Introduction to Statistical Modeling with SAS/STAT Software SAS/STAT

Introduction to Statistical Modeling with SAS/STAT Software SAS/STAT
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SAS/STAT 9.2 User’s Guide
Introduction to Statistical
Modeling with SAS/STAT
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Chapter 3
Introduction to Statistical Modeling with
SAS/STAT Software
Contents
Overview: Statistical Modeling . . . . . . . . . . . . . . . . . . . . . . .
Statistical Models . . . . . . . . . . . . . . . . . . . . . . . . . .
Classes of Statistical Models . . . . . . . . . . . . . . . . . . . . .
Linear and Nonlinear Models . . . . . . . . . . . . . . . .
Regression Models and Models with Classification Effects
Univariate and Multivariate Models . . . . . . . . . . . .
Fixed, Random, and Mixed Models . . . . . . . . . . . .
Generalized Linear Models . . . . . . . . . . . . . . . . .
Latent Variable Models . . . . . . . . . . . . . . . . . . .
Bayesian Models . . . . . . . . . . . . . . . . . . . . . .
Classical Estimation Principles . . . . . . . . . . . . . . . . . . .
Least Squares . . . . . . . . . . . . . . . . . . . . . . . .
Likelihood . . . . . . . . . . . . . . . . . . . . . . . . .
Inference Principles for Survey Data . . . . . . . . . . . .
Statistical Background . . . . . . . . . . . . . . . . . . . . . . . . . . .
Hypothesis Testing and Power . . . . . . . . . . . . . . . . . . . .
Important Linear Algebra Concepts . . . . . . . . . . . . . . . . .
Expectations of Random Variables and Vectors . . . . . . . . . . .
Mean Squared Error . . . . . . . . . . . . . . . . . . . . . . . . .
Linear Model Theory . . . . . . . . . . . . . . . . . . . . . . . .
Finding the Least Squares Estimators . . . . . . . . . . .
Analysis of Variance . . . . . . . . . . . . . . . . . . . .
Estimating the Error Variance . . . . . . . . . . . . . . .
Maximum Likelihood Estimation . . . . . . . . . . . . .
Estimable Functions . . . . . . . . . . . . . . . . . . . .
Test of Hypotheses . . . . . . . . . . . . . . . . . . . . .
Residual Analysis . . . . . . . . . . . . . . . . . . . . . .
Sweep Operator . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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28 F Chapter 3: Introduction to Statistical Modeling with SAS/STAT Software
Overview: Statistical Modeling
There are more than 70 procedures in SAS/STAT software, and the majority of them are dedicated
to solving problems in statistical modeling. The goal of this chapter is to provide a roadmap to
statistical models and to modeling tasks, enabling you to make informed choices about the appropriate modeling context and tool. This chapter also introduces important terminology, notation,
and concepts used throughout this documentation. Subsequent introductory chapters discuss model
families and related procedures.
It is difficult to capture the complexity of statistical models in a simple scheme, so the classification
used here is necessarily incomplete. It is most practical to classify models in terms of simple criteria,
such as the presence of random effects, the presence of nonlinearity, characteristics of the data, and
so on. That is the approach used here. After a brief introduction to statistical modeling in general
terms, the chapter describes a number of model classifications and relates them to modeling tools
in SAS/STAT software.
Statistical Models
Deterministic and Stochastic Models
Purely mathematical models, in which the relationships between inputs and outputs are captured
entirely in deterministic fashion, can be important theoretical tools but are impractical for describing
observational, experimental, or survey data. For such phenomena, researchers usually allow the
model to draw on stochastic as well as deterministic elements. When the uncertainty of realizations
leads to the inclusion of random components, the resulting models are called stochastic models.
A statistical model, finally, is a stochastic model that contains parameters, which are unknown
constants that need to be estimated based on assumptions about the model and the observed data.
There are many reasons why statistical models are preferred over deterministic models. For example:
Randomness is often introduced into a system in order to achieve a certain balance or representativeness. For example, random assignment of treatments to experimental units allows
unbiased inferences about treatment effects. As another example, selecting individuals for a
survey sample by random mechanisms ensures a representative sample.
Even if a deterministic model can be formulated for the phenomenon under study, a stochastic
model can provide a more parsimonious and more easily comprehended description. For
example, it is possible in principle to capture the result of a coin toss with a deterministic
model, taking into account the properties of the coin, the method of tossing, conditions of the
medium through which the coin travels and of the surface on which it lands, and so on. A
very complex model is required to describe the simple outcome—heads or tails. Alternatively,
you can describe the outcome quite simply as the result of a stochastic process, a Bernoulli
variable that results in heads with a certain probability.
Statistical Models F 29
It is often sufficient to describe the average behavior of a process, rather than each particular
realization. For example, a regression model might be developed to relate plant growth to
nutrient availability. The explicit aim of the model might be to describe how the average
growth changes with nutrient availability, not to predict the growth of an individual plant.
The support for the notion of averaging in a model lies in the nature of expected values,
describing typical behavior in the presence of randomness. This, in turn, requires that the
model contain stochastic components.
The defining characteristic of statistical models is their dependence on parameters and the incorporation of stochastic terms. The properties of the model and the properties of quantities derived from
it must be studied in a long-run, average sense through expectations, variances, and covariances.
The fact that the parameters of the model must be estimated from the data introduces a stochastic
element in applying a statistical model: because the model is not deterministic but includes randomness, parameters and related quantities derived from the model are likewise random. The properties
of parameter estimators can often be described only in an asymptotic sense, imagining that some
aspect of the data increases without bound (for example, the number of observations or the number
of groups).
The process of estimating the parameters in a statistical model based on your data is called fitting
the model. For many classes of statistical models there are a number of procedures in SAS/STAT
software that can perform the fitting. In many cases, different procedures solve identical estimation problems—that is, their parameter estimates are identical. In some cases, the same model
parameters are estimated by different statistical principles, such as least squares versus maximum
likelihood estimation. Parameter estimates obtained by different methods typically have different statistical properties—distribution, variance, bias, and so on. The choice between competing
estimation principles is often made on the basis of properties of the estimators. Distinguishing
properties might include (but are not necessarily limited to) computational ease, interprative ease,
bias, variance, mean squared error, and consistency.
Model-Based and Design-Based Randomness
A statistical model is a description of the data-generating mechanism, not a description of the specific data to which it is applied. The aim of a model is to capture those aspects of a phenomenon
that are relevant to inquiry and to explain how the data could have come about as a realization of
a random experiment. These relevant aspects might include the genesis of the randomness and the
stochastic effects in the phenomenon under study. Different schools of thought can lead to different
model formulations, different analytic strategies, and different results. Coarsely, you can distinguish
between a viewpoint of innate randomness and one of induced randomness. This distinction leads
to model-based and design-based inference approaches.
In a design-based inference framework, the random variation in the observed data is induced by
random selection or random assignment. Consider the case of a survey sample from a finite population of size N ; suppose that FN D fyi W i 2 UN g denotes the finite set of possible values and UN
is the index set UN D f1; 2; : : : ; N g. Then a sample S , a subset of UN , is selected by probability
rules. The realization of the random experiment is the selection of a particular set S ; the associated
values selected from FN are considered fixed. If properties of a design-based sampling estimator
30 F Chapter 3: Introduction to Statistical Modeling with SAS/STAT Software
are evaluated, such as bias, variance, and mean squared error, they are evaluated with respect to the
distribution induced by the sampling mechanism.
Design-based approaches also play an important role in the analysis of data from controlled experiments by randomization tests. Suppose that k treatments are to be assigned to kr homogeneous
experimental units. If you form k sets of r units with equal probability, and you assign the j th treatment to the tth set, a completely randomized experimental design (CRD) results. A design-based
view treats the potential response of a particular treatment for a particular experimental unit as a
constant. The stochastic nature of the error-control design is induced by randomly selecting one of
the potential responses.
Statistical models are often used in the design-based framework. In a survey sample the model is
used to motivate the choice of the finite population parameters and their sample-based estimators.
In an experimental design, an assumption of additivity of the contributions from treatments, experimental units, observational errors, and experimental errors leads to a linear statistical model. The
approach to statistical inference where statistical models are used to construct estimators and their
properties are evaluated with respect to the distribution induced by the sample selection mechanism
is known as model-assisted inference (Särndal, Swensson, and Wretman 1992).
In a purely model-based framework, the only source of random variation for inference comes from
the unknown variation in the responses. Finite population values are thought of as a realization of a
superpopulation model that describes random variables Y1 ; Y2 ; . The observed values y1 ; y2 ; are realizations of these random variables. A model-based framework does not imply that there
is only one source of random variation in the data. For example, mixed models might contain
random terms that represent selection of effects from hierarchical (super-) populations at different
granularity. The analysis takes into account the hierarchical structure of the random variation, but it
continues to be model based.
A design-based approach is implicit in SAS/STAT procedures whose name commences with SURVEY, such as the SURVEYFREQ, SURVEYMEANS, SURVEYREG, and SURVEYLOGISTIC
procedures. Inferential approaches are model based in other SAS/STAT procedures. For more information about analyzing survey data with SAS/STAT software, see Chapter 14, “Introduction to
Survey Procedures.”
Model Specification
If the model is accepted as a description of the data-generating mechanism, then its parameters
are estimated using the data at hand. Once the parameter estimates are available, you can apply
the model to answer questions of interest about the study population. In other words, the model
becomes the lens through which you view the problem itself, in order to ask and answer questions
of interest. For example, you might use the estimated model to derive new predictions or forecasts,
to test hypotheses, to derive confidence intervals, and so on.
Obviously, the model must be “correct” to the extent that it sufficiently describes the data-generating
mechanism. Model selection, diagnosis, and discrimination are important steps in the modelbuilding process. This is typically an iterative process, starting with an initial model and refining
it. The first important step is thus to formulate your knowledge about the data-generating process
and to express the real observed phenomenon in terms of a statistical model. A statistical model describes the distributional properties of one or more variables, the response variables. The extent of
Classes of Statistical Models F 31
the required distributional specification depends on the model, estimation technique, and inferential
goals. This description often takes the simple form of a model with additive error structure:
response = mean + error
In mathematical notation this simple model equation becomes
Y D f .x1 ; ; xk I ˇ1 ; ; ˇp / C In this equation Y is the response variable, often also called the dependent variable or the outcome variable. The terms x1 ; ; xk denote the values of k regressor variables, often termed the
covariates or the “independent” variables. The terms ˇ1 ; ; ˇp denote parameters of the model,
unknown constants that are to be estimated. The term denotes the random disturbance of the
model; it is also called the residual term or the error term of the model.
In this simple model formulation, stochastic properties are usually associated only with the term.
The covariates x1 ; ; xk are usually known values, not subject to random variation. Even if the
covariates are measured with error, so that their values are in principle random, they are considered
fixed in most models fit by SAS/STAT software. In other words, stochastic properties under the
model are derived conditional on the xs. If is the only stochastic term in the model, and if the
errors have a mean of zero, then the function f ./ is the mean function of the statistical model. More
formally,
EŒY  D f .x1 ; ; xk I ˇ1 ; ; ˇp /
where EŒ denotes the expectation operator.
In many applications, a simple model formulation is inadequate. It might be necessary to specify
not only the stochastic properties of a single error term, but also how model errors associated with
different observations relate to each other. A simple additive error model is typically inappropriate
to describe the data-generating mechanism if the errors do not have zero mean or if the variance of
observations depends on their means. For example, if Y is a Bernoulli random variable that takes
on the values 0 and 1 only, a regression model with additive error is not meaningful. Models for
such data require more elaborate formulations involving probability distributions.
Classes of Statistical Models
Linear and Nonlinear Models
A statistical estimation problem is nonlinear if the estimating equations—the equations whose solution yields the parameter estimates—depend on the parameters in a nonlinear fashion. Such estimation problems typically have no closed-form solution and must be solved by iterative, numerical
techniques.
Nonlinearity in the mean function is often used to distinguish between linear and nonlinear models.
A model has a nonlinear mean function if the derivative of the mean function with respect to the
32 F Chapter 3: Introduction to Statistical Modeling with SAS/STAT Software
parameters depends on at least one other parameter. Consider, for example, the following models
that relate a response variable Y to a single regressor variable x:
EŒY jx D ˇ0 C ˇ1 x
EŒY jx D ˇ0 C ˇ1 x C ˇ2 x 2
EŒY jx D ˇ C x=˛
In these expressions, EŒY jx denotes the expected value of the response variable Y at the fixed
value of x. (The conditioning on x simply indicates that the predictor variables are assumed to be
non-random. Conditioning is often omitted for brevity in this and subsequent chapters.)
The first model in the previous list is a simple linear regression (SLR) model. It is linear in the
parameters ˇ0 and ˇ1 since the model derivatives do not depend on unknowns:
@
.ˇ0 C ˇ1 x/ D 1
ˇ0
@
.ˇ0 C ˇ1 x/ D x
ˇ1
The model is also linear in its relationship with x (a straight line). The second model is also linear
in the parameters, since
@
ˇ0 C ˇ1 x C ˇ2 x 2 D 1
ˇ0
@
ˇ0 C ˇ1 x C ˇ2 x 2 D x
ˇ1
@
ˇ0 C ˇ1 x C ˇ2 x 2 D x 2
ˇ2
However, this second model is curvilinear, since it exhibits a curved relationship when plotted
against x. The third model, finally, is a nonlinear model since
@
.ˇ C x=˛/ D 1
ˇ
@
x
.ˇ C x=˛/ D
˛
˛2
The second of these derivatives depends on a parameter ˛. A model is nonlinear if it is not linear
in at least one parameter. Only the third model is a nonlinear model. A graph of EŒY  versus the
regressor variable thus does not indicate whether a model is nonlinear. A curvilinear relationship in
this graph can be achieved by a model that is linear in the parameters.
Nonlinear mean functions lead to nonlinear estimation. It is important to note, however, that nonlinear estimation arises also because of the estimation principle or because the model structure contains
nonlinearity in other parts, such as the covariance structure. For example, fitting a simple linear regression model by minimizing the sum of the absolute residuals leads to a nonlinear estimation
problem despite the fact that the mean function is linear.
Classes of Statistical Models F 33
Regression Models and Models with Classification Effects
A linear regression model in the broad sense has the form
Y D Xˇ C where Y is the vector of response values, X is the matrix of regressor effects, ˇ is the vector of
regression parameters, and is the vector of errors or residuals. A regression model in the narrow
sense—as compared to a classification model—is a linear model in which all regressor effects are
continuous variables. In other words, each effect in the model contributes a single column to the X
matrix and a single parameter to the overall model. For example, a regression of subjects’ weight
(Y ) on the regressors age (x1 ) and body mass index (bmi, x2 ) is a regression model in this narrow
sense. In symbolic notation you can write this regression model as
weight = age + bmi + error
This symbolic notation expands into the statistical model
Yi D ˇ0 C ˇ1 xi1 C ˇ2 xi2 C i
Single parameters are used to model the effects of age .ˇ1 / and bmi .ˇ2 /, respectively.
