI
MI: P-value based on a mixture of chi-squares.
Note that the 2 Log Like reported in Output 38.14.5 agrees with the value reported in the “Fit
Statistics” table for the Poisson model (Output 38.14.2) and that the estimate of the random intercept
under the null hypothesis agrees with the “Covariance Parameter Estimates” table in Output 38.14.2.
Because the null hypothesis places the parameter (or ) on the boundary of the parameter space, a
mixture correction is applied in the p-value calculation. Because of the magnitude of the likelihood
ratio statistic (144:64), this correction has no effect on the displayed p-value.
2404 F Chapter 38: The GLIMMIX Procedure
Example 38.15: Comparing Multiple B-Splines
This example uses simulated data to demonstrate the use of the nonpositional syntax (see the section “Positional and Nonpositional Syntax for Contrast Coefficients” on page 2262 for details) in
combination with the experimental EFFECT statement to produce interesting predictions and comparisons in models containing fixed spline effects. Consider the data in the following DATA step.
Each of the 100 observations for the continuous response variable y is associated with one of two
groups.
data spline;
input group y @@;
x = _n_;
datalines;
1
-.020 1
0.199
2
-.397 1
0.065
1
0.253 2
-.460
1
0.379 1
0.971
2
0.574 2
0.755
2
1.088 2
0.607
1
0.629 2
1.237
2
1.002 2
1.201
1
1.329 1
1.580
2
1.052 2
1.108
2
1.726 2
1.179
2
2.105 2
1.828
1
1.984 2
1.867
2
1.522 2
2.200
1
2.769 1
2.534
1
2.873 1
2.678
1
2.893 1
3.023
2
2.549 1
2.836
1
3.727 1
3.806
1
2.948 2
1.954
1
3.744 2
2.431
2
1.996 2
2.028
2
2.337 1
4.516
2
2.474 2
2.221
1
5.253 2
3.024
;
2
2
2
1
1
2
2
1
2
2
2
2
1
1
2
1
1
2
1
2
2
2
2
1
2
-1.36
-.861
0.195
0.712
0.316
0.959
0.734
1.520
1.098
1.257
1.338
1.368
2.771
2.562
1.969
3.135
3.050
2.375
3.269
2.326
2.040
2.321
2.326
4.867
2.403
1
1
2
2
2
1
2
1
1
2
1
1
1
1
1
2
2
2
1
2
1
2
2
2
1
-.026
0.251
-.108
0.811
0.961
0.653
0.299
1.105
1.613
2.005
1.707
2.252
2.052
2.517
2.460
1.705
2.273
1.841
3.533
2.017
3.995
2.479
2.144
2.453
5.498
The following statements produce a scatter plot of the response variable by group (Output 38.15.1):
proc sgplot data=spline;
scatter y=y x=x / group=group name="data";
keylegend "data" / title="Group";
run;
Example 38.15: Comparing Multiple B-Splines F 2405
Output 38.15.1 Scatter Plot of Observed Data by Group
The trends in the two groups exhibit curvature, but the type of curvature is not the same in the
groups. Also, there appear to be ranges of x values where the groups are similar and areas where
the point scatters separate. To model the trends in the two groups separately and with flexibility,
you might want to allow for some smooth trends in x that vary by group. Consider the following
PROC GLIMMIX statements:
proc glimmix data=spline outdesign=x;
class group;
effect spl = spline(x);
model y = group spl*group / s noint;
output out=gmxout pred=p;
run;
The EFFECT statement defines a constructed effect named spl by expanding the x into a spline with
seven columns. The group main effect creates separate intercepts for the groups, and the interaction
of the group variable with the spline effect creates separate trends. The NOINT option suppresses
the intercept. This is not necessary and is done here only for convenience of interpretation. The
OUTPUT statement computes predicted values.
The “Parameter Estimates” table contains the estimates of the group-specific “intercepts,” the spline
coefficients varied by group, and the residual variance (“Scale,” Output 38.15.2).
2406 F Chapter 38: The GLIMMIX Procedure
Output 38.15.2 Parameter Estimates in Two-Group Spline Model
The GLIMMIX Procedure
Parameter Estimates
Effect
group
group
spl*group
spl*group
spl*group
spl*group
spl*group
spl*group
spl*group
spl*group
spl*group
spl*group
spl*group
spl*group
spl*group
spl*group
Scale
spl
group
Estimate
Standard
Error
DF
t Value
Pr > |t|
1
1
2
2
3
3
4
4
5
5
6
6
7
7
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
9.7027
6.3062
-11.1786
-20.1946
-9.5327
-5.8565
-8.9612
-5.5567
-7.2615
-4.3678
-6.4462
-4.0380
-4.6382
-4.3029
0
0
0.07352
3.1342
2.6299
3.7008
3.9765
3.2576
2.7906
3.0718
2.5717
3.2437
2.7247
2.9617
2.4589
3.7095
3.0479
.
.
0.01121
86
86
86
86
86
86
86
86
86
86
86
86
86
86
.
.
.
3.10
2.40
-3.02
-5.08
-2.93
-2.10
-2.92
-2.16
-2.24
-1.60
-2.18
-1.64
-1.25
-1.41
.
.
.
0.0026
0.0187
0.0033
<.0001
0.0044
0.0388
0.0045
0.0335
0.0278
0.1126
0.0323
0.1042
0.2146
0.1616
.
.
.
Because the B-spline coefficients for an observation sum to 1 and the model contains group-specific
constants, the last spline coefficient in each group is zero. In other words, you can achieve exactly
the same fit with the MODEL statement
model y = spl*group / noint;
or
model y = spl*group;
The following statements graph the observed and fitted values in the two groups (Output 38.15.3):
proc sgplot data=gmxout;
series y=p x=x / group=group name="fit";
scatter y=y x=x / group=group;
keylegend "fit" / title="Group";
run;
Example 38.15: Comparing Multiple B-Splines F 2407
Output 38.15.3 Observed and Predicted Values by Group
Suppose that you are interested in estimating the mean response at particular values of x and in
performing comparisons of predicted values. The following program uses ESTIMATE statements
with nonpositional syntax to accomplish this:
proc glimmix data=spline;
class group;
effect spl = spline(x);
model y = group spl*group / s noint;
estimate ’Group 1, x=20’ group 1
group*spl [1,1 20] / e;
estimate ’Group 2, x=20’ group 0 1 group*spl [1,2 20];
estimate ’Diff at x=20 ’ group 1 -1 group*spl [1,1 20] [-1,2 20];
run;
The first ESTIMATE statement predicts the mean response at x D 20 in group 1. The E option
requests the coefficient vector for this linear combination of the parameter estimates. The coefficient
for the group effect is entered with positional (standard) syntax. The coefficients for the group*spl
effect are formed based on nonpositional syntax. Because this effect comprises the interaction of a
standard effect (group) with a constructed effect, the values and levels for the standard effect must
precede those for the constructed effect. A similar statement produces the predicted mean at x D 20
in group 2.
2408 F Chapter 38: The GLIMMIX Procedure
The GLIMMIX procedure interprets the syntax
group*spl [1,2 20]
as follows: construct the spline basis at x D 20 as appropriate for group 2; then multiply the
resulting coefficients for these columns of the L matrix with 1.
The final ESTIMATE statement represents the difference between the predicted values; it is a group
comparison at x D 20.
Output 38.15.4 Coefficients from First ESTIMATE Statement
The GLIMMIX Procedure
Coefficients for Estimate
Group 1, x=20
Effect
group
group
spl*group
spl*group
spl*group
spl*group
spl*group
spl*group
spl*group
spl*group
spl*group
spl*group
spl*group
spl*group
spl*group
spl*group
spl
group
1
1
2
2
3
3
4
4
5
5
6
6
7
7
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
Row1
1
0.0021
0.3035
0.619
0.0754
The “Coefficients” table shows how the value 20 supplied in the ESTIMATE statement was expanded into the appropriate spline basis (Output 38.15.4). There is no significant difference between
the group means at x D 20 (p D 0:8346, Output 38.15.5).
Output 38.15.5 Results from ESTIMATE Statements
Estimates
Label
Estimate
Standard
Error
DF
t Value
Pr > |t|
Group 1, x=20
Group 2, x=20
Diff at x=20
0.6915
0.7175
-0.02602
0.09546
0.07953
0.1243
86
86
86
7.24
9.02
-0.21
<.0001
<.0001
0.8346
Example 38.15: Comparing Multiple B-Splines F 2409
The group comparisons you can achieve in this way are comparable to slices of interaction effects
with classification effects. There are, however, no preset number of levels at which to perform the
comparisons because x is continuous. If you add further x values for the comparisons, a multiplicity
correction is in order to control the familywise Type I error. The following statements compare
the groups at values x D 0; 5; 10; ; 80 and compute simulation-based step-down-adjusted pvalues. The results appear in Output 38.15.6. (The numeric results for simulation-based p-value
adjustments depend slightly on the value of the random number seed.)
ods select Estimates;
proc glimmix data=spline;
class group;
effect spl = spline(x);
model y = group spl*group / s;
estimate ’Diff at x= 0’ group 1 -1 group*spl
’Diff at x= 5’ group 1 -1 group*spl
’Diff at x=10’ group 1 -1 group*spl
’Diff at x=15’ group 1 -1 group*spl
’Diff at x=20’ group 1 -1 group*spl
’Diff at x=25’ group 1 -1 group*spl
’Diff at x=30’ group 1 -1 group*spl
’Diff at x=35’ group 1 -1 group*spl
’Diff at x=40’ group 1 -1 group*spl
’Diff at x=45’ group 1 -1 group*spl
’Diff at x=50’ group 1 -1 group*spl
’Diff at x=55’ group 1 -1 group*spl
’Diff at x=60’ group 1 -1 group*spl
’Diff at x=65’ group 1 -1 group*spl
’Diff at x=70’ group 1 -1 group*spl
’Diff at x=75’ group 1 -1 group*spl
’Diff at x=80’ group 1 -1 group*spl
adjust=sim(seed=1) stepdown;
run;
[1,1
[1,1
[1,1
[1,1
[1,1
[1,1
[1,1
[1,1
[1,1
[1,1
[1,1
[1,1
[1,1
[1,1
[1,1
[1,1
[1,1
0]
5]
10]
15]
20]
25]
30]
35]
40]
45]
50]
55]
60]
65]
70]
75]
80]
[-1,2
[-1,2
[-1,2
[-1,2
[-1,2
[-1,2
[-1,2
[-1,2
[-1,2
[-1,2
[-1,2
[-1,2
[-1,2
[-1,2
[-1,2
[-1,2
[-1,2
0],
5],
10],
15],
20],
25],
30],
35],
40],
45],
50],
55],
60],
65],
70],
75],
80] /
2410 F Chapter 38: The GLIMMIX Procedure
Output 38.15.6 Estimates with Multiplicity Adjustments
The GLIMMIX Procedure
Estimates
Adjustment for Multiplicity: Holm-Simulated
Label
Diff
Diff
Diff
Diff
Diff
Diff
Diff
Diff
Diff
Diff
Diff
Diff
Diff
Diff
Diff
Diff
Diff
at
at
at
at
at
at
at
at
at
at
at
at
at
at
at
at
at
x= 0
x= 5
x=10
x=15
x=20
x=25
x=30
x=35
x=40
x=45
x=50
x=55
x=60
x=65
x=70
x=75
x=80
Estimate
Standard
Error
DF
t Value
Pr > |t|
Adj P
12.4124
1.0376
0.3778
0.05822
-0.02602
0.02014
0.1023
0.1924
0.2883
0.3877
0.4885
0.5903
0.7031
0.8401
1.0147
1.2400
1.5237
4.2130
0.1759
0.1540
0.1481
0.1243
0.1312
0.1378
0.1236
0.1114
0.1195
0.1308
0.1231
0.1125
0.1203
0.1348
0.1326
0.1281
86
86
86
86
86
86
86
86
86
86
86
86
86
86
86
86
86
2.95
5.90
2.45
0.39
-0.21
0.15
0.74
1.56
2.59
3.24
3.74
4.79
6.25
6.99
7.52
9.35
11.89
0.0041
<.0001
0.0162
0.6952
0.8346
0.8783
0.4600
0.1231
0.0113
0.0017
0.0003
<.0001
<.0001
<.0001
<.0001
<.0001
<.0001
0.0206
<.0001
0.0545
0.9101
0.9565
0.9565
0.7418
0.2925
0.0450
0.0096
0.0024
<.0001
<.0001
<.0001
<.0001
<.0001
<.0001
There are significant differences at the low end and high end of the x range. Notice that without the
multiplicity adjustment you would have concluded at the 0.05 level that the groups are significantly
different at x D 10. At the 0.05 level, the groups separate significantly for x < 10 and x > 40.
