The GLM Procedure SAS/STAT 13.1 User’s Guide ®

The GLM Procedure SAS/STAT 13.1 User’s Guide ®
®
SAS/STAT 13.1 User’s Guide
The GLM Procedure
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Chapter 44
The GLM Procedure
Contents
Overview: GLM Procedure . . . . . . . . . . . . . . . . .
PROC GLM Features . . . . . . . . . . . . . . . .
PROC GLM Contrasted with Other SAS Procedures
Getting Started: GLM Procedure . . . . . . . . . . . . . .
PROC GLM for Unbalanced ANOVA . . . . . . . .
PROC GLM for Quadratic Least Squares Regression
Syntax: GLM Procedure . . . . . . . . . . . . . . . . . .
PROC GLM Statement . . . . . . . . . . . . . . . .
ABSORB Statement . . . . . . . . . . . . . . . . .
BY Statement . . . . . . . . . . . . . . . . . . . .
CLASS Statement . . . . . . . . . . . . . . . . . .
CODE Statement . . . . . . . . . . . . . . . . . . .
CONTRAST Statement . . . . . . . . . . . . . . .
ESTIMATE Statement . . . . . . . . . . . . . . . .
FREQ Statement . . . . . . . . . . . . . . . . . . .
ID Statement . . . . . . . . . . . . . . . . . . . . .
LSMEANS Statement . . . . . . . . . . . . . . . .
MANOVA Statement . . . . . . . . . . . . . . . . .
MEANS Statement . . . . . . . . . . . . . . . . . .
MODEL Statement . . . . . . . . . . . . . . . . . .
OUTPUT Statement . . . . . . . . . . . . . . . . .
RANDOM Statement . . . . . . . . . . . . . . . .
REPEATED Statement . . . . . . . . . . . . . . . .
STORE Statement . . . . . . . . . . . . . . . . . .
TEST Statement . . . . . . . . . . . . . . . . . . .
WEIGHT Statement . . . . . . . . . . . . . . . . .
Details: GLM Procedure . . . . . . . . . . . . . . . . . .
Statistical Assumptions for Using PROC GLM . . .
Specification of Effects . . . . . . . . . . . . . . .
Using PROC GLM Interactively . . . . . . . . . . .
Parameterization of PROC GLM Models . . . . . .
Hypothesis Testing in PROC GLM . . . . . . . . .
Effect Size Measures for F Tests in GLM . . . . . .
Absorption . . . . . . . . . . . . . . . . . . . . . .
Specification of ESTIMATE Expressions . . . . . .
Comparing Groups . . . . . . . . . . . . . . . . . .
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3434
3435
3436
3437
3437
3440
3446
3448
3454
3455
3455
3457
3457
3459
3460
3461
3461
3470
3474
3480
3484
3487
3488
3492
3493
3494
3494
3494
3495
3497
3498
3502
3508
3513
3515
3517
3434 F Chapter 44: The GLM Procedure
Means versus LS-Means . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Multiple Comparisons . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Simple Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Homogeneity of Variance in One-Way Models . . . . . . . . . . . . . . . .
Weighted Means . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Construction of Least Squares Means . . . . . . . . . . . . . . . . . . . . .
Multivariate Analysis of Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Repeated Measures Analysis of Variance . . . . . . . . . . . . . . . . . . . . . . . .
Random-Effects Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Missing Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Computational Resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Computational Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Output Data Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Displayed Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ODS Table Names . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ODS Graphics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Examples: GLM Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Example 44.1: Randomized Complete Blocks with Means Comparisons and Contrasts
Example 44.2: Regression with Mileage Data . . . . . . . . . . . . . . . . . . . . .
Example 44.3: Unbalanced ANOVA for Two-Way Design with Interaction . . . . . .
Example 44.4: Analysis of Covariance . . . . . . . . . . . . . . . . . . . . . . . . .
Example 44.5: Three-Way Analysis of Variance with Contrasts . . . . . . . . . . . .
Example 44.6: Multivariate Analysis of Variance . . . . . . . . . . . . . . . . . . . .
Example 44.7: Repeated Measures Analysis of Variance . . . . . . . . . . . . . . . .
Example 44.8: Mixed Model Analysis of Variance with the RANDOM Statement . .
Example 44.9: Analyzing a Doubly Multivariate Repeated Measures Design . . . . .
Example 44.10: Testing for Equal Group Variances . . . . . . . . . . . . . . . . . .
Example 44.11: Analysis of a Screening Design . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3517
3519
3530
3532
3533
3533
3536
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3552
3552
3554
3555
3558
3560
3560
3565
3569
3574
3581
3585
3593
3598
3602
3607
3612
3617
Overview: GLM Procedure
The GLM procedure uses the method of least squares to fit general linear models. Among the statistical
methods available in PROC GLM are regression, analysis of variance, analysis of covariance, multivariate
analysis of variance, and partial correlation.
PROC GLM analyzes data within the framework of general linear models. PROC GLM handles models
relating one or several continuous dependent variables to one or several independent variables. The independent variables can be either classification variables, which divide the observations into discrete groups,
or continuous variables. Thus, the GLM procedure can be used for many different analyses, including the
following:
• simple regression
PROC GLM Features F 3435
• multiple regression
• analysis of variance (ANOVA), especially for unbalanced data
• analysis of covariance
• response surface models
• weighted regression
• polynomial regression
• partial correlation
• multivariate analysis of variance (MANOVA)
• repeated measures analysis of variance
PROC GLM Features
The following list summarizes the features in PROC GLM:
• PROC GLM enables you to specify any degree of interaction (crossed effects) and nested effects. It
also provides for polynomial, continuous-by-class, and continuous-nesting-class effects.
• Through the concept of estimability, the GLM procedure can provide tests of hypotheses for the effects
of a linear model regardless of the number of missing cells or the extent of confounding. PROC GLM
displays the sum of squares (SS) associated with each hypothesis tested and, upon request, the form
of the estimable functions employed in the test. PROC GLM can produce the general form of all
estimable functions.
• The REPEATED statement enables you to specify effects in the model that represent repeated measurements on the same experimental unit for the same response, providing both univariate and multivariate
tests of hypotheses.
• The RANDOM statement enables you to specify random effects in the model; expected mean squares
are produced for each Type I, Type II, Type III, Type IV, and contrast mean square used in the analysis.
Upon request, F tests that use appropriate mean squares or linear combinations of mean squares as
error terms are performed.
• The ESTIMATE statement enables you to specify an L vector for estimating a linear function of the
parameters Lˇ.
• The CONTRAST statement enables you to specify a contrast vector or matrix for testing the hypothesis
that Lˇ D 0. When specified, the contrasts are also incorporated into analyses that use the MANOVA
and REPEATED statements.
• The MANOVA statement enables you to specify both the hypothesis effects and the error effect to use
for a multivariate analysis of variance.
3436 F Chapter 44: The GLM Procedure
• PROC GLM can create an output data set containing the input data set in addition to predicted values,
residuals, and other diagnostic measures.
• PROC GLM can be used interactively. After you specify and fit a model, you can execute a variety of
statements without recomputing the model parameters or sums of squares.
• For analysis involving multiple dependent variables but not the MANOVA or REPEATED statements,
a missing value in one dependent variable does not eliminate the observation from the analysis for
other dependent variables. PROC GLM automatically groups together those variables that have the
same pattern of missing values within the data set or within a BY group. This ensures that the analysis
for each dependent variable brings into use all possible observations.
• The GLM procedure automatically produces graphs as part of its ODS output. For general information
about ODS Graphics, see the section “ODS Graphics” on page 3558 and Chapter 21, “Statistical
Graphics Using ODS.”
PROC GLM Contrasted with Other SAS Procedures
As described previously, PROC GLM can be used for many different analyses and has many special features
not available in other SAS procedures. However, for some types of analyses, other procedures are available.
As discussed in the sections “PROC GLM for Unbalanced ANOVA” on page 3437 and “PROC GLM for
Quadratic Least Squares Regression” on page 3440, sometimes these other procedures are more efficient
than PROC GLM. The following procedures perform some of the same analyses as PROC GLM:
ANOVA
performs analysis of variance for balanced designs. The ANOVA procedure is generally
more efficient than PROC GLM for these designs.
MIXED
fits mixed linear models by incorporating covariance structures in the model fitting process.
Its RANDOM and REPEATED statements are similar to those in PROC GLM but offer
different functionalities.
NESTED
performs analysis of variance and estimates variance components for nested random
models. The NESTED procedure is generally more efficient than PROC GLM for these
models.
NPAR1WAY
performs nonparametric one-way analysis of rank scores. This can also be done using the
RANK procedure and PROC GLM.
REG
performs simple linear regression. The REG procedure allows several MODEL statements
and gives additional regression diagnostics, especially for detection of collinearity.
RSREG
performs quadratic response surface regression, and canonical and ridge analysis. The
RSREG procedure is generally recommended for data from a response surface experiment.
Getting Started: GLM Procedure F 3437
TTEST
compares the means of two groups of observations. Also, tests for equality of variances
for the two groups are available. The TTEST procedure is usually more efficient than
PROC GLM for this type of data.
VARCOMP
estimates variance components for a general linear model.
Getting Started: GLM Procedure
PROC GLM for Unbalanced ANOVA
Analysis of variance, or ANOVA, typically refers to partitioning the variation in a variable’s values into
variation between and within several groups or classes of observations. The GLM procedure can perform
simple or complicated ANOVA for balanced or unbalanced data.
This example discusses the analysis of variance for the unbalanced 2 2 data shown in Table 44.1. The
experimental design is a full factorial, in which each level of one treatment factor occurs at each level of the
other treatment factor. Note that there is only one value for the cell with A=‘A2’ and B=‘B2’. Since one cell
contains a different number of values from the other cells in the table, this is an unbalanced design.
Table 44.1 Unbalanced Two-Way Data
B1
B2
A1
12, 14
11, 9
A2
20, 18
17
The following statements read the data into a SAS data set and then invoke PROC GLM to produce the
analysis.
title 'Analysis of Unbalanced 2-by-2 Factorial';
data exp;
input A $ B $ Y @@;
datalines;
A1 B1 12 A1 B1 14
A1 B2 11 A1 B2 9
A2 B1 20 A2 B1 18
A2 B2 17
;
proc glm data=exp;
class A B;
model Y=A B A*B;
run;
Both treatments are listed in the CLASS statement because they are classification variables. A*B denotes the
interaction of the A effect and the B effect. The results are shown in Figure 44.1 and Figure 44.2.
3438 F Chapter 44: The GLM Procedure
Figure 44.1 Class Level Information
Analysis of Unbalanced 2-by-2 Factorial
The GLM Procedure
Class Level Information
Class
Levels
Values
A
2
A1 A2
B
2
B1 B2
Number of Observations Read
Number of Observations Used
7
7
Figure 44.1 displays information about the classes as well as the number of observations in the data set.
Figure 44.2 shows the ANOVA table, simple statistics, and tests of effects.
Figure 44.2 ANOVA Table and Tests of Effects
Analysis of Unbalanced 2-by-2 Factorial
The GLM Procedure
Dependent Variable: Y
DF
Sum of
Squares
Mean Square
F Value
Pr > F
Model
3
91.71428571
30.57142857
15.29
0.0253
Error
3
6.00000000
2.00000000
Corrected Total
6
97.71428571
Source
Source
A
B
A*B
R-Square
Coeff Var
Root MSE
Y Mean
0.938596
9.801480
1.414214
14.42857
DF
Type I SS
Mean Square
F Value
Pr > F
1
1
1
80.04761905
11.26666667
0.40000000
80.04761905
11.26666667
0.40000000
40.02
5.63
0.20
0.0080
0.0982
0.6850
PROC GLM for Unbalanced ANOVA F 3439
Figure 44.2 continued
Source
A
B
A*B
DF
Type III SS
Mean Square
F Value
Pr > F
1
1
1
67.60000000
10.00000000
0.40000000
67.60000000
10.00000000
0.40000000
33.80
5.00
0.20
0.0101
0.1114
0.6850
The degrees of freedom can be used to check your data. The Model degrees of freedom for a 2 2 factorial
design with interaction are .ab 1/, where a is the number of levels of A and b is the number of levels of B;
in this case, .2 2 1/ D 3. The Corrected Total degrees of freedom are always one less than the number of
observations used in the analysis; in this case, 7 – 1 = 6.
The overall F test is significant .F D 15:29; p D 0:0253/, indicating strong evidence that the means for the
four different AB cells are different. You can further analyze this difference by examining the individual
tests for each effect.
Four types of estimable functions of parameters are available for testing hypotheses in PROC GLM. For data
with no missing cells, the Type III and Type IV estimable functions are the same and test the same hypotheses
that would be tested if the data were balanced. Type I and Type III sums of squares are typically not equal
when the data are unbalanced; Type III sums of squares are preferred in testing effects in unbalanced cases
because they test a function of the underlying parameters that is independent of the number of observations
per treatment combination.
According to a significance level of 5% .˛ D 0:05/, the A*B interaction is not significant .F D 0:20; p D
0:6850/. This indicates that the effect of A does not depend on the level of B and vice versa. Therefore, the
tests for the individual effects are valid, showing a significant A effect .F D 33:80; p D 0:0101/ but no
significant B effect .F D 5:00; p D 0:1114/.
If ODS Graphics is enabled, GLM also displays by default an interaction plot for this analysis. The following
statements, which are the same as in the previous analysis but with ODS Graphics enabled, additionally
produce Figure 44.3.
ods graphics on;
proc glm data=exp;
class A B;
model Y=A B A*B;
run;
ods graphics off;
3440 F Chapter 44: The GLM Procedure
Figure 44.3 Plot of Y by A and B
The insignificance of the A*B interaction is reflected in the fact that two lines in Figure 44.3 are nearly
parallel. For more information about the graphics that GLM can produce, see the section “ODS Graphics” on
page 3558.