A classification effect, on the other hand, is associated with possibly more than one column of
the X matrix. Classification with respect to a variable is the process by which each observation is
associated with one of k levels; the process of determining these k levels is referred to as levelization
of the variable. Classification variables are used in models to identify experimental conditions,
group membership, treatments, and so on. The actual values of the classification variable are not
important, and the variable can be a numeric or a character variable. What is important is the
association of discrete values or levels of the classification variable with groups of observations.
For example, in the previous illustration, if the regression also takes into account the subjects’
gender, this can be incorporated in the model with a two-level classification variable. Suppose that
the values of the gender variable are coded as “F” and “M,” respectively. In symbolic notation the
model
weight = age + bmi + gender + error
expands into the statistical model
Yi D ˇ0 C ˇ1 xi1 C ˇ2 xi2 C 1 I.gender D “F”/ C 2 I.gender D “M”/ C i
where I.gender D“F”/ is the indicator function that returns 1 if the value of the gender variable is
“F” and 0 otherwise. Parameters 1 and 2 are associated with the gender classification effect. This
form of parameterizing the gender effect in the model is only one of several different methods of
incorporating the levels of a classification variable in the model. This form, the so-called singular
parameterization, is the most general approach, and it is used in the GLM, MIXED, and GLIMMIX
procedures. Alternatively, classification effects with various forms of nonsingular parameterizations are available in such procedures as GENMOD and LOGISTIC. See the documentation for
the individual SAS/STAT procedures on their respective facilities for parameterizing classification
34 F Chapter 3: Introduction to Statistical Modeling with SAS/STAT Software
variables and the section “Parameterization of Model Effects” on page 368 in Chapter 18, “Shared
Concepts and Topics,” for general details.
Models that contain only classification effects are often identified with analysis of variance
(ANOVA) models, because ANOVA methods are frequently used in their analysis. This is particularly true for experimental data where the model effects comprise effects of the treatment and
error-control design. However, classification effects appear more widely than in models to which
analysis of variance methods are applied. For example, many mixed models, where parameters
are estimated by restricted maximum likelihood, consist entirely of classification effects but do not
permit the sum of squares decomposition typical for ANOVA techniques.
Many models contain both continuous and classification effects. For example, a continuous-byclass effect consists of at least one continuous variable and at least one classification variable. Such
effects are convenient, for example, to vary slopes in a regression model by the levels of a classification variable. Also, recent enhancements to linear modeling syntax in some SAS/STAT procedures
(including GLIMMIX and GLMSELECT) enable you to construct sets of columns in X matrices
from a single continuous variable. An example is modeling with splines where the values of a continuous variable x are expanded into a spline basis that occupies multiple columns in the X matrix.
For purposes of the analysis you can treat these columns as a single unit or as individual, unrelated columns. For more details, see the section “Constructed Effects and the EFFECT Statement
(Experimental)” on page 377 in Chapter 18, “Shared Concepts and Topics.”
Univariate and Multivariate Models
A multivariate statistical model is a model in which multiple response variables are modeled jointly.
Suppose, for example, that your data consist of heights .hi / and weights .wi / of children, collected
over several years .ti /. The following separate regressions represent two univariate models:
wi D ˇw0 C ˇw1 ti C wi
hi D ˇh0 C ˇh1 ti C hi
In the univariate setting, no information about the children’s heights “flows” to the model about
their weights and vice versa. In a multivariate setting, the heights and weights would be modeled
jointly. For example:
wi
wi
Yi D
D Xˇ C
hi
hi
D Xˇ C i
2
1 12
i 0;
12 22
The vectors Yi and i collect the responses and errors for the two observation that belong to the
same subject. The errors from the same child now have the correlation
CorrŒwi ; hi  D q
12
12 22
and it is through this correlation that information about heights “flows” to the weights and vice
versa. This simple example shows only one approach to modeling multivariate data, through the
Classes of Statistical Models F 35
use of covariance structures. Other techniques involve seemingly unrelated regressions, systems of
linear equations, and so on.
Multivariate data can be coarsely classified into three types. The response vectors of homogeneous
multivariate data consist of observations of the same attribute. Such data are common in repeated
measures experiments and longitudinal studies, where the same attribute is measured repeatedly
over time. Homogeneous multivariate data also arise in spatial statistics where a set of geostatistical
data is the incomplete observation of a single realization of a random experiment that generates a
two-dimensional surface. One hundred measurements of soil electrical conductivity collected in a
forest stand compose a single observation of a 100-dimensional homogeneous multivariate vector.
Heterogeneous multivariate observations arise when the responses that are modeled jointly refer
to different attributes, such as in the previous example of children’s weights and heights. There
are two important subtypes of heterogeneous multivariate data. In homocatanomic multivariate
data the observations come from the same distributional family. For example, the weights and
heights might both be assumed to be normally distributed. With heterocatanomic multivariate data
the observations can come from different distributional families. The following are examples of
heterocatanomic multivariate data:
For each patient you observe blood pressure (a continuous outcome), the number of prior
episodes of an illness (a count variable), and whether the patient has a history of diabetes in
the family (a binary outcome). A multivariate model that models the three attributes jointly
might assume a lognormal distribution for the blood pressure measurements, a Poisson distribution for the count variable and a Bernoulli distribution for the family history.
In a study of HIV/AIDS survival, you model jointly a patients CD4 cell count over time—
itself a homogeneous multivariate outcome—and the survival of the patient (event-time data).
Fixed, Random, and Mixed Models
Each term in a statistical model represents either a fixed effect or a random effect. Models in
which all effects are fixed are called fixed-effects models. Similarly, models in which all effects are
random—apart from possibly an overall intercept term—are called random-effects models. Mixed
models, then, are those models that have fixed-effects and random-effects terms. In matrix notation,
the linear fixed, linear random, and linear mixed model are represented by the following model
equations, respectively:
Y D Xˇ
YD
C
Z C Y D Xˇ C Z C In these expressions, X and Z are design or regressor matrices associated with the fixed and random
effects, respectively. The vector ˇ is a vector of fixed-effects parameters, and the vector represents
the random effects. The mixed modeling procedures in SAS/STAT software assume that the random
effects follow a normal distribution with variance-covariance matrix G and, in most cases, that
the random effects have mean zero.
Random effects are often associated with classification effects, but this is not necessary. As an
example of random regression effects, you might want to model the slopes in a growth model
36 F Chapter 3: Introduction to Statistical Modeling with SAS/STAT Software
as consisting of two components: an overall (fixed-effects) slope that represents the slope of the
average individual, and individual-specific random deviations from the overall slope. The X and Z
matrix would then have column entries for the regressor variable associated with the slope. You are
modeling fixed and randomly varying regression coefficients.
Having random effects in your model has a number of important consequences:
Some observations are no longer uncorrelated but instead have a covariance that depends on
the variance of the random effects.
You can and should distinguish between the inference spaces; inferences can be drawn in a
broad, intermediate, and narrow inference space. In the narrow inference space, conclusions
are drawn about the particular values of the random effects selected in the study. The broad
inference space applies if inferences are drawn with respect to all possible levels of the random effects. The intermediate inference space can be applied for effects consisting of more
than one random term, when inferences are broad with respect to some factors and narrow
with respect to others. In fixed-effects models, there is no corresponding concept to the broad
and intermediate inference spaces.
Depending on the structure of G and VarŒ and also subject to the balance in your data, there
might be no closed-form solution for the parameter estimates. Although the model is linear
in ˇ, iterative estimation methods might be required to estimate all parameters of the model.
Certain concepts, such as least squares means and Type III estimable functions, are meaningful only for fixed effects.
By using random effects, you are modeling variation through variance. Variation in data
simply implies that things are not equal. Variance, on the other hand, describes a feature of a
random variable. Random effects in your model are random variables: they model variation
through variance.
It is important to properly determine the nature of the model effects as fixed or random. An effect
is either fixed or random by its very nature; it is improper to consider it fixed in one analysis
and random in another depending on what type of results you want to produce. If, for example,
a treatment effect is random and you are interested in comparing treatment means, and only the
levels selected in the study are of interest, then it is not appropriate to model the treatment effect
as fixed so that you can draw on least squares mean analysis. The appropriate strategy is to model
the treatment effect as random and to compare the solutions for the treatment effects in the narrow
inference space.
In determining whether an effect is fixed or random, it is helpful to inquire about the genesis of
the effect. If the levels of an effect are randomly sampled, then the effect is a random effect. The
following are examples:
In a large clinical trial, drugs A, B, and C are applied to patients in various clinical centers. If
the clinical centers are selected at random from a population of possible clinics, their effect
on the response is modeled with a random effect.
In repeated measures experiments with people or animals as subjects, subjects are declared
to be random because they are selected from the larger population to which you want to
generalize.
Classes of Statistical Models F 37
Fertilizers could be applied at a number of levels. Three levels are randomly selected for an
experiment to represent the population of possible levels. The fertilizer effects are random
effects.
Quite often it is not possible to select effects at random, or it is not known how the values in the
data became part of the study. For example, suppose you are presented with a data set consisting
of student scores in three school districts, with four to ten schools in each district and two to three
classrooms in each school. How do you decide which effects are fixed and which are random? As
another example, in an agricultural experiment conducted in successive years at two locations, how
do you decide whether location and year effects are fixed or random? In these situations, the fixed
or random nature of the effect might be debatable, bearing out the adage that “one modeler’s fixed
effect is another modeler’s random effect.” However, this fact does not constitute license to treat as
random those effects that are clearly fixed, or vice versa.
When an effect cannot be randomized or it is not known whether its levels have been randomly selected, it can be a random effect if its impact on the outcome variable is of a stochastic nature—that
is, if it is the realization of a random process. Again, this line of thinking relates to the genesis of the
effect. A random year, location, or school district effect is a placeholder for different environments
that cannot be selected at random but whose effects are the cumulative result of many individual
random processes. Note that this argument does not imply that effects are random because the experimenter does not know much about them. The key notion is that effects represent something,
whether or not that something is known to the modeler. Broadening the inference space beyond the
observed levels is thus possible, although you might not be able to articulate what the realizations
of the random effects represent.
A consequence of having random effects in your model is that some observations are no longer
uncorrelated but instead have a covariance that depends on the variance of the random effect. In
fact, in some modeling applications random effects might be used not only to model heterogeneity
in the parameters of a model, but also to induce correlations among observations. The typical
assumption about random effects in SAS/STAT software is that the effects are normally distributed.
For more information about mixed modeling tools in SAS/STAT software, see Chapter 6,
“Introduction to Mixed Modeling Procedures.”
Generalized Linear Models
A class of models that has gained increasing importance in the past several decades is the class
of generalized linear models. The theory of generalized linear models originated with Nelder and
Wedderburn (1972) and Wedderburn (1974), and was subsequently made popular in the monograph
by McCullagh and Nelder (1989). This class of models extends the theory and methods of linear
models to data with nonnormal responses. Before this theory was developed, modeling of nonnormal data typically relied on transformations of the data, and the transformations were chosen to
improve symmetry, homogeneity of variance, or normality. Such transformations have to be performed with care because they also have implications for the error structure of the model, Also,
back-transforming estimates or predicted values can introduce bias.
Generalized linear models also apply a transformation, known as the link function, but it is applied
to a deterministic component, the mean of the data. Furthermore, generalized linear model take the
38 F Chapter 3: Introduction to Statistical Modeling with SAS/STAT Software
distribution of the data into account, rather than assuming that a transformation of the data leads to
normally distributed data to which standard linear modeling techniques can be applied.
To put this generalization in place requires a slightly more sophisticated model setup than that
required for linear models for normal data:
The systematic component is a linear predictor similar to that in linear models, D x0 ˇ. The
linear predictor is a linear function in the parameters. In contrast to the linear model, does
not represent the mean function of the data.
The link function g. / relates the linear predictor to the mean, g./ D . The link function
is a monotonic, invertible function. The mean can thus be expressed as the inversely linked
linear predictor, D g 1 ./. For example, a common link function for binary and binomial
data is the logit link, g.t / D logft=.1 t/g. The mean function of a generalized linear model
with logit link and a single regressor can thus be written as
log
D ˇ0 C ˇ1 x
1 1
D
1 C expf ˇ0 ˇ1 xg
This is known as a logistic regression model.
The random component of a generalized linear model is the distribution of the data, assumed
to be a member of the exponential family of distributions. Discrete members of this family
include the Bernoulli (binary), binomial, Poisson, geometric, and negative binomial (for a
given value of the scale parameter) distribution. Continuous members include the normal
(Gaussian), beta, gamma, inverse gaussian, and exponential distribution.
The standard linear model with normally distributed error is a special case of a generalized linear
model; the link function is the identity function and the distribution is normal.
Latent Variable Models
Latent variable modeling involves variables that are not observed directly in your research. It has
a relatively long history, dating back from the measure of general intelligence by common factor
analysis (Spearman 1904) to the emergence of modern-day structural equation modeling (Jöreskog
1973; Keesling, 1972; Wiley, 1973).
Latent variables are involved in almost all kinds of regression models. In a broad sense, all additive error terms in regression models are latent variables simply because they are not measured in
research. Hereafter, however, a narrower sense of latent variables is used when referring to latent
variable models. Latent variables are systematic unmeasured variables that are also referred to as
factors. For example, in the following diagram a simple relation between Emotional Intelligence and
Career Achievement is shown:
Emotional Intelligence
ˇ
Career Achievement
ı
Classes of Statistical Models F 39
In the diagram, both Emotional Intelligence and Career Achievement are treated as latent factors. They
are hypothetical constructs in your model. You hypothesize that Emotional Intelligence is a “causal
factor” or predictor of Career Achievement. The symbol ˇ represents the regression coefficient or
the effect of Emotional Intelligence on Career Achievement. However, the “causal relationship” or
prediction is not perfect. There is an error term ı, which accounts for the unsystematic part of the
prediction. You can represent the preceding diagram by using the following linear equation:
CA D ˇEI C ı
where CA represents Career Achievement and EI represents Emotional Intelligence. The means of the
latent factors in the linear model are arbitrary, and so they are assumed to be zero. The error variable
ı also has a zero mean with an unknown variance. This equation represents the so-called “structural
model,” where the “true” relationships among latent factors are theorized.
In order to model this theoretical model with latent factors, some observed variables must somehow
relate to these factors. This calls for the measurement models for latent factors. For example, Emotional Intelligence could be measured by some established tests. In these tests, individuals are asked
to respond to certain special situations that involve stressful decision making, personal confrontations, and so on. Their responses to these situations are then rated by experts or a standardized
scoring system. Suppose there are three such tests and the test scores are labeled as X1, X2 and
X3, respectively. The measurement model for the latent factor Emotional Intelligence is specified as
follows:
X1 D a1 EI C e1
X2 D a2 EI C e2
X3 D a3 EI C e3
where a1 , a2 , and a3 are regression coefficients and e1 , e2 , and e3 are measurement errors. Measurement errors are assumed to be independent of the latent factors EI and CA. In the measurement
model, X1, X2, and X3 are called the indicators of the latent variable EI. These observed variables
are assumed to be centered in the model, and therefore no intercept terms are needed. Each of the
indicators is a scaled measurement of the latent factor EI plus a unique error term.