Example 38.16: Diallel Experiment with Multimember Random Effects
Cockerham and Weir (1977) apply variance component models in the analysis of reciprocal crosses.
In these experiments it is of interest to separate genetically determined variation from variation
determined by parentage. We analyze here the data for the diallel experiment in Cockerham and
Weir (1977, Appendix C). A diallel is a mating design that consists of all possible crosses of a set
of parental lines. It includes reciprocal crossings, but not self-crossings.
The basic model for a cross is Yij k D ˇ C ˛ij C ij k , where Yij k is the observation for offspring k
from maternal parent i and paternal parent j . The various models in Cockerham and Weir (1977)
are different decompositions of the term ˛ij , the total effect that is due to the parents. Their “bio
model” (model (c)) decomposes ˛ij into
˛ij D i C j C i C j C ./ij C ij
where i and j are contributions of the female and male parents, respectively. The term ./ij
captures the interaction between maternal and paternal effects. In contrast to usual interaction
effects, this term must obey a symmetry because of the reciprocals: ./ij D ./j i . The terms
Example 38.16: Diallel Experiment with Multimember Random Effects F 2411
i and j in the decomposition are extranuclear maternal and paternal effects, and the remaining
interactions are captured by the ij term.
The following DATA step creates a SAS data set for the diallel example in Appendix C of Cockerham and Weir (1977):
data diallel;
label time = ’Flowering time in days’;
do p = 1 to 8;
do m = 1 to 8;
if (m ne p) then do;
sym = trim(left(min(m,p))) || ’,’ || trim(left(max(m,p)));
do block = 1 to 2;
input time @@;
output;
end;
end;
end;
end;
datalines;
14.4 16.2 27.2 30.8 17.2 27.0 18.3 20.2 16.2 16.8 18.6 14.4 16.4 16.0
15.4 16.5 14.8 14.6 18.6 18.6 15.2 15.3 17.0 15.2 14.4 14.8 10.8 13.2
31.8 30.4 21.0 23.0 24.6 25.4 19.2 20.0 29.8 28.4 12.8 14.2 13.0 14.4
16.2 17.8 11.4 13.0 16.8 16.3 12.4 14.2 16.8 14.8 12.6 12.2 9.6 11.2
14.6 18.8 12.2 13.6 15.2 15.4 15.2 13.8 18.0 16.0 10.4 12.2 13.4 20.0
20.2 23.4 14.2 14.0 18.6 14.8 22.2 17.0 14.3 17.3 9.0 10.2 11.8 12.8
14.0 16.6 12.2 9.2 13.6 16.2 13.8 14.4 15.6 15.6 15.6 11.0 13.0 9.8
15.2 17.2 10.0 11.6 17.0 18.2 20.8 20.8 20.0 17.4 17.0 12.6 13.0 9.8
;
The observations represent mean flowering times of Nicotiana rustica (Aztec tobacco) from crosses
of inbred varieties grown in two blocks. The variables p and m identify the eight paternal and
maternal lines, respectively. The variable sym is used to model the interaction between the parents,
subject to the symmetry condition ./ij D ./j i . For example, the first two observations, 14.4
and 16.2 days, represent the observations from blocks 1 and 2 where paternal line 1 was crossed
with maternal line 2.
The following PROC GLIMMIX statements fit the “bio model” in Cockerham and Weir (1977):
proc glimmix data=diallel outdesign(z)=zmat;
class block sym p m;
effect line = mm(p m);
model time = block;
random line sym p m p*m;
run;
The EFFECT statement defines the nuclear parental contributions as a multimember effect based on
the CLASS variables p and m. Each observation has two nonzero entries in the design matrix for the
effect that identifies the paternal and maternal lines. The terms in the RANDOM statement model
2
the variance components as follows: line ! n2 , sym ! ./
, p ! 2 , m ! 2 , p*m ! 2 . The
OUTDESIGN= option in the PROC GLIMMIX statement writes the Z matrix to the SAS data set
zmat. The EFFECT statement alleviates the need for complex coding, as in Section 2.3 of Saxton
(2004).
2412 F Chapter 38: The GLIMMIX Procedure
Output 38.16.1 displays the “Class Level Information” table of the diallel model. Because the
interaction terms are symmetric, there are only 8 7=2 D 28 levels for the 8 lines. The estimates of
the variance components and the residual variance in Output 38.16.1 agree with the results in Table
7 of Cockerham and Weir (1977).
Output 38.16.1 Class Levels and Covariance Parameter Estimates in Diallel Example
The GLIMMIX Procedure
Class Level Information
Class
Levels
block
sym
2
28
p
m
8
8
Values
1 2
1,2
2,7
5,6
1 2
1 2
1,3
2,8
5,7
3 4
3 4
1,4
3,4
5,8
5 6
5 6
1,5 1,6 1,7 1,8 2,3 2,4 2,5 2,6
3,5 3,6 3,7 3,8 4,5 4,6 4,7 4,8
6,7 6,8 7,8
7 8
7 8
Covariance Parameter Estimates
Cov Parm
Estimate
Standard
Error
line
sym
p
m
p*m
Residual
5.1047
2.3856
3.3080
1.9134
4.0196
3.6225
4.0021
1.9025
3.4053
2.9891
1.8323
0.6908
The following statements print the Z matrix columns that correspond to the multimember line effect
for the first 10 observations in block 1 (Output 38.16.2). For each observation there are two nonzero
entries, and their column index corresponds to the index of the paternal and maternal line.
proc print data=zmat(where=(block=1) obs=10);
var p m time _z1-_z8;
run;
Output 38.16.2 Z Matrix for Line Effect of the First 10 Observations in Block 1
Obs
p
m
time
_Z1
_Z2
_Z3
_Z4
_Z5
_Z6
_Z7
_Z8
1
3
5
7
9
11
13
15
17
19
1
1
1
1
1
1
1
2
2
2
2
3
4
5
6
7
8
1
3
4
14.4
27.2
17.2
18.3
16.2
18.6
16.4
15.4
14.8
18.6
1
1
1
1
1
1
1
1
0
0
1
0
0
0
0
0
0
1
1
1
0
1
0
0
0
0
0
0
1
0
0
0
1
0
0
0
0
0
0
1
0
0
0
1
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
1
0
0
0
Example 38.17: Linear Inference Based on Summary Data F 2413
Example 38.17: Linear Inference Based on Summary Data
The GLIMMIX procedure has facilities for multiplicity-adjusted inference through the ADJUST=
and STEPDOWN options in the ESTIMATE, LSMEANS, and LSMESTIMATE statements. You
can employ these facilities to test linear hypotheses among parameters even in situations where the
quantities were obtained outside the GLIMMIX procedure. This example demonstrates the process.
The basic idea is to prepare a data set containing the estimates of interest and a data set containing
their covariance matrix. These are then passed to the GLIMMIX procedure, preventing updating of
the parameters, essentially moving directly into the post-processing stage as if estimates with this
covariance matrix had been produced by the GLIMMIX procedure.
The final documentation example in Chapter 60, “The NLIN Procedure,” in the SAS/STAT User’s
Guide discusses a nonlinear first-order compartment pharmacokinetic model for theophylline concentration. The data are derived by collapsing and averaging the subject-specific data from Pinheiro
and Bates (1995) in a particular—yet unimportant—way that leads to two groups for comparisons.
The following DATA step creates these data:
data theop;
input time dose conc @@;
if (dose = 4) then group=1;
datalines;
0.00
4 0.1633 0.25
4
0.27
4
4.4 0.30
4
0.35
4
1.89 0.37
4
0.50
4
3.96 0.57
4
0.58
4
6.9 0.60
4
0.63
4
9.03 0.77
4
1.00
4
7.82 1.02
4
1.05
4
7.14 1.07
4
1.12
4
10.5 2.00
4
2.02
4
7.93 2.05
4
2.13
4
8.38 3.50
4
3.52
4
9.75 3.53
4
3.55
4
10.21 3.62
4
3.82
4
8.58 5.02
4
5.05
4
9.18 5.07
4
5.08
4
6.2 5.10
4
7.02
4
5.78 7.03
4
7.07
4
5.945 7.08
4
7.17
4
4.24 8.80
4
9.00
4
4.9 9.02
4
9.03
4
6.11 9.05
4
9.38
4
7.14 11.60
4
11.98
4
4.19 12.05
4
12.10
4
5.68 12.12
4
12.15
4
3.7 23.70
4
24.15
4
1.17 24.17
4
24.37
4
3.28 24.43
4
24.65
4
1.15 0.00
5
0.25
5
2.92 0.27
5
else group=2;
2.045
7.37
2.89
6.57
4.6
5.22
7.305
8.6
9.72
7.83
7.54
5.66
7.5
6.275
8.57
8.36
7.47
8.02
4.11
5.33
6.89
3.16
4.57
5.94
2.42
1.05
1.12
0.025
1.505
2414 F Chapter 38: The GLIMMIX Procedure
0.30
0.52
0.98
1.02
1.92
2.02
3.48
3.53
3.60
5.02
6.98
7.02
7.15
9.03
9.10
12.00
12.10
23.85
24.12
24.30
;
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
2.02
5.53
7.655
5.02
8.33
7.8233
7.09
6.59
5.87
6.2867
5.25
7.09
4.73
3.62
5.9
3.69
2.89
0.92
1.25
0.9
0.50
0.58
1.00
1.15
1.98
2.03
3.50
3.57
5.00
5.05
7.00
7.03
9.00
9.07
9.22
12.05
12.12
24.08
24.22
24.35
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
4.795
3.08
9.855
6.44
6.81
6.32
7.795
5.53
5.8
5.88
4.02
4.925
4.47
4.57
3.46
3.53
2.69
0.86
1.15
1.57
In terms of two fixed treatment groups, the nonlinear model for these data can be written as
Cit D
Dkei kai
Œexp. kei t/
C li .kai kei /
exp. kai t/ C i t
where Cit is the observed concentration in group i at time t , D is the dose of theophylline, kei is
the elimination rate constant in group i , kai is the absorption rate in group i , C li is the clearance in
group i , and it denotes the model error. Because the rates and the clearance must be positive, you
can parameterize the model in terms of log rates and the log clearance:
C li D expfˇ1i g
kai D expfˇ2i g
kei D expfˇ3i g
In this parameterization the model contains six parameters, and the rates and clearance vary by
group. The following PROC NLIN statements fit the model and obtain the group-specific parameter
estimates:
proc nlin data=theop outest=cov;
parms beta1_1=-3.22 beta2_1=0.47 beta3_1=-2.45
beta1_2=-3.22 beta2_2=0.47 beta3_2=-2.45;
if (group=1) then do;
cl
= exp(beta1_1);
ka
= exp(beta2_1);
ke
= exp(beta3_1);
end; else do;
cl
= exp(beta1_2);
ka
= exp(beta2_2);
Example 38.17: Linear Inference Based on Summary Data F 2415
ke
= exp(beta3_2);
end;
mean = dose*ke*ka*(exp(-ke*time)-exp(-ka*time))/cl/(ka-ke);
model conc = mean;
ods output ParameterEstimates=ests;
run;
The conditional programming statements determine the clearance, elimination, and absorption rates
depending on the value of the group variable. The OUTEST= option in the PROC NLIN statement
saves estimates and their covariance matrix to the data set cov. The ODS OUTPUT statement saves
the “Parameter Estimates” table to the data set ests.