PROC GLM for Quadratic Least Squares Regression
In polynomial regression, the values of a dependent variable (also called a response variable) are described
or predicted in terms of polynomial terms involving one or more independent or explanatory variables. An
example of quadratic regression in PROC GLM follows. These data are taken from Draper and Smith (1966,
p. 57). Thirteen specimens of 90/10 Cu-Ni alloys are tested in a corrosion-wheel setup in order to examine
corrosion. Each specimen has a certain iron content. The wheel is rotated in salt sea water at 30 ft/sec for 60
days. Weight loss is used to quantify the corrosion. The fe variable represents the iron content, and the loss
variable denotes the weight loss in milligrams/square decimeter/day in the following DATA step.
title 'Regression in PROC GLM';
data iron;
input fe loss @@;
PROC GLM for Quadratic Least Squares Regression F 3441
datalines;
0.01 127.6
0.48 124.0
1.19 101.5
0.01 130.1
0.71 113.1
1.96 83.7
1.96 86.2
;
0.71 110.8
0.48 122.0
0.01 128.0
0.95 103.9
1.44 92.3
1.44 91.4
The SGSCATTER procedure is used in the following statements to request a scatter plot of the response
variable versus the independent variable.
ods graphics on;
proc sgscatter data=iron;
plot loss*fe;
run;
ods graphics off;
The plot in Figure 44.4 displays a strong negative relationship between iron content and corrosion resistance,
but it is not clear whether there is curvature in this relationship.
Figure 44.4 Plot of Observed Corrosion Resistance by Iron Content
3442 F Chapter 44: The GLM Procedure
The following statements fit a quadratic regression model to the data. This enables you to estimate the
linear relationship between iron content and corrosion resistance and to test for the presence of a quadratic
component. The intercept is automatically fit unless the NOINT option is specified.
proc glm data=iron;
model loss=fe fe*fe;
run;
The CLASS statement is omitted because a regression line is being fitted. Unlike PROC REG, PROC GLM
allows polynomial terms in the MODEL statement.
PROC GLM first displays preliminary information, shown in Figure 44.5, telling you that the GLM procedure
has been invoked and stating the number of observations in the data set. If the model involves classification
variables, they are also listed here, along with their levels.
Figure 44.5 Data Information
Regression in PROC GLM
The GLM Procedure
Number of Observations Read
Number of Observations Used
13
13
Figure 44.6 shows the overall ANOVA table and some simple statistics. The degrees of freedom can be
used to check that the model is correct and that the data have been read correctly. The Model degrees of
freedom for a regression is the number of parameters in the model minus 1. You are fitting a model with
three parameters in this case,
loss D ˇ0 C ˇ1 .fe/ C ˇ2 .fe/2 C error
so the degrees of freedom are 3 1 D 2. The Corrected Total degrees of freedom are always one less than
the number of observations used in the analysis.
Figure 44.6 ANOVA Table
Regression in PROC GLM
The GLM Procedure
Dependent Variable: loss
DF
Sum of
Squares
Mean Square
F Value
Pr > F
Model
2
3296.530589
1648.265295
164.68
<.0001
Error
10
100.086334
10.008633
Corrected Total
12
3396.616923
Source
PROC GLM for Quadratic Least Squares Regression F 3443
Figure 44.6 continued
R-Square
Coeff Var
Root MSE
loss Mean
0.970534
2.907348
3.163642
108.8154
The R square indicates that the model accounts for 97% of the variation in LOSS. The coefficient of variation
(Coeff Var), Root MSE (Mean Square for Error), and mean of the dependent variable are also listed.
The overall F test is significant .F D 164:68; p < 0:0001/, indicating that the model as a whole accounts
for a significant amount of the variation in LOSS. Thus, it is appropriate to proceed to testing the effects.
Figure 44.7 contains tests of effects and parameter estimates. The latter are displayed by default when the
model contains only continuous variables.
Figure 44.7 Tests of Effects and Parameter Estimates
Source
fe
fe*fe
Source
fe
fe*fe
DF
Type I SS
Mean Square
F Value
Pr > F
1
1
3293.766690
2.763899
3293.766690
2.763899
329.09
0.28
<.0001
0.6107
DF
Type III SS
Mean Square
F Value
Pr > F
1
1
356.7572421
2.7638994
356.7572421
2.7638994
35.64
0.28
0.0001
0.6107
Parameter
Estimate
Standard
Error
t Value
Pr > |t|
Intercept
fe
fe*fe
130.3199337
-26.2203900
1.1552018
1.77096213
4.39177557
2.19828568
73.59
-5.97
0.53
<.0001
0.0001
0.6107
The t tests provided are equivalent to the Type III F tests. The quadratic term is not significant (p = 0.6107)
and thus can be removed from the model; the linear term is significant .p < 0:0001/. This suggests that there
is indeed a straight-line relationship between loss and fe.
Finally, if ODS Graphics is enabled, PROC GLM also displays by default a scatter plot of the original data, as
in Figure 44.4, with the quadratic fit overlaid. The following statements, which are the same as the previous
analysis but with ODS Graphics enabled, additionally produce Figure 44.8.
ods graphics on;
proc glm data=iron;
model loss=fe fe*fe;
run;
ods graphics off;
3444 F Chapter 44: The GLM Procedure
Figure 44.8 Plot of Observed and Fit Corrosion Resistance by Iron Content, Quadratic Model
The insignificance of the quadratic term in the model is reflected in the fact that the fit is nearly linear.
Fitting the model without the quadratic term provides more accurate estimates for ˇ0 and ˇ1 . PROC GLM
allows only one MODEL statement per invocation of the procedure, so the PROC GLM statement must be
issued again. The following statements are used to fit the linear model.
proc glm data=iron;
model loss=fe;
run;
Figure 44.9 displays the output produced by these statements. The linear term is still significant .F D
352:27; p < 0:0001/. The estimated model is now
loss D 129:79
24:02 fe
PROC GLM for Quadratic Least Squares Regression F 3445
Figure 44.9 Linear Model Output
Regression in PROC GLM
The GLM Procedure
Dependent Variable: loss
DF
Sum of
Squares
Mean Square
F Value
Pr > F
Model
1
3293.766690
3293.766690
352.27
<.0001
Error
11
102.850233
9.350021
Corrected Total
12
3396.616923
Source
R-Square
Coeff Var
Root MSE
loss Mean
0.969720
2.810063
3.057780
108.8154
Source
fe
Source
fe
DF
Type I SS
Mean Square
F Value
Pr > F
1
3293.766690
3293.766690
352.27
<.0001
DF
Type III SS
Mean Square
F Value
Pr > F
1
3293.766690
3293.766690
352.27
<.0001
Parameter
Estimate
Standard
Error
t Value
Pr > |t|
Intercept
fe
129.7865993
-24.0198934
1.40273671
1.27976715
92.52
-18.77
<.0001
<.0001
3446 F Chapter 44: The GLM Procedure
Syntax: GLM Procedure
The following statements are available in the GLM procedure:
PROC GLM < options > ;
CLASS variable < (REF= option) > . . . < variable < (REF= option) > > < / global-options > ;
MODEL dependent-variables = independent-effects < / options > ;
ABSORB variables ;
BY variables ;
CODE < options > ;
FREQ variable ;
ID variables ;
WEIGHT variable ;
CONTRAST ’label’ effect values < . . . effect values > < / options > ;
ESTIMATE ’label’ effect values < . . . effect values > < / options > ;
LSMEANS effects < / options > ;
MANOVA < test-options > < / detail-options > ;
MEANS effects < / options > ;
OUTPUT < OUT=SAS-data-set > keyword=names < . . . keyword=names > < / option > ;
RANDOM effects < / options > ;
REPEATED factor-specification < / options > ;
STORE < OUT= >item-store-name < / LABEL='label' > ;
TEST < H=effects > E=effect < / options > ;
Although there are numerous statements and options available in PROC GLM, many applications use only
a few of them. Often you can find the features you need by looking at an example or by quickly scanning
through this section.
To use PROC GLM, the PROC GLM and MODEL statements are required. You can specify only one
MODEL statement (in contrast to the REG procedure, for example, which allows several MODEL statements
in the same PROC REG run). If your model contains classification effects, the classification variables must
be listed in a CLASS statement, and the CLASS statement must appear before the MODEL statement. In
addition, if you use a CONTRAST statement in combination with a MANOVA, RANDOM, REPEATED, or
TEST statement, the CONTRAST statement must be entered first in order for the contrast to be included in
the MANOVA, RANDOM, REPEATED, or TEST analysis.
Table 44.2 summarizes the positional requirements for the statements in the GLM procedure.
Syntax: GLM Procedure F 3447
Table 44.2 Positional Requirements for PROC GLM Statements
Statement
ABSORB
Must Precede. . .
First RUN statement
BY
First RUN statement
CLASS
MODEL statement
CONTRAST
MANOVA, REPEATED,
or RANDOM statement
ESTIMATE
Must Follow. . .
MODEL statement
MODEL statement
FREQ
First RUN statement
ID
First RUN statement
LSMEANS
MODEL statement
MANOVA
CONTRAST or
MODEL statement
MEANS
MODEL statement
MODEL
CONTRAST, ESTIMATE,
LSMEANS, or MEANS
statement
CLASS statement
OUTPUT
MODEL statement
RANDOM
CONTRAST or
MODEL statement
REPEATED
CONTRAST, MODEL,
or TEST statement
TEST
MANOVA or
REPEATED statement
WEIGHT
First RUN statement
MODEL statement
Table 44.3 summarizes the function of each statement (other than the PROC statement) in the GLM procedure.
Table 44.3 Statements in the GLM Procedure
Statement
ABSORB
BY
CLASS
CODE
CONTRAST
ESTIMATE
FREQ
ID
LSMEANS
MANOVA
Description
Absorbs classification effects in a model
Specifies variables to define subgroups for the analysis
Declares classification variables
Requests that the procedure write SAS DATA step code to a file or
catalog entry for computing predicted values according to the fitted
model
Constructs and tests linear functions of the parameters
Estimates linear functions of the parameters
Specifies a frequency variable
Identifies observations on output
Computes least squares (marginal) means
Performs a multivariate analysis of variance
3448 F Chapter 44: The GLM Procedure
Table 44.3 continued
Statement
MEANS
MODEL
OUTPUT
Description
Computes and optionally compares arithmetic means
Defines the model to be fit
Requests an output data set containing diagnostics for each observation
Declares certain effects to be random and computes expected mean
squares
Performs multivariate and univariate repeated measures analysis of
variance
Requests that the procedure save the context and results of the
statistical analysis into an item store
Constructs tests that use the sums of squares for effects and the
error term you specify
Specifies a variable for weighting observations
RANDOM
REPEATED
STORE
TEST
WEIGHT
The rest of this section provides detailed syntax information for each of these statements, beginning with the
PROC GLM statement. The remaining statements are covered in alphabetical order.
The STORE and CODE statements are also used by many other procedures. A summary description of
functionality and syntax for these statements is also shown after the PROC GLM statement in alphabetical
order, but you can find full documentation about them in the section “STORE Statement” on page 508 in
Chapter 19, “Shared Concepts and Topics.”
PROC GLM Statement
PROC GLM < options > ;
The PROC GLM statement invokes the GLM procedure. Table 44.4 summarizes the options available in the
PROC GLM statement.
Table 44.4 PROC GLM Statement Options
Option
Description
ALPHA=
DATA=
MANOVA
Specifies the level of significance for confidence intervals
Names the SAS data set used by the GLM procedure
Requests the multivariate mode of eliminating observations with missing
values
Requests that the input data set be reread when necessary, instead of using a
utility file
Specifies the length of effect names
Suppresses the normal display of results
Specifies the order in which to sort classification variables
Names an output data set for information and statistics on each model effect
Controls the plots produced through ODS Graphics
MULTIPASS
NAMELEN=
NOPRINT
ORDER=
OUTSTAT=
PLOTS
PROC GLM Statement F 3449
You can specify the following options in the PROC GLM statement.
ALPHA=p
specifies the level of significance p for 100.1 p/% confidence intervals. The value must be between 0
and 1; the default value of p = 0.05 results in 95% intervals. This value is used as the default confidence
level for limits computed by the following options.
Statement
LSMEANS
Options
CL
MEANS
CLM CLDIFF
MODEL
CLI CLM CLPARM
OUTPUT
UCL= LCL= UCLM= LCLM=
You can override the default in each of these cases by specifying the ALPHA= option for each statement
individually.
DATA=SAS-data-set
names the SAS data set used by the GLM procedure. By default, PROC GLM uses the most recently
created SAS data set.
MANOVA
requests the multivariate mode of eliminating observations with missing values. If any of the dependent
variables have missing values, the procedure eliminates that observation from the analysis. The
MANOVA option is useful if you use PROC GLM in interactive mode and plan to perform a multivariate
analysis.
MULTIPASS
requests that PROC GLM reread the input data set when necessary, instead of writing the necessary
values of dependent variables to a utility file. This option decreases disk space usage at the expense
of increased execution times, and is useful only in rare situations where disk space is at an absolute
premium.
NAMELEN=n
specifies the length of effect names in tables and output data sets to be n characters long, where n is a
value between 20 and 200 characters. The default length is 20 characters.
NOPRINT
suppresses the normal display of results. The NOPRINT option is useful when you want only to create
one or more output data sets with the procedure. Note that this option temporarily disables the Output
Delivery System (ODS); see Chapter 20, “Using the Output Delivery System,” for more information.
3450 F Chapter 44: The GLM Procedure
ORDER=DATA | FORMATTED | FREQ | INTERNAL
specifies the sort order for the levels of the classification variables (which are specified in the CLASS
statement). This ordering determines which parameters in the model correspond to each level in
the data, so the ORDER= option can be useful when you specify the CONTRAST or ESTIMATE
statement.
This option applies to the levels for all classification variables, except when you use the (default)
ORDER=FORMATTED option with numeric classification variables that have no explicit format. In
that case, the levels of such variables are ordered by their internal value.