Similarly, you need to have a measurement model for the latent factor CA. Suppose that there are
four observed indicators Y1, Y2, Y3, and Y4 (for example, Job Status) for this latent factor. The
measurement model for CA is specified as follows:
Y1 D a4 CA C e4
Y2 D a5 CA C e5
Y3 D a6 CA C e6
Y4 D a7 CA C e7
where a4 , a5 , a6 , and a7 are regression coefficients and e4 , e5 , e6 , and e7 are error terms. Again,
the error terms are assumed to be independent of the latent variables EI and CA, and Y1, Y2, Y3, and
Y4 are centered in the equations.
Given the data for the measured variables, you analyze the structural and measurement models
simultaneously by the structural equation modeling techniques. In other words, estimation of ˇ,
a1 –a7 , and other parameters in the model are carried out simultaneously in the modeling.
40 F Chapter 3: Introduction to Statistical Modeling with SAS/STAT Software
Modeling involving the use of latent factors is quite common in social and behavioral sciences,
personality assessment, and marketing research. Hypothetical constructs, although not observable,
are very important in building theories in these areas.
Another use of latent factors in modeling is to “purify” the predictors in regression analysis. A
common assumption in linear regression models is that predictors are measured without errors.
That is, in the following linear equation x is assumed to have been measured without errors:
y D ˛ C ˇx C However, if x has been contaminated with measurement errors that cannot be ignored, the estimate
of ˇ might be biased severely so that the true relationship between x and y would be masked.
A measurement model for x provides a solution to such a problem. Let Fx be a “purified” version of
x. That is, Fx is the “true” measure of x without measurement errors, as described in the following
equation:
x D Fx C ı
where ı represents a random measurement error term. Now, the linear relationship of interest is
specified in the following new linear regression equation:
y D ˛ C ˇFx C In this equation, Fx , which is now free from measurement errors, replaces x in the original equation.
With measurement errors taken into account in the simultaneous fitting of the measurement and the
new regression equations, estimation of ˇ is unbiased; hence it reflects the true relationship much
better.
Certainly, introducing latent factors in models is not a “free lunch.” You must pay attention to
the identification issues induced by the latent variable methodology. That is, in order to estimate
the parameters in structural equation models with latent variables, you must set some identification
constraints in these models. There are some established rules or conventions that would lead to
proper model identification and estimation. See Chapter 17, “Introduction to Structural Equation
Modeling with Latent Variables,” for examples and general details.
In addition, because of the nature of latent variables, estimation in structural equation modeling
with latent variables does not follow the same form as that of linear regression analysis. Instead of
defining the estimators in terms of the data matrices, most estimation methods in structural equation modeling use the fitting of the first- and second- order moments. Hence, estimation principles
described in the section “Classical Estimation Principles” on page 42 do not apply to structural
equation modeling. However, you can see the section “Estimation Criteria” on page 6880 in Chapter 88, “The TCALIS Procedure (Experimental),” for details about estimation in structural equation
modeling with latent variables.
Bayesian Models
Statistical models based on the classical (or frequentist) paradigm treat the parameters of the model
as fixed, unknown constants. They are not random variables, and the notion of probability is derived
in an objective sense as a limiting relative frequency. The Bayesian paradigm takes a different
Classes of Statistical Models F 41
approach. Model parameters are random variables, and the probability of an event is defined in
a subjective sense as the degree to which you believe that the event is true. This fundamental
difference in philosophy leads to profound differences in the statistical content of estimation and
inference. In the frequentist framework, you use the data to best estimate the unknown value of
a parameter; you are trying to pinpoint a value in the parameter space as well as possible. In the
Bayesian framework, you use the data to update your beliefs about the behavior of the parameter to
assess its distributional properties as well as possible.
Suppose you are interested in estimating from data Y D ŒY1 ; ; Yn  by using a statistical model
described by a density p.yj /. Bayesian philosophy states that cannot be determined exactly, and
uncertainty about the parameter is expressed through probability statements and distributions. You
can say, for example, that follows a normal distribution with mean 0 and variance 1, if you believe
that this distribution best describes the uncertainty associated with the parameter.
The following steps describe the essential elements of Bayesian inference:
1. A probability distribution for is formulated as ./, which is known as the prior distribution, or just the prior. The prior distribution expresses your beliefs, for example, on the mean,
the spread, the skewness, and so forth, about the parameter prior to examining the data.
2. Given the observed data Y, you choose a statistical model p.yj/ to describe the distribution
of Y given .
3. You update your beliefs about by combining information from the prior distribution and the
data through the calculation of the posterior distribution, p. jy/.
The third step is carried out by using Bayes’ theorem, from which this branch of statistical philosophy derives its name. The theorem enables you to combine the prior distribution and the model in
the following way:
p.; y/
p.yj /. /
p.yj/./
D
DR
p.y/
p.y/
p.yj/./d
R
The quantity p.y/ D p.yj /. / d is the normalizing constant of the posterior distribution. It is
also the marginal distribution of Y, and it is sometimes called the marginal distribution of the data.
p. jy/ D
The likelihood function of is any function proportional to p.yj/—that is, L./ / p.yj/. Another way of writing Bayes’ theorem is
p. jy/ D R
L. /. /
L. /. / d
The marginal distribution p.y/ is an integral; therefore, provided that it is finite, the particular value
of the integral does not yield any additional information about the posterior distribution. Hence,
p. jy/ can be written up to an arbitrary constant, presented here in proportional form, as
p. jy/ / L. /. /
Bayes’ theorem instructs you how to update existing knowledge with new information. You start
from a prior belief . /, and, after learning information from data y, you change or update the
42 F Chapter 3: Introduction to Statistical Modeling with SAS/STAT Software
belief on and obtain p. jy/. These are the essential elements of the Bayesian approach to data
analysis.
In theory, Bayesian methods offer a very simple alternative to statistical inference—all inferences
follow from the posterior distribution p. jy/. However, in practice, only the most elementary problems enable you to obtain the posterior distribution analytically. Most Bayesian analyses require
sophisticated computations, including the use of simulation methods. You generate samples from
the posterior distribution and use these samples to estimate the quantities of interest.
Both Bayesian and classical analysis methods have their advantages and disadvantages. Your choice
of method might depend on the goals of your data analysis. If prior information is available, such
as in the form of expert opinion or historical knowledge, and you want to incorporate this information into the analysis, then you might consider Bayesian methods. In addition, if you want
to communicate your findings in terms of probability notions that can be more easily understood
by nonstatisticians, Bayesian methods might be appropriate. The Bayesian paradigm can provide a
framework for answering specific scientific questions that a single point estimate cannot sufficiently
address. On the other hand, if you are interested in estimating parameters and in formulating inferences based on the properties of the parameter estimators, then there is no need to use Bayesian
analysis. When the sample size is large, Bayesian inference often provides results for parametric
models that are very similar to the results produced by classical, frequentist methods.
For more information, see Chapter 7, “Introduction to Bayesian Analysis Procedures.”
Classical Estimation Principles
An estimation principle captures the set of rules and procedures by which parameter estimates
are derived. When an estimation principle “meets” a statistical model, the result is an estimation
problem, the solution of which are the parameter estimates. For example, if you apply the estimation
principle of least squares to the SLR model Yi D ˇ0 C ˇ1 xi C i , the estimation problem is to find
those values b
ˇ 0 and b
ˇ 1 that minimize
n
X
.yi
ˇ0
ˇ1 xi /2
i D1
The solutions are the least squares estimators.
The two most important classes of estimation principles in statistical modeling are the least squares
principle and the likelihood principle. All principles have in common that they provide a metric by
which you measure the distance between the data and the model. They differ in the nature of the
metric; least squares relies on a geometric measure of distance, while likelihood inference is based
on a distance that measures plausability.
Classical Estimation Principles F 43
Least Squares
The idea of the ordinary least squares (OLS) principle is to choose parameter estimates that minimize the squared distance between the data and the model. In terms of the general, additive model,
Yi D f .xi1 ; ; xik I ˇ1 ; ; ˇp / C i
the OLS principle minimizes
SSE D
n
X
yi
2
f .xi1 ; ; xik I ˇ1 ; ; ˇp /
i D1
The least squares principle is sometimes called “nonparametric” in the sense that it does not require
the distributional specification of the response or the error term, but it might be better termed “distributionally agnostic.” In an additive-error model it is only required that the model errors have zero
mean. For example, the specification
Yi D ˇ0 C ˇ1 xi C i
EŒi  D 0
is sufficient to derive ordinary least squares (OLS) estimators for ˇ0 and ˇ1 and to study a number
of their properties. It is easy to show that the OLS estimators in this SLR model are
!
n
n
n
.X
X
X
b
.xi x/
.xi x/2
ˇ1 D
Yi Y
i D1
b
ˇ0 D Y
i D1
i D1
b
ˇ1x
Based on the assumption of a zero mean of the model errors, you can show that these estimators are
unbiased, EŒb
ˇ 1  D ˇ1 , EŒb
ˇ 0  D ˇ0 . However, without further assumptions about the distribution
of the i , you cannot derive the variability of the least squares estimators or perform statistical inferences such as hypothesis tests or confidence intervals. In addition, depending on the distribution
of the i , other forms of least squares estimation can be more efficient than OLS estimation.
The conditions for which ordinary least squares estimation is efficient are zero mean, homoscedastic, uncorrelated model errors. Mathematically,
EŒi  D 0
VarŒi  D 2
CovŒi ; j  D 0 if i 6D j
The second and third assumption are met if the errors have an iid distribution—that is, if they are
independent and identically distributed. Note, however, that the notion of stochastic independence
is stronger than that of absence of correlation. Only if the data are normally distributed does the
latter implies the former.
The various other forms of the least squares principle are motivated by different extensions of these
assumptions in order to find more efficient estimators.
44 F Chapter 3: Introduction to Statistical Modeling with SAS/STAT Software
Weighted Least Squares
The objective function in weighted least squares (WLS) estimation is
SSEw D
n
X
2
f .xi1 ; ; xi k I ˇ1 ; ; ˇp /
wi Yi
i D1
where wi is a weight associated with the i th observation. A situation where WLS estimation is appropriate is when the errors are uncorrelated but not homoscedastic. If the weights for the observations are proportional to the reciprocals of the error variances, VarŒi  D 2 =wi , then the weighted
least squares estimates are best linear unbiased estimators (BLUE). Suppose that the weights wi are
collected in the diagonal matrix W and that the mean function has the form of a linear model. The
weighted sum of squares criterion then can be written as
SSEw D .Y
Xˇ/0 W .Y
Xˇ/
which gives rise to the weighted normal equations
.X0 WX/ˇ D X0 WY
The resulting WLS estimator of ˇ is
b̌w D X0 WX X0 WY
Iteratively Reweighted Least Squares
If the weights in a least squares problem depend on the parameters, then a change in the parameters
also changes the weight structure of the model. Iteratively reweighted least squares (IRLS) estimation is an iterative technique that solves a series of weighted least squares problems, where the
weights are recomputed between iterations. IRLS estimation can be used, for example, to derive
maximum likelihood estimates in generalized linear models.
Generalized Least Squares
The previously discussed least squares methods have in common that the observations are assumed
to be uncorrelated—that is, CovŒi ; j  D 0, whenever i 6D j . The weighted least squares estimation problem is a special case of a more general least squares problem, where the model errors have
a general convariance matrix, VarŒ D †. Suppose again that the mean function is linear, so that
the model becomes
Y D Xˇ C .0; †/
The generalized least squares (GLS) principle is to minimize the generalized error sum of squares
SSEg D .Y
Xˇ/0 †
1
.Y
Xˇ/
This leads to the generalized normal equations
.X0 †
1
X/ˇ D X0 †
1
Y
Classical Estimation Principles F 45
and the GLS estimator
b̌g D X0 † 1 X X0 †
1
Y
Obviously, WLS estimation is a special case of GLS estimation, where † D 2 W
model is
Y D Xˇ C 0; 2 W 1
1 —that
is, the
Likelihood
There are several forms of likelihood estimation and a large number of offshoot principles derived
from it, such as pseudo-likelihood, quasi-likelihood, composite likelihood, etc. The basic likelihood
principle is maximum likelihood, which asks to estimate the model parameters by those quantities
that maximize the likelihood function of the data. The likelihood function is the joint distribution
of the data, but in contrast to a probability mass or density function, it is thought of as a function of
the parameters, given the data. The heuristic appeal of the maximum likelihood estimates (MLE) is
that these are the values that make the observed data “most likely.” Especially for discrete response
data, the value of the likelihood function is the ordinate of a probability mass function, even if the
likelihood is not a probability function. Since a statistical model is thought of as a representation of
the data-generating mechanism, what could be more preferable as parameter estimates than those
values that make it most likely that the data at hand will be observed?
Maximum likelihood estimates, if they exist, have appealing statistical properties. Under fairly mild
conditions, they are best-asymptotic-normal (BAN) estimates—that is, their asymptotic distribution
is normal, and no other estimator has a smaller asymptotic variance. However, their statistical
behavior in finite samples is often difficult to establish, and you have to appeal to the asymptotic
results that hold as the sample size tends to infinity. For example, maximum likelihood estimates
are often biased estimates and the bias disappears as the sample size grows. A famous example is
random sampling from a normal distribution. The corresponding statistical model is
Yi D C i
i iid N.0; 2 /
where the symbol is read as “is distributed as” and iid is read as “independent and identically
distributed.” Under the normality assumption, the density function of yi is
1
1 yi 2
2
f .yi I ; / D p
exp
2
2 2
and the likelihood for a random sample of size n is
n
Y
1
1 yi 2
2
L.; I y/ D
exp
p
2
2 2
i D1
Maximizing the likelihood function L.; 2 I y/ is equivalent to maximizing the log-likelihood
function log L D l.; 2 I y/,
n
X
1
.yi /2
2
2
l.; I y/ D
logf2g C
C logf g
2
2
i D1
!
n
X
1
.yi /2 = 2
D
n logf2g C n logf 2 g C
2
i D1
46 F Chapter 3: Introduction to Statistical Modeling with SAS/STAT Software
The maximum likelihood estimators of and 2 are thus
n
1X
b
D
Yi D Y
n
i D1
n
1X
.Yi
b
D
n
2
b
/2
i D1
The MLE of the mean is the sample mean, and it is an unbiased estimator of . However, the
MLE of the variance 2 is not an unbiased estimator. It has bias
2
E b
2 D
1 2
n
As the sample size n increases, the bias vanishes.
For certain classes of models, special forms of likelihood estimation have been developed to maintain the appeal of likelihood-based statistical inference and to address specific properties that are
believed to be shortcomings:
The bias in maximum likelihood parameter estimators of variances and covariances has led
to the development of restricted (or residual) maximum likelihood (REML) estimators that
play an important role in mixed models.