Output 38.17.1 displays the analysis of variance table and the parameter estimates from this NLIN
run. Note that the confidence levels in the “Parameter Estimates” table are based on 92 degrees of
freedom, corresponding to the residual degrees of freedom in the analysis of variance table.
Output 38.17.1 Analysis of Variance and Parameter Estimates for Nonlinear Model
The NLIN Procedure
NOTE: An intercept was not specified for this model.
Source
DF
Sum of
Squares
Mean
Square
Model
Error
Uncorrected Total
6
92
98
3247.9
138.9
3386.8
541.3
1.5097
Parameter
beta1_1
beta2_1
beta3_1
beta1_2
beta2_2
beta3_2
Estimate
Approx
Std Error
-3.5671
0.4421
-2.6230
-3.0111
0.3977
-2.4442
0.0864
0.1349
0.1265
0.1061
0.1987
0.1618
F Value
Approx
Pr > F
358.56
<.0001
Approximate 95% Confidence
Limits
-3.7387
0.1742
-2.8742
-3.2219
0.00305
-2.7655
-3.3956
0.7101
-2.3718
-2.8003
0.7924
-2.1229
The following DATA step extracts the part of the cov data set that contains the covariance matrix of
the parameter estimates in Output 38.17.1 and renames the variables as Col1–Col6. Output 38.17.2
shows the result of the DATA step.
data covb;
set cov(where=(_type_=’COVB’));
rename beta1_1=col1 beta2_1=col2 beta3_1=col3
beta1_2=col4 beta2_2=col5 beta3_2=col6;
row = _n_;
Parm = 1;
keep parm row beta:;
run;
proc print data=covb;
run;
2416 F Chapter 38: The GLIMMIX Procedure
Output 38.17.2 Covariance Matrix of NLIN Parameter Estimates
Obs
col1
col2
col3
col4
col5
col6
row
1
2
3
4
5
6
0.007462
-0.005222
0.010234
0.000000
0.000000
0.000000
-0.005222
0.018197
-0.010590
0.000000
0.000000
0.000000
0.010234
-0.010590
0.015999
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.011261
-0.009096
0.015785
0.000000
0.000000
0.000000
-0.009096
0.039487
-0.019996
0.000000
0.000000
0.000000
0.015785
-0.019996
0.026172
1
2
3
4
5
6
Parm
1
1
1
1
1
1
The reason for this transformation of the data is to use the resulting data set to define a covariance
structure in PROC GLIMMIX. The following statements reconstitute a model in which the parameter estimates from PROC NLIN are the observations and in which the covariance matrix of the
“observations” matches the covariance matrix of the NLIN parameter estimates:
proc glimmix data=ests order=data;
class Parameter;
model Estimate = Parameter / noint df=92 s;
random _residual_ / type=lin(1) ldata=covb v;
parms (1) / noiter;
lsmeans parameter / cl;
lsmestimate Parameter
’beta1 eq. across groups’ 1 0 0 -1,
’beta2 eq. across groups’ 0 1 0 0 -1,
’beta3 eq. across groups’ 0 0 1 0 0 -1 /
adjust=bon stepdown ftest(label=’Homogeneity’);
run;
In other words, you are using PROC GLIMMIX to set up a linear statistical model
Y D I˛ C .0; A/
where the covariance matrix A is given by
2
0:007
0:005
0:010
6 0:005
0:018
0:011
6
6 0:010
0:011
0:016
AD6
6 0
0
0
6
4 0
0
0
0
0
0
0
0
0
0:011
0:009
0:016
0
0
0
0:009
0:039
0:019
0
0
0
0:016
0:019
0:026
3
7
7
7
7
7
7
5
The generalized least squares estimate for ˛ in this saturated model reproduces the observations:
1
I
1
1
b
˛ D I0 A
D A
Dy
1 0
IA
A
1
y
1
y
Example 38.17: Linear Inference Based on Summary Data F 2417
The ORDER=DATA option in the PROC GLIMMIX statement requests that the sort order of the
Parameter variable be identical to the order in which it appeared in the “Parameter Estimates” table
of the NLIN procedure (Output 38.17.1). The MODEL statement uses the Estimate and Parameter
variables from that table to form a model in which the X matrix is the identity; hence the NOINT
option. The DF=92 option sets the degrees of freedom equal to the value used in the NLIN procedure. The RANDOM statement specifies a linear covariance structure with a single component and
supplies the values for the structure through the LDATA= data set. This structure models the covariance matrix as VarŒY D A, where the A matrix is given previously. Essentially, the TYPE=LIN(1)
structure forces an unstructured covariance matrix onto the data. To make this work, the parameter
is held fixed at 1 in the PARMS statement.
Output 38.17.3 displays the parameter estimates and least squares means for this model. Note
that estimates and least squares means are identical, since the X matrix is the identity. Also, the
confidence limits agree with the values reported by PROC NLIN (see Output 38.17.1).
Output 38.17.3 Parameter Estimates and LS-Means from Summary Data
The GLIMMIX Procedure
Solutions for Fixed Effects
Effect
Parameter
Parameter
Parameter
Parameter
Parameter
Parameter
Parameter
beta1_1
beta2_1
beta3_1
beta1_2
beta2_2
beta3_2
Estimate
Standard
Error
DF
t Value
Pr > |t|
-3.5671
0.4421
-2.6230
-3.0111
0.3977
-2.4442
0.08638
0.1349
0.1265
0.1061
0.1987
0.1618
92
92
92
92
92
92
-41.29
3.28
-20.74
-28.37
2.00
-15.11
<.0001
0.0015
<.0001
<.0001
0.0483
<.0001
Parameter Least Squares Means
Parameter
beta1_1
beta2_1
beta3_1
beta1_2
beta2_2
beta3_2
Estimate
Standard
Error
DF
t Value
Pr > |t|
Alpha
-3.5671
0.4421
-2.6230
-3.0111
0.3977
-2.4442
0.08638
0.1349
0.1265
0.1061
0.1987
0.1618
92
92
92
92
92
92
-41.29
3.28
-20.74
-28.37
2.00
-15.11
<.0001
0.0015
<.0001
<.0001
0.0483
<.0001
0.05
0.05
0.05
0.05
0.05
0.05
Parameter Least Squares Means
Parameter
beta1_1
beta2_1
beta3_1
beta1_2
beta2_2
beta3_2
Lower
Upper
-3.7387
0.1742
-2.8742
-3.2219
0.003050
-2.7655
-3.3956
0.7101
-2.3718
-2.8003
0.7924
-2.1229
2418 F Chapter 38: The GLIMMIX Procedure
The (marginal) covariance matrix of the data is shown in Output 38.17.4 to confirm that it matches
the A matrix given earlier.
Output 38.17.4 R-Side Covariance Matrix
Estimated V Matrix for Subject 1
Row
Col1
Col2
Col3
1
2
3
4
5
6
0.007462
-0.00522
0.01023
-0.00522
0.01820
-0.01059
0.01023
-0.01059
0.01600
Col4
Col5
Col6
0.01126
-0.00910
0.01579
-0.00910
0.03949
-0.02000
0.01579
-0.02000
0.02617
The LSMESTIMATE statement specifies three linear functions. These set equal the ˇ parameters
from the groups. The step-down Bonferroni adjustment requests a multiplicity adjustment for the
family of three tests. The FTEST option requests a joint test of the three estimable functions; it is a
global test of parameter homogeneity across groups.
Output 38.17.5 displays the result from the LSMESTIMATE statement. The joint test is highly
significant (F D 30:52, p < 0:0001). From the p-values associated with the individual rows of
the estimates, you can see that the lack of homogeneity is due to group differences for ˇ1 , the log
clearance.