The ORDER= option can take the following values:
Value of ORDER=
Levels Sorted By
DATA
Order of appearance in the input data set
FORMATTED
External formatted value, except for numeric variables with
no explicit format, which are sorted by their unformatted
(internal) value
FREQ
Descending frequency count; levels with the most observations come first in the order
INTERNAL
Unformatted value
By default, ORDER=FORMATTED. For ORDER=FORMATTED and ORDER=INTERNAL, the sort
order is machine-dependent.
For more information about sort order, see the chapter on the SORT procedure in the Base SAS
Procedures Guide and the discussion of BY-group processing in SAS Language Reference: Concepts.
OUTSTAT=SAS-data-set
names an output data set that contains sums of squares, degrees of freedom, F statistics, and probability
levels for each effect in the model, as well as for each CONTRAST that uses the overall residual or error
mean square (MSE) as the denominator in constructing the F statistic. If you use the CANONICAL
option in the MANOVA statement and do not use an M= specification in the MANOVA statement, the
data set also contains results of the canonical analysis.
See the section “Output Data Sets” on page 3552 for more information.
PLOTS < (global-plot-options) > < =plot-request < (options) > >
PLOTS < (global-plot-options) > < =(plot-request < (options) > < ... plot-request < (options) > >) >
controls the plots produced through ODS Graphics. When you specify only one plot-request , you can
omit the parentheses from around the plot-request . For example:
PLOTS=NONE
PLOTS=(DIAGNOSTICS RESIDUALS)
PLOTS(UNPACK)=RESIDUALS
PLOT=MEANPLOT(CLBAND)
ODS Graphics must be enabled before plots can be requested. For example:
PROC GLM Statement F 3451
ods graphics on;
proc glm data=iron;
model loss=fe fe*fe;
run;
ods graphics off;
For more information about enabling and disabling ODS Graphics, see the section “Enabling and
Disabling ODS Graphics” on page 606 in Chapter 21, “Statistical Graphics Using ODS.”
If ODS Graphics is enabled but you do not specify the PLOTS= option, then PROC GLM produces a
default set of plots, which might be different for different models, as discussed in the following.
• If you specify a one-way analysis of variance model, with just one CLASS variable, the GLM
procedure produces a grouped box plot of the response values versus the CLASS levels. For
an example of the box plot, see the section “One-Way Layout with Means Comparisons” on
page 946 in Chapter 26, “The ANOVA Procedure.”
• If you specify a two-way analysis of variance model, with just two CLASS variables, the
GLM procedure produces an interaction plot of the response values, with horizontal position
representing one CLASS variable and marker style representing the other; and with predicted
response values connected by lines representing the two-way analysis. For an example of the
interaction plot, see the section “PROC GLM for Unbalanced ANOVA” on page 3437.
• If you specify a model with a single continuous predictor, the GLM procedure produces a fit plot
of the response values versus the covariate values, with a curve representing the fitted relationship
and a band representing the confidence limits for individual mean values. For an example of the
fit plot, see the section “PROC GLM for Quadratic Least Squares Regression” on page 3440.
• If you specify a model with two continuous predictors and no CLASS variables, the GLM
procedure produces a contour fit plot, overlaying a scatter plot of the data and a contour plot of
the predicted surface.
• If you specify an analysis of covariance model, with one or two CLASS variables and one
continuous variable, the GLM procedure produces an analysis of covariance plot of the response
values versus the covariate values, with lines representing the fitted relationship within each
classification level. For an example of the analysis of covariance plot, see Example 44.4.
• If you specify an LSMEANS statement with the PDIFF option, the GLM procedure produces a
plot appropriate for the type of LS-means comparison. For PDIFF=ALL (which is the default if
you specify only PDIFF), the procedure produces a diffogram, which displays all pairwise LSmeans differences and their significance. The display is also known as a “mean-mean scatter plot”
(Hsu 1996). For PDIFF=CONTROL, the procedure produces a display of each noncontrol LSmean compared to the control LS-mean, with two-sided confidence intervals for the comparison.
For PDIFF=CONTROLL and PDIFF=CONTROLU a similar display is produced, but with
one-sided confidence intervals. Finally, for the PDIFF=ANOM option, the procedure produces
an “analysis of means” plot, comparing each LS-mean to the average LS-mean.
• If you specify a MEANS statement, the GLM procedure produces a grouped box plot of the
response values versus the effect for which means are being calculated.
3452 F Chapter 44: The GLM Procedure
The global-plot-options include the following:
MAXPOINTS=NONE | number
specifies that plots with elements that require processing of more than number points be
suppressed. The default is MAXPOINTS=5000. This limit is ignored if you specify MAXPOINTS=NONE.
ONLY
suppresses the default plots. Only plots specifically requested are displayed.
UNPACKPANEL
UNPACK
suppresses paneling. By default, multiple plots can appear in some output panels. Specify
UNPACKPANEL to get each plot in a separate panel. You can specify PLOTS(UNPACKPANEL)
to just unpack the default plots. You can also specify UNPACKPANEL as a suboption with
DIAGNOSTICS and RESIDUALS.
The following individual plots and plot options are available. If you specify only one plot , then you
can omit the parentheses.
ALL
produces all appropriate plots. You can specify other options with ALL; for example, to request
all plots and unpack just the residuals, specify: PLOTS=(ALL RESIDUALS(UNPACK)).
ANCOVAPLOT< (CLM CLI LIMITS) >
modifies the analysis of covariance plot produced by default when you have an analysis of covariance model, with one or two CLASS variables and one continuous variable. By default the plot
does not show confidence limits around the predicted values. The PLOTS=ANCOVAPLOT(CLM)
option adds limits for the expected predicted values, and PLOTS=ANCOVAPLOT(CLI) adds
limits for new predictions. Use PLOTS=ANCOVAPLOT(LIMITS) to add both kinds of limits.
ANOMPLOT
requests an analysis of means display, in which least squares means are compared against an
average least squares mean (Ott 1967; Nelson 1982, 1991, 1993). LS-mean ANOM plots are
produced only if you also specify PDIFF=ANOM or ADJUST=NELSON in the LSMEANS
statement, and in this case they are produced by default.
BOXPLOT< (NPANELPOS=n) >
modifies the plot produced by default for the model effect in a one-way analysis of variance
model, or for an effect specified in the MEANS statement. Suppose the effect has m levels. By
default, or if you specify PLOTS=BOXPLOT(NPANELPOS=0), all m levels of the effect are
displayed in a single plot. Specifying a nonzero value of n will result in P panels, where P is the
integer part of m=n C 1. If n > 0, then the levels will be approximately balanced across the P
panels; whereas if n < 0, precisely jnj levels will be displayed on each panel except possibly the
last.
CONTOURFIT< (OBS=obs-options) >
modifies the contour fit plot produced by default when you have a model involving only two
continuous predictors. The plot displays a contour plot of the predicted surface overlaid with
a scatter plot of the observed data. You can use the following obs-options to control how the
observations are displayed:
PROC GLM Statement F 3453
GRADIENT
specifies that observations are displayed as circles colored by the observed response. The
same color gradient is used to display the fitted surface and the observations. Observations
where the predicted response is close to the observed response have similar colors: the
greater the contrast between the color of an observation and the surface, the larger the
residual is at that point.
NONE
suppresses the observations.
OUTLINE
specifies that observations are displayed as circles with a border but with a completely
transparent fill.
OUTLINEGRADIENT
is the same as OBS=GRADIENT except that a border is shown around each observation. This
option is useful to identify the location of observations where the residuals are small, since
at these points the color of the observations and the color of the surface are indistinguishable.
OBS=OUTLINEGRADIENT is the default if you do not specify any obs-options.
CONTROLPLOT
requests a display in which least squares means are compared against a reference level. LS-mean
control plots are produced only when you specify PDIFF=CONTROL or ADJUST=DUNNETT
in the LSMEANS statement, and in this case they are produced by default.
DIAGNOSTICS< (LABEL UNPACK) >
requests that a panel of summary diagnostics for the fit be displayed. The panel displays scatter
plots of residuals, absolute residuals, studentized residuals, and observed responses by predicted
values; studentized residuals by leverage; Cook’s D by observation; a Q-Q plot of residuals;
a residual histogram; and a residual-fit spread plot. The LABEL option displays labels on
observations satisfying RSTUDENT > 2, LEVERAGE > 2p=n, and on the Cook’s D plot,
COOKSD > 4=n, where n is the number of observations used in fitting the model, and p is
the number of parameters in the model. The label is the first ID variable if the ID statement is
specified; otherwise, it is the observation number. The UNPACK option unpanels the diagnostic
display and produces the series of individual plots that form the paneled display.
DIFFPLOT< (ABS NOABS CENTER NOLINES) >
modifies the plot produced by an LSMEANS statement with the PDIFF=ALL option (or just
PDIFF, since ALL is the default argument). The ABS and NOABS options determine the
positioning of the line segments in the plot. When the ABS option is in effect, and this is the
default, all line segments are shown on the same side of the reference line. The NOABS option
separates comparisons according to the sign of the difference. The CENTER option marks the
center point for each comparison. This point corresponds to the intersection of two least squares
means. The NOLINES option suppresses the display of the line segments that represent the
confidence bounds for the differences of the least squares means. The NOLINES option implies
the CENTER option. The default is to draw line segments in the upper portion of the plot area
without marking the center point.
3454 F Chapter 44: The GLM Procedure
FITPLOT< (NOCLM NOCLI NOLIMITS) >
modifies the fit plot produced by default when you have a model with a single continuous
predictor. By default the plot includes confidence limits for both the expected predicted values
and individual new predictions. The PLOTS=FITPLOT(NOCLM) option removes the limits
on the expected values and the PLOTS=FITPLOT(NOCLI) option removes the limits on new
predictions. The PLOTS=FITPLOT(NOLIMITS) option removes both kinds of confidence limits.
INTPLOT< (CLM CLI LIMITS) >
modifies the interaction plot produced by default when you have a two-way analysis of variance model, with just two CLASS variables. By default the plot does not show confidence
limits around the predicted values. The PLOTS=INTPLOT(CLM) option adds limits for the
expected predicted values and PLOTS=INTPLOT(CLI) adds limits for new predictions. Use
PLOTS=INTPLOT(LIMITS) to add both kinds of limits.
MEANPLOT< (CL CLBAND CONNECT ASCENDING DESCENDING) >
modifies the grouped box plot produced by an MEANS statement. Upper and lower confidence
limits are plotted when the CL option is used. When the CLBAND option is in effect, confidence
limits are shown as bands and the means are connected. By default, means are not joined by lines.
You can achieve that effect with the CONNECT option. Means are displayed in the same order as
they appear in the “Means” table. You can change that order for plotting with the ASCENDING
and DESCENDING options.
NONE
specifies that no graphics be displayed.
RESIDUALS< (SMOOTH UNPACK) >
requests that scatter plots of the residuals against each continuous covariate be displayed. The
SMOOTH option overlays a Loess smooth on each residual plot. Note that if a WEIGHT variable
is specified, then it is not used to weight the smoother. See Chapter 57, “The LOESS Procedure,”
for more information. The UNPACK option unpanels the residual display and produces a series
of individual plots that form the paneled display.
ABSORB Statement
ABSORB variables ;
Absorption is a computational technique that provides a large reduction in time and memory requirements for
certain types of models. The variables are one or more variables in the input data set.
For a main-effect variable that does not participate in interactions, you can absorb the effect by naming it in
an ABSORB statement. This means that the effect can be adjusted out before the construction and solution
of the rest of the model. This is particularly useful when the effect has a large number of levels.
Several variables can be specified, in which case each one is assumed to be nested in the preceding variable
in the ABSORB statement.
N OTE : When you use the ABSORB statement, the data set (or each BY group, if a BY statement appears)
must be sorted by the variables in the ABSORB statement. The GLM procedure cannot produce predicted
values or least squares means (LS-means) or create an output data set of diagnostic values if an ABSORB
BY Statement F 3455
statement is used. If the ABSORB statement is used, it must appear before the first RUN statement; otherwise,
it is ignored.
When you use an ABSORB statement and also use the INT option in the MODEL statement, the procedure
ignores the option but computes the uncorrected total sum of squares (SS) instead of the corrected total sums
of squares.
See the section “Absorption” on page 3513 for more information.
BY Statement
BY variables ;
You can specify a BY statement with PROC GLM to obtain separate analyses of observations in groups that
are defined by the BY variables. When a BY statement appears, the procedure expects the input data set to be
sorted in order of the BY variables. If you specify more than one BY statement, only the last one specified is
used.
If your input data set is not sorted in ascending order, use one of the following alternatives:
• Sort the data by using the SORT procedure with a similar BY statement.
• Specify the NOTSORTED or DESCENDING option in the BY statement for the GLM procedure. The
NOTSORTED option does not mean that the data are unsorted but rather that the data are arranged
in groups (according to values of the BY variables) and that these groups are not necessarily in
alphabetical or increasing numeric order.
• Create an index on the BY variables by using the DATASETS procedure (in Base SAS software).
Since sorting the data changes the order in which PROC GLM reads observations, the sort order for the
levels of the classification variables might be affected if you also specify ORDER=DATA in the PROC GLM
statement. This, in turn, affects specifications in the CONTRAST and ESTIMATE statements.
If you specify the BY statement, it must appear before the first RUN statement; otherwise, it is ignored.
When you use a BY statement, the interactive features of PROC GLM are disabled.
When both the BY and ABSORB statements are used, observations must be sorted first by the variables in
the BY statement, and then by the variables in the ABSORB statement.
For more information about BY-group processing, see the discussion in SAS Language Reference: Concepts.
For more information about the DATASETS procedure, see the discussion in the Base SAS Procedures Guide.
CLASS Statement
CLASS variable < (REF= option) > . . . < variable < (REF= option) > > < / global-options > ;
The CLASS statement names the classification variables to be used in the model. Typical classification
variables are Treatment, Sex, Race, Group, and Replication. If you use the CLASS statement, it must appear
before the MODEL statement.