Quasi-likelihood methods do not require that the joint distribution of the data be specified.
These methods derive estimators based on only the first two moments (mean and variance) of
the joint distributions and play an important role in the analysis of correlated data.
The idea of composite likelihood is applied in situations where the likelihood of the vector
of responses is intractable but the likelihood of components or functions of the full-data likelihood are tractable. For example, instead of the likelihood of Y, you might consider the
likelihood of pairwise differences Yi Yj .
The pseudo-likelihood concept is also applied when the likelihood function is intractable, but
the likelihood of a related, simpler model is available. An important difference between quasilikelihood and pseudo-likelihood techniques is that the latter make distributional assumptions
to obtain a likelihood function in the pseudo-model. Quasi-likelihood methods do not specify
the distributional family.
The penalized likelihood principle is applied when additional constraints and conditions need
to be imposed on the parameter estimates or the resulting model fit. For example, you might
augment the likelihood with conditions that govern the smoothness of the predictions or that
prevent overfitting of the model.
Classical Estimation Principles F 47
Least Squares or Likelihood
For many statistical modeling problems, you have a choice between a least squares principle and
the maximum likelihood principle. Table 3.1 compares these two basic principles.
Table 3.1
Least Squares and Maximum Likelihood
Criterion
Least Squares
Maximum Likelihood
Requires specification of joint
distribution of
data
No, but in order to perform confirmatory
inference (tests, confidence intervals), a
distributional assumption is needed, or an
appeal to asymptotics.
Yes, no progress can be made with
the genuine likelihood principle without
knowing the distribution of the data.
All parameters
of the model are
estimated
No. In the additive-error type models, least squares provides estimates of
only the parameters in the mean function.
The residual variance, for example, must
be estimated by some other method—
typically by using the mean squared error
of the model.
Yes
Estimates
always exist
Yes, but they might not be unique, such
as when the X matrix is singular.
No, maximum likelihood estimates do
not exist for all estimation problems.
Estimators
biased
are
Unbiased, provided that the model is
correct—that is, the errors have zero
mean.
Often biased,
unbiased
but
asymptotically
Estimators
consistent
are
Not necessarily, but often true. Sometimes estimators are consistent even in a
misspecified model, such as when misspecification is in the covariance structure.
Almost always
Estimators are
best linear unbiased estimates
(BLUE)
Typically, if the least squares assumptions are met.
Not necessarily: estimators are often nonlinear in the data and are often biased.
Asymptotically
most efficient
Not necessarily
Typically
Easy to compute
Yes
No
Inference Principles for Survey Data
Design-based and model-assisted statistial inference for survey data requires that the randomness
due to the selection mechanism be taken into account. This can require special estimation principles
and techniques.
48 F Chapter 3: Introduction to Statistical Modeling with SAS/STAT Software
The SURVEYMEANS, SURVEYFREQ, SURVEYREG, and SURVEYLOGISTIC procedures support design-based and/or model-assisted inference for sample surveys. Suppose i is the selection
probability for unit i in sample S . The inverse of the inclusion probability is known as sampling
weight and is denoted by wi . Briefly, the idea is to apply a relationship that exists in the population
to the sample and to take
P into account the sampling weights. For example, to estimate the finite
population total TN D i 2UN yi based on the sample S , you can accumulate the sampled values
P
b D
b
while properly weighting: T
i 2S wi yi . It is easy to verify that T is design-unbiased in the
b
sense that EŒT jFN  D TN (see Cochran 1997).
When a statistical model is present, similar ideas apply. For example, if ˇN 0 and ˇN1 are finite
population quantities for a simple linear regression working model that minimize the sum of squares
X
.yi ˇ0N ˇ1N xi /2
i 2UN
in the population, then the sample-based estimators b
ˇ 0S and b
ˇ 1S are obtained by minimizing the
weighted sum of squares
X
wi .yi ˇO0S ˇO1S xi /2
i 2S
in the sample, taking into account the inclusion probabilities.
In model-assisted inference, weighted least squares or pseudo-maximum likelihood estimators are
commonly used to solve such estimation problems. Maximum pseudo-likelihood or weighted maximum likelihood estimators for survey data maximize a sample-based estimator of the population
likelihood. Assume a working model with uncorrelated responses such that the finite population
log-likelihood is
X
l.1N ; : : : ; pN I yi /;
i 2UN
where 1N ; : : : ; pN are finite population quantities. For independent sampling, one possible
sample-based estimator of the population log likelihood is
X
wi l.1N ; : : : ; pN I yi /
i 2S
Sample-based estimators b
1S ; : : : ; b
pS are obtained by maximizing this expression.
Design-based and model-based statistical analysis might employ the same statistical model (for example, a linear regression) and the same estimation principle (for example, weighted least squares),
and arrive at the same estimates. The design-based estimation of the precision of the estimators
differs from the model-based estimation, however. For complex surveys, design-based variance estimates are in general different from their model-based counterpart. The SAS/STAT procedures for
survey data (SURVEYMEANS, SURVEYFREQ, SURVEYREG, and SURVEYLOGISTIC procedures) compute design-based variance estimates for complex survey data. See the section “Variance
Estimation” on page 267, in Chapter 14, “Introduction to Survey Procedures,” for details about
design-based variance estimation.
Statistical Background F 49
Statistical Background
Hypothesis Testing and Power
In statistical hypothesis testing, you typically express the belief that some effect exists in a population by specifying an alternative hypothesis H1 . You state a null hypothesis H0 as the assertion
that the effect does not exist and attempt to gather evidence to reject H0 in favor of H1 . Evidence
is gathered in the form of sample data, and a statistical test is used to assess H0 . If H0 is rejected
but there really is no effect, this is called a Type I error. The probability of a Type I error is usually
designated “alpha” or ˛, and statistical tests are designed to ensure that ˛ is suitably small (for
example, less than 0.05).
If there is an effect in the population but H0 is not rejected in the statistical test, then a Type II
error has been committed. The probability of a Type II error is usually designated “beta” or ˇ. The
probability 1 ˇ of avoiding a Type II error—that is, correctly rejecting H0 and achieving statistical
significance, is called the power of the test.
An important goal in study planning is to ensure an acceptably high level of power. Sample size
plays a prominent role in power computations because the focus is often on determining a sufficient
sample size to achieve a certain power, or assessing the power for a range of different sample sizes.
There are several tools available in SAS/STAT software for power and sample size analysis. PROC
POWER covers a variety of analyses such as t tests, equivalence tests, confidence intervals, binomial proportions, multiple regression, one-way ANOVA, survival analysis, logistic regression,
and the Wilcoxon rank-sum test. PROC GLMPOWER supports more complex linear models. The
Power and Sample Size application provides a user interface and implements many of the analyses
supported in the procedures.
Important Linear Algebra Concepts
A matrix A is a rectangular array of numbers. The order of a matrix with n rows and k columns is
.n k/. The element in row i , column j of A is denoted as aij , and the notation aij is sometimes
used to refer to the two-dimensional row-column array
2
3
a11 a12 a13 a1k
6 a21 a22 a23 a2k 7
6
7 6
7
A D 6 a31 a32 a33 a3k 7 D aij
6 ::
::
::
:
::
:: 7
4 :
5
:
:
:
an1 an2 an3 ank
A vector is a one-dimensional array of numbers. A column vector has a single column (k D 1). A
row vector has a single row (n D 1). A scalar is a matrix of order .1 1/—that is, a single number.
50 F Chapter 3: Introduction to Statistical Modeling with SAS/STAT Software
A square matrix has the same row and column order, n D k. A diagonal matrix is a square matrix
where all off-diagonal elements are zero, aij D 0 if i 6D j . The identity matrix I is a diagonal
matrix with ai i D 1 for all i . The unit vector 1 is a vector where all elements are 1. The unit matrix
J is a matrix of all 1s. Similarly, the elements of the null vector and the null matrix are all 0.
Basic matrix operations are as follows:
Addition
If A and B are of the same order, then A C B is the matrix of elementwise sums,
A C B D aij C bij
Subtraction
If A and B are of the same order, then A
differences,
A B D aij bij
Dot product
The dot product of two n-vectors a and b is the sum of their elementwise products,
abD
n
X
B is the matrix of elementwise
ai bi
i D1
The dot product is also known as the inner product of a and b. Two vectors are
said to be orthogonal if their dot product is zero.
Multiplication
Matrices A and B are said to be conformable for AB multiplication if the number
of columns in A equals the number of rows in B. Suppose that A is of order
.n k/ and that B is of order .k p/. The product AB is then defined as the
.n p/ matrix of the dot products of the i th row of A and the j th column of B,
AB D ai bj np
Transposition
The transpose of the .n k/ matrix A is denoted as A0 or AT or AT and is
obtained by interchanging the rows and columns,
2
3
a11 a21 a31 an1
6 a12 a22 a23 an2 7
6
7 6
7
A0 D 6 a13 a23 a33 an3 7 D aj i
6 ::
::
::
:: 7
::
4 :
:
:
:
: 5
a1k a2k a3k ank
A symmetric matrix is equal to its transpose, A D A0 . The inner product of two
.n 1/ column vectors a and b is a b D a0 b.
Matrix Inversion
Regular Inverses
The right inverse of a matrix A is the matrix that yields the identity when A is postmultiplied by
it. Similarly, the left inverse of A yields the identity if A is premultiplied by it. A is said to be
Important Linear Algebra Concepts F 51
invertible and B is said to be the inverse of A, if B is its right and left inverse, BA D AB D I.
This requires A to be square and nonsingular. The inverse of a matrix A is commonly denoted as
A 1 . The following results are useful in manipulating inverse matrices (assuming both A and C are
invertible):
AA
A0
1
1
.AC/
1
A
1
1
1
DA
D A
ADI
1 0
DA
DC
1
A
rank.A/ D rank A
1
1
If D is a diagonal matrix with nonzero entries on the diagonal—that is, D D diag .d1 ; ; dn /—
then D 1 D diag .1=d1 ; ; 1=dn /. If D is a block-diagonal matrix whose blocks are invertible,
then
2
3
2
3
D1 0
0
0
D1 1 0
0
0
6 0
6 0
7
D2 0
0 7
D2 1 0
0
6
7
6
7
1
6 0
7
6
7
0
D3 0 7
0
D3
0
DD6
D 1D6 0
7
6 ::
6 ::
7
::
::
::
::
: : :: 7
: : ::
4 :
5
4
5
: :
: :
:
:
:
:
:
1
0
0
0
Dn
0
0
0
Dn
In statistical applications the following two results are particularly important, because they can
significantly reduce the computational burden in working with inverse matrices.
Partitioned Matrix Suppose A is a nonsingular matrix that is partitioned as
A11 A12
AD
A21 A22
Then, provided that all the inverses exist, the inverse of A is given by
B11 B12
1
A D
B21 B22
where B11 D
A11
A12 A221 A21
A221 A21 B11 , and B22 D A22
Patterned Sum
1
, B12 D
1
A21 A111 A12
.
B11 A12 A221 , B21 D
Suppose R is .n n/ nonsingular, G is .k k/ nonsingular, and B and C are
.n k/ and .k n/ matrices, respectively. Then the inverse of R C BGC is given
by
1
.R C BGC/ 1 D R 1 R 1 B G 1 C CR 1 B
CR 1
This formula is particularly useful if k << n and R has a simple form that is
easy to invert. This case arises, for example, in mixed models where R might be
a diagonal or block-diagonal matrix, and B D C0 .
Another situation where this formula plays a critical role is in the computation
of regression diagnostics, such as in determining the effect of removing an observation from the analysis. Suppose that A D X0 X represents the crossproduct
52 F Chapter 3: Introduction to Statistical Modeling with SAS/STAT Software
matrix in the linear model EŒY D Xˇ. If x0i is the i th row of the X matrix,
then .X0 X xi x0i / is the crossproduct matrix in the same model with the i th
observation removed. Identifying B D xi , C D x0i , and G D I in the preceding inversion formula, you can obtain the expression for the inverse of the
crossproduct matrix:
0
XX
xi x0i
1
0
DXXC
X0 X
1
1
xi x0i X0 X
1
xi
x0i X0 X
1
This expression for the inverse of the reduced data crossproduct matrix enables
you to compute “leave-one-out” deletion diagnostics in linear models without
refitting the model.
Generalized Inverse Matrices
If A is rectangular (not square) or singular, then it is not invertible and the matrix A
exist. Suppose you want to find a solution to simultaneous linear equations of the form
1
does not
Ab D c
If A is square and nonsingular, then the unique solution is b D A 1 c. In statistical applications, the
case where A is .nk/ rectangular is less important than the case where A is a .k k/ square matrix
of rank less than k. For example, the normal equations in ordinary least squares (OLS) estimation
in the model Y D Xˇ C are
X0 X ˇ D X0 Y
A generalized inverse matrix is a matrix A such that A c is a solution
to the linear system. In the
0
0
0
OLS example, a solution can be found as X X X Y, where X X is a generalized inverse of
X0 X.
The following four conditions are often associated with generalized inverses. For the square or
rectangular matrix A there exist matrices G that satisfy
.i/
.ii/
.iii/
.iv/
AGA
GAG
.AG/0
.GA/0
D
D
D
D
A
G
AG
GA
The matrix G that satisfies all four conditions is unique and is called the Moore-Penrose inverse,
after the first published work on generalized inverses by Moore (1920) and the subsequent definition
by Penrose (1955). Only the first condition is required, however, to provide a solution to the linear
system above.
Pringle and Rayner (1971) introduced a numbering system to distinguish between different types
of generalized inverses. A matrix that satisfies only condition (i) is a g1 -inverse. The g2 -inverse
satistifes conditions (i) and (ii). It is also called a reflexive generalized inverse. Matrices satisfying
conditions (i)–(iii) or conditions (i), (ii), and (iv) are g3 -inverses. Note that a matrix that satisfies
the first three conditions is a right generalized inverse, and a matrix that satisfies conditions (i), (ii),
1 0
and (iv) is a left generalized inverse. For example, if B is .n k/ of rank k, then B0 B
B is
Important Linear Algebra Concepts F 53
a left generalized inverse of B. The notation g4 -inverse for the Moore-Penrose inverse, satisfying
conditions (i)–(iv), is often used by extension, but note that Pringle and Rayner (1971) do not use
it; rather, they call such a matrix “the” generalized inverse.
If the .n k/ matrix X is rank-deficient—that is, rank.X/ < minfn; kg—then the system of equations
X0 X ˇ D X0 Y
does not have a unique solution. A particular solution depends on the choice of the generalized
inverse. However, some aspects of the statistical inference are invariant to the choice of the generalized inverse. If G is a generalized inverse of X0 X, then XGX0 is invariant to the choice of G. This
result comes into play, for example, when you are computing predictions in an OLS model with a
rank-deficient X matrix, since it implies that the predicted values
Xb̌ D X X0 X X0 y
are invariant to the choice of X0 X .