Output 38.17.5 Test of Parameter Homogeneity across Groups
Least Squares Means Estimates
Adjustment for Multiplicity: Holm
Effect
Label
Parameter
Parameter
Parameter
beta1 eq. across groups
beta2 eq. across groups
beta3 eq. across groups
Estimate
Standard
Error
DF
t Value
Pr > |t|
-0.5560
0.04443
-0.1788
0.1368
0.2402
0.2054
92
92
92
-4.06
0.18
-0.87
0.0001
0.8537
0.3862
Least Squares Means Estimates
Adjustment for Multiplicity: Holm
Effect
Label
Adj P
Parameter
Parameter
Parameter
beta1 eq. across groups
beta2 eq. across groups
beta3 eq. across groups
0.0003
0.8537
0.7725
Least Squares Means Ftest
Label
Homogeneity
Num
DF
Den
DF
F Value
Pr > F
3
92
30.52
<.0001
References F 2419
An alternative method to set up this model is given by the following statements, where the data set
pdata contains the covariance parameters:
random _residual_ / type=un;
parms / pdata=pdata noiter
The following DATA step creates an appropriate PDATA= data set from the data set covb constructed earlier:
data pdata; set covb;
array col{6};
do i=1 to _n_;
estimate = col{i};
output;
end;
keep estimate;
run;
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Subject Index
adaptive Gaussian quadrature
GLIMMIX procedure, 2101
Akaike’s information criterion
GLIMMIX procedure, 2097
Akaike’s information criterion (finite sample
corrected version)
GLIMMIX procedure, 2097
alpha level
GLIMMIX procedure, 2128, 2134, 2143,
2155, 2162, 2178, 2185
anisotropic power covariance structure
GLIMMIX procedure, 2200
anisotropic spatial power structure
GLIMMIX procedure, 2200
ANOM adjustment
GLIMMIX procedure, 2142
anom plot
GLIMMIX procedure, 2286
ANTE(1) structure
GLIMMIX procedure, 2191
ante-dependence structure
GLIMMIX procedure, 2191
AR(1) structure
GLIMMIX procedure, 2192
asymptotic covariance
GLIMMIX procedure, 2093, 2097
automatic variables
GLIMMIX procedure, 2138, 2207
autoregressive moving-average structure
GLIMMIX procedure, 2192
autoregressive structure
GLIMMIX procedure, 2192
banded Toeplitz structure
GLIMMIX procedure, 2201
Bernoulli distribution
GLIMMIX procedure, 2166
beta distribution
GLIMMIX procedure, 2166
bias
GLIMMIX procedure, 2230
binary distribution
GLIMMIX procedure, 2166
binomial distribution
GLIMMIX procedure, 2166
BLUP
GLIMMIX procedure, 2190, 2207, 2208,
2276
Bonferroni adjustment
GLIMMIX procedure, 2142
boundary constraints
GLIMMIX procedure, 2180, 2183
box plots
GLIMMIX procedure, 2280
centering
GLIMMIX procedure, 2170
chi-square mixture
GLIMMIX procedure, 2130
chi-square test
GLIMMIX procedure, 2124, 2155, 2162,
2165
Cholesky
covariance structure (GLIMMIX), 2193
method (GLIMMIX), 2094
root (GLIMMIX), 2193, 2195
class level
GLIMMIX procedure, 2106, 2272
classification variables
GLIMMIX procedure, 2120
comparing splines
GLIMMIX procedure, 2404
compound symmetry structure
GLIMMIX procedure, 2194
computed variables
GLIMMIX procedure, 2138
confidence limits
adjusted (GLIMMIX), 2133, 2142, 2154,
2355
adjusted, simulated (GLIMMIX), 2142
and isotronic contrasts (GLIMMIX), 2355
and step-down (GLIMMIX), 2136, 2152,
2157
covariance parameters (GLIMMIX), 2128
estimate, lower (GLIMMIX), 2137
estimate, upper (GLIMMIX), 2138
estimated likelihood (GLIMMIX), 2129
estimates (GLIMMIX), 2134
exponentiated (GLIMMIX), 2156
fixed effects (GLIMMIX), 2162
in mean plot (GLIMMIX), 2112, 2149
inversely linked (GLIMMIX), 2145, 2277
least squares mean estimate (GLIMMIX),
2154
least squares mean estimate, lower
(GLIMMIX), 2157
least squares mean estimate, upper
(GLIMMIX), 2159
least squares means (GLIMMIX), 2090,
2142, 2144
least squares means estimates (GLIMMIX),
2155
likelihood-based, details (GLIMMIX), 2238
odds ratios (GLIMMIX), 2107, 2116, 2171,
2255
profile likelihood (GLIMMIX), 2129
random-effects solution (GLIMMIX), 2185
truncation (GLIMMIX), 2277
vs. prediction limits (GLIMMIX), 2276
Wald (GLIMMIX), 2130
constraints
boundary (GLIMMIX), 2180, 2183
constructed effects
GLIMMIX procedure, 2132, 2410
containment method
GLIMMIX procedure, 2164
contrast-specification
GLIMMIX procedure, 2121, 2132
contrasts
GLIMMIX procedure, 2121
control plot
GLIMMIX procedure, 2286
convergence criterion
GLIMMIX procedure, 2093, 2103, 2109,
2267, 2274, 2301, 2347
MIXED procedure, 2267
convergence status
GLIMMIX procedure, 2274
covariance
parameters (GLIMMIX), 2082, 2083, 2087,
2093
parameters, confidence interval
(GLIMMIX), 2125
parameters, testing (GLIMMIX), 2125
covariance parameter estimates
GLIMMIX procedure, 2100, 2109, 2275
covariance structure
anisotropic power (GLIMMIX), 2200
ante-dependence (GLIMMIX), 2191
autoregressive (GLIMMIX), 2082, 2192
autoregressive moving-average
(GLIMMIX), 2192
banded (GLIMMIX), 2200
Cholesky type (GLIMMIX), 2193
compound symmetry (GLIMMIX), 2194
equi-correlation (GLIMMIX), 2194
examples (GLIMMIX), 2202
exponential (GLIMMIX), 2199
factor-analytic (GLIMMIX), 2195
G-side (GLIMMIX), 2081, 2125, 2191
gaussian (GLIMMIX), 2199
general (GLIMMIX), 2079
general linear (GLIMMIX), 2195
heterogeneous autoregressive (GLIMMIX),
2192
heterogeneous compound symmetry
(GLIMMIX), 2195
heterogeneous Toeplitz (GLIMMIX), 2201
Huynh-Feldt (GLIMMIX), 2195
Matérn (GLIMMIX), 2199
misspecified (GLIMMIX), 2079
parameter reordering (GLIMMIX), 2180
penalized B-spline (GLIMMIX), 2187,
2189, 2196
positive (semi-)definite, 2193
power (GLIMMIX), 2200
R-side (GLIMMIX), 2081, 2125, 2180,
2184, 2190, 2191
R-side with profiled scale (GLIMMIX),
2180
radial smooth (GLIMMIX), 2187, 2197
simple (GLIMMIX), 2199
spatial (GLIMMIX), 2186, 2199
spherical (GLIMMIX), 2200
Toeplitz (GLIMMIX), 2200
unstructured (GLIMMIX), 2201
unstructured, correlation (GLIMMIX), 2201
variance components (GLIMMIX), 2202
with second derivatives (GLIMMIX), 2165
working, independence (GLIMMIX), 2096
crossed effects
GLIMMIX procedure, 2259
default estimation technique
GLIMMIX procedure, 2271
default output
GLIMMIX procedure, 2271
degrees of freedom
between-within method (GLIMMIX), 2164
chi-square mixture (GLIMMIX), 2130
containment method (GLIMMIX), 2164
GLIMMIX procedure, 2123, 2124, 2133,
2134, 2144, 2155, 2163, 2164, 2239
infinite (GLIMMIX), 2134, 2144, 2155,
2165
Kenward-Roger method (GLIMMIX), 2165
method (GLIMMIX), 2164
residual method (GLIMMIX), 2165
Satterthwaite method (GLIMMIX), 2165
diagnostic plots
GLIMMIX procedure, 2280
diffogram
GLIMMIX procedure, 2286
dimension information
GLIMMIX procedure, 2273
dispersion parameter
GLIMMIX procedure, 2213
doubly iterative algorithm
GLIMMIX procedure, 2269
Dunnett’s adjustment
GLIMMIX procedure, 2142
EBE
GLIMMIX procedure, 2191, 2276
EBLUP
GLIMMIX procedure, 2190
effect
name length (GLIMMIX), 2106
empirical Bayes estimates
GLIMMIX procedure, 2102, 2191, 2276
empirical Bayes estimation
GLIMMIX procedure, 2102
empirical estimator
GLIMMIX procedure, 2094, 2241, 2244
estimability
GLIMMIX procedure, 2122
estimates
GLIMMIX procedure, 2132
multiple comparison adjustment
(GLIMMIX), 2133
estimation methods
GLIMMIX procedure, 2100
examples, GLIMMIX
k-d tree information, 2250
_LINP_, 2207–2209
_LOGL_, 2400
_MU_, 2208, 2209
_VARIANCE_, 2209, 2333
adding computed variables to output data
set, 2139
analysis of means, ANOM, 2295
analysis of summary data, 2416
anom plot, 2147
anom plots, 2295
binary data, 2337
binary data, GLMM, 2312
binary data, pseudo-likelihood, 2312
binary data, sort order, 2108
binomial data, 2085
binomial data, GLM, 2299
binomial data, GLMM, 2304
binomial data, overdispersed, 2328
binomial data, spatial covariance, 2308
bivariate data; Poisson, binary, 2339
blotch incidence data, 2328
box plots, 2284
bucket size in k-d tree, 2250
central t distribution, 2168
Cholesky covariance structure, 2357
collection effect, 2261
computed variables, 2320, 2345
constructed random effect, 2411
containment hierarchy, 2102, 2247
contrast, among covariance parameters,
2127, 2128, 2366
contrast, differences of splines, 2265
contrast, nonpositional syntax, 2122, 2262,
2407
contrast, positional syntax, 2121, 2262
contrast, with groups, 2124
contrast, with spline effects, 2264
control plot, 2148, 2293
covariance structure, 2202
covariates in LS-mean construction, 2143
COVTEST statement, 2131
COVTEST with keywords, 2127
COVTEST with no restrictions, 2235
COVTEST with specified values, 2127
cow weight data, 2343
diallel experiment, 2411
diffogram, 2148, 2290, 2291, 2316
diffplot, 2148
empirical Bayes estimates, 2377
ENSO data, 2370
epileptic seizure data, 2383
equivalent models, TYPE=VC, 2245
equivalent models, with and without subject,
2245
estimate, multi-row, 2135
estimate, with groups, 2135
estimate, with varied divisors, 2135
ferrite cores data, 2355
FIRSTORDER option for Kenward-Roger
method, 2360
Fisher’s iris data, 2362
foot shape data, 2376
FREQ statement, 2377
G-side spatial covariance, 2186
GEE-type model, 2096, 2232, 2386
generalized logit, 2123, 2134
generalized logit with random effects, 2266
generalized Poisson distribution, 2400
getting started, 2084
GLM mode, 2232
GLMM mode, 2232
graphics, anom plots, 2295
graphics, box plots, 2284
graphics, control plot, 2293
graphics, custom template, 2324
graphics, diffogram, 2290, 2291, 2316
graphics, mean plots, 2286
graphics, Pearson residual panel, 2332
graphics, predicted profiles, 2349
graphics, residual panel, 2280, 2281
graphics, studentized residual panel, 2323
group option in contrast, 2124
group option in estimate, 2135
group-specific smoothing, 2350
grouped analysis, 2377
groups in RANDOM statement, 2126, 2363
herniorrhaphy data, 2336
Hessian fly data, 2298
holding covariance parameters fixed, 2179
homogeneity of covariance parameters,
2126, 2127, 2363
identity model, 2416
infinite degrees of freedom, 2384
inverse linking, 2090, 2157
isotonic contrast, 2355
joint model (DIST=BYOBS), 2167
joint model, independent, 2339
joint model, marginal correlation, 2341
joint model, shared random effect, 2340
Kenward-Roger method, 2357
knot construction, k-d tree, 2188, 2250,
2345
knot construction, equal, 2188
knot construction, optimization, 2189
Laplace approximation, 2224, 2225, 2229,
2393
LDATA= option, 2196
least squares mean estimate, 2153, 2355
least squares mean estimate, multi-row, 2156
least squares mean estimate, with varied