3456 F Chapter 44: The GLM Procedure
Classification variables can be either character or numeric. By default, class levels are determined from the
entire set of formatted values of the CLASS variables.
N OTE : Prior to SAS 9, class levels were determined by using no more than the first 16 characters of the
formatted values. To revert to this previous behavior, you can use the TRUNCATE option in the CLASS
statement.
In any case, you can use formats to group values into levels. See the discussion of the FORMAT procedure
in the Base SAS Procedures Guide and the discussions of the FORMAT statement and SAS formats in SAS
Formats and Informats: Reference. You can adjust the order of CLASS variable levels with the ORDER=
option in the PROC GLM statement.
The GLM procedure displays a table that summarizes the CLASS variables and their levels. You can use
this table to check the ordering of levels and, hence, of the corresponding parameters for main effects.
If you need to check the ordering of parameters for interaction effects, use the E option in the MODEL,
CONTRAST, ESTIMATE, or LSMEANS statement. See the section “Parameterization of PROC GLM
Models” on page 3498 for more information.
You can specify the following REF= option to indicate how the levels of an individual classification variable
are to be ordered by enclosing it in parentheses after the variable name:
REF=’level’ | FIRST | LAST
specifies a level of the classification variable to be put at the end of the list of levels. This level thus
corresponds to the reference level in the usual interpretation of the estimates with PROC GLM’s
singular parameterization. You can specify the level of the variable to use as the reference level; specify
a value that corresponds to the formatted value of the variable if a format is assigned. Alternatively, you
can specify REF=FIRST to designate that the first ordered level serve as the reference, or REF=LAST to
designate that the last ordered level serve as the reference. To specify that REF=FIRST or REF=LAST
be used for all classification variables, use the REF= global-option after the slash (/) in the CLASS
statement.
You can specify the following global-options in the CLASS statement after a slash (/):
REF=FIRST | LAST
specifies a level of all classification variables to be put at the end of the list of levels. This level
thus corresponds to the reference level in the usual interpretation of the estimates with PROC GLM’s
singular parameterization. Specify REF=FIRST to designate that the first ordered level for each
classification variable serve as the reference. Specify REF=LAST to designate that the last ordered
level serve as the reference. This option applies to all the variables specified in the CLASS statement. To
specify different reference levels for different classification variables, use REF= options for individual
variables.
TRUNCATE
specifies that class levels be determined by using only up to the first 16 characters of the formatted
values of CLASS variables. When formatted values are longer than 16 characters, you can use this
option to revert to the levels as determined in releases prior to SAS 9.
CONTRAST Statement F 3457
CODE Statement
CODE < options > ;
The CODE statement writes SAS DATA step code for computing predicted values of the fitted model either
to a file or to a catalog entry. This code can then be included in a DATA step to score new data.
Table 44.5 summarizes the options available in the CODE statement.
Table 44.5 CODE Statement Options
Option
Description
CATALOG=
DUMMIES
ERROR
FILE=
FORMAT=
GROUP=
IMPUTE
Names the catalog entry where the generated code is saved
Retains the dummy variables in the data set
Computes the error function
Names the file where the generated code is saved
Specifies the numeric format for the regression coefficients
Specifies the group identifier for array names and statement labels
Imputes predicted values for observations with missing or invalid
covariates
Specifies the line size of the generated code
Specifies the algorithm for looking up CLASS levels
Computes residuals
LINESIZE=
LOOKUP=
RESIDUAL
For details about the syntax of the CODE statement, see the section “CODE Statement” on page 395 in
Chapter 19, “Shared Concepts and Topics.”
CONTRAST Statement
CONTRAST ’label’ effect values < . . . effect values > < / options > ;
The CONTRAST statement enables you to perform custom hypothesis tests by specifying an L vector or
matrix for testing the univariate hypothesis Lˇ D 0 or the multivariate hypothesis LBM D 0. Thus, to
use this feature you must be familiar with the details of the model parameterization that PROC GLM uses.
For more information, see the section “Parameterization of PROC GLM Models” on page 3498. All of the
elements of the L vector might be given, or if only certain portions of the L vector are given, the remaining
elements are constructed by PROC GLM from the context (in a manner similar to rule 4 discussed in the
section “Construction of Least Squares Means” on page 3533).
There is no limit to the number of CONTRAST statements you can specify, but they must appear after the
MODEL statement. In addition, if you use a CONTRAST statement and a MANOVA, REPEATED, or TEST
statement, appropriate tests for contrasts are carried out as part of the MANOVA, REPEATED, or TEST
analysis. If you use a CONTRAST statement and a RANDOM statement, the expected mean square of the
contrast is displayed. As a result of these additional analyses, the CONTRAST statement must appear before
the MANOVA, REPEATED, RANDOM, or TEST statement.
3458 F Chapter 44: The GLM Procedure
In the CONTRAST statement,
label
identifies the contrast on the output. A label is required for every contrast specified. Labels
must be enclosed in quotes.
effect
identifies an effect that appears in the MODEL statement, or the INTERCEPT effect. The
INTERCEPT effect can be used when an intercept is fitted in the model. You do not need
to include all effects that are in the MODEL statement.
values
are constants that are elements of the L vector associated with the effect.
You can specify the following options in the CONTRAST statement after a slash (/).
E
displays the entire L vector. This option is useful in confirming the ordering of parameters for
specifying L.
E=effect
specifies an error term, which must be one of the effects in the model. The procedure uses this effect
as the denominator in F tests in univariate analysis. In addition, if you use a MANOVA or REPEATED
statement, the procedure uses the effect specified by the E= option as the basis of the E matrix. By
default, the procedure uses the overall residual or error mean square (MSE) as an error term.
ETYPE=n
specifies the type (1, 2, 3, or 4, corresponding to a Type I, II, III, or IV test, respectively) of the E=
effect. If the E= option is specified and the ETYPE= option is not, the procedure uses the highest type
computed in the analysis.
SINGULAR=number
tunes the estimability checking. If ABS.L LH/ > C number for any row in the contrast, then L is
declared nonestimable. H is the .X0 X/ X0 X matrix, and C is ABS.L/ except for rows where L is zero,
and then it is 1. The default value for the SINGULAR= option is 10 4 . Values for the SINGULAR=
option must be between 0 and 1.
As stated previously, the CONTRAST statement enables you to perform custom hypothesis tests. If
the hypothesis is testable in the univariate case, SS(H0 W Lˇ D 0) is computed as
.Lb/0 .L.X0 X/ L0 /
1
.Lb/
where b D .X0 X/ X0 y. This is the sum of squares displayed on the analysis-of-variance table.
For multivariate testable hypotheses, the usual multivariate tests are performed using
H D M0 .LB/0 .L.X0 X/ L0 /
1
.LB/M
where B D .X0 X/ X0 Y and Y is the matrix of multivariate responses or dependent variables. The
degrees of freedom associated with the hypothesis are equal to the row rank of L. The sum of squares
computed in this situation is equivalent to the sum of squares computed using an L matrix with any
row deleted that is a linear combination of previous rows.
Multiple-degrees-of-freedom hypotheses can be specified by separating the rows of the L matrix with
commas.
For example, for the model
ESTIMATE Statement F 3459
proc glm;
class A B;
model Y=A B;
run;
with A at 5 levels and B at 2 levels, the parameter vector is
. ˛1 ˛2 ˛3 ˛4 ˛5 ˇ1 ˇ2 /
To test the hypothesis that the pooled A linear and A quadratic effect is zero, you can use the following
L matrix:
0
2
1
0
1 2 0 0
LD
0
2
1
2
1 2 0 0
The corresponding CONTRAST statement is
contrast 'A LINEAR & QUADRATIC'
a -2 -1 0 1 2,
a 2 -1 -2 -1 2;
If the first level of A is a control level and you want a test of control versus others, you can use this
statement:
contrast 'CONTROL VS OTHERS'
a -1 0.25 0.25 0.25 0.25;
See the following discussion of the ESTIMATE statement and the section “Specification of ESTIMATE
Expressions” on page 3515 for rules on specification, construction, distribution, and estimability in the
CONTRAST statement.
ESTIMATE Statement
ESTIMATE ’label’ effect values < . . . effect values > < / options > ;
The ESTIMATE statement enables you to estimate linear functions of the parameters by multiplying the
vector L by the parameter estimate vector b, resulting in Lb. All of the elements of the L vector might be
given, or, if only certain portions of the L vector are given, the remaining elements are constructed by PROC
GLM from the context (in a manner similar to rule 4 discussed in the section “Construction of Least Squares
Means” on page 3533).
The linear function is checked for estimability.
The estimate Lb, where b D .X0 X/ X0 y, is displayed along
p
0
with its associated standard error, L.X X/ L0 s 2 , and t test. If you specify the CLPARM option in the
MODEL statement (see page 3482), confidence limits for the true value are also displayed.
There is no limit to the number of ESTIMATE statements that you can specify, but they must appear after the
MODEL statement. In the ESTIMATE statement,
3460 F Chapter 44: The GLM Procedure
label
identifies the estimate on the output. A label is required for every contrast specified.
Labels must be enclosed in quotes.
effect
identifies an effect that appears in the MODEL statement, or the INTERCEPT effect. The
INTERCEPT effect can be used as an effect when an intercept is fitted in the model. You
do not need to include all effects that are in the MODEL statement.
values
are constants that are the elements of the L vector associated with the preceding effect.
For example,
estimate 'A1 VS A2' A
1 -1;
forms an estimate that is the difference between the parameters estimated for the first and
second levels of the CLASS variable A.
You can specify the following options in the ESTIMATE statement after a slash (/):
DIVISOR=number
specifies a value by which to divide all coefficients so that fractional coefficients can be entered as
integer numerators. For example, you can use
estimate '1/3(A1+A2) - 2/3A3' a 1 1 -2 / divisor=3;
instead of
estimate '1/3(A1+A2) - 2/3A3' a 0.33333 0.33333 -0.66667;
E
displays the entire L vector. This option is useful in confirming the ordering of parameters for
specifying L.
SINGULAR=number
tunes the estimability checking. If ABS.L LH/ > C number , then the L vector is declared
nonestimable. H is the .X0 X/ X0 X matrix, and C is ABS.L/ except for rows where L is zero, and then
it is 1. The default value for the SINGULAR= option is 10 4 . Values for the SINGULAR= option
must be between 0 and 1.
See also the section “Specification of ESTIMATE Expressions” on page 3515.
FREQ Statement
FREQ variable ;
The FREQ statement names a variable that provides frequencies for each observation in the DATA= data set.
Specifically, if n is the value of the FREQ variable for a given observation, then that observation is used n
times.
The analysis produced using a FREQ statement reflects the expanded number of observations. For example,
means and total degrees of freedom reflect the expanded number of observations. You can produce the same
ID Statement F 3461
analysis (without the FREQ statement) by first creating a new data set that contains the expanded number
of observations. For example, if the value of the FREQ variable is 5 for the first observation, the first 5
observations in the new data set are identical. Each observation in the old data set is replicated ni times in
the new data set, where ni is the value of the FREQ variable for that observation.
If the value of the FREQ variable is missing or is less than 1, the observation is not used in the analysis. If
the value is not an integer, only the integer portion is used.
If you specify the FREQ statement, it must appear before the first RUN statement or it is ignored.
ID Statement
ID variables ;
When predicted values are requested as a MODEL statement option, values of the variables given in the ID
statement are displayed beside each observed, predicted, and residual value for identification. Although there
are no restrictions on the length of ID variables, PROC GLM might truncate the number of values listed in
order to display them on one line. The GLM procedure displays a maximum of five ID variables.
If you specify the ID statement, it must appear before the first RUN statement or it is ignored.
LSMEANS Statement
LSMEANS effects < / options > ;
Table 44.6 summarizes the options available in the LSMEANS statement.
Table 44.6
LSMEANS Statement Options
Option
Description
ADJUST=
ALPHA=
AT
BYLEVEL
CL
COV
E
E=
ETYPE=
LINES
Requests a multiple comparison adjustment
Specifies the level of significance
Enables you to modify the values of the covariates
Processes the OM data set by each level of the LS-mean effect
Requests confidence limits
Includes variances and covariances of the LS-means
Displays the coefficients of the linear functions
Specifies an effect in the model to use as an error term
Specifies the type of the E= effect
Uses connecting lines to indicate nonsignificantly different subsets of LSmeans
Suppresses the normal display of results
Specifies a potentially different weighting scheme
Creates an output data set
Requests that p-values for differences
Requests graphics related to least squares means
Specifies effects within which to test for differences
Tunes the estimability checking
NOPRINT
OBSMARGINS
OUT=
PDIFF
PLOT=
SLICE=
SINGULAR=
3462 F Chapter 44: The GLM Procedure
Table 44.6 continued
Option
Description
STDERR
TDIFF
Produces the standard error
Produces the t values
Least squares means (LS-means) are computed for each effect listed in the LSMEANS statement. You can
specify only classification effects in the LSMEANS statement—that is, effects that contain only classification
variables. You can also specify options to perform multiple comparisons. In contrast to the MEANS statement,
the LSMEANS statement performs multiple comparisons on interactions as well as main effects.
LS-means are predicted population margins; that is, they estimate the marginal means over a balanced
population. In a sense, LS-means are to unbalanced designs as class and subclass arithmetic means are to
balanced designs. Each LS-mean is computed as L0 b for a certain column vector L, where b is the vector of
parameter estimates—that is, the solution of the normal equations. For further information, see the section
“Construction of Least Squares Means” on page 3533.
Multiple effects can be specified in one LSMEANS statement, or multiple LSMEANS statements can be
used, but they must all appear after the MODEL statement. For example:
proc glm;
class A B;
model Y=A B A*B;
lsmeans A B A*B;
run;
LS-means are displayed for each level of the A, B, and A*B effects.