Matrix Differentiation
Taking the derivative of expressions involving matrices is a frequent task in statistical estimation.
Objective functions that are to be minimized or maximized are usually written in terms of model
matrices and/or vectors whose elements depend on the unknowns of the estimation problem. Suppose that A and B are real matrices whose elements depend on the scalar quantities ˇ and —that
is, A D aij .ˇ; / , and similarly for B.
The following are useful results in finding the derivative of elements of a matrix and of functions
involving a matrix. For more in-depth discussion of matrix differentiation and matrix calculus, see,
for example, Magnus and Neudecker (1999) and Harville (1997).
P ˇ and is the matrix of the first derivatives of the
The derivative of A with respect to ˇ is denoted A
elements of A:
@aij .ˇ; /
@
P
Aˇ D
AD
@ˇ
@ˇ
Similarly, the second derivative of A with respect to ˇ and is the matrix of the second derivatives
2
2
R ˇ D @ A D @ aij .ˇ; /
A
@ˇ@
@ˇ@
54 F Chapter 3: Introduction to Statistical Modeling with SAS/STAT Software
The following are some basic results involving sums, products, and traces of matrices:
@
Pˇ
c1 A D c1 A
@ˇ
@
P ˇ C BP ˇ
.A C B/ D A
@ˇ
@
.c1 A C c2 B/ D
@ˇ
@
AB D
@ˇ
@
trace.A/ D
@ˇ
@
trace.AB/ D
@ˇ
Pˇ
P ˇ C c2 B
c1 A
Pˇ C A
P ˇB
AB
Pˇ
trace A
P ˇB
trace ABP ˇ C trace A
The next set of results is useful in finding the derivative of elements of A and of functions of A, if
A is a nonsingular matrix:
@ 0 1
xA xD
@ˇ
@
A 1D
@ˇ
@
jAj D
@ˇ
@
log fjAjg D
@ˇ
@2
A 1D
@ˇ@
@2
log fjAjg D
@ˇ@
x0 A
1
P ˇA
A
P ˇA
A
1
jAj trace A
1
A
1
1
x
Pˇ
A
1 @
A D trace A
jAj @ˇ
A
1
R ˇ A
A
trace A
1
1
R ˇ
A
1
Pˇ
A
P ˇA
A
1
P A
A
1
CA
trace A
1
P ˇA
A
1
P
A
CA
1
1
P A
A
1
P ˇA
A
1
Now suppose that a and b are column vectors that depend on ˇ and/or and that x is a vector
of constants. The following results are useful for manipulating derivatives of linear and quadratic
forms:
@ 0
axDa
@x
@
Bx D B
@x0
@ 0
x Bx D B C B0 x
@x
@2 0
x Bx D B C B0
@[email protected]
Important Linear Algebra Concepts F 55
Matrix Decompositions
To decompose a matrix is to express it as a function—typically a product—of other matrices
that have particular properties such as orthogonality, diagonality, triangularity. For example, the
Cholesky decomposition of a symmetric positive definite matrix A is CC0 D A, where C is a lowertriangular matrix. The spectral decomposition of a symmetric matrix is A D PDP0 , where D is a
diagonal matrix and P is an orthogonal matrix.
Matrix decomposition play an important role in statistical theory as well as in statistical computations. Calculations in terms of decompositions can have greater numerical stability. Decompositions are often necessary to extract information about matrices, such as matrix rank, eigenvalues,
or eigenvectors. Decompositions are also used to form special transformations of matrices, such
as to form a “square-root” matrix. This section briefly mentions several decompositions that are
particularly prevalent and important.
LDU, LU, and Cholesky Decomposition
Every square matrix A, whether it is positive definite or not, can be expressed in the form A D LDU,
where L is a unit lower-triangular matrix, D is a diagonal matrix, and U is a unit upper-triangular
matrix. (The diagonal elements of a unit triangular matrix are 1.) Because of the arrangement of the
matrices, the decomposition is called the LDU decomposition. Since you can absorb the diagonal
matrix into the triangular matrices, the decomposition
A D LD1=2 D1=2 U D L U
is also referred to as the LU decomposition of A.
If the matrix A is positive definite, then the diagonal elements of D are positive and the LDU
decomposition is unique. Furthermore, we can add more specificity to this result in that for a
symmetric, positive definite matrix, there is a unique decomposition A D U0 DU, where U is unit
upper-triangular and D is diagonal with positive elements. Absorbing the square root of D into
U, C D D1=2 U, the decomposition is known as the Cholesky decomposition of a positive-definite
matrix:
B D U0 D1=2 D1=2 U D C0 C
where C is upper triangular.
If B is .n n/ symmetric nonnegative definite of rank k, then we can extend the Cholesky decomposition as follows. Let C denote the lower-triangular matrix such that
Ckk 0
C D
0
0
Then B D CC0 .
Spectral Decomposition
Suppose that A is an .n n/ symmetric matrix. Then there exists an orthogonal matrix Q and a
diagonal matrix D such that A D QDQ0 . Of particular importance is the case where the orthogonal
56 F Chapter 3: Introduction to Statistical Modeling with SAS/STAT Software
matrix is also orthonormal—that is, its column vectors have unit norm. Denote this orthonormal
matrix as P. Then the corresponding diagonal matrix—ƒ D diag.i ; ; n /, say—contains the
eigenvalues of A. The spectral decomposition of A can be written as
A D PƒP0 D
n
X
i pi p0i
i D1
where pi denotes the i th column vector of P. The right-side expression decomposes A into a sum
of rank-1 matrices, and the weight of each contribution is equal to the eigenvalue associated with
the i th eigenvector. The sum furthermore emphasizes that the rank of A is equal to the number of
nonzero eigenvalues.
Harville (1997, p. 538) refers to the spectral decomposition of A as the decomposition that takes
the previous sum one step further and accumulates contributions
with the distinct eigenP associated
0
pi pi , where the sum is taken over
values. If i ; ; k are the distinct eigenvalues and Ej D
the set of columns for which i D j , then
AD
k
X
j Ej
i D1
You can employ the spectral decomposition of a nonnegative definite symmetric matrix to form a
“square-root” matrix of A. Suppose that ƒ1=2 is the diagonal matrix containing the square roots of
the i . Then B D Pƒ1=2 P0 is a square-root matrix of A in the sense that BB D A, because
BB D Pƒ1=2 P0 Pƒ1=2 P0 D Pƒ1=2 ƒ1=2 P0 D PƒP0
Generating the Moore-Penrose inverse of a matrix based on the spectral decomposition is also simple. Denote as  the diagonal matrix with typical element
1=i i 6D 0
ıi D
0
i D 0
P
Then the matrix PP0 D ıi pi p0i is the Moore-Penrose (g4 -generalized) inverse of A.
Singular-Value Decomposition
The singular-value decomposition is related to the spectral decomposition of a matrix, but it is more
general. The singular-value decomposition can be applied to any matrix. Let B be an .n p/ matrix
of rank k. Then there exist orthogonal matrices P and Q of order .n n/ and .p p/, respectively,
and a diagonal matrix D such that
D1 0
0
P BQ D D D
0 0
where D1 is a diagonal matrix of order k. The diagonal elements of D1 are strictly positive. As
with the spectral decomposition, this result can be written as a decomposition of B into a weighted
sum of rank-1 matrices
BD
PDQ0 D
n
X
i D1
di pi q0i
Expectations of Random Variables and Vectors F 57
The scalars d1 ; ; dk are called the singular values of the matrix B. They are the positive square
roots of the nonzero eigenvalues of the matrix B0 B. If the singular-value decomposition is applied
to a symmetric, nonnegative definite matrix A, then the singular values d1 ; ; dn are the nonzero
eigenvalues of A and the singular-value decomposition is the same as the spectral decomposition.
As with the spectral decomposition, you can use the results of the singular-value decomposition to
generate the Moore-Penrose inverse of a matrix. If B is .n p/ with singular-value decomposition
PDQ0 , and if  is a diagonal matrix with typical element
1=di jdi j 6D 0
ıi D
0
di D 0
then QP0 is the g4 -generalized inverse of B.
Expectations of Random Variables and Vectors
If Y is a discrete random variable with mass function p.y/ and support (possible values) y1 ; y2 ; ,
then the expectation (expected value) of Y is defined as
EŒY  D
1
X
yj p.yj /
j D1
P
provided that
jyj jp.yj / < 1, otherwise the sum in the definition
P is not well-defined. The
expected value of a function h.y/ is similarly defined: provided that jh.yj /jp.yj / < 1,
EŒh.Y / D
1
X
h.yj / p.yj /
j D1
For continuous random variables, similar definitions apply, but summation is replaced by integration
over the support
of the random variable. If X is a continuous random variable with density function
R
f .x/, and jxjf .x/dx < 1, then the expectation of X is defined as
Z 1
EŒX  D
xf .x/ dx
1
The expected value of a random variable is also called its mean or its first moment. A particularly
important function of a random variable is h.Y / D .Y
EŒY /2 . The expectation of h.Y / is
called the variance of Y or the second central moment of Y . When you study the properties of
multiple random variables, then you might be interested in aspects of their joint distribution. The
covariance between random variables Y and X is defined as the expected value of the function
.Y EŒY /.X EŒX /, where the expectation is taken under the bivariate joint distribution of Y
and X:
Z Z
CovŒY; X  D EŒ.Y EŒY /.X EŒX / D EŒYX EŒY EŒX D
x yf .x; y/ dxdy EŒY EŒX
The covariance between a random variable and itself is the variance, CovŒY; Y  D VarŒY .
58 F Chapter 3: Introduction to Statistical Modeling with SAS/STAT Software
In statistical applications and formulas, random variables are often collected into vectors. For example, a random sample of size n from the distribution of Y generates a random vector of order
.n 1/,
2
3
Y1
6 Y2 7
6
7
YD6 : 7
:
4 : 5
Yn
The expected value of the .n 1/ random vector Y is the vector of the means of the elements of Y:
2
3
EŒY1 
6 EŒY2  7
6
7
EŒY D ŒE ŒYi  D 6 : 7
4 :: 5
EŒYn 
It is often useful to directly apply rules about working with means, variances, and covariances of
random vectors. To develop these rules, suppose that Y and U denote two random vectors with typical elements Y1 ; ; Yn and U1 ; ; Uk . Further suppose that A and B are constant (nonstochastic)
matrices, that a is a constant vector, and that the ci are scalar constants.
The following rules enable you to derive the mean of a linear function of a random vector:
EŒA D A
EŒY C a D EŒY
EŒAY C a D AEŒY C a
EŒY C U D EŒY C EŒU
The covariance matrix of Y and U is the .n k/ matrix whose typical element in row i , column
j is the covariance between Yi and Uj . The covariance matrix between two random vectors is
frequently denoted with the Cov “operator.”
CovŒY; U D CovŒYi ; Uj 
D E .Y EŒY/ .U EŒU/0 D E YU0
EŒYEŒU0
2
6
6
6
D6
6
4
CovŒY1 ; U3  CovŒY2 ; U3  CovŒY3 ; U3  ::
::
:
:
CovŒYn ; U1  CovŒYn ; U2  CovŒYn ; U3  CovŒY1 ; U1 
CovŒY2 ; U1 
CovŒY3 ; U1 
::
:
CovŒY1 ; U2 
CovŒY2 ; U2 
CovŒY3 ; U2 
::
:
CovŒY1 ; Uk 
CovŒY2 ; Uk 
CovŒY3 ; Uk 
::
:
CovŒYn ; Uk 
3
7
7
7
7
7
5
Expectations of Random Variables and Vectors F 59
The variance matrix of a random vector Y is the covariance matrix between Y and itself. The
variance matrix is frequently denoted with the Var “operator.”
VarŒY D CovŒY; Y D CovŒYi ; Yj 
D E .Y EŒY/ .Y EŒY/0 D E YY0
EŒYEŒY0
2
6
6
6
D6
6
4
2
6
6
6
D6
6
4
CovŒY1 ; Y3  CovŒY2 ; Y3  CovŒY3 ; Y3  ::
::
:
:
CovŒYn ; Y1  CovŒYn ; Y2  CovŒYn ; Y3  CovŒY1 ; Yn 
CovŒY2 ; Yn 
CovŒY3 ; Yn 
::
:
CovŒY1 ; Y3  CovŒY2 ; Y3  VarŒY3 
::
::
:
:
CovŒYn ; Y1  CovŒYn ; Y2  CovŒYn ; Y3  CovŒY1 ; Yn 
CovŒY2 ; Yn 
CovŒY3 ; Yn 
::
:
CovŒY1 ; Y1 
CovŒY2 ; Y1 
CovŒY3 ; Y1 
::
:
VarŒY1 
CovŒY2 ; Y1 
CovŒY3 ; Y1 
::
:
CovŒY1 ; Y2 
CovŒY2 ; Y2 
CovŒY3 ; Y2 
::
:
CovŒY1 ; Y2 
VarŒY2 
CovŒY3 ; Y2 
::
:
3
7
7
7
7
7
5
CovŒYn ; Yn 
3
7
7
7
7
7
5
VarŒYn 
Because the variance matrix contains variances on the diagonal and covariances in the off-diagonal
positions, it is also referred to as the variance-covariance matrix of the random vector Y.
If the elements of the covariance matrix CovŒY; U are zero, the random vectors are uncorrelated.
If Y and U are normally distributed, then a zero covariance matrix implies that the vectors are
stochastically independent. If the off-diagonal elements of the variance matrix VarŒY are zero, the
elements of the random vector Y are uncorrelated. If Y is normally distributed, then a diagonal
variance matrix implies that its elements are stochastically independent.
Suppose that A and B are constant (nonstochastic) matrices and that ci denotes a scalar constant.
The following results are useful in manipulating covariance matrices:
CovŒAY; U D ACovŒY; U
CovŒY; BU D CovŒY; UB0
CovŒAY; BU D ACovŒY; UB0
CovŒc1 Y1 C c2 U1 ; c3 Y2 C c4 U2  D c1 c3 CovŒY1 ; Y2  C c1 c4 CovŒY1 ; U2 
C c2 c3 CovŒU1 ; Y2  C c2 c4 CovŒU1 ; U2 
Since CovŒY; Y D VarŒY, these results can be applied to produce the following results, useful in
manipulating variances of random vectors:
VarŒA D 0
VarŒAY D AVarŒYA0
VarŒY C x D VarŒY
VarŒx0 Y D x0 VarŒYx
VarŒc1 Y D c12 VarŒY
VarŒc1 Y C c2 U D c12 VarŒY C c22 VarŒU C 2c1 c2 CovŒY; U
60 F Chapter 3: Introduction to Statistical Modeling with SAS/STAT Software
Another area where expectation rules are helpful is quadratic forms in random variables. These
forms arise particularly in the study of linear statistical models and in linear statistical inference.