divisors, 2156
least squares means, 2090
least squares means, AT option, 2143
least squares means, covariate, 2143
least squares means, differences against
control, 2145
least squares means, slice, 2150
least squares means, slice differences, 2150
linear combination of LS-means, 2153
linear covariance structure, 2196, 2416
logistic model with random effects,
binomial data, 2304
logistic model, binomial data, 2299
logistic regression with random intercepts,
2084
logistic regression, binary data, 2255, 2337
logistic regression, binomial data, 2269
marginal variance matrix, 2204
mean plot, sliced interaction, 2149
mean plot, three-way, 2149
mean plots, 2286
MIVQUE0 estimates, 2182
multicenter clinical trial, 2388
multimember effect, 2261, 2411
multinomial data, 2123, 2134, 2266, 2377,
2393
multiple local minima, 2374
multiple plot requests, 2114
multiplicity adjustment, 2355, 2390, 2393,
2409
multivariate distributions, 2167
multivariate normal model, 2363
nesting v. crossing, 2247
NLIN procedure, 2414
NLOPTIONS statement, 2345
NOFIT option, 2250
NOITER option for covariance parameters,
2181
nonlinear regression, 2414
NOPROFILE option, 2371
odds ratio, 2171, 2256
odds ratio, all pairwise differences, 2172,
2257
odds ratio, with interactions, 2171, 2257
odds ratio, with reference value, 2171, 2257
odds ratio, with specified units, 2172, 2257
ordinal data, 2377, 2393
OUTDESIGN option, 2405, 2411
output statistics, 2175, 2207, 2320, 2345,
2405
overdispersion, 2082, 2096, 2184, 2232,
2328
parallel shifted smooths, 2352
Pearson residual panel, 2332
penalized B-spline, 2197
Poisson model with offset, 2320, 2384
Poisson model with random effects, 2398
Poisson regression, 2338
Pothoff-Roy repeated measures data, 2356
proportional odds model with random effect,
2377, 2393
quadrature approximation, 2227, 2377, 2398
quasi-likelihood, 2333
R-side covariance structure, 2357, 2386
R-side covariance, binomial data, 2308
radial smooth, with parallel shifts, 2352
radial smoothing, 2250, 2345, 2371
radial smoothing, group-specific, 2350
REPEATED in MIXED vs RANDOM in
GLIMMIX, 2267
residual panel, 2280, 2281
row-wise adjustment of LS-mean
differences, 2141
salamander data, 2311
Satterthwaite method, 2141
saturated model, 2416
Scottish lipcancer data, 2319
SGPANEL procedure, 2349
SGPLOT procedure, 2373, 2374, 2381,
2404, 2406
SGRENDER procedure, 2324
simple differences, 2150, 2316
simple differences with control, 2151
simulated p-values, 2390, 2393, 2409
slice F test, 2150
slice differences, 2150, 2316
slice differences with control, 2151
space-filling design, 2189
spatial covariance, binomial data, 2308
specifying lower bounds, 2180
specifying values for degrees of freedom,
2163
spline differences, 2409
spline effect, 2261, 2405
splines in interactions, 2409
standardized mortlity rate, 2320
starting values, 2374
starting values and BY groups, 2183
starting values from data set, 2183
step-down p-values, 2390, 2393, 2409
studentized maximum modulus, 2141
studentized residual panel, 2323
subject processing, 2102, 2245
subject processing, containment, 2247
subject processing, crossed effects, 2247
subject processing, nested effects, 2247
subject-processing, asymptotics, 2224, 2225
syntax, differences to MIXED, 2267, 2268
test for independence, 2380
test for Poisson distribution, 2400
testing covariance parameters, 2126, 2363,
2366, 2371
theophylline data, 2413
TYPE=CS and TYPE=VC equivalence,
2194
user-defined log-likelihood function, 2400
user-defined variance function, 2229
user-specified link function, 2208, 2209
user-specified variance function, 2209, 2333
working independence, 2096, 2232
expansion locus
theory (GLIMMIX), 2222
exponential covariance structure
GLIMMIX procedure, 2199
exponential distribution
GLIMMIX procedure, 2166
factor-analytic structure
GLIMMIX procedure, 2195
finite differences
theory (GLIMMIX), 2228
Fisher’s scoring method
GLIMMIX procedure, 2093, 2117
fit statistics
GLIMMIX procedure, 2274
fitting information
GLIMMIX procedure, 2274
fixed effects
GLIMMIX procedure, 2081
frequency variable
GLIMMIX procedure, 2138
G matrix
GLIMMIX procedure, 2184, 2186
G-side random effect
GLIMMIX procedure, 2081
gamma distribution
GLIMMIX procedure, 2166
gaussian covariance structure
GLIMMIX procedure, 2199
gaussian distribution
GLIMMIX procedure, 2166
GEE, see also generalized estimating equations
general linear covariance structure
GLIMMIX procedure, 2195
generalized estimating equations
compound symmetry (GLIMMIX), 2386
working independence (GLIMMIX), 2096,
2232
generalized linear mixed model, see also
GLIMMIX procedure
generalized linear mixed model (GLIMMIX)
least squares means, 2153
theory, 2216
generalized linear model, see also GLIMMIX
procedure
generalized linear model (GLIMMIX)
theory, 2210
generalized logit
example (GLIMMIX), 2123, 2134
generalized Poisson distribution
GLIMMIX procedure, 2396
geometric distribution
GLIMMIX procedure, 2166
GLIMMIX procedure
adaptive Gaussian quadrature, 2101
Akaike’s information criterion, 2097
Akaike’s information criterion (finite sample
corrected version), 2097
alpha level, 2128, 2134, 2143, 2155, 2162,
2178, 2185
anisotropic power covariance structure, 2200
anisotropic spatial power structure, 2200
ANOM adjustment, 2142
anom plot, 2286
ANTE(1) structure, 2191
ante-dependence structure, 2191
AR(1) structure, 2192
asymptotic covariance, 2093, 2097
automatic variables, 2138, 2207
autoregressive moving-average structure,
2192
autoregressive structure, 2192
banded Toeplitz structure, 2201
Bernoulli distribution, 2166
beta distribution, 2166
between-within method, 2164
bias of estimates, 2230
binary distribution, 2166
binomial distribution, 2166
BLUP, 2190, 2207, 2208, 2276
Bonferroni adjustment, 2142
boundary constraints, 2180, 2183
box plots, 2280
BYLEVEL processing of LSMEANS, 2144,
2147, 2155
centering, 2170
chi-square mixture, 2130
chi-square test, 2124, 2155, 2162, 2165
Cholesky covariance structure, 2193
Cholesky method, 2094
Cholesky root, 2193, 2195
class level, 2106, 2272
classification variables, 2120
comparing splines, 2404
comparison with the MIXED procedure,
2266
compound symmetry structure, 2194
computed variables, 2138
confidence interval, 2136, 2157, 2185
confidence limits, 2134, 2144, 2155, 2162,
2185
confidence limits, covariance parameters,
2128
constrained covariance parameters, 2131
constructed effects, 2132, 2410
containment method, 2164
continuous effects, 2191
contrast-specification, 2121, 2132
contrasts, 2121
control plot, 2286
convergence criterion, 2093, 2103, 2109,
2267, 2274, 2301, 2347
convergence status, 2274
correlations of least squares means, 2144
correlations of least squares means
contrasts, 2155
covariance parameter estimates, 2100, 2275
covariance parameters, 2082
covariance structure, 2191, 2202
covariances of least squares means, 2144
covariances of least squares means
contrasts, 2155
covariate values for LSMEANS, 2143, 2155
crossed effects, 2259
default estimation technique, 2271
default output, 2271
default variance function, 2206
degrees of freedom, 2123, 2124, 2130,
2133, 2134, 2140, 2141, 2144, 2154,
2155, 2163, 2164, 2173, 2239, 2260
diagnostic plots, 2280
diffogram, 2148, 2286, 2290, 2291, 2316
dimension information, 2273
dispersion parameter, 2213
doubly iterative algorithm, 2269
Dunnett’s adjustment, 2142
EBE, 2191, 2276
EBLUP, 2190
effect name length, 2106
empirical Bayes estimates, 2102, 2191, 2276
empirical Bayes estimation, 2102
empirical estimator, 2244
estimability, 2122, 2124, 2136, 2139, 2173,
2260
estimated-likelihood interval, 2129
estimates, 2132
estimation methods, 2100
estimation modes, 2231
examples, see also examples, GLIMMIX,
2298
expansion locus, 2222
exponential covariance structure, 2199
exponential distribution, 2166
factor-analytic structure, 2195
finite differences, 2228
Fisher’s scoring method, 2093, 2117
fit statistics, 2274
fitting information, 2274
fixed effects, 2081
fixed-effects parameters, 2173
G matrix, 2184, 2186
G-side random effect, 2081
gamma distribution, 2166
gaussian covariance structure, 2199
gaussian distribution, 2166
general linear covariance structure, 2195
generalized linear mixed model theory, 2216
generalized linear model theory, 2210
generalized Poisson distribution, 2396
geometric distribution, 2166
GLM mode, 2096, 2213, 2231, 2232
GLMM mode, 2096, 2232
grid search, 2179
group effect, 2186
Hannan-Quinn information criterion, 2097
Hessian matrix, 2093, 2097
heterogeneous AR(1) structure, 2192
heterogeneous autoregressive structure,
2192
heterogeneous compound symmetry
structure, 2195
heterogeneous Toeplitz structure, 2201
Hsu’s adjustment, 2142
Huynh-Feldt covariance structure, 2195
infinite degrees of freedom, 2124, 2134,
2144, 2155, 2165
information criteria, 2097
initial values, 2179
input data sets, 2094
integral approximation, 2216
interaction effects, 2259
intercept, 2259
intercept random effect, 2184
introductory example, 2084
inverse gaussian distribution, 2166
iteration details, 2099
iteration history, 2273
iterations, 2273
Kackar-Harville-Jeske adjusted estimator,
2244
Kenward-Roger method, 2165
knot selection, 2249
KR adjusted estimator, 2244
L matrices, 2121, 2139
lag functionality, 2206
Laplace approximation, 2100, 2222
least squares means, 2139, 2145, 2151
likelihood ratio test, 2125, 2232
linear covariance structure, 2195
linearization, 2216, 2218
link function, 2081, 2169
log-normal distribution, 2166
Matérn covariance structure, 2199
maximum likelihood, 2100, 2271
missing level combinations, 2260
MIVQUE0 estimation, 2182, 2273
mixed model smoothing, 2187, 2189, 2196,
2197, 2248
model information, 2272
multimember effect, 2410
multimember example, 2410
multinomial distribution, 2166
multiple comparisons of estimates, 2133
multiple comparisons of least squares
means, 2141, 2142, 2145, 2151, 2154
multiplicity adjustment, 2133, 2136, 2141,
2142, 2152, 2154, 2157
negative binomial distribution, 2166
Nelson’s adjustment, 2142
nested effects, 2259
non-full-rank parameterization, 2260
non-positional syntax, 2122, 2262, 2404
normal distribution, 2166
notation, 2081
number of observations, 2272