You can specify the following options in the LSMEANS statement after a slash (/):
ADJUST=BON
ADJUST=DUNNETT
ADJUST=NELSON
ADJUST=SCHEFFE
ADJUST=SIDAK
ADJUST=SIMULATE < (simoptions) >
ADJUST=SMM | GT2
ADJUST=TUKEY
ADJUST=T
requests a multiple comparison adjustment for the p-values and confidence limits for the differences
of LS-means. The ADJUST= option modifies the results of the TDIFF and PDIFF options; thus, if
you omit the TDIFF or PDIFF option then the ADJUST= option implies PDIFF. By default, PROC
GLM analyzes all pairwise differences. If you specify ADJUST=DUNNETT, PROC GLM analyzes
all differences with a control level. If you specify the ADJUST=NELSON option, PROC GLM
analyzes all differences with the average LS-mean. The default is ADJUST=T, which really signifies
no adjustment for multiple comparisons.
The BON (Bonferroni) and SIDAK adjustments involve correction factors described in the section
“Multiple Comparisons” on page 3519 and in Chapter 65, “The MULTTEST Procedure.” When
LSMEANS Statement F 3463
you specify ADJUST=TUKEY and your data are unbalanced, PROC GLM uses the approximation
described in Kramer (1956) and identifies the adjustment as “Tukey-Kramer” in the results. Similarly,
when you specify either ADJUST=DUNNETT or the ADJUST=NELSON option and the LS-means
are correlated, PROC GLM uses the factor-analytic covariance approximation described in Hsu (1992)
and identifies the adjustment in the results as “Dunnett-Hsu” or “Nelson-Hsu,” respectively. The
preceding references also describe the SCHEFFE and SMM adjustments.
The SIMULATE adjustment computes the adjusted p-values from the simulated distribution of the
maximum or maximum absolute value of a multivariate t random vector. The simulation estimates q,
the true .1 ˛/ quantile, where 1 ˛ is the confidence coefficient. The default ˛ is the value of the
ALPHA= option in the PROC GLM statement or 0.05 if that option is not specified. You can change
this value with the ALPHA= option in the LSMEANS statement.
The number of samples for the SIMULATE adjustment is set so that the tail area for the simulated q is
within a certain accuracy radius of 1 ˛ with an accuracy confidence of 100.1 /%. In equation
form,
P .jF .q/
O
.1
˛/j / D 1
where qO is the simulated q and F is the true distribution function of the maximum; see Edwards and
Berry (1987) for details. By default, = 0.005 and = 0.01, so that the tail area of qO is within 0.005 of
0.95 with 99% confidence.
You can specify the following simoptions in parentheses after the ADJUST=SIMULATE option.
ACC=value
specifies the target accuracy radius of a 100.1 /% confidence interval for the true probability
content of the estimated .1 ˛/ quantile. The default value is ACC=0.005. Note that, if you also
specify the CVADJUST simoption, then the actual accuracy radius will probably be substantially
less than this target.
CVADJUST
specifies that the quantile should be estimated by the control variate adjustment method of Hsu
and Nelson (1998) instead of simply as the quantile of the simulated sample. Specifying the
CVADJUST option typically has the effect of significantly reducing the accuracy radius of
a 100 .1 /% confidence interval for the true probability content of the estimated .1 ˛/
quantile. The control-variate-adjusted quantile estimate takes roughly twice as long to compute,
but it is typically much more accurate than the sample quantile.
EPS=value
specifies the value for a 100 .1 /% confidence interval for the true probability content of the
estimated .1 ˛/ quantile. The default value for the accuracy confidence is 99%, corresponding
to EPS=0.01.
NSAMP=n
specifies the sample size for the simulation. By default, n is set based on the values of the target
accuracy radius and accuracy confidence 100 .1 /% for an interval for the true probability
content of the estimated .1 ˛/ quantile. With the default values for , , and ˛ (0.005, 0.01,
and 0.05, respectively), NSAMP=12604 by default.
3464 F Chapter 44: The GLM Procedure
REPORT
specifies that a report on the simulation should be displayed, including a listing of the parameters,
such as , , and ˛, as well as an analysis of various methods for estimating or approximating the
quantile.
SEED=number
specifies an integer used to start the pseudo-random number generator for the simulation. If you
do not specify a seed, or specify a value less than or equal to zero, the seed is by default generated
from reading the time of day from the computer’s clock.
THREADS
specifies that the computational work for the simulation be divided into parallel threads, where
the number of threads is the value of the SAS system option CPUCOUNT=. For large simulations
(as specified directly using the NSAMP= simoption or indirectly using the ACC= or EPS=
simoptions), parallel processing can markedly speed up the computation of adjusted p-values and
confidence intervals. However, because the parallel processing has different pseudo-random number streams, the precise results are different from the default ones, which are computed in sequence
rather than in parallel. This option overrides the SAS system option THREADS | NOTHREADS.
NOTHREADS
specifies that the computational work for the simulation be performed in sequence rather
than in parallel. NOTHREADS is the default. This option overrides the SAS system option
THREADS | NOTHREADS.
ALPHA=p
specifies the level of significance p for 100.1 p/% confidence intervals. This option is useful only if
you also specify the CL option, and, optionally, the PDIFF option. By default, p is equal to the value
of the ALPHA= option in the PROC GLM statement or 0.05 if that option is not specified, This value
is used to set the endpoints for confidence intervals for the individual means as well as for differences
between means.
AT variable = value
AT (variable-list)= (value-list)
AT MEANS
enables you to modify the values of the covariates used in computing LS-means. By default, all
covariate effects are set equal to their mean values for computation of standard LS-means. The
AT option enables you to set the covariates to whatever values you consider interesting. For more
information, see the section “Setting Covariate Values” on page 3534.
BYLEVEL
requests that PROC GLM process the OM data set by each level of the LS-mean effect in question. For
more details, see the entry for the OM option in this section.
CL
requests confidence limits for the individual LS-means. If you specify the PDIFF option, confidence
limits for differences between means are produced as well. You can control the confidence level with
the ALPHA= option. Note that, if you specify an ADJUST= option, the confidence limits for the
differences are adjusted for multiple inference but the confidence intervals for individual means are
not adjusted.
LSMEANS Statement F 3465
COV
includes variances and covariances of the LS-means in the output data set specified in the OUT= option
in the LSMEANS statement. Note that this is the covariance matrix for the LS-means themselves,
not the covariance matrix for the differences between the LS-means, which is used in the PDIFF
computations. If you omit the OUT= option, the COV option has no effect. When you specify the
COV option, you can specify only one effect in the LSMEANS statement.
E
displays the coefficients of the linear functions used to compute the LS-means.
E=effect
specifies an effect in the model to use as an error term. The procedure uses the mean square for
the effect as the error mean square when calculating estimated standard errors (requested with the
STDERR option) and probabilities (requested with the STDERR, PDIFF, or TDIFF option). Unless
you specify STDERR, PDIFF or TDIFF, the E= option is ignored. By default, if you specify the
STDERR, PDIFF, or TDIFF option and do not specify the E= option, the procedure uses the error
mean square for calculating standard errors and probabilities.
ETYPE=n
specifies the type (1, 2, 3, or 4, corresponding to a Type I, II, III, or IV test, respectively) of the E=
effect. If you specify the E= option but not the ETYPE= option, the highest type computed in the
analysis is used. If you omit the E= option, the ETYPE= option has no effect.
LINES
presents results of comparisons between all pairs of means (specified by the PDIFF=ALL option) by
listing the means in descending order and indicating nonsignificant subsets by line segments beside
the corresponding means. When all differences have the same variance, these comparison lines are
guaranteed to accurately reflect the inferences based on the corresponding tests, made by comparing
the respective p-values to the value of the ALPHA= option (0.05 by default). However, equal variances
are rarely the case for differences between LS-means. If the variances are not all the same, then the
comparison lines might be conservative, in the sense that if you base your inferences on the lines alone,
you will detect fewer significant differences than the tests indicate. If there are any such differences, a
note is appended to the table that lists the pairs of means that are inferred to be significantly different
by the tests but not by the comparison lines. Note, however, that in many cases, even though the
variances are unbalanced, they are near enough that the comparison lines in fact accurately reflect the
test inferences.
NOPRINT
suppresses the normal display of results from the LSMEANS statement. This option is useful when an
output data set is created with the OUT= option in the LSMEANS statement.
OBSMARGINS
OM
specifies a potentially different weighting scheme for computing LS-means coefficients. The standard
LS-means have equal coefficients across classification effects; however, the OM option changes these
coefficients to be proportional to those found in the input data set. For more information, see the
section “Changing the Weighting Scheme” on page 3535.
The BYLEVEL option modifies the observed-margins LS-means. Instead of computing the margins
across the entire data set, the procedure computes separate margins for each level of the LS-mean
3466 F Chapter 44: The GLM Procedure
effect in question. The resulting LS-means are actually equal to raw means in this case. If you specify
the BYLEVEL option, it disables the AT option.
OUT=SAS-data-set
creates an output data set that contains the values, standard errors, and, optionally, the covariances (see
the COV option) of the LS-means.
For more information, see the section “Output Data Sets” on page 3552.
PDIFF< =difftype >
requests that p-values for differences of the LS-means be produced. The optional difftype specifies
which differences to display. Possible values for difftype are ALL, CONTROL, CONTROLL, CONTROLU, and ANOM. The ALL value requests all pairwise differences, and it is the default. The
CONTROL value requests the differences with a control that, by default, is the first level of each of
the specified LS-mean effects. The ANOM value requests differences between each LS-mean and
the average LS-mean, as in the analysis of means (Ott 1967). The average is computed as a weighted
mean of the LS-means, the weights being inversely proportional to the variances. Note that the ANOM
procedure in SAS/QC software implements both tables and graphics for the analysis of means with a
variety of response types. For one-way designs, the PDIFF=ANOM computations are equivalent to
the results of PROC ANOM. See the section “Analysis of Means: Comparing Each Treatments to the
Average” on page 3525 for more details.
To specify which levels of the effects are the controls, list the quoted formatted values in parentheses
after the keyword CONTROL. For example, if the effects A, B, and C are CLASS variables, each
having two levels, ’1’ and ’2’, the following LSMEANS statement specifies the ’1’ ’2’ level of A*B
and the ’2’ ’1’ level of B*C as controls:
lsmeans A*B B*C / pdiff=control('1' '2', '2' '1');
For multiple-effect situations such as this one, the ordering of the list is significant, and you should
check the output to make sure that the controls are correct.
Two-tailed tests and confidence limits are associated with the CONTROL difftype. For one-tailed
results, use either the CONTROLL or CONTROLU difftype.
• CONTROLL tests whether the noncontrol levels are less than the control; you declare a noncontrol level to be significantly less than the control if the associated upper confidence limit for
the noncontrol level minus the control is less than zero, and you ignore the associated lower
confidence limits (which are set to minus infinity).
• CONTROLU tests whether the noncontrol levels are greater than the control; you declare a
noncontrol level to be significantly greater than the control if the associated lower confidence
limit for the noncontrol level minus the control is greater than zero, and you ignore the associated
upper confidence limits (which are set to infinity).
The default multiple comparisons adjustment for each difftype is shown in the following table.
LSMEANS Statement F 3467
difftype
Default ADJUST=
Not specified
T
ALL
TUKEY
CONTROL
CONTROLL
CONTROLU
DUNNETT
ANOM
NELSON
If no difftype is specified, the default for the ADJUST= option is T (that is, no adjustment); for
PDIFF=ALL, ADJUST=TUKEY is the default; for PDIFF=CONTROL, PDIFF=CONTROLL, or
PDIFF=CONTROLU, the default value for the ADJUST= option is DUNNETT. For PDIFF=ANOM,
ADJUST=NELSON is the default. If there is a conflict between the PDIFF= and ADJUST= options,
the ADJUST= option takes precedence.
For example, in order to compute one-sided confidence limits for differences with a control, adjusted
according to Dunnett’s procedure, the following statements are equivalent:
lsmeans Treatment / pdiff=controll cl;
lsmeans Treatment / pdiff=controll cl adjust=dunnett;
PLOT | PLOTS< =plot-request< (options) > >
PLOT | PLOTS< =(plot-request< (options) > < . . . plot-request< (options) > >) >
requests that graphics related to least squares means be produced via ODS Graphics, provided that
ODS Graphics is enabled and the plot-request does not conflict with other options in the LSMEANS
statement. For general information about ODS Graphics, see Chapter 21, “Statistical Graphics Using
ODS.”
The available options and suboptions are as follows:
ALL
requests that the default plots that correspond to this LSMEANS statement be produced. The
default plot depends on the options in the statement.
ANOMPLOT
ANOM
requests an analysis-of-means display in which least squares means are compared to an average
least squares mean. Least squares mean ANOM plots are produced only for those model effects
that are listed in LSMEANS statements and have options that do not contradict with the display.
For example, the following statements produce analysis-of-mean plots for effects A and C:
lsmeans A / diff=anom plot=anom;
lsmeans B / diff
plot=anom;
lsmeans C /
plot=anom;
The PDIFF option in the second LSMEANS statement implies all pairwise differences.
3468 F Chapter 44: The GLM Procedure
CONTROLPLOT
CONTROL
requests a display in which least squares means are visually compared against a reference level.
These plots are produced only for statements with options that are compatible with control
differences. For example, the following statements produce control plots for effects A and C:
lsmeans A / diff=control('1') plot=control;
lsmeans B / diff
plot=control;
lsmeans C
plot=control;
The PDIFF option in the second LSMEANS statement implies all pairwise differences.
DIFFPLOT< (diffplot-options) >
DIFFOGRAM< (diffplot-options) >
DIFF< (diffplot-options) >
requests a display of all pairwise least squares mean differences and their significance. The
display is also known as a “mean-mean scatter plot” when it is based on arithmetic means (Hsu
1996; Hsu and Peruggia 1994). For each comparison a line segment, centered at the LS-means in
the pair, is drawn. The length of the segment corresponds to the projected width of a confidence
interval for the least squares mean difference. Segments that fail to cross the 45-degree reference
line correspond to significant least squares mean differences.