Linear inference is statistical inference about linear function of random variables, even if those
random variables are defined through nonlinear models. For example, the parameter estimator b
might be derived in a nonlinear model, but this does not prevent statistical questions from being
raised that can be expressed through linear functions of ; for example,
1 22 D 0
H0 W
2 3 D 0
if A is a matrix of constants and Y is a random vector, then
EŒY0 AY D trace.AVarŒY/ C EŒY0 AEŒY
Mean Squared Error
The mean squared error is arguably the most important criterion used to evaluate the performance of
a predictor or an estimator. (The subtle distinction between predictors and estimators is that random
variables are predicted and constants are estimated.) The mean squared error is also useful to relay
the concepts of bias, precision, and accuracy in statistical estimation. In order to examine a mean
squared error, you need a target of estimation or prediction, and a predictor or estimator that is a
function of the data. Suppose that the target, whether a constant or a random variable, is denoted as
U . The mean squared error of the estimator or predictor T .Y/ for U is
h
i
MSE ŒT .Y/I U  D E .T .Y/ U /2
The reason for using a squared difference to measure the “loss” between T .Y/ and U is mostly convenience; properties of squared differences involving random variables are more easily examined
than, say, absolute differences. The reason for taking an expectation is to remove the randomness
of the squared difference by averaging over the distribution of the data.
Consider first the case where the target U is a constant—say, the parameter ˇ—and denote the mean
of the estimator T .Y/ as T . The mean squared error can then be decomposed as
h
i
MSEŒT .YI ˇ D E .T .Y/ ˇ/2
h
i
h
i
D E .T .Y/ T /2
E .ˇ T /2
D VarŒT .Y/ C .ˇ
T /2
The mean squared error thus comprises the variance of the estimator and the squared bias. The two
components can be associated with an estimator’s precision (small variance) and its accuracy (small
bias).
If T .Y/ is an unbiased estimator of ˇ—that is, if EŒT .Y/ D ˇ—then the mean squared error is
simply the variance of the estimator. By choosing an estimator that has minimum variance, you also
choose an estimator that has minimum mean squared error among all unbiased estimators. However,
as you can see from the previous expression, bias is also an “average” property; it is defined as an
Linear Model Theory F 61
expectation. It is quite possible to find estimators in some statistical modeling problems that have
smaller mean squared error than a minimum variance unbiased estimator; these are estimators that
permit a certain amount of bias but improve on the variance. For example, in models where regressors are highly collinear, the ordinary least squares estimator continues to be unbiased. However,
the presence of collinearity can induce poor precision and lead to an erratic estimator. Ridge regression stabilizes the regression estimates in this situation, and the coefficient estimates are somewhat
biased, but the bias is more than offset by the gains in precision.
When the target U is a random variable, you need to carefully define what an unbiased prediction
means. If the statistic and the target have the same expectation, EŒU  D EŒT .Y/, then
MSE ŒT .Y/I U  D VarŒT .Y/ C VarŒU 
2CovŒT .Y/; U 
In many instances the target U is a new observation that was not part of the analysis. If the data
are uncorrelated, then it is reasonable to assume in that instance that the new observation is also not
correlated with the data. The mean squared error then reduces to the sum of the two variances. For
example, in a linear regression model where U is a new observation Y0 and T .Y/ is the regression
estimator
b0 D x00 X0 X 1 X0 Y
Y
with variance VarŒY0  D 2 x00 X0 X
h
i
bI Y0 D 2 x00 X0 X
MSE Y
1
x0 , the mean squared prediction error for Y0 is
1
x0 C 1
and the mean squared prediction error for predicting the mean EŒY0  is
h
i
bI EŒY0  D 2 x00 X0 X 1 x0
MSE Y
Linear Model Theory
This section presents some basic statistical concepts and results for the linear model with homoscedastic, uncorrelated errors in which the parameters are estimated by ordinary least squares.
The model can be written as
Y D Xˇ C .0; 2 I/
where Y is an .n 1/ vector and X is an .n k/ matrix of known constants. The model equation
implies the following expected values:
EŒY D Xˇ
2
VarŒY D I , CovŒYi ; Yj  D
2 i D j
0
otherwise
62 F Chapter 3: Introduction to Statistical Modeling with SAS/STAT Software
Finding the Least Squares Estimators
Finding the least squares estimator of ˇ can be motivated as a calculus problem or by considering
the geometry of least squares. The former approach simply states that the OLS estimator is the
vector b̌ that minimizes the objective function
SSE D .Y
Xˇ/0 .Y
Xˇ/
Applying the differentiation rules from the section “Matrix Differentiation” on page 53 leads to
@
@
SSE D
.Y0 Y 2Y0 Xˇ C ˇ 0 X0 Xˇ/
@ˇ
@ˇ
D 0 2X0 Y C 2X0 Xˇ
@2
SSE D X0 X
@ˇ@ˇ
@
Consequently, the solution to the normal equations, X0 Xˇ D X0 Y, solves @ˇ
SSE D 0, and the fact
that the second derivative is nonnegative definite guarantees that this solution minimizes SSE. The
geometric argument to motivate ordinary least squares estimation is as follows. Assume that X is
Q the following identity holds:
of rank k. For any value of ˇ, such as ˇ,
Y D XˇQ C Y XˇQ
Q is a point
The vector XˇQ is a point in a k-dimensional subspace of Rn , and the residual .Y Xˇ/
in an .n k/-dimensional subspace. The OLS estimator is the value b̌ that minimizes the distance
of XˇQ from Y, implying that Xb̌ and .Y Xb̌/ are orthogonal to each other; that is,
.Y
Xb̌/0 Xb̌ D 0. This in turn implies that b̌ satisfies the normal equations, since
b̌0 X0 Y D b̌0 X0 Xb̌ , X0 Xb̌ D X0 Y
Full-Rank Case
If X is of full column rank, the OLS estimator is unique and given by
b̌ D X0 X 1 X0 Y
The OLS estimator is an unbiased estimator of ˇ—that is,
h
1 0 i
EŒb̌ DE X0 X
XY
1 0
1 0
X EŒY D X0 X
X Xˇ D ˇ
D X0 X
Note that this result holds if EŒY D Xˇ; in other words, the condition that the model errors have
mean zero is sufficient for the OLS estimator to be unbiased. If the errors are homoscedastic and
uncorrelated, the OLS estimator is indeed the best linear unbiased estimator (BLUE) of ˇ—that is,
no other estimator that is a linear function of Y has a smaller mean squared error. The fact that the
estimator is unbiased implies that no other linear estimator has a smaller variance. If, furthermore,
the model errors are normally distributed, then the OLS estimator has minimum variance among all
unbiased estimators of ˇ, whether they are linear or not. Such an estimator is called a uniformly
minimum variance unbiased estimator, or UMVUE.
Linear Model Theory F 63
Rank-Deficient Case
In the case of a rank-deficient X matrix, a generalized inverse is used to solve the normal equations:
b̌ D X0 X X0 Y
Although a g1 -inverse is sufficient to solve a linear system, computational expedience and interpretation of the results often dictate the use of a generalized inverse with reflexive properties (that is,
a g2 -inverse; see the section “Generalized Inverse Matrices” on page 52 for details). Suppose, for
example, that the X matrix is partitioned as X D ŒX1 X2 , where X1 is of full column rank and each
column in X2 is a linear combination of the columns of X1 . The matrix
#
"
1
1 0
X1 X2
X01 X1
X01 X1
G1 D
1
0
X02 X1 X01 X1
is a g1 -inverse of X0 X and
"
#
1
X01 X1
0
G2 D
0
0
is a g2 -inverse. If the least squares solution is computed with the g1 -inverse, then computing the
variance of the estimator requires additional matrix operations and storage. On the other hand, the
variance of the solution that uses a g2 -inverse is proportional to G2 .
Var G1 X0 Y D 2 G1 X0 XG1
Var G2 X0 Y D 2 G2 X0 XG2 D 2 G2
If a generalized inverse G of X0 X is used to solve the normal equations, then the resulting solution
0
is a biased estimator
h i of ˇ (unless X X is of full rank, in which case the generalized inverse is “the”
inverse), since E b̌ D GX0 Xˇ, which is not in general equal to ˇ.
If you think of estimation as “estimation without bias,” then b̌ is the estimator of something, namely
GXˇ. Since this is not a quantity of interest and since it is not unique—it depends on your choice of
G—Searle (1971, p. 169) cautions that in the less-than-full-rank case, b̌ is a solution to the normal
equations and “nothing more.”
Analysis of Variance
The identity
Y D XˇQ C Y
XˇQ
Q but only for the least squares solution is the residual .Y Xb̌/ orthogonal
holds for all vectors ˇ,
to the predicted value Xb̌. Because of this orthogonality, the additive identity holds not only for the
vectors themselves, but also for their lengths (Pythagorean theorem):
jjYjj2 D jjXb̌jj2 C jj.Y
Xb̌/jj2
64 F Chapter 3: Introduction to Statistical Modeling with SAS/STAT Software
1 0
Note that Xb̌ D X X0 X
X Y = HY and note that Y Xb̌ D .I H/Y D MY. The matrices H
and M D I H play an important role in the theory of linear models and in statistical computations.
Both are projection matrices—that is, they are symmetric and idempotent. (An idempotent matrix
A is a square matrix that satisfies AA D A. The eigenvalues of an idempotent matrix take on the
values 1 and 0 only.) The matrix H projects onto the subspace of Rn that is spanned by the columns
of X. The matrix M projects onto the orthogonal complement of that space. Because of these
properties you have H0 D H, HH D H, M0 D M, MM D M, HM D 0.
The Pythagorean relationship now can be written in terms of H and M as follows:
jjYjj2 D Y0 Y D jjHYjj2 C jjMYjj2 D Y0 H0 HY C Y0 M0 MY D Y0 HY C Y0 MY
If X0 X is deficient in rank and a generalized inverse is used
to solve the normal equations, then you
0
0
work instead with the projection matrices H D X X X X . Note that if G is a generalized inverse
of X0 X, then XGX0 , and hence also H and M, are invariant to the choice of G.
The matrix H is sometimes referred to as the “hat” matrix because when you premultiply the vector
of observations with H, you produce the fitted values, which are commonly denoted by placing a
“hat” over the Y vector, b
Y D HY.
The term Y0 Y is the uncorrected total sum of squares (SST) of the linear model, Y0 MY is the error
(residual) sum of squares (SSR), and Y0 HY is the uncorrected model sum of squares. This leads to
the analysis of variance table shown in Table 3.2.
Table 3.2
Analysis of Variance with Uncorrected Sums of Squares
Source
df
Sum of Squares
Model
Residual
rank.X/
n rank.X/
Uncorr. Total
n
SSM D Y0 HY D b̌0 X0 Y
SSR D Y0 MY D Y0 Y b̌X0 Y D
2
Pn b
Y
Y
i
i
i D1
P
SST D Y0 Y D niD1 Yi2
When the model contains an intercept term, then the analysis of variance is usually corrected for
the mean, as shown in Table 3.3.
Table 3.3
Analysis of Variance with Corrected Sums of Squares
Source
df
Sum of Squares
Model
rank.X/
Residual
n
rank.X/
Corrected Total
n
1
1
2
P
2
bi Y i
nY D niD1 Y
SSR D Y0 MY D Y0 Y b̌X0 Y D
2
Pn bi
Y
Y
i
i D1
2
P
2
SSTc D Y0 Y nY D niD1 Yi Y
SSMc D b̌0 X0 Y
Linear Model Theory F 65
The coefficient of determination, also called the R-square statistic, measures the proportion of the
total variation explained by the linear model. In models with intercept, it is defined as the ratio
2
Pn b
Y
Y
i
i
i D1
SSR
R2 D 1
D1
2
P
n
SSTc
Yi Y
i D1
In models without intercept, the R-square statistic is a ratio of the uncorrected sums of squares
2
Pn bi
Y
Y
i
i D1
SSR
R2 D 1
D1
Pn
2
SST
i D1 Yi
Estimating the Error Variance
The least squares principle does not provide for a parameter estimator for 2 . The usual approach
is to use a method-of-moments estimator that is based on the sum of squared residuals. If the model
is correct, then the mean square for error, defined to be SSR divided by its degrees of freedom,
0 1
b
2 D
Y Xb̌ Y Xb̌
n rank.X/
D SSR=.n rank.X//
is an unbiased estimator of 2 .
Maximum Likelihood Estimation
To estimate the parameters in a linear model with mean function EŒY D Xˇ by maximum likelihood, you need to specify the distribution of the response vector Y. In the linear model with a
continuous response variable, it is commonly assumed that the response is normally distributed. In
that case, the estimation problem is completely defined by specifying the mean and variance of Y
in addition to the normality assumption. The model can be written as Y N.Xˇ; 2 I/, where
the notation N.a; V/ indicates a multivariate normal distribution with mean vector a and variance
matrix V. The log likelihood for Y then can be written as
l.ˇ; 2 I y/ D
n
logf2g
2
n
logf 2 g
2
1
.y
2 2
Xˇ/0 .y
Xˇ/
This function is maximized in ˇ when the sum of squares .y Xˇ/0 .y Xˇ/ is minimized. The
maximum likelihood estimator of ˇ is thus identical to the ordinary least squares estimator. To
maximize l.ˇ; 2 I y/ with respect to 2 , note that
@l.ˇ; 2 I y/
D
@ 2
n
1
C 4 .y
2
2
2
Xˇ/0 .y
Xˇ/
Hence the MLE of 2 is the estimator
0 1
b
2M D
Y Xb̌ Y Xb̌
n
D SSR=n
This is a biased estimator of 2 , with a bias that decreases with n.
66 F Chapter 3: Introduction to Statistical Modeling with SAS/STAT Software
Estimable Functions
A function Lˇ is said to be estimable if there exists a linear combination of the expected value of
Y, such as KEŒY, that equals Lˇ. Since EŒY D Xˇ, the definition of estimability implies that Lˇ
is estimable if there is a matrix K such that L D KX. Another way of looking at this result is that
the rows of X form a generating set from which all estimable functions can be constructed.
The concept of estimability of functions is important in the theory and application of linear models
because hypotheses of interest are often expressed as linear combinations of the parameter estimates
(for example, hypotheses of equality between parameters, ˇ1 D ˇ2 , ˇ1 ˇ2 D 0). Since
estimability is not related to the particular value of the parameter estimate, but to the row space of
X, you can test only hypotheses that consist of estimable functions. Further, because estimability is
not related to the value of ˇ (Searle 1971, p. 181), the choice of the generalized inverse in a situation
with rank-deficient X0 X matrix is immaterial, since
Lb̌ D KXb̌ D KX X0 X X0 Y
where X X0 X X is invariant to the choice of generalized inverse.
Lˇ is estimable if and only if L.X0 X/ .X0 X/ D L (see, for example, Searle 1971, p. 185). If X
is of full rank, then the Hermite matrix .X0 X/ .X0 X/ is the identity, which implies that all linear
functions are estimable in the full-rank case.
See Chapter 15, “The Four Types of Estimable Functions,” for many details about the various forms
of estimable functions in SAS/STAT.