numerical integration, 2225
odds estimation, 2254
odds ratio estimation, 2254
odds ratios, 2107
ODS graph names, 2297
ODS Graphics, 2110, 2280
ODS table names, 2278
offset, 2172, 2207, 2320, 2321, 2384
optimization, 2173
optimization information, 2273
ordering of effects, 2108
output statistics, 2276
overdispersion, 2396
P-spline, 2196
parameterization, 2259
penalized B-spline, 2196
Poisson distribution, 2166
Poisson mixture, 2396
population average, 2222
positive definiteness, 2193
power covariance structure, 2200
profile-likelihood interval, 2129
profiling residual variance, 2107, 2117
programming statements, 2205
pseudo-likelihood, 2100, 2271
quadrature approximation, 2100, 2225
quasi-likelihood, 2271
R-side random effect, 2081, 2190
radial smoother structure, 2197
radial smoothing, 2187, 2197, 2248
random effects, 2081, 2184
random-effects parameter, 2190
reference category, 2265
residual effect, 2184
residual likelihood, 2100
residual maximum likelihood, 2271
residual method, 2165
residual plots, 2280
response level ordering, 2161, 2265
response profile, 2265, 2272
response variable options, 2161
restricted maximum likelihood, 2271
sandwich estimator, 2244
Satterthwaite method, 2165, 2239
scale parameter, 2081–2083, 2098, 2107,
2117, 2129, 2179, 2180, 2183, 2206,
2210, 2213–2215, 2219, 2220, 2222,
2226, 2237, 2268, 2273, 2275, 2313,
2329, 2333, 2346, 2350, 2379, 2397
Schwarz’s Bayesian information criterion,
2097
scoring, 2093
Sidak’s adjustment, 2142
simple covariance matrix, 2199
simple effects, 2150
simple effects differences, 2150
simulation-based adjustment, 2142
singly iterative algorithm, 2269
spatial covariance structure, 2199
spatial exponential structure, 2199
spatial gaussian structure, 2199
spatial Matérn structure, 2199
spatial power structure, 2200
spatial spherical structure, 2200
spherical covariance structure, 2200
spline comparisons, 2404
spline smoothing, 2196, 2197
standard error adjustment, 2094, 2165
statistical graphics, 2280, 2297
subject effect, 2191
subject processing, 2245
subject-specific, 2222
t distribution, 2166
table names, 2278
test-specification for covariance parameters,
2125
testing covariance parameters, 2125, 2232
tests of fixed effects, 2276
thin plate spline (approx.), 2248
Toeplitz structure, 2200
Tukey’s adjustment, 2142
Type I testing, 2169
Type II testing, 2169
Type III testing, 2169
unstructured covariance, 2201
unstructured covariance matrix, 2195
user-defined link function, 2206
V matrix, 2203
Wald test, 2276
Wald tests of covariance parameters, 2132
weighting, 2204
GLM, see also GLIMMIX procedure
GLMM, see also GLIMMIX procedure
group effect
GLIMMIX procedure, 2186
Hannan-Quinn information criterion
GLIMMIX procedure, 2097
Hessian matrix
GLIMMIX procedure, 2093, 2097
heterogeneous AR(1) structure
GLIMMIX procedure, 2192
heterogeneous autoregressive structure
GLIMMIX procedure, 2192
heterogeneous compound symmetry structure
GLIMMIX procedure, 2195
heterogeneous Toeplitz structure
GLIMMIX procedure, 2201
Hsu’s adjustment
GLIMMIX procedure, 2142
Huynh-Feldt
stucture (GLIMMIX), 2195
infinite degrees of freedom
GLIMMIX procedure, 2124, 2134, 2144,
2155, 2165
information criteria
GLIMMIX procedure, 2097
initial values
GLIMMIX procedure, 2179
integral approximation
theory (GLIMMIX), 2216
interaction effects
GLIMMIX procedure, 2259
intercept
GLIMMIX procedure, 2259
inverse gaussian distribution
GLIMMIX procedure, 2166
iteration details
GLIMMIX procedure, 2099
iteration history
GLIMMIX procedure, 2273
iterations
history (GLIMMIX), 2273
Kenward-Roger method
GLIMMIX procedure, 2165
knot selection
GLIMMIX procedure, 2249
L matrices
GLIMMIX procedure, 2121, 2139
mixed model (GLIMMIX), 2121, 2139
lag functionality
GLIMMIX procedure, 2206
Laplace approximation
GLIMMIX procedure, 2100
theory (GLIMMIX), 2222
least squares means
Bonferroni adjustment (GLIMMIX), 2142
BYLEVEL processing (GLIMMIX), 2144,
2147, 2155
comparison types (GLIMMIX), 2145, 2151
covariate values (GLIMMIX), 2143, 2155
Dunnett’s adjustment (GLIMMIX), 2142
generalized linear mixed model
(GLIMMIX), 2139, 2153
Hsu’s adjustment (GLIMMIX), 2142
multiple comparison adjustment
(GLIMMIX), 2141, 2142, 2154
Nelson’s adjustment (GLIMMIX), 2142
observed margins (GLIMMIX), 2146, 2157
Scheffe’s adjustment (GLIMMIX), 2142
Sidak’s adjustment (GLIMMIX), 2142
simple effects (GLIMMIX), 2150
simple effects differences (GLIMMIX),
2150
simulation-based adjustment (GLIMMIX),
2142
Tukey’s adjustment (GLIMMIX), 2142
likelihood ratio test
GLIMMIX procedure, 2125, 2232
linear covariance structure
GLIMMIX procedure, 2195
linearization
theory (GLIMMIX), 2216, 2218
link function
GLIMMIX procedure, 2081, 2169
user-defined (GLIMMIX), 2206
log-normal distribution
GLIMMIX procedure, 2166
Matérn covariance structure
GLIMMIX procedure, 2199
maximum likelihood
GLIMMIX procedure, 2100, 2271
MBN adjusted sandwich estimators
GLIMMIX procedure, 2242
missing level combinations
GLIMMIX procedure, 2260
MIVQUE0 estimation
GLIMMIX procedure, 2182, 2273
mixed model (GLIMMIX)
parameterization, 2259
mixed model smoothing
GLIMMIX procedure, 2187, 2189, 2196,
2197, 2248
MIXED procedure
comparison with the GLIMMIX procedure,
2266
convergence criterion, 2267
mixture
chi-square (GLIMMIX), 2125, 2130
chi-square, weights (GLIMMIX), 2132
Poisson (GLIMMIX), 2396
model
information (GLIMMIX), 2272
multimember effect
GLIMMIX procedure, 2410
multimember example
GLIMMIX procedure, 2410
multinomial distribution
GLIMMIX procedure, 2166
multiple comparison adjustment (GLIMMIX)
estimates, 2133
least squares means, 2141, 2142, 2154
multiple comparisons of estimates
GLIMMIX procedure, 2133
multiple comparisons of least squares means
GLIMMIX procedure, 2141, 2142, 2145,
2151, 2154
multiplicity adjustment
Bonferroni (GLIMMIX), 2142, 2154
Dunnett (GLIMMIX), 2142
estimates (GLIMMIX), 2133
GLIMMIX procedure, 2133
Hsu (GLIMMIX), 2142
least squares means (GLIMMIX), 2142
least squares means estimates (GLIMMIX),
2154
Nelson (GLIMMIX), 2142
row-wise (GLIMMIX), 2133, 2141
Scheffe (GLIMMIX), 2142, 2154
Sidak (GLIMMIX), 2142, 2154
Simulate (GLIMMIX), 2154
simulation-based (GLIMMIX), 2142
step-down p-values (GLIMMIX), 2136,
2152, 2157
T (GLIMMIX), 2154
Tukey (GLIMMIX), 2142
negative binomial distribution
GLIMMIX procedure, 2166
Nelson’s adjustment
GLIMMIX procedure, 2142
nested effects
GLIMMIX procedure, 2259
non-full-rank parameterization
GLIMMIX procedure, 2260
non-positional syntax
GLIMMIX procedure, 2122, 2262, 2404
normal distribution
GLIMMIX procedure, 2166
notation
GLIMMIX procedure, 2081
number of observations
GLIMMIX procedure, 2272
numerical integration
theory (GLIMMIX), 2225
odds estimation
GLIMMIX procedure, 2254
odds ratio estimation
GLIMMIX procedure, 2254
ODS graph names
GLIMMIX procedure, 2297
ODS Graphics
GLIMMIX procedure, 2110, 2280
offset
GLIMMIX procedure, 2172, 2207, 2320,
2321, 2384
optimization
GLIMMIX procedure, 2173
optimization information
GLIMMIX procedure, 2273
options summary
LSMEANS statement, (GLIMMIX), 2140
MODEL statement (GLIMMIX), 2159
PROC GLIMMIX statement, 2092
RANDOM statement (GLIMMIX), 2184
output statistics
GLIMMIX procedure, 2276
overdispersion
GLIMMIX procedure, 2396
P-spline
GLIMMIX procedure, 2196
parameterization
GLIMMIX procedure, 2259
mixed model (GLIMMIX), 2259
penalized B-spline
GLIMMIX procedure, 2196
Poisson distribution
GLIMMIX procedure, 2166
Poisson mixture
GLIMMIX procedure, 2396
positive definiteness
GLIMMIX procedure, 2193
power covariance structure
GLIMMIX procedure, 2200
probability distributions
GLIMMIX procedure, 2166
PROC GLIMMIX procedure
residual variance tolerance, 2118
programming statements
GLIMMIX procedure, 2205
pseudo-likelihood
GLIMMIX procedure, 2100, 2271
quadrature approximation
GLIMMIX procedure, 2100
theory (GLIMMIX), 2225
quasi-likelihood
GLIMMIX procedure, 2271
R-side random effect
GLIMMIX procedure, 2081
radial smoother structure
GLIMMIX procedure, 2197
radial smoothing
GLIMMIX procedure, 2187, 2197, 2248
random effects
GLIMMIX procedure, 2081, 2184
reference category
GLIMMIX procedure, 2265
residual likelihood
GLIMMIX procedure, 2100
residual plots
GLIMMIX procedure, 2280
residual-based sandwich estimators
GLIMMIX procedure, 2241
response level ordering
GLIMMIX procedure, 2161, 2265
response profile
GLIMMIX procedure, 2265, 2272
response variable options
GLIMMIX procedure, 2161
restricted maximum likelihood
GLIMMIX procedure, 2271
reverse response level ordering
GLIMMIX procedure, 2161
sandwich estimator, see also empirical estimator
GLIMMIX procedure, 2094, 2241, 2244
Satterthwaite method
GLIMMIX procedure, 2165, 2239
scale parameter
GLIMMIX compared to GENMOD, 2214
GLIMMIX procedure, 2081–2083, 2098,
2107, 2117, 2129, 2179, 2180, 2183,
2206, 2210, 2213, 2215, 2219, 2220,
2222, 2226, 2237, 2268, 2273, 2275,
2313, 2329, 2333, 2346, 2350, 2379,
2397
Schwarz’s Bayesian information criterion
GLIMMIX procedure, 2097
scoring
GLIMMIX procedure, 2093
Sidak’s adjustment
GLIMMIX procedure, 2142
simple covariance matrix
GLIMMIX procedure, 2199
simple effects
GLIMMIX procedure, 2150
simple effects differences
GLIMMIX procedure, 2150
simulation-based adjustment
GLIMMIX procedure, 2142
singly iterative algorithm
GLIMMIX procedure, 2269
spatial covariance structure
GLIMMIX procedure, 2199
spatial exponential structure
GLIMMIX procedure, 2199
spatial gaussian structure
GLIMMIX procedure, 2199
spatial Matérn structure
GLIMMIX procedure, 2199
spatial power structure
GLIMMIX procedure, 2200
spatial spherical structure
GLIMMIX procedure, 2200
spherical covariance structure
GLIMMIX procedure, 2200
spline comparisons
GLIMMIX procedure, 2404
spline smoothing
GLIMMIX procedure, 2196, 2197
statistical graphics
GLIMMIX procedure, 2280, 2297
subject effect
GLIMMIX procedure, 2191
subject processing
GLIMMIX procedure, 2245
t distribution
GLIMMIX procedure, 2166
table names
GLIMMIX procedure, 2278
test-specification for covariance parameters
GLIMMIX procedure, 2125
testing covariance parameters
GLIMMIX procedure, 2125, 2232
tests of fixed effects
GLIMMIX procedure, 2276
theophylline data
examples, GLIMMIX, 2413
thin plate spline (approx.)