LS-mean difference plots are produced only for statements with options that are compatible with
the display. For example, the following statements request differences against a control level for
the A effect, all pairwise differences for the B effect, and the least squares means for the C effect:
lsmeans A / diff=control('1') plot=diff;
lsmeans B / diff
plot=diff;
lsmeans C
plot=diff;
The PDIFF= type in the first statement is incompatible with a display of all pairwise differences.
You can specify the following diffplot-options:
ABS
determines the positioning of the line segments in the plot. This is the default diffplot-options.
When the ABS option is in effect, all line segments are shown on the same side of the
reference line.
NOABS
determines the positioning of the line segments in the plot. The NOABS option separates
comparisons according to the sign of the difference.
CENTER
marks the center point for each comparison. This point corresponds to the intersection of
two least squares means.
NOLINES
suppresses the display of the line segments that represent the confidence bounds for the
differences of the least squares means. The NOLINES option implies the CENTER option.
The default is to draw line segments in the upper portion of the plot area without marking
the center point.
LSMEANS Statement F 3469
MEANPLOT< (meanplot-options) >
requests displays of the least squares means.
The following meanplot-options control the display of the least squares means.
ASCENDING
displays the least squares means in ascending order. This option has no effect if means are
displayed in separate plots.
CL
displays upper and lower confidence limits for the least squares means. By default, 95%
limits are drawn. You can change the confidence level with the ALPHA= option. Confidence
limits are drawn by default if the CL option is specified in the LSMEANS statement.
CLBAND
displays confidence limits as bands. This option implies the JOIN option.
DESCENDING
displays the least squares means in descending order. This option has no effect if means are
displayed in separate plots.
ILINK
requests that means (and confidence limits) be displayed on the inverse linked scale.
JOIN
CONNECT
connects the least squares means with lines. This option is implied by the CLBAND option.
NONE
requests that no plots be produced.
When LS-mean calculations are adjusted for multiplicity by using the ADJUST= option, the plots are
adjusted accordingly.
SLICE=fixed-effect | (fixed-effects)
specifies effects within which to test for differences between interaction LS-mean effects. This can
produce what are known as tests of simple effects (Winer 1971). For example, suppose that A*B is
significant and you want to test for the effect of A within each level of B. The appropriate LSMEANS
statement is
lsmeans A*B / slice=B;
This statement tests for the simple main effects of A for B, which are calculated by extracting the
appropriate rows from the coefficient matrix for the A*B LS-means and using them to form an F test
as performed by the CONTRAST statement.
SINGULAR=number
tunes the estimability checking. If ABS.L LH/ > C number for any row, then L is declared
nonestimable. H is the .X0 X/ X0 X matrix, and C is ABS.L/ except for rows where L is zero, and then
it is 1. The default value for the SINGULAR= option is 10 4 . Values for the SINGULAR= option
must be between 0 and 1.
3470 F Chapter 44: The GLM Procedure
STDERR
produces the standard error of the LS-means and the probability level for the hypothesis
H0 W LS-mean D 0.
TDIFF
produces the t values for all hypotheses H0 W LS-mean.i / D LS-mean.j / and the corresponding
probabilities.
MANOVA Statement
MANOVA < test-options > < / detail-options > ;
If the MODEL statement includes more than one dependent variable, you can perform multivariate analysis of
variance with the MANOVA statement. The test-options define which effects to test, while the detail-options
specify how to execute the tests and what results to display. Table 44.7 summarizes the options available in
the MANOVA statement.
Table 44.7 MANOVA Statement Options
Option
Description
Test Options
H=
E=
M=
MNAMES=
PREFIX=
Specifies hypothesis effects
Specifies the error effect
Specifies a transformation matrix for the dependent variables
Provides names for the transformed variables
Alternatively identifies the transformed variables
Detail Options
CANONICAL
ETYPE=
HTYPE=
MSTAT=
ORTH
PRINTE
PRINTH
SUMMARY
Displays a canonical analysis of the H and E matrices
Specifies the type of the E matrix
Specifies the type of the H matrix
Specifies the method of evaluating the multivariate test statistics
Orthogonalizes the rows of the transformation matrix
Displays the error SSCP matrix E
Displays the hypothesis SSCP matrix H
Produces analysis-of-variance tables for each dependent variable
When a MANOVA statement appears before the first RUN statement, PROC GLM enters a multivariate
mode with respect to the handling of missing values; in addition to observations with missing independent
variables, observations with any missing dependent variables are excluded from the analysis. If you want to
use this mode of handling missing values and do not need any multivariate analyses, specify the MANOVA
option in the PROC GLM statement.
If you use both the CONTRAST and MANOVA statements, the MANOVA statement must appear after the
CONTRAST statement.
MANOVA Statement F 3471
Test Options
The following options can be specified in the MANOVA statement as test-options in order to define which
multivariate tests to perform.
H=effects | INTERCEPT | _ALL_
specifies effects in the preceding model to use as hypothesis matrices. For each H matrix (the SSCP
matrix associated with an effect), the H= specification displays the characteristic roots and vectors
of E 1 H (where E is the matrix associated with the error effect), along with the Hotelling-Lawley
trace, Pillai’s trace, Wilks’ lambda, and Roy’s greatest root. By default, these statistics are tested with
approximations based on the F distribution. To test them with exact (but computationally intensive)
calculations, use the MSTAT=EXACT option.
Use the keyword INTERCEPT to produce tests for the intercept. To produce tests for all effects listed
in the MODEL statement, use the keyword _ALL_ in place of a list of effects.
For background and further details, see the section “Multivariate Analysis of Variance” on page 3536.
E=effect
specifies the error effect. If you omit the E= specification, the GLM procedure uses the error SSCP
(residual) matrix from the analysis.
M=equation,. . . ,equation | (row-of-matrix,. . . ,row-of-matrix)
specifies a transformation matrix for the dependent variables listed in the MODEL statement. The
equations in the M= specification are of the form
c1 dependent-variable ˙ c2 dependent-variable
˙ cn dependent-variable
where the ci values are coefficients for the various dependent-variables. If the value of a given ci is
1, it can be omitted; in other words 1 Y is the same as Y. Equations should involve two or more
dependent variables. For sample syntax, see the section “Examples” on page 3473.
Alternatively, you can input the transformation matrix directly by entering the elements of the matrix
with commas separating the rows and parentheses surrounding the matrix. When this alternate form
of input is used, the number of elements in each row must equal the number of dependent variables.
Although these combinations actually represent the columns of the M matrix, they are displayed by
rows.
When you include an M= specification, the analysis requested in the MANOVA statement is carried out
for the variables defined by the equations in the specification, not the original dependent variables. If
you omit the M= option, the analysis is performed for the original dependent variables in the MODEL
statement.
If an M= specification is included without either the MNAMES= or PREFIX= option, the variables
are labeled MVAR1, MVAR2, and so forth, by default. For further information, see the section
“Multivariate Analysis of Variance” on page 3536.
3472 F Chapter 44: The GLM Procedure
MNAMES=names
provides names for the variables defined by the equations in the M= specification. Names in the list
correspond to the M= equations or to the rows of the M matrix (as it is entered).
PREFIX=name
is an alternative means of identifying the transformed variables defined by the M= specification. For
example, if you specify PREFIX=DIFF, the transformed variables are labeled DIFF1, DIFF2, and so
forth.
Detail Options
You can specify the following options in the MANOVA statement after a slash (/) as detail-options.
CANONICAL
displays a canonical analysis of the H and E matrices (transformed by the M matrix, if specified)
instead of the default display of characteristic roots and vectors.
ETYPE=n
specifies the type (1, 2, 3, or 4, corresponding to a Type I, II, III, or IV test, respectively) of the
E matrix, the SSCP matrix associated with the E= effect. You need this option if you use the E=
specification to specify an error effect other than residual error and you want to specify the type of
sums of squares used for the effect. If you specify ETYPE=n, the corresponding test must have been
performed in the MODEL statement, either by options SSn, En, or the default Type I and Type III tests.
By default, the procedure uses an ETYPE= value corresponding to the highest type (largest n) used in
the analysis.
HTYPE=n
specifies the type (1, 2, 3, or 4, corresponding to a Type I, II, III, or IV test, respectively) of the H
matrix. See the ETYPE= option for more details.
MSTAT=FAPPROX | EXACT
specifies the method of evaluating the multivariate test statistics. The default is MSTAT=FAPPROX,
which specifies that the multivariate tests are evaluated using the usual approximations based on
the F distribution, as discussed in the section “Multivariate Tests” in Chapter 4, “Introduction to
Regression Procedures.” Alternatively, you can specify MSTAT=EXACT to compute exact p-values
for three of the four tests (Wilks’ lambda, the Hotelling-Lawley trace, and Roy’s greatest root) and
an improved F approximation for the fourth (Pillai’s trace). While MSTAT=EXACT provides better
control of the significance probability for the tests, especially for Roy’s greatest root, computations for
the exact p-values can be appreciably more demanding, and are in fact infeasible for large problems
(many dependent variables). Thus, although MSTAT=EXACT is more accurate for most data, it is
not the default method. For more information about the results of MSTAT=EXACT, see the section
“Multivariate Analysis of Variance” on page 3536.
ORTH
requests that the transformation matrix in the M= specification of the MANOVA statement be orthonormalized by rows before the analysis.
MANOVA Statement F 3473
PRINTE
displays the error SSCP matrix E. If the E matrix is the error SSCP (residual) matrix from the analysis,
the partial correlations of the dependent variables given the independent variables are also produced.
For example, the statement
manova / printe;
displays the error SSCP matrix and the partial correlation matrix computed from the error SSCP matrix.
PRINTH
displays the hypothesis SSCP matrix H associated with each effect specified by the H= specification.
SUMMARY
produces analysis-of-variance tables for each dependent variable. When no M matrix is specified, a
table is displayed for each original dependent variable from the MODEL statement; with an M matrix
other than the identity, a table is displayed for each transformed variable defined by the M matrix.
Examples
The following statements provide several examples of using a MANOVA statement.
proc glm;
class A B;
model Y1-Y5=A B(A) / nouni;
manova h=A e=B(A) / printh printe htype=1 etype=1;
manova h=B(A) / printe;
manova h=A e=B(A) m=Y1-Y2,Y2-Y3,Y3-Y4,Y4-Y5
prefix=diff;
manova h=A e=B(A) m=(1 -1 0 0 0,
0 1 -1 0 0,
0 0 1 -1 0,
0 0 0 1 -1) prefix=diff;
run;
Since this MODEL statement requests no options for type of sums of squares, the procedure uses Type I
and Type III sums of squares. The first MANOVA statement specifies A as the hypothesis effect and B(A)
as the error effect. As a result of the PRINTH option, the procedure displays the hypothesis SSCP matrix
associated with the A effect; and, as a result of the PRINTE option, the procedure displays the error SSCP
matrix associated with the B(A) effect. The option HTYPE=1 specifies a Type I H matrix, and the option
ETYPE=1 specifies a Type I E matrix.
The second MANOVA statement specifies B(A) as the hypothesis effect. Since no error effect is specified,
PROC GLM uses the error SSCP matrix from the analysis as the E matrix. The PRINTE option displays
this E matrix. Since the E matrix is the error SSCP matrix from the analysis, the partial correlation matrix
computed from this matrix is also produced.
The third MANOVA statement requests the same analysis as the first MANOVA statement, but the analysis is
carried out for variables transformed to be successive differences between the original dependent variables.
The option PREFIX=DIFF labels the transformed variables as DIFF1, DIFF2, DIFF3, and DIFF4.
3474 F Chapter 44: The GLM Procedure
Finally, the fourth MANOVA statement has the identical effect as the third, but it uses an alternative form of
the M= specification. Instead of specifying a set of equations, the fourth MANOVA statement specifies rows
of a matrix of coefficients for the five dependent variables.
As a second example of the use of the M= specification, consider the following:
proc glm;
class group;
model dose1-dose4=group / nouni;
manova h = group
m = -3*dose1 dose2 +
dose3 + 3*dose4,
dose1 dose2 dose3 +
dose4,
-dose1 + 3*dose2 - 3*dose3 +
dose4
mnames = Linear Quadratic Cubic
/ printe;
run;
The M= specification gives a transformation of the dependent variables dose1 through dose4 into orthogonal
polynomial components, and the MNAMES= option labels the transformed variables LINEAR, QUADRATIC,
and CUBIC, respectively. Since the PRINTE option is specified and the default residual matrix is used as an
error term, the partial correlation matrix of the orthogonal polynomial components is also produced.
MEANS Statement
MEANS effects < / options > ;
Within each group corresponding to each effect specified in the MEANS statement, PROC GLM computes
the arithmetic means and standard deviations of all continuous variables in the model (both dependent and
independent). You can specify only classification effects in the MEANS statement—that is, effects that
contain only classification variables.
Note that the arithmetic means are not adjusted for other effects in the model; for adjusted means, see the
section “LSMEANS Statement” on page 3461.
If you use a WEIGHT statement, PROC GLM computes weighted means; see the section “Weighted Means”
on page 3533.
You can also specify options to perform multiple comparisons. However, the MEANS statement performs
multiple comparisons only for main-effect means; for multiple comparisons of interaction means, see the
section “LSMEANS Statement” on page 3461.
You can use any number of MEANS statements, provided that they appear after the MODEL statement. For
example, suppose A and B each have two levels. Then, if you use the statements
proc glm;
class A B;
model Y=A B A*B;
means A B / tukey;
means A*B;
run;
the means, standard deviations, and Tukey’s multiple comparisons tests are displayed for each level of
the main effects A and B, and just the means and standard deviations are displayed for each of the four
MEANS Statement F 3475
combinations of levels for A*B. Since multiple comparisons tests apply only to main effects, the single
MEANS statement
means A B A*B / tukey;
produces the same results.