Test of Hypotheses
Consider a general linear hypothesis of the form H W Lˇ D d, where L is a .k p/ matrix. It is
assumed that d is such that this hypothesis is linearly consistent—that is, that there exists some ˇ
for which Lˇ D d. This is always the case if d is in the column space of L, if L has full row
rank, or if d D 0; the latter is the most common case. Since many linear models have a rankdeficient X matrix, the question arises whether the hypothesis is testable. The idea of testability of a
hypothesis is—not surprisingly—connected to the concept of estimability as introduced previously.
The hypothesis H W Lˇ D d is testable if it consists of estimable functions.
There are two important approaches to testing hypotheses in statistical applications—the reduction
principle and the linear inference approach. The reduction principle states that the validity of the
hypothesis can be inferred by comparing a suitably chosen summary statistic between the model
at hand and a reduced model in which the constraint Lˇ D d is imposed. The linear inference
approach relies on the fact that b̌ is an estimator of ˇ and its stochastic properties are known, at least
approximately. A test statistic can then be formed using b̌, and its behavior under the restriction
Lˇ D d can be ascertained.
The two principles lead to identical results in certain—for example, least squares estimation in
the classical linear model. In more complex situations the two approaches lead to similar but not
identical results. This is the case, for example, when weights or unequal variances are involved, or
when b̌ is a nonlinear estimator.
Linear Model Theory F 67
Reduction Tests
The two main reduction principles are the sum of squares reduction test and the likelihood ratio
test. The test statistic in the former is proportional to the difference of the residual sum of squares
between the reduced model and the full model. The test statistic in the likelihood ratio test is
proportional to the difference of the log likelihoods between the full and reduced models. To fix
these ideas, suppose that you are fitting the model Y D Xˇ C , where N.0; 2 I/. Suppose
that SSR denotes the residual sum of squares in this model and that SSRH is the residual sum of
squares in the model for which Lˇ D d holds. Then under the hypothesis the ratio
.SSRH
SSR/= 2
follows a chi-square distribution with degrees of freedom equal to the rank of L. Maybe surprisingly,
the residual sum of squares in the full model is distributed independently of this quantity, so that
under the hypothesis,
F D
.SSRH SSR/=rank.L/
SSR=.n rank.X//
follows an F distribution with rank.L/ numerator and n rank.X/ denominator degrees of freedom.
Note that the quantity in the denominator of the F statistic is a particular estimator of 2 —namely,
the unbiased moment-based estimator that is customarily associated with least squares estimation.
It is also the restricted maximum likelihood estimator of 2 if Y is normally distributed.
In the case of the likelihood ratio test, suppose that l.b̌; b
2 I y/ denotes the log likelihood evaluated
2
b̌
at the ML estimators. Also suppose that l. H ; b
H I y/ denotes the log likelihood in the model for
which Lˇ D d holds. Then under the hypothesis the statistic
D 2 l.b̌; b
2 I y/ l.b̌H ; b
2H I y/
follows approximately a chi-square distribution with degrees of freedom equal to the rank of L. In
the case of a normally distributed response, the log-likelihood function can be profiled with respect
to ˇ. The resulting profile log likelihood is
n
n
l.b
2 I y/ D
logfb
2g
logf2g
2
2
and the likelihood ratio test statistic becomes
˚ 2 ˚ 2 D n log b
H
log b
D n .log fSSRH g
log fSSRg/ D n .log fSSRH =SSRg/
The preceding expressions show that, in the case of normally distributed data, both reduction principles lead to simple functions of the residual sums of squares in two models. As Pawitan (2001, p.
151) puts it, there is, however, an important difference not in the computations but in the statistical
content. The least squares principle, where sum of squares reduction tests are widely used, does not
require a distributional specification. Assumptions about the distribution of the data are added to
provide a framework for confirmatory inferences, such as the testing of hypotheses. This framework
stems directly from the assumption about the data’s distribution, or from the sampling distribution
of the least squares estimators. The likelihood principle, on the other hand, requires a distributional
specification at the outset. Inference about the parameters is implicit in the model; it is the result
of further computations following the estimation of the parameters. In the least squares framework,
inference about the parameters is the result of further assumptions.
68 F Chapter 3: Introduction to Statistical Modeling with SAS/STAT Software
Linear Inference
The principle of linear inference is to formulate a test statistic for H W Lˇ D d that builds on the
linearity of the hypothesis about ˇ. For many models that have linear components, the estimator
Lb̌ is also linear in Y. It is then simple to establish the distributional properties of Lb̌ based on the
distributional assumptions about Y or based on large-sample arguments. For example, b̌ might be
a nonlinear estimator, but it is known to asymptotically follow a normal distribution; this is the case
in many nonlinear and generalized linear models.
If the sampling distribution or the asymptotic distribution of b̌ is normal, then one can easily derive
quadratic forms with known distributional properties. For example, if the random vector U is distributed as N.; †/, then U0 AU follows a chi-square distribution with rank.A/ degrees of freedom
and noncentrality parameter 1=20 A, provided that A†A† D A†.
In the classical linear model, suppose that X is deficient in rank and that b̌ D X0 X X0 Y is a
solution to the normal equations. Then, if the errors are normally distributed,
b̌ N X0 X X0 Xˇ; 2 X0 X X0 X X0 X
Because H W Lˇ D d is testable, Lˇ is estimable, and thus L.X0 X/ X0 X D L, as established in the
previous section. Hence,
Lb̌ N Lˇ; 2 L X0 X L0
The conditions for a chi-square distribution of the quadratic form
.Lb̌ d/0 L.X0 X/ L0 .Lb̌ d/
are thus met, provided that
.L.X0 X/ L0 / L.X0 X/ L0 .L.X0 X/ L0 / L.X0 X/ L0 D .L.X0 X/ L0 / L.X0 X/ L0
This condition is obviously met if L.X0 X/ L0 is of full rank. The condition is also met if
L.X0 X/ L0 is a reflexive inverse (a g2 -inverse) of L.X0 X/ L.
The test statistic to test the linear hypothesis H W Lˇ D d is thus
.Lb̌ d/0 L.X0 X/ L0 .Lb̌ d/=rank.L/
F D
S SR=.n rank.X//
and it follows an F distribution with rank.L/ numerator and n
freedom under the hypothesis.
rank.X/ denominator degrees of
This test statistic looks very similar to the F statistic for the sum of squares reduction test. This
is no accident. If the model is linear and parameters are estimated
by ordinary least squares, then
0
0
0
b̌
b̌
you can show that the quadratic form .L
d/ L.X X/ L .L
d/ equals the differences in
the residual sum of squares, SSRH SSR, where SSRH is obtained as the residual sum of squares
from OLS estimation in a model that satisfies Lˇ D d. However, this correspondence between the
two test formulations does not apply when a different estimation principle is used. For example,
assume that N.0; V/ and that ˇ is estimated by generalized least squares:
b̌g D X0 V 1 X X0 V 1 Y
Linear Model Theory F 69
The construction of L matrices associated with hypotheses in SAS/STAT software is frequently
based on the properties of the X matrix, not of X0 V X. In other words, the construction of the L
matrix is governed only by the design. A sum of squares reduction test for H W Lˇ D 0 that uses the
generalized residual sum of squares .Y b̌g /0 V 1 .Y b̌g / is not identical to a linear hypothesis
test with the statistic
b̌0g L0 L X0 V 1 X L0 Lb̌g
F D
rank.L/
Furthermore, V is usually unknown and must be estimated as well. The estimate for V depends
on the model, and imposing a constraint on the model would change the estimate. The asymptotic
distribution of the statistic F is a chi-square distribution. However, in practical applications the F
distribution with rank.L/ numerator and denominator degrees of freedom is often used because it
provides a better approximation to the sampling distribution of F in finite samples. The computation of the denominator degrees of freedom , however, is a matter of considerable discussion. A
number of methods have been proposed and are implemented in various forms in SAS/STAT (see,
for example, the degrees-of-freedom methods in the MIXED and GLIMMIX procedures).
Residual Analysis
The model errors D Y Xˇ are unobservable. Yet important features of the statistical model
are connected to them, such as the distribution of the data, the correlation among observations, and
the constancy of variance. It is customary to diagnose and investigate features of the model errors
through the fitted residuals b
DY b
Y D Y HY D MY. These residuals are projections of the
data onto the null space of X and are also referred to as the “raw” residuals to contrast them with
other forms of residuals that are transformations of b
. For the classical linear model, the statistical
properties of b
are affected by the features of that projection and can be summarized as follows:
EŒb
 D 0
VarŒb
 D 2 M
rank.M/ D n
rank.X/
Furthermore, if N.0; 2 I/, then b
N.0; 2 M/.
Because M D I H, and the “hat” matrix H satisfies @b
[email protected], the hat matrix is also the leverage
matrix of the model. If hi i denotes the i th diagonal element of H (the leverage of observation
i ), then the leverages are bounded in a model with intercept, 1=n hi i 1. Consequently,
the variance of a raw residual is less than that of an observation: VarŒb
i  D 2 .1 hi i / < 2 .
In applications where the variability of the data is estimated from fitted residuals, the estimate is
invariably biased low. An example is the computation of an empirical semivariogram based on fitted
(detrended) residuals.
More important, the diagonal entries of H are not necessarily identical; the residuals are heteroscedastic. The “hat” matrix is also not a diagonal matrix; the residuals are correlated. In summary, the only property that the fitted residuals b
share with the model errors is a zero mean. It is
thus commonplace to use transformations of the fitted residuals for diagnostic purposes.
70 F Chapter 3: Introduction to Statistical Modeling with SAS/STAT Software
Raw and Studentized Residuals
A standardized residual is a raw residual that is divided by its standard deviation:
Yi
b
i D q
bi
Y
VarŒYi
b
i
Dp
2
.1 hi i /
bi 
Y
Because 2 is unknown, residual standardization is usually not practical. A studentized residual is
a raw residual that is divided by its estimated standard deviation. If the estimate of the standard
deviation is based on the same data that were used in fitting the model, the residual is also called an
internally studentized residual:
Yi
b
is D q
bi
Y
c i
VarŒY
b
i
Dp
b
2 .1 hi i /
bi 
Y
If the estimate of the residual’s variance does not involve the i th observation, it is called an externally studentized residual. Suppose that b
2 i denotes the estimate of the residual variance obtained
without the ith observation; then the externally studentized residual is
b
i
b
ir D q
b
2 i .1
hi i /
Scaled Residuals
A scaled residual is simply a raw residual divided by a scalar quantity that is not an estimate of the
variance of the residual. For example, residuals divided by the standard deviation of the response
variable are scaled and referred to as Pearson or Pearson-type residuals:
bi
Yi Y
b
ic D q
c i
VarŒY
In generalized linear models, where the variance of an observation is a function of the mean and
possibly of an extra scale parameter, VarŒY  D a./, the Pearson residual is
Yi b
i
b
iP D p
a.b
/
because the sum of the squared Pearson residuals equals the Pearson X 2 statistic:
2
X D
n
X
b
2iP
i D1
When the scale parameter participates in the scaling, the residual is also referred to as a Pearsontype residual:
Yi b
i
b
iP D p
a.b
/
Linear Model Theory F 71
Other Residuals
You might encounter other residuals in SAS/STAT software. A “leave-one-out” residual is the
difference between the observed value and the residual obtained from fitting a model in which the
bi is the predicted value of the i th observation and
observation in question did not participate. If Y
b
Y i; i is the predicted value if Yi is removed from the analysis, then the “leave-one-out” residual is
b
i;
i
D Yi
bi;
Y
i
Since the sum of the squared “leave-one-out” residuals is the PRESS statistic (prediction sum of
squares; Allen 1974), b
i; i is also called the PRESS residual. The concept of the PRESS residual
can be generalized if the deletion residual can be based on the removal of sets of observations. In
the classical linear model, the PRESS residual for case deletion has a particularly simple form:
b
i;
i
D Yi
bi;
Y
i
D
b
i
1 hi i
That is, the PRESS residual is simply a scaled form of the raw residual, where the scaling factor is
a function of the leverage of the observation.
When data are correlated, VarŒY D V, you can scale the vector of residuals rather than scale each
residual separately. This takes the covariances among the observations into account. This form
of scaling is accomplished by forming the Cholesky root C0 C D V, where C0 is a lower-triangular
matrix. Then C0 1 Y is a vector of uncorrelated variables with unit variance. The Cholesky residuals
in the model Y D Xˇ C are
b
C D C0 1 Y Xb̌
In generalized linear models, the fit of a model can be measured by the scaled deviance statistic
D . It measures the difference between the log likelihood under the model and the maximum log
likelihood
that is achievable. In models with a scale parameter , the deviance is D D D D
Pn
i D1 di . The deviance residuals are the signed square roots of the contributions to the deviance
statistic:
p
b
id D signfyi b
i g di
Sweep Operator
The sweep operator (Goodnight 1979) is closely related to Gauss-Jordan elimination and the Forward Doolittle procedure. The fact that a sweep operation can produce a generalized inverse by
in-place mapping with minimal storage and that its application invariably leads to some form of
matrix inversion is important, but this observation does not do justice to the pervasive relevance of
sweeping to statistical computing. In this section the sweep operator is discussed as a conceptual
tool for further insight into linear model operations. Consider the nonnegative definite, symmetric,
partitioned matrix
A11 A12
AD
A012 A22
Sweeping a matrix consists of performing a series of row operations akin to Gauss-Jordan elimination. Basic row operations are the multiplication of a row by a constant and the addition of a
72 F Chapter 3: Introduction to Statistical Modeling with SAS/STAT Software
multiple of one row to another. The sweep operator restricts row operations to pivots on the diagonal elements of a matrix; further details about the elementary operations can be found in Goodnight
(1979). The process of sweeping the matrix A on its leading partition is denoted as Sweep.A; A11 /
and leads to
A11 A12
A11
Sweep.A; A11 / D
A012 A11 A22 A012 A11 A12
If the kth row and column are set to zero when the pivot is zero (or in practice, less than some
singularity tolerance), the generalized inverse in the leading position of the swept matrix is a reflexive, g2 -inverse. Suppose that the crossproduct matrix of the linear model is augmented with a
“Y-border” as follows:
0
X X X0 Y
CD
Y0 X Y0 Y
Then the result of sweeping on the rows of X is
X0 X X0 X X0 Y
Sweep.C; X/ D
Y0 X X0 X
Y0 Y Y0 X X0 X X0 Y
"
# "
#
b̌
b̌
X0 X
X0 X
D
D
b̌
b̌
Y0 MY
SSR
The “Y-border” has been transformed into the least squares solution and the residual sum of squares.