GLIMMIX procedure, 2248
Toeplitz structure
GLIMMIX procedure, 2200
Tukey’s adjustment
GLIMMIX procedure, 2142
Type I testing
GLIMMIX procedure, 2169
Type II testing
GLIMMIX procedure, 2169
Type III testing
GLIMMIX procedure, 2169
unstructured covariance
GLIMMIX procedure, 2201
unstructured covariance matrix
GLIMMIX procedure, 2195
V matrix
GLIMMIX procedure, 2203
variance function
GLIMMIX procedure, 2206
user-defined (GLIMMIX), 2206
Wald test
GLIMMIX procedure, 2276
Wald tests of covariance parameters
GLIMMIX procedure, 2132
weighting
GLIMMIX procedure, 2204
Syntax Index
ABSPCONV option
PROC GLIMMIX statement, 2093
ADJDFE= option
ESTIMATE statement (GLIMMIX), 2133
LSMEANS statement (GLIMMIX), 2141
LSMESTIMATE statement (GLIMMIX),
2154
ADJUST= option
ESTIMATE statement (GLIMMIX), 2133
LSMEANS statement (GLIMMIX), 2142
LSMESTIMATE statement (GLIMMIX),
2154
ALLSTATS option
OUTPUT statement (GLIMMIX), 2177
ALPHA= option
ESTIMATE statement (GLIMMIX), 2134
LSMEANS statement (GLIMMIX), 2143
LSMESTIMATE statement (GLIMMIX),
2155
OUTPUT statement (GLIMMIX), 2178
RANDOM statement (GLIMMIX), 2185
ASYCORR option
PROC GLIMMIX statement, 2093
ASYCOV option
PROC GLIMMIX statement, 2093
AT MEANS option
LSMEANS statement (GLIMMIX), 2143
LSMESTIMATE statement (GLIMMIX),
2155
AT option
LSMEANS statement (GLIMMIX), 2143,
2144
LSMESTIMATE statement (GLIMMIX),
2155
BUCKET= suboption
RANDOM statement (GLIMMIX), 2187
BY statement
GLIMMIX procedure, 2119
BYCAT option
CONTRAST statement (GLIMMIX), 2123
ESTIMATE statement (GLIMMIX), 2134
BYCATEGORY option
CONTRAST statement (GLIMMIX), 2123
ESTIMATE statement (GLIMMIX), 2134
BYLEVEL option
LSMEANS statement (GLIMMIX), 2144
LSMESTIMATE statement (GLIMMIX),
2155
CHISQ option
CONTRAST statement (GLIMMIX), 2124
LSMESTIMATE statement (GLIMMIX),
2155
MODEL statement (GLIMMIX), 2162
CHOL option
PROC GLIMMIX statement, 2094
CHOLESKY option
PROC GLIMMIX statement, 2094
CL option
COVTEST statement (GLIMMIX), 2128
ESTIMATE statement (GLIMMIX), 2134
LSMEANS statement (GLIMMIX), 2144
LSMESTIMATE statement (GLIMMIX),
2155
MODEL statement (GLIMMIX), 2162
RANDOM statement (GLIMMIX), 2185
CLASS statement
GLIMMIX procedure, 2120
CLASSICAL option
COVTEST statement (GLIMMIX), 2130
CONTRAST statement
GLIMMIX procedure, 2121
CORR option
LSMEANS statement (GLIMMIX), 2144
LSMESTIMATE statement (GLIMMIX),
2155
CORRB option
MODEL statement (GLIMMIX), 2162
COV option
LSMEANS statement (GLIMMIX), 2144
LSMESTIMATE statement (GLIMMIX),
2155
COVB option
MODEL statement (GLIMMIX), 2162
COVBI option
MODEL statement (GLIMMIX), 2163
COVTEST statement
GLIMMIX procedure, 2125
DATA= option
OUTPUT statement (GLIMMIX), 2175
PROC GLIMMIX statement, 2094
DDF= option
MODEL statement (GLIMMIX), 2163
DDFM= option
MODEL statement (GLIMMIX), 2164
DER option
OUTPUT statement (GLIMMIX), 2178
DERIVATIVES option
OUTPUT statement (GLIMMIX), 2178
DESCENDING option
MODEL statement, 2161
DF= option
CONTRAST statement (GLIMMIX), 2124
COVTEST statement (GLIMMIX), 2130
ESTIMATE statement (GLIMMIX), 2134
LSMEANS statement (GLIMMIX), 2144
LSMESTIMATE statement (GLIMMIX),
2155
MODEL statement (GLIMMIX), 2163
DIFF option
LSMEANS statement (GLIMMIX), 2145
DIST= option
MODEL statement (GLIMMIX), 2166
DISTRIBUTION= option
MODEL statement (GLIMMIX), 2166
DIVISOR= option
ESTIMATE statement (GLIMMIX), 2134
LSMESTIMATE statement (GLIMMIX),
2156
E option
CONTRAST statement (GLIMMIX), 2124
ESTIMATE statement (GLIMMIX), 2135
LSMEANS statement (GLIMMIX), 2145
LSMESTIMATE statement (GLIMMIX),
2156
MODEL statement (GLIMMIX), 2168
E1 option
MODEL statement (GLIMMIX), 2169
E2 option
MODEL statement (GLIMMIX), 2169
E3 option
MODEL statement (GLIMMIX), 2169
EFFECT statement
GLIMMIX procedure, 2132
ELSM option
LSMESTIMATE statement (GLIMMIX),
2156
EMPIRICAL= option
PROC GLIMMIX statement, 2094
ERROR= option
MODEL statement (GLIMMIX), 2166
ESTIMATE statement
GLIMMIX procedure, 2132
ESTIMATES option
COVTEST statement (GLIMMIX), 2131
EXP option
ESTIMATE statement (GLIMMIX), 2135
LSMESTIMATE statement (GLIMMIX),
2156
EXPHESSIAN option
PROC GLIMMIX statement, 2097
FDIGITS= option
PROC GLIMMIX statement, 2097
FREQ statement
GLIMMIX procedure, 2138
FTEST option
LSMESTIMATE statement (GLIMMIX),
2156
G option
RANDOM statement (GLIMMIX), 2186
GC option
RANDOM statement (GLIMMIX), 2186
GCI option
RANDOM statement (GLIMMIX), 2186
GCOORD= option
RANDOM statement (GLIMMIX), 2186
GCORR option
RANDOM statement (GLIMMIX), 2186
GI option
RANDOM statement (GLIMMIX), 2186
GLIMMIX procedure, 2091
BY statement, 2119
CLASS statement, 2120
CONTRAST statement, 2121
COVTEST statement, 2125
EFFECT statement, 2132
ESTIMATE statement, 2132
FREQ statement, 2138
ID statement, 2138
LSMEANS statement, 2139
LSMESTIMATE statement, 2153
MODEL statement, 2159
NLOPTIONS statement, 2173
OUTPUT statement, 2174
PARMS statement, 2179
PROC GLIMMIX statement, 2092
Programming statements, 2205
RANDOM statement, 2184
syntax, 2091
WEIGHT statement, 2204
GLIMMIX procedure, BY statement, 2119
GLIMMIX procedure, CLASS statement, 2120
TRUNCATE option, 2120
GLIMMIX procedure, CONTRAST statement,
2121
BYCAT option, 2123
BYCATEGORY option, 2123
CHISQ option, 2124
DF= option, 2124
E option, 2124
GROUP option, 2124
SINGULAR= option, 2124
SUBJECT option, 2124
GLIMMIX procedure, COVTEST statement,
2125
CL option, 2128
CLASSICAL option, 2130
ESTIMATES option, 2131
MAXITER= option, 2131
PARMS option, 2131
RESTART option, 2131
TOLERANCE= option, 2132
WALD option, 2132
WGHT= option, 2132
GLIMMIX procedure, DF= statement
CLASSICAL option, 2130
GLIMMIX procedure, EFFECT statement, 2132
GLIMMIX procedure, ESTIMATE statement,
2132
ADJDFE= option, 2133
ADJUST= option, 2133
ALPHA= option, 2134
BYCAT option, 2134
BYCATEGORY option, 2134
CL option, 2134
DF= option, 2134
DIVISOR= option, 2134
E option, 2135
EXP option, 2135
GROUP option, 2135
ILINK option, 2135
LOWERTAILED option, 2136
SINGULAR= option, 2136
STEPDOWN option, 2136
SUBJECT option, 2137
UPPERTAILED option, 2138
GLIMMIX procedure, FREQ statement, 2138
GLIMMIX procedure, ID statement, 2138
GLIMMIX procedure, LSMEANS statement,
2139
ADJUST= option, 2142
ALPHA= option, 2143
AT MEANS option, 2143
AT option, 2143, 2144
BYLEVEL option, 2144
CL option, 2144
CORR option, 2144
COV option, 2144
DF= option, 2144
DIFF option, 2145
E option, 2145
ILINK option, 2145
LINES option, 2146
OBSMARGINS option, 2146
ODDS option, 2146
ODDSRATIO option, 2146
OM option, 2146
PDIFF option, 2145, 2147
PLOT option, 2147
PLOTS option, 2147
SIMPLEDIFF= option, 2150
SIMPLEDIFFTYPE option, 2151
SINGULAR= option, 2150
SLICE= option, 2150
SLICEDIFF= option, 2150
SLICEDIFFTYPE option, 2151
STEPDOWN option, 2152
GLIMMIX procedure, LSMESTIMATE
statement, 2153
ADJUST= option, 2154
ALPHA= option, 2155
AT MEANS option, 2155
AT option, 2155
BYLEVEL option, 2155
CHISQ option, 2155
CL option, 2155
CORR option, 2155
COV option, 2155
DF= option, 2155
DIVISOR= option, 2156
E option, 2156
ELSM option, 2156
EXP option, 2156
FTEST option, 2156
ILINK option, 2156
LOWERTAILED option, 2157
OBSMARGINS option, 2157
OM option, 2157
SINGULAR= option, 2157
STEPDOWN option, 2157
UPPERTAILED option, 2159
GLIMMIX procedure, MODEL statement, 2159
CHISQ option, 2162
CL option, 2162
CORRB option, 2162
COVB option, 2162
COVBI option, 2163
DDF= option, 2163
DDFM= option, 2164
DESCENDING option, 2161
DF= option, 2163
DIST= option, 2166
DISTRIBUTION= option, 2166
E option, 2168
E1 option, 2169
E2 option, 2169
E3 option, 2169
ERROR= option, 2166
HTYPE= option, 2169
INTERCEPT option, 2169
LINK= option, 2169
LWEIGHT= option, 2170
NOCENTER option, 2170
NOINT option, 2171, 2259
ODDSRATIO option, 2171
OFFSET= option, 2172
ORDER= option, 2161
REFLINP= option, 2173
SOLUTION option, 2173, 2260
STDCOEF option, 2173
ZETA= option, 2173
GLIMMIX procedure, OUTPUT statement, 2174
ALLSTATS option, 2177
ALPHA= option, 2178
DATA= option, 2175
DER option, 2178
DERIVATIVES option, 2178
keyword= option, 2175
NOMISS option, 2178
NOUNIQUE option, 2178
NOVAR option, 2178
OBSCAT option, 2178
OUT= option, 2175
SYMBOLS option, 2178
GLIMMIX procedure, PARMS statement, 2179
HOLD= option, 2179
LOWERB= option, 2180
NOBOUND option, 2181