PROC GLM does not compute means for interaction effects containing continuous variables. Thus, if you
have the model
class A;
model Y=A X A*X;
then the effects X and A*X cannot be used in the MEANS statement. However, if you specify the effect A in
the means statement
means A;
then PROC GLM, by default, displays within-A arithmetic means of both Y and X. You can use the DEPONLY
option to display means of only the dependent variables.
means A / deponly;
If you use a WEIGHT statement, PROC GLM computes weighted means and estimates their variance as
inversely proportional to the corresponding sum of weights (see the section “Weighted Means” on page 3533).
However, note that the statistical interpretation of multiple comparison tests for weighted means is not well
understood. See the section “Multiple Comparisons” on page 3519 for formulas. Table 44.8 summarizes the
options available in the MEANS statement.
Table 44.8 MEANS Statement Options
Option
Description
Modify output
DEPONLY
Displays only means for the dependent variables
Perform multiple comparison tests
BON
Performs Bonferroni t tests
DUNCAN
Performs Duncan’s multiple range test
DUNNETT
Performs Dunnett’s two-tailed t test
DUNNETTL Performs Dunnett’s lower one-tailed t test
DUNNETTU Performs Dunnett’s upper one-tailed t test
GABRIEL
Performs Gabriel’s multiple-comparison procedure
REGWQ
Performs the Ryan-Einot-Gabriel-Welsch multiple range test
SCHEFFE
Performs Scheffé’s multiple-comparison procedure
SIDAK
Performs pairwise t tests on differences between means
SMM or GT2 Performs pairwise comparisons based on the studentized maximum modulus
and Sidak’s uncorrelated-t inequality
SNK
Performs the Student-Newman-Keuls multiple range test
T or LSD
Performs pairwise t tests
TUKEY
Performs Tukey’s studentized range test (HSD)
WALLER
Performs the Waller-Duncan k-ratio t test
Specify additional details for multiple comparison tests
ALPHA=
Specifies the level of significance
3476 F Chapter 44: The GLM Procedure
Table 44.8 continued
Option
Description
CLDIFF
CLM
E=
ETYPE=
HTYPE=
KRATIO=
LINES
Presents confidence intervals for all pairwise differences between means
Presents results as intervals for the mean of each level of the variables
Specifies the error mean square used in the multiple comparisons
Specifies the type of mean square for the error effect
Specifies the MS type for the hypothesis MS
Specifies the Type 1/Type 2 error seriousness ratio
Lists the means in descending order and indicating nonsignificant subsets
by line segments
Prevents the means from being sorted into descending order
NOSORT
Test for homogeneity of variances
HOVTEST
Requests a homogeneity of variance test
Compensate for heterogeneous variances
WELCH
Requests the variance-weighted one-way ANOVA of Welch (1951)
The options available in the MEANS statement are described in the following list.
ALPHA=
ALPHA=p specifies the level of significance for comparisons among the means. By default, p is equal
to the value of the ALPHA= option in the PROC GLM statement or 0.05 if that option is not specified.
You can specify any value greater than 0 and less than 1.
BON
performs Bonferroni t tests of differences between means for all main-effect means in the MEANS
statement. See the CLDIFF and LINES options for a discussion of how the procedure displays results.
CLDIFF
presents results of the BON, GABRIEL, SCHEFFE, SIDAK, SMM, GT2, T, LSD, and TUKEY options
as confidence intervals for all pairwise differences between means, and the results of the DUNNETT,
DUNNETTU, and DUNNETTL options as confidence intervals for differences with the control. The
CLDIFF option is the default for unequal cell sizes unless the DUNCAN, REGWQ, SNK, or WALLER
option is specified.
CLM
presents results of the BON, GABRIEL, SCHEFFE, SIDAK, SMM, T, and LSD options as intervals
for the mean of each level of the variables specified in the MEANS statement. For all options except
GABRIEL, the intervals are confidence intervals for the true means. For the GABRIEL option, they
are comparison intervals for comparing means pairwise: in this case, if the intervals corresponding to
two means overlap, then the difference between them is insignificant according to Gabriel’s method.
DEPONLY
displays only means for the dependent variables. By default, PROC GLM produces means for all
continuous variables, including continuous independent variables.
MEANS Statement F 3477
DUNCAN
performs Duncan’s multiple range test on all main-effect means given in the MEANS statement. See
the LINES option for a discussion of how the procedure displays results.
DUNNETT < (formatted-control-values) >
performs Dunnett’s two-tailed t test, testing if any treatments are significantly different from a single
control for all main-effect means in the MEANS statement.
To specify which level of the effect is the control, enclose the formatted value in quotes and parentheses
after the keyword . If more than one effect is specified in the MEANS statement, you can use a list of
control values within the parentheses. By default, the first level of the effect is used as the control. For
example:
means A
/ dunnett('CONTROL');
where CONTROL is the formatted control value of A. As another example:
means A B C / dunnett('CNTLA' 'CNTLB' 'CNTLC');
where CNTLA, CNTLB, and CNTLC are the formatted control values for A, B, and C, respectively.
DUNNETTL < (formatted-control-value) >
performs Dunnett’s one-tailed t test, testing if any treatment is significantly less than the control.
Control level information is specified as described for the DUNNETT option.
DUNNETTU < (formatted-control-value) >
performs Dunnett’s one-tailed t test, testing if any treatment is significantly greater than the control.
Control level information is specified as described for the DUNNETT option.
E=effect
specifies the error mean square used in the multiple comparisons. By default, PROC GLM uses the
overall residual or error mean square (MS). The effect specified with the E= option must be a term in
the model; otherwise, the procedure uses the residual MS.
ETYPE=n
specifies the type of mean square for the error effect. When you specify E=effect , you might need to
indicate which type (1, 2, 3, or 4) of MS is to be used. The n value must be one of the types specified
in or implied by the MODEL statement. The default MS type is the highest type used in the analysis.
GABRIEL
performs Gabriel’s multiple-comparison procedure on all main-effect means in the MEANS statement.
See the CLDIFF and LINES options for discussions of how the procedure displays results.
GT2
See the SMM option.
3478 F Chapter 44: The GLM Procedure
HOVTEST
HOVTEST=BARTLETT
HOVTEST=BF
HOVTEST=LEVENE < ( TYPE= ABS | SQUARE ) >
HOVTEST=OBRIEN < ( W=number ) >
requests a homogeneity of variance test for the groups defined by the MEANS effect. You can
optionally specify a particular test; if you do not specify a test, Levene’s test (Levene 1960) with
TYPE=SQUARE is computed. Note that this option is ignored unless your MODEL statement specifies
a simple one-way model.
The HOVTEST=BARTLETT option specifies Bartlett’s test (Bartlett 1937), a modification of the
normal-theory likelihood ratio test.
The HOVTEST=BF option specifies Brown and Forsythe’s variation of Levene’s test (Brown and
Forsythe 1974).
The HOVTEST=LEVENE option specifies Levene’s test (Levene 1960), which is widely considered to
be the standard homogeneity of variance test. You can use the TYPE= option in parentheses to specify
whether to use the absolute residuals (TYPE=ABS) or the squared residuals (TYPE=SQUARE) in
Levene’s test. TYPE=SQUARE is the default.
The HOVTEST=OBRIEN option specifies O’Brien’s test (O’Brien 1979), which is basically a modification of HOVTEST=LEVENE(TYPE=SQUARE). You can use the W= option in parentheses to
tune the variable to match the suspected kurtosis of the underlying distribution. By default, W=0.5, as
suggested by O’Brien (1979, 1981).
See the section “Homogeneity of Variance in One-Way Models” on page 3532 for more details on these
methods. Example 44.10 illustrates the use of the HOVTEST and WELCH options in the MEANS
statement in testing for equal group variances and adjusting for unequal group variances in a one-way
ANOVA.
HTYPE=n
specifies the MS type for the hypothesis MS. The HTYPE= option is needed only when the WALLER
option is specified. The default HTYPE= value is the highest type used in the model.
KRATIO=value
specifies the Type 1/Type 2 error seriousness ratio for the Waller-Duncan test. Reasonable values for
the KRATIO= option are 50, 100, 500, which roughly correspond for the two-level case to ALPHA
levels of 0.1, 0.05, and 0.01, respectively. By default, the procedure uses the value of 100.
LINES
presents results of the BON, DUNCAN, GABRIEL, REGWQ, SCHEFFE, SIDAK, SMM, GT2, SNK,
T, LSD, TUKEY, and WALLER options by listing the means in descending order and indicating
nonsignificant subsets by line segments beside the corresponding means. The LINES option is
appropriate for equal cell sizes, for which it is the default. The LINES option is also the default if
the DUNCAN, REGWQ, SNK, or WALLER option is specified, or if there are only two cells of
unequal size. The LINES option cannot be used in combination with the DUNNETT, DUNNETTL, or
DUNNETTU option. In addition, the procedure has a restriction that no more than 24 overlapping
groups of means can exist. If a mean belongs to more than 24 groups, the procedure issues an error
message. You can either reduce the number of levels of the variable or use a multiple comparison test
that allows the CLDIFF option rather than the LINES option.
MEANS Statement F 3479
N OTE : If the cell sizes are unequal, the harmonic mean of the cell sizes is used to compute the critical
ranges. This approach is reasonable if the cell sizes are not too different, but it can lead to liberal tests
if the cell sizes are highly disparate. In this case, you should not use the LINES option for displaying
multiple comparisons results; use the TUKEY and CLDIFF options instead.
LSD
See the T option.
NOSORT
prevents the means from being sorted into descending order when the CLDIFF or CLM option is
specified.
REGWQ
performs the Ryan-Einot-Gabriel-Welsch multiple range test on all main-effect means in the MEANS
statement. See the LINES option for a discussion of how the procedure displays results.
SCHEFFE
erforms Scheffé’s multiple-comparison procedure on all main-effect means in the MEANS statement.
See the CLDIFF and LINES options for discussions of how the procedure displays results.
SIDAK
performs pairwise t tests on differences between means with levels adjusted according to Sidak’s
inequality for all main-effect means in the MEANS statement. See the CLDIFF and LINES options for
discussions of how the procedure displays results.
SMM
GT2
performs pairwise comparisons based on the studentized maximum modulus and Sidak’s uncorrelated-t
inequality, yielding Hochberg’s GT2 method when sample sizes are unequal, for all main-effect means
in the MEANS statement. See the CLDIFF and LINES options for discussions of how the procedure
displays results.
SNK
performs the Student-Newman-Keuls multiple range test on all main-effect means in the MEANS
statement. See the LINES option for discussions of how the procedure displays results.
T
LSD
performs pairwise t tests, equivalent to Fisher’s least significant difference test in the case of equal cell
sizes, for all main-effect means in the MEANS statement. See the CLDIFF and LINES options for
discussions of how the procedure displays results.
TUKEY
performs Tukey’s studentized range test (HSD) on all main-effect means in the MEANS statement.
(When the group sizes are different, this is the Tukey-Kramer test.) See the CLDIFF and LINES
options for discussions of how the procedure displays results.
WALLER
performs the Waller-Duncan k-ratio t test on all main-effect means in the MEANS statement. See the
KRATIO= and HTYPE= options for information about controlling details of the test, and the LINES
option for a discussion of how the procedure displays results.
3480 F Chapter 44: The GLM Procedure
WELCH
requests the variance-weighted one-way ANOVA of Welch (1951). This alternative to the usual analysis
of variance for a one-way model is robust to the assumption of equal within-group variances. This
option is ignored unless your MODEL statement specifies a simple one-way model.
Note that using the WELCH option merely produces one additional table consisting of Welch’s
ANOVA. It does not affect all of the other tests displayed by the GLM procedure, which still require
the assumption of equal variance for exact validity.
See the section “Homogeneity of Variance in One-Way Models” on page 3532 for more details on
Welch’s ANOVA. Example 44.10 illustrates the use of the HOVTEST and WELCH options in the
MEANS statement in testing for equal group variances and adjusting for unequal group variances in a
one-way ANOVA.
MODEL Statement
MODEL dependent-variables = independent-effects < / options > ;
The MODEL statement names the dependent variables and independent effects. The syntax of effects is
described in the section “Specification of Effects” on page 3495. For any model effect involving classification
variables (interactions as well as main effects), the number of levels cannot exceed 32,767. If no independent
effects are specified, only an intercept term is fit. You can specify only one MODEL statement (in contrast to
the REG procedure, for example, which allows several MODEL statements in the same PROC REG run).
Table 44.9 summarizes the options available in the MODEL statement.
Table 44.9 MODEL Statement Options
Option
Description
Produce effect size information
EFFECTSIZE Adds measures of effect size to each analysis of variance table
Produce tests for the intercept
INTERCEPT
Produces the hypothesis tests associated with the intercept
Omit the intercept parameter from model
NOINT
Omits the intercept parameter from the model
Produce parameter estimates
SOLUTION
Produces parameter estimates
Produce tolerance analysis
TOLERANCE Displays the tolerances used in the SWEEP routine
Suppress univariate tests and output
NOUNI
Suppresses the display of univariate statistics
Display estimable functions
E
Displays the general form of all estimable functions
E1
Displays the Type I estimable functions and computes sums of squares
E2
Displays the Type II estimable functions and computes sums of squares
E3
Displays the Type III estimable functions and computes sums of squares
E4
Displays the Type IV estimable functions computes sums of squares
MODEL Statement F 3481
Table 44.9 continued
Option
Description
ALIASING
Specifies that the estimable functions should be displayed as an aliasing
structure
Control hypothesis tests performed
SS1
Displays Type I sum of squares
SS2
Displays Type II sum of squares
SS3
Displays Type III sum of squares
SS4
Displays Type IV sum of squares
Produce confidence intervals
ALPHA=
Specifies the level of significance
CLI
Produces confidence limits for individual predicted values
CLM
Produces confidence limits for a mean predicted value
CLPARM
Produces confidence limits for the parameter estimates
Display predicted and residual values
P
Displays observed, predicted, and residual values
Display intermediate calculations
INVERSE
Displays the augmented inverse (or generalized inverse) X0 X matrix
XPX
Displays the augmented X0 X crossproducts matrix
Tune sensitivity
SINGULAR= Tunes the sensitivity of the regression routine to linear dependencies
Tunes the sensitivity of the check for estimability for Type III and Type IV
ZETA=
functions
The options available in the MODEL statement are described in the following list.