Partial sweeps are common in model selection. Suppose that the X matrix is partitioned as ŒX1 X2 ,
and consider the augmented crossproduct matrix
3
2 0
X1 X1 X01 X2 X01 Y
C D 4 X02 X1 X02 X2 X02 Y 5
Y0 X1 Y0 X2 Y0 Y
Sweeping on the X1 partition yields
2
X01 X1
Sweep.C; X1 / D 4 X02 X1 X01 X1
Y0 X1 X01 X1
X01 X1 X01 X2
X02 M1 X2
Y0 M1 X2
3
X01 X1 X01 Y
5
X02 M1 Y
0
Y M1 Y
where M1 D I X1 X01 X1 X01 . The entries in the first row of this partition are the generalized
inverse of X0 X, the coefficients for regressing X2 on X1 , and the coefficients for regressing Y on
X1 . The diagonal entries X02 M1 X2 and Y0 M1 Y are the sum of squares and crossproduct matrices
for regressing X2 on X1 and for regressing Y on X1 , respectively. As you continue to sweep the
matrix, the last cell in the partition contains the residual sum of square of a model in which Y is
regressed on all columns swept up to that point.
The sweep operator is not only useful to conceptualize the computation of least squares solutions,
Type I and Type II sums of squares, and generalized inverses. It can also be used to obtain other
statistical information. For example, adding the logarithms of the pivots of the rows that are swept
yields the log determinant of the matrix.
References F 73
References
Allen, D. M. (1974), “The Relationship between Variable Selection and Data Augmentation and a
Method of Prediction,” Technometrics, 16, 125–127.
Cochran, W. G. (1997), Sampling Techniques, Third Edition, New York: John Wiley & Sons.
Goodnight, J. H. (1979), “A Tutorial on the Sweep Operator,” The American Statistician, 33, 149–
158.
Harville, D. A. (1997), Matrix Algebra from a Statistician’s Perspective, New York: Springer-Verlag
Hastie, T., Tibshirani, R., and Friedman, J. (2001), The Elements of Statistical Learning, New York:
Springer-Verlag.
Jöreskog, K. G. (1973), “A General Method for Estimating a Linear Structural Equation System,”
in Structural Equation Models in the Social Sciences, ed. A. S. Goldberger and O. D. Duncan, New
York: Seminar Press.
Keesling, J. W. (1972), “Maximum Likelihood Approaches to Causal Analysis,” Ph. D. dissertation,
University of Chicago, 1972.
Magnus, J. R., and Neudecker, H. (1999), Matrix Differential Calculus with Applications in Statistics and Econometrics, Second Edition, New York: John Wiley & Sons.
McCullagh, P. and Nelder. J. A. (1989), Generalized Linear Models, Second Edition, London:
Chapman & Hall.
Moore, E. H. (1920), “On the Reciprocal of the General Algebraic Matrix,” Bulletin of the American
Mathematical Society, 26, 394–395.
Nelder, J. A. and Wedderburn, R. W. M. (1972), “Generalized Linear Models,” Journal of the Royal
Statistical Society A, 135, 370–384.
Pawitan, Y. (2001), In All Likelihood: Statistical Modelling and Inference Using Likelihood, Oxford: Clarendon Press.
Penrose, R. A. (1955), “A Generalized Inverse for Matrices,” Proceedings of the Cambridge Philosophical Society, 51, 406–413.
Pringle, R. M. and Rayner, A. A. (1971), Generalized Inverse Matrices with Applications to Statistics, New York: Hafner Publishing.
Särndal, C. E., Swensson, B., and Wretman, J. (1992), Model Assisted Survey Sampling, New York:
Springer-Verlag.
Searle, S. R. (1971), Linear Models, New York: John Wiley & Sons.
Spearman, C. (1904), “General Intelligence, Objectively Determined and Measured.” American
Journal of Psychology, 15, 201–293.
74 F Chapter 3: Introduction to Statistical Modeling with SAS/STAT Software
Wedderburn, R. W. M. (1974), “Quasilikelihood Functions, Generalized Linear Models and the
Gauss-Newton Method,” Biometrika, 61, 439–447.
Wiley, D. E. (1973), “The Identification Problem for Structural Equation Models with Unmeasured
Variables,” in Structural Equation Models in the Social Sciences, ed. A. S. Goldberger and O. D.
Duncan, New York: Seminar Press, 69–83.
Index
analysis of variance
corrected total sum of squares (Introduction
to Modeling), 64
geometry (Introduction to Modeling), 63
model (Introduction to Modeling), 34
sum of squares (Introduction to Modeling),
34
uncorrected total sum of squares
(Introduction to Modeling), 64
Bayesian models
Introduction to Modeling, 40
classification effect
Introduction to Modeling, 33
coefficient of determination
definition (Introduction to Modeling), 64
covariance
matrix, definition (Introduction to
Modeling), 58
of random variables (Introduction to
Modeling), 57
estimability
definition (Introduction to Modeling), 66
estimable function
definition (Introduction to Modeling), 66
estimating equations
Introduction to Modeling, 31
expected value
definition (Introduction to Modeling), 57
of vector (Introduction to Modeling), 58
exponential family
Introduction to Modeling, 38
function
estimable, definition (Introduction to
Modeling), 66
generalized linear model
Introduction to Modeling, 37, 68, 70, 71
heteroscedasticity
Introduction to Modeling, 69
homoscedasticity
Introduction to Modeling, 61
hypothesis testing
Introduction to Modeling, 66
indepdendent
random variables (Introduction to
Modeling), 59
inference
design-based (Introduction to Modeling), 29
model-based (Introduction to Modeling), 29
Introduction to Modeling
additive error, 31
analysis of variance, 34, 63, 64
augmented crossproduct matrix, 72
Bayesian models, 40
Cholesky decomposition, 55, 71
Cholesky residual, 71
classification effect, 33
coefficient of determination, 64
column space, 66
covariance, 57
covariance matrix, 58
crossproduct matrix, 72
curvilinear models, 32
deletion residual, 71
dependent variable, 31
deviance residual, 71
diagonal matrix, 50, 69
effect genesis, 36
estimable, 66
estimating equations, 31
expectation operator, 31
expected value, 57
expected value of vector, 58
exponential family, 38
externally studentized residual, 70
fitted residual, 69
fixed effect, 35
fixed-effects model, 35
g1-inverse, 52
g2-inverse, 52, 68, 72
generalized inverse, 52, 66, 72
generalized least squares, 44, 68
generalized linear model, 37, 68, 70, 71
hat matrix, 64, 69
heterocatanomic data, 35
heterogeneous multivariate data, 35
heteroscedasticity, 69
homocatanomic data, 35
homogeneous multivariate data, 35
homoscedasticity, 61
hypothesis testing, 66
idempotent matrix, 64
independent random variables, 59
independent variable, 31
inner product of vectors, 50
internally studentized residual, 70
inverse of matrix, 50
inverse of partitioned matrix, 51
inverse of patterned sum of matrices, 51
inverse, generalized, 52, 66, 72
iteratively reweighted least squares, 44
latent variable models, 38
LDU decomposition, 55
least squares, 43
leave-one-out residual, 71
levelization, 33
leverage, 69, 71
likelihood, 45
likelihood ratio test, 67
linear hypothesis, 66
linear inference, 66, 68
linear model theory, 61
linear regression, 33
link function, 37
LU decomposition, 55
matrix addition, 50
matrix decomposition, Cholesky, 55, 71
matrix decomposition, LDU, 55
matrix decomposition, LU, 55
matrix decomposition, singular-value, 56
matrix decomposition, spectral, 55
matrix decompositions, 55
matrix differentiation, 53
matrix dot product, 50
matrix inverse, 50
matrix inverse, g1, 52
matrix inverse, g2, 52, 68, 72
matrix inverse, Moore-Penrose, 52, 56, 57
matrix inverse, partitioned, 51
matrix inverse, patterned sum, 51
matrix inverse, reflexive, 52, 68, 72
matrix multiplication, 50
matrix order, 49
matrix partition, 72
matrix subtraction, 50
matrix transposition, 50
matrix, column space, 66
matrix, diagonal, 50, 69
matrix, idempotent, 64
matrix, projection, 64
matrix, rank deficient, 66
matrix, square, 50
matrix, sweeping, 71
mean function, 31
mean squared error, 60
model fitting, 29
model-based v. design-based, 29
Moore-Penrose inverse, 52, 56, 57
multivariate model, 34
nonlinear model, 31, 68
outcome variable, 31
parameter, 28
Pearson-type residual, 70
power, 49
PRESS statistic, 71
projected residual, 69
projection matrix, 64
pseudo-likelihood, 45
quadratic forms, 60
quasi-likelihood, 45
R-square, 64
random effect, 35
random-effects model, 35
rank deficient matrix, 66
raw residual, 69
reduction principle, testing, 66
reflexive inverse, 52, 68, 72
residual analysis, 69
residual, Cholesky, 71
residual, deletion, 71
residual, deviance, 71
residual, externally studentized, 70
residual, fitted, 69
residual, internally studentized, 70
residual, leave-one-out, 71
residual, Pearson-type, 70
residual, PRESS, 71
residual, projected, 69
residual, raw, 69
residual, scaled, 70
residual, standardized, 70
residual, studentized, 70
response variable, 31
sample size, 49
scaled residual, 70
singular-value decomposition, 56
spectral decomposition, 55
square matrix, 50
standardized residual, 70
statistical model, 28
stochastic model, 28
studentized residual, 70
sum of squares reduction test, 67, 68
sweep, elementary operations, 72
sweep, log determinant, 72
sweep, operator, 71
sweep, pivots, 72
testable hypothesis, 66, 68
testing hypotheses, 66
uncorrelated random variables, 59
univariate model, 34
variance, 57
variance matrix, 58
variance-covariance matrix, 59
weighted least squares, 44
latent variable models
Introduction to Modeling, 38
least squares
definition (Introduction to Modeling), 43
generalized (Introduction to Modeling), 44,
68
iteratively reweighted (Introduction to
Modeling), 44
weighted (Introduction to Modeling), 44
likelihood
function (Introduction to Modeling), 45
Introduction to Modeling, 45
likelihood ratio test
Introduction to Modeling, 67
linear hypothesis
consistency (Introduction to Modeling), 66
definition (Introduction to Modeling), 66
Introduction to Modeling, 66
linear inference principle (Introduction to
Modeling), 66, 68
reduction principle (Introduction to
Modeling), 66
testable (Introduction to Modeling), 66, 68
testing (Introduction to Modeling), 66
testing, linear inference (Introduction to
Modeling), 66, 68
testing, reduction principle (Introduction to
Modeling), 66
linear model theory
Introduction to Modeling, 61
linear regression
Introduction to Modeling, 33
link function
Introduction to Modeling, 37
matrix
addition (Introduction to Modeling), 50
Choleksy decomposition (Introduction to
Modeling), 55, 71
column space (Introduction to Modeling), 66
crossproduct (Introduction to Modeling), 72
crossproduct, augmented (Introduction to
Modeling), 72
decomposition, Cholesky (Introduction to
Modeling), 55, 71
decomposition, LDU (Introduction to
Modeling), 55
decomposition, LU (Introduction to
Modeling), 55
decomposition, singular-value (Introduction
to Modeling), 56
decomposition, spectral (Introduction to
Modeling), 55
decompositions (Introduction to Modeling),
55
determinant, by sweeping (Introduction to
Modeling), 72
diagonal (Introduction to Modeling), 50, 69
differentiation (Introduction to Modeling),
53
dot product (Introduction to Modeling), 50
g1-inverse (Introduction to Modeling), 52
g2-inverse (Introduction to Modeling), 52,
68, 72
generalized inverse (Introduction to
Modeling), 52, 66, 72
hat (Introduction to Modeling), 64, 69
idempotent (Introduction to Modeling), 64
inner product (Introduction to Modeling), 50
inverse (Introduction to Modeling), 50
inverse, g1 (Introduction to Modeling), 52
inverse, g2 (Introduction to Modeling), 52,
68, 72
inverse, generalized (Introduction to
Modeling), 52, 66, 72
inverse, Moore-Penrose (Introduction to
Modeling), 52, 56, 57
inverse, partitioned (Introduction to
Modeling), 51
inverse, patterned (Introduction to
Modeling), 51
inverse, reflexive (Introduction to
Modeling), 52, 68, 72
LDU decomposition (Introduction to
Modeling), 55
leverage (Introduction to Modeling), 69
LU decomposition (Introduction to
Modeling), 55
Moore-Penrose inverse (Introduction to
Modeling), 52, 56, 57
multiplication (Introduction to Modeling),
50
order (Introduction to Modeling), 49
partition (Introduction to Modeling), 72
projection (Introduction to Modeling), 64
rank deficient (Introduction to Modeling), 66
reflexive inverse (Introduction to Modeling),
52, 68, 72
singular-value decomposition (Introduction
to Modeling), 56
spectral decomposition (Introduction to
Modeling), 55
square (Introduction to Modeling), 50
subtraction (Introduction to Modeling), 50
sweep (Introduction to Modeling), 71
transposition (Introduction to Modeling), 50
mean function
linear (Introduction to Modeling), 31
nonlinear (Introduction to Modeling), 31
mean squared error
Introduction to Modeling, 60
multivariate data
heterocatanomic (Introduction to Modeling),
35
heterogeneous (Introduction to Modeling),
35
homocatanomic (Introduction to Modeling),
35
homogeneous (Introduction to Modeling),
35
nonlinear model
Introduction to Modeling, 31
parameter
definition (Introduction to Modeling), 28
power
Introduction to Modeling, 49
quadratic forms
Introduction to Modeling, 60
R-square
definition (Introduction to Modeling), 64
residuals
Cholesky (Introduction to Modeling), 71
deletion (Introduction to Modeling), 71
deviance (Introduction to Modeling), 71
externally studentized (Introduction to
Modeling), 70
fitted (Introduction to Modeling), 69
internally studentized (Introduction to
Modeling), 70
leave-one-out (Introduction to Modeling),
71
Pearson-type (Introduction to Modeling), 70
PRESS (Introduction to Modeling), 71
projected, (Introduction to Modeling), 69
raw (Introduction to Modeling), 69
scaled (Introduction to Modeling), 70
standardized (Introduction to Modeling), 70
studentized (Introduction to Modeling), 70
studentized, external (Introduction to
Modeling), 70
studentized, internal (Introduction to
Modeling), 70
sample size
Introduction to Modeling, 49
statistical model
definition (Introduction to Modeling), 28
stochastic model
definition (Introduction to Modeling), 28
sum of squares
corrected total (Introduction to Modeling),
64
uncorrected total (Introduction to
Modeling), 64
sum of squares reduction test
Introduction to Modeling, 67, 68
Sweep operator
and generalized inverse (Introduction to
Modeling), 71
and log determinant (Introduction to
Modeling), 72
elementary operations (Introduction to
Modeling), 72
Gauss-Jordan elimination (Introduction to
Modeling), 71
pivots (Introduction to Modeling), 72
row operations (Introduction to Modeling),
71
testable hypothesis
Introduction to Modeling, 66, 68
testing hypotheses
Introduction to Modeling, 66
uncorrelated
random variables (Introduction to
Modeling), 59
variance
matrix, definition (Introduction to
Modeling), 58
of random variable (Introduction to
Modeling), 57
variance-covariance matrix
definition (Introduction to Modeling), 59
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