NOITER option, 2181
PARMSDATA= option, 2182
PDATA= option, 2182
UPPERB= option, 2183
GLIMMIX procedure, PROC GLIMMIX
statement, 2092
ABSPCONV option, 2093
ASYCORR option, 2093
ASYCOV option, 2093
CHOL option, 2094
CHOLESKY option, 2094
DATA= option, 2094
EMPIRICAL= option, 2094
EXPHESSIAN option, 2097
FDIGITS= option, 2097
GRADIENT option, 2097
HESSIAN option, 2097
IC= option, 2097
INFOCRIT= option, 2097
INITGLM option, 2099
INITITER option, 2099
ITDETAILS option, 2099
LIST option, 2099
MAXLMMUPDATE option, 2099
MAXOPT option, 2099
METHOD= option, 2100
NAMELEN= option, 2106
NOBOUND option, 2106
NOBSDETAIL option, 2106
NOCLPRINT option, 2106
NOFIT option, 2106
NOINITGLM option, 2107
NOITPRINT option, 2107
NOPROFILE option, 2107
NOREML option, 2107
ODDSRATIO option, 2107
ORDER= option, 2108
OUTDESIGN option, 2109
PCONV option, 2109
PLOT option, 2110
PLOTS option, 2110
PROFILE option, 2117
SCOREMOD option, 2117
SCORING= option, 2117
SINGCHOL= option, 2118
SINGULAR= option, 2118
STARTGLM option, 2118
SUBGRADIENT option, 2118
GLIMMIX procedure, Programming statements,
2205
ABORT statement, 2205
CALL statement, 2205
DELETE statement, 2205
DO statement, 2205
GOTO statement, 2205
IF statement, 2205
LINK statement, 2205
PUT statement, 2205
RETURN statement, 2205
SELECT statement, 2205
STOP statement, 2205
SUBSTR statement, 2205
WHEN statement, 2205
GLIMMIX procedure, RANDOM statement,
2184
ALPHA= option, 2185
CL option, 2185
G option, 2186
GC option, 2186
GCI option, 2186
GCOORD= option, 2186
GCORR option, 2186
GI option, 2186
GROUP= option, 2186
KNOTINFO option, 2187
KNOTMAX= option, 2187
KNOTMETHOD= option, 2187
KNOTMIN= option, 2189
LDATA= option, 2189
NOFULLZ option, 2190
RESIDUAL option, 2190
RSIDE option, 2190
SOLUTION option, 2190
SUBJECT= option, 2191
TYPE= option, 2191
V option, 2203
VC option, 2204
VCI option, 2204
VCORR option, 2204
VI option, 2204
GLIMMIX procedure, WEIGHT statement, 2204
GRADIENT option
PROC GLIMMIX statement, 2097
GROUP option
CONTRAST statement (GLIMMIX), 2124
ESTIMATE statement (GLIMMIX), 2135
GROUP= option
RANDOM statement (GLIMMIX), 2186
HESSIAN option
PROC GLIMMIX statement, 2097
HOLD= option
PARMS statement (GLIMMIX), 2179
HTYPE= option
MODEL statement (GLIMMIX), 2169
IC= option
PROC GLIMMIX statement, 2097
ID statement
GLIMMIX procedure, 2138
ILINK option
ESTIMATE statement (GLIMMIX), 2135
LSMEANS statement (GLIMMIX), 2145
LSMESTIMATE statement (GLIMMIX),
2156
INFOCRIT= option
PROC GLIMMIX statement, 2097
INITGLM option
PROC GLIMMIX statement, 2099
INITITER option
PROC GLIMMIX statement, 2099
INTERCEPT option
MODEL statement (GLIMMIX), 2169
ITDETAILS option
PROC GLIMMIX statement, 2099
keyword= option
OUTPUT statement (GLIMMIX), 2175
KNOTINFO option
RANDOM statement (GLIMMIX), 2187
KNOTMAX= option
RANDOM statement (GLIMMIX), 2187
KNOTMETHOD= option
RANDOM statement (GLIMMIX), 2187
KNOTMIN= option
RANDOM statement (GLIMMIX), 2189
KNOTTYPE= suboption
RANDOM statement (GLIMMIX), 2188
LDATA= option
RANDOM statement (GLIMMIX), 2189
LINES option
LSMEANS statement (GLIMMIX), 2146
LINK= option
MODEL statement (GLIMMIX), 2169
LIST option
PROC GLIMMIX statement, 2099
LOWERB= option
PARMS statement (GLIMMIX), 2180
LOWERTAILED option
ESTIMATE statement (GLIMMIX), 2136
LSMESTIMATE statement (GLIMMIX),
2157
LSMEANS statement
GLIMMIX procedure, 2139
LSMESTIMATE statement
GLIMMIX procedure, 2153
LWEIGHT= option
MODEL statement (GLIMMIX), 2170
MAXITER= option
COVTEST statement (GLIMMIX), 2131
MAXLMMUPDATE option
PROC GLIMMIX statement, 2099
MAXOPT option
PROC GLIMMIX statement, 2099
METHOD= option
PROC GLIMMIX statement, 2100
MODEL statement
GLIMMIX procedure, 2159
NAMELEN= option
PROC GLIMMIX statement, 2106
NEAREST suboption
RANDOM statement (GLIMMIX), 2188
NLOPTIONS statement
GLIMMIX procedure, 2173
NOBOUND option
PARMS statement (GLIMMIX), 2181
PROC GLIMMIX statement, 2106
NOBSDETAIL option
PROC GLIMMIX statement, 2106
NOCENTER option
MODEL statement (GLIMMIX), 2170
NOCLPRINT option
PROC GLIMMIX statement, 2106
NOFIT option
PROC GLIMMIX statement, 2106
NOFULLZ option
RANDOM statement (GLIMMIX), 2190
NOINITGLM option
PROC GLIMMIX statement, 2107
NOINT option
MODEL statement (GLIMMIX), 2171,
2259
NOITER option
PARMS statement (GLIMMIX), 2181
NOITPRINT option
PROC GLIMMIX statement, 2107
NOMISS option
OUTPUT statement (GLIMMIX), 2178
NOPROFILE option
PROC GLIMMIX statement, 2107
NOREML option
PROC GLIMMIX statement, 2107
NOUNIQUE option
OUTPUT statement (GLIMMIX), 2178
NOVAR option
OUTPUT statement (GLIMMIX), 2178
OBSCAT option
OUTPUT statement (GLIMMIX), 2178
OBSMARGINS option
LSMEANS statement (GLIMMIX), 2146
LSMESTIMATE statement (GLIMMIX),
2157
ODDS option
LSMEANS statement (GLIMMIX), 2146
ODDSRATIO option
LSMEANS statement (GLIMMIX), 2146
MODEL statement (GLIMMIX), 2171
PROC GLIMMIX statement, 2107
OFFSET= option
MODEL statement (GLIMMIX), 2172
OM option
LSMEANS statement (GLIMMIX), 2146
LSMESTIMATE statement (GLIMMIX),
2157
ORDER= option
MODEL statement, 2161
PROC GLIMMIX statement, 2108
OUT= option
OUTPUT statement (GLIMMIX), 2175
OUTDESIGN option
PROC GLIMMIX statement, 2109
OUTPUT statement
GLIMMIX procedure, 2174
PARMS option
COVTEST statement (GLIMMIX), 2131
PARMS statement
GLIMMIX procedure, 2179
PARMSDATA= option
PARMS statement (GLIMMIX), 2182
PCONV option
PROC GLIMMIX statement, 2109
PDATA= option
PARMS statement (GLIMMIX), 2182
PDIFF option
LSMEANS statement (GLIMMIX), 2145,
2147
PLOT option
LSMEANS statement (GLIMMIX), 2147
PROC GLIMMIX statement, 2110
PLOTS option
LSMEANS statement (GLIMMIX), 2147
PROC GLIMMIX statement, 2110
PROC GLIMMIX procedure, PROC GLIMMIX
statement
SINGRES= option, 2118
PROC GLIMMIX statement, see GLIMMIX
procedure
GLIMMIX procedure, 2092
PROFILE option
PROC GLIMMIX statement, 2117
Programming statements
GLIMMIX procedure, 2205
RANDOM statement
GLIMMIX procedure, 2184
RANDOM statement (GLIMMIX)
BUCKET= suboption, 2187
KNOTTYPE= suboption, 2188
NEAREST suboption, 2188
TREEINFO suboption, 2188
REFLINP= option
MODEL statement (GLIMMIX), 2173
RESIDUAL option
RANDOM statement (GLIMMIX), 2190
RESTART option
COVTEST statement (GLIMMIX), 2131
RSIDE option
RANDOM statement (GLIMMIX), 2190
SCOREMOD option
PROC GLIMMIX statement, 2117
SCORING= option
PROC GLIMMIX statement, 2117
SIMPLEDIFFTYPE option
LSMEANS statement (GLIMMIX), 2151
SIMPLEEDIFF= option
LSMEANS statement (GLIMMIX), 2150
SINGCHOL= option
PROC GLIMMIX statement, 2118
SINGRES= option
PROC GLIMMIX statement (GLIMMIX),
2118
SINGULAR= option
CONTRAST statement (GLIMMIX), 2124
ESTIMATE statement (GLIMMIX), 2136
LSMEANS statement (GLIMMIX), 2150
LSMESTIMATE statement (GLIMMIX),
2157
PROC GLIMMIX statement, 2118
SLICE= option
LSMEANS statement (GLIMMIX), 2150
SLICEDIFF= option
LSMEANS statement (GLIMMIX), 2150
SLICEDIFFTYPE option
LSMEANS statement (GLIMMIX), 2151
SOLUTION option
MODEL statement (GLIMMIX), 2173,
2260
RANDOM statement (GLIMMIX), 2190
STARTGLM option
PROC GLIMMIX statement, 2118
STDCOEF option
MODEL statement (GLIMMIX), 2173
STEPDOWN option
ESTIMATE statement (GLIMMIX), 2136
LSMEANS statement (GLIMMIX), 2152
LSMESTIMATE statement (GLIMMIX),
2157
SUBGRADIENT option
PROC GLIMMIX statement, 2118
SUBJECT option
CONTRAST statement (GLIMMIX), 2124
ESTIMATE statement (GLIMMIX), 2137
SUBJECT= option
RANDOM statement (GLIMMIX), 2191
SYMBOLS option
OUTPUT statement (GLIMMIX), 2178
syntax
GLIMMIX procedure, 2091
TOLERANCE= option
COVTEST statement (GLIMMIX), 2132
TREEINFO suboption
RANDOM statement (GLIMMIX), 2188
TRUNCATE option
CLASS statement (GLIMMIX), 2120
TYPE= option
RANDOM statement (GLIMMIX), 2191
UPPERB= option
PARMS statement (GLIMMIX), 2183
UPPERTAILED option
ESTIMATE statement (GLIMMIX), 2138
LSMESTIMATE statement (GLIMMIX),
2159
V option
RANDOM statement (GLIMMIX), 2203
VC option
RANDOM statement (GLIMMIX), 2204
VCI option
RANDOM statement (GLIMMIX), 2204
VCORR option
RANDOM statement (GLIMMIX), 2204
VI option
RANDOM statement (GLIMMIX), 2204
WALD option
COVTEST statement (GLIMMIX), 2132
WEIGHT statement
GLIMMIX procedure, 2204
WGHT= option
COVTEST statement (GLIMMIX), 2132
ZETA= option
MODEL statement (GLIMMIX), 2173
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