ALIASING
specifies that the estimable functions should be displayed as an aliasing structure, for which each row
says which linear combination of the parameters is estimated by each estimable function; also, this
option adds a column of the same information to the table of parameter estimates, giving for each
parameter the expected value of the estimate associated with that parameter. This option is most useful
in fractional factorial experiments that can be analyzed without a CLASS statement.
ALPHA=p
specifies the level of significance p for 100.1 p/% confidence intervals. By default, p is equal to the
value of the ALPHA= option in the PROC GLM statement, or 0.05 if that option is not specified. You
can use values between 0 and 1.
CLI
produces confidence limits for individual predicted values for each observation. The CLI option is
ignored if the CLM option is also specified.
CLM
produces confidence limits for a mean predicted value for each observation.
3482 F Chapter 44: The GLM Procedure
CLPARM
produces confidence limits for the parameter estimates (if the SOLUTION option is also specified) and
for the results of all ESTIMATE statements.
E
displays the general form of all estimable functions. This is useful for determining the order of
parameters when you are writing CONTRAST and ESTIMATE statements.
E1
displays the Type I estimable functions for each effect in the model and computes the corresponding
sums of squares.
E2
displays the Type II estimable functions for each effect in the model and computes the corresponding
sums of squares.
E3
displays the Type III estimable functions for each effect in the model and computes the corresponding
sums of squares.
E4
displays the Type IV estimable functions for each effect in the model and computes the corresponding
sums of squares.
EFFECTSIZE
adds measures of effect size to each analysis of variance table displayed by the procedure, except for
those displayed by the TEST option in the RANDOM statement, by CONTRAST and TEST statements
with the E= option, and by MANOVA and REPEATED statements. The effect size measures include
the intraclass correlation and both estimates and confidence intervals for the noncentrality for the
F test, as well as for the semipartial and partial correlation ratios. For more information about the
computation and interpretation of these measures, see the section “Effect Size Measures for F Tests in
GLM” on page 3508.
INTERCEPT
INT
produces the hypothesis tests associated with the intercept as an effect in the model. By default, the
procedure includes the intercept in the model but does not display associated tests of hypotheses.
Except for producing the uncorrected total sum of squares instead of the corrected total sum of squares,
the INT option is ignored when you use an ABSORB statement.
INVERSE
I
displays the augmented inverse (or generalized inverse) X0 X matrix:
.X0 X/
.X0 X/ X0 y
y0 X.X0 X/
y0 y y0 X.X0 X/ X0 y
The upper-left corner is the generalized inverse of X0 X, the upper-right corner is the parameter
estimates, and the lower-right corner is the error sum of squares.
MODEL Statement F 3483
NOINT
omits the intercept parameter from the model. The NOINT option is ignored when you use an ABSORB
statement.
NOUNI
suppresses the display of univariate statistics. You typically use the NOUNI option with a multivariate
or repeated measures analysis of variance when you do not need the standard univariate results.
The NOUNI option in a MODEL statement does not affect the univariate output produced by the
REPEATED statement.
P
displays observed, predicted, and residual values for each observation that does not contain missing
values for independent variables. The Durbin-Watson statistic is also displayed when the P option is
specified. The PRESS statistic is also produced if either the CLM or CLI option is specified.
SINGULAR=number
tunes the sensitivity of the regression routine to linear dependencies in the design. If a diagonal pivot
element is less than C number as PROC GLM sweeps the X0 X matrix, the associated design column
is declared to be linearly dependent with previous columns, and the associated parameter is zeroed.
The C value adjusts the check to the relative scale of the variable. The C value is equal to the corrected
sum of squares for the variable, unless the corrected sum of squares is 0, in which case C is 1. If you
specify the NOINT option but not the ABSORB statement, PROC GLM uses the uncorrected sum of
squares instead.
The default value of the SINGULAR= option, 10 7 , might be too small, but this value is necessary in
order to handle the high-degree polynomials used in the literature to compare regression routines.
SOLUTION
produces a solution to the normal equations (parameter estimates). PROC GLM displays a solution by
default when your model involves no classification variables, so you need this option only if you want
to see the solution for models with classification effects.
SS1
displays the sum of squares associated with Type I estimable functions for each effect. These are also
displayed by default.
SS2
displays the sum of squares associated with Type II estimable functions for each effect.
SS3
displays the sum of squares associated with Type III estimable functions for each effect. These are also
displayed by default.
SS4
displays the sum of squares associated with Type IV estimable functions for each effect.
TOLERANCE
displays the tolerances used in the SWEEP routine. The tolerances are of the form C/USS or C/CSS,
as described in the discussion of the SINGULAR= option. The tolerance value for the intercept is not
divided by its uncorrected sum of squares.
3484 F Chapter 44: The GLM Procedure
XPX
displays the augmented X0 X crossproducts matrix:
0
X X X0 y
y0 X y0 y
ZETA=value
tunes the sensitivity of the check for estimability for Type III and Type IV functions. Any element
in the estimable function basis with an absolute value less than the ZETA= option is set to zero. The
default value for the ZETA= option is 10 8 .
Although it is possible to generate data for which this absolute check can be defeated, the check suffices
in most practical examples. Additional research is needed in order to make this check relative rather
than absolute.
OUTPUT Statement
OUTPUT < OUT=SAS-data-set > keyword=names < . . . keyword=names > < option > ;
The OUTPUT statement creates a new SAS data set that saves diagnostic measures calculated after fitting the
model. At least one specification of the form keyword=names is required.
All the variables in the original data set are included in the new data set, along with variables created in the
OUTPUT statement. These new variables contain the values of a variety of diagnostic measures that are
calculated for each observation in the data set. If you want to create a SAS data set in a permanent library,
you must specify a two-level name. For more information about permanent libraries and SAS data sets, see
SAS Language Reference: Concepts.
Details on the specifications in the OUTPUT statement follow.
keyword=names
specifies the statistics to include in the output data set and provides names to the new variables that
contain the statistics. Specify a keyword for each desired statistic (see the following list of keywords),
an equal sign, and the variable or variables to contain the statistic.
In the output data set, the first variable listed after a keyword in the OUTPUT statement contains that
statistic for the first dependent variable listed in the MODEL statement; the second variable contains
the statistic for the second dependent variable in the MODEL statement, and so on. The list of variables
following the equal sign can be shorter than the list of dependent variables in the MODEL statement.
In this case, the procedure creates the new names in order of the dependent variables in the MODEL
statement. See the section “Examples” on page 3486.
The keywords allowed and the statistics they represent are as follows:
COOKD
Cook’s D influence statistic
OUTPUT Statement F 3485
COVRATIO
standard influence of observation on covariance of parameter estimates
DFFITS
standard influence of observation on predicted value
H
leverage, hi D xi .X0 X/
1x0
i
LCL
lower bound of a 100(1 - p)% confidence interval for an individual prediction. The p-level is
equal to the value of the ALPHA= option in the OUTPUT statement or, if this option is not
specified, to the ALPHA= option in the PROC GLM statement. If neither of these options is set,
then p = 0.05 by default, resulting in the lower bound for a 95% confidence interval. The interval
also depends on the variance of the error, as well as the variance of the parameter estimates. For
the corresponding upper bound, see the UCL keyword.
LCLM
lower bound of a 100(1 - p)% confidence interval for the expected value (mean) of the predicted
value. The p-level is equal to the value of the ALPHA= option in the OUTPUT statement or, if
this option is not specified, to the ALPHA= option in the PROC GLM statement. If neither of
these options is set, then p = 0.05 by default, resulting in the lower bound for a 95% confidence
interval. For the corresponding upper bound, see the UCLM keyword.
PREDICTED | P
predicted values
PRESS
residual for the ith observation that results from dropping it and predicting it on the basis of all
other observations. This is the residual divided by .1 hi /, where hi is the leverage, defined
previously.
RESIDUAL | R
residuals, calculated as ACTUAL – PREDICTED
RSTUDENT
a studentized residual with the current observation deleted
STDI
standard error of the individual predicted value
STDP
standard error of the mean predicted value
STDR
standard error of the residual
STUDENT
studentized residuals, the residual divided by its standard error
3486 F Chapter 44: The GLM Procedure
UCL
upper bound of a 100(1 - p)% confidence interval for an individual prediction. The p-level is
equal to the value of the ALPHA= option in the OUTPUT statement or, if this option is not
specified, to the ALPHA= option in the PROC GLM statement. If neither of these options is set,
then p = 0.05 by default, resulting in the upper bound for a 95% confidence interval. The interval
also depends on the variance of the error, as well as the variance of the parameter estimates. For
the corresponding lower bound, see the LCL keyword.
UCLM
upper bound of a 100(1 - p)% confidence interval for the expected value (mean) of the predicted
value. The p-level is equal to the value of the ALPHA= option in the OUTPUT statement or, if
this option is not specified, to the ALPHA= option in the PROC GLM statement. If neither of
these options is set, then p = 0.05 by default, resulting in the upper bound for a 95% confidence
interval. For the corresponding lower bound, see the LCLM keyword.
OUT=SAS-data-set
gives the name of the new data set. By default, the procedure uses the DATAn convention to name the
new data set.
The following option is available in the OUTPUT statement and is specified after a slash (/):
ALPHA=p
specifies the level of significance p for 100.1 p/% confidence intervals. By default, p is equal to the
value of the ALPHA= option in the PROC GLM statement or 0.05 if that option is not specified. You
can use values between 0 and 1.
See Chapter 4, “Introduction to Regression Procedures,” and the section “Influence Statistics” on
page 7108 in Chapter 83, “The REG Procedure,” for details on the calculation of these statistics.
Examples
The following statements show the syntax for creating an output data set with a single dependent variable.
proc glm;
class a b;
model y=a b a*b;
output out=new p=yhat r=resid stdr=eresid;
run;
These statements create an output data set named new. In addition to all the variables from the original data
set, new contains the variable yhat, with values that are predicted values of the dependent variable y; the
variable resid, with values that are the residual values of y; and the variable eresid, with values that are the
standard errors of the residuals.
The following statements show a situation with five dependent variables.
proc glm;
by group;
class a;
model y1-y5=a x(a);
output out=pout predicted=py1-py5;
run;
RANDOM Statement F 3487
The data set pout contains five new variables, py1 through py5. The values of py1 are the predicted values of
y1; the values of py2 are the predicted values of y2; and so on.
For more information about the data set produced by the OUTPUT statement, see the section “Output Data
Sets” on page 3552.
RANDOM Statement
RANDOM effects < / options > ;
When some model effects are random (that is, assumed to be sampled from a normal population of effects),
you can specify these effects in the RANDOM statement in order to compute the expected values of mean
squares for various model effects and contrasts and, optionally, to perform random effects analysis of variance
tests. You can use as many RANDOM statements as you want, provided that they appear after the MODEL
statement. If you use a CONTRAST statement with a RANDOM statement and you want to obtain the
expected mean squares for the contrast hypothesis, you must enter the CONTRAST statement before the
RANDOM statement.
N OTE : PROC GLM uses only the information pertaining to expected mean squares when you specify the
TEST option in the RANDOM statement and, even then, only in the extra F tests produced by the RANDOM
statement. Other features in the GLM procedure—including the results of the LSMEANS and ESTIMATE
statements—assume that all effects are fixed, so that all tests and estimability checks for these statements are
based on a fixed-effects model, even when you use a RANDOM statement. Therefore, you should use the
MIXED procedure to compute tests involving these features that take the random effects into account; see the
section “PROC GLM versus PROC MIXED for Random-Effects Analysis” on page 3545 and Chapter 63,
“The MIXED Procedure,” for more information.
When you use the RANDOM statement, by default the GLM procedure produces the Type III expected
mean squares for model effects and for contrasts specified before the RANDOM statement in the program
statements. In order to obtain expected values for other types of mean squares, you need to specify which
types of mean squares are of interest in the MODEL statement. See the section “Computing Type I, II, and
IV Expected Mean Squares” on page 3548 for more information.
The list of effects in the RANDOM statement should contain one or more of the pure classification effects
specified in the MODEL statement (that is, main effects, crossed effects, or nested effects involving only classification variables). The coefficients corresponding to each effect specified are assumed to be normally and
independently distributed with common variance. Levels in different effects are assumed to be independent.
You can specify the following options in the RANDOM statement after a slash (/):
Q
displays all quadratic forms in the fixed effects that appear in the expected mean squares. For some
designs, such as large mixed-level factorials, the Q option might generate a substantial amount of
output.
TEST
performs hypothesis tests for each effect specified in the model, using appropriate error terms as
determined by the expected mean squares.
C AUTION : PROC GLM does not automatically declare interactions to be random when the effects in
the interaction are declared random. For example,
3488 F Chapter 44: The GLM Procedure
random a b / test;
does not produce the same expected mean squares or tests as
random a b a*b / test;
To ensure correct tests, you need to list all random interactions and random main effects in the
RANDOM statement.
See the section “Random-Effects Analysis” on page 3545 for more information about the calculation of
expected mean squares and tests and on the similarities and differences between the GLM and MIXED
procedures. See Chapter 5, “Introduction to Analysis of Variance Procedures,” and Chapter 63, “The
MIXED Procedure,” for more information about random effects.
REPEATED Statement
REPEATED factor-specification < / options > ;
When values of the dependent variables in the MODEL statement represent repeated measurements on the
same experimental unit, the REPEATED statement enables you to test hypotheses about the measurement
factors (often called within-subject factors) as well as the interactions of within-subject factors with independent variables in the MODEL statement (often called between-subject factors). The REPEATED statement
provides multivariate and univariate tests as well as hypothesis tests for a variety of single-degree-of-freedom
contrasts. There is no limit to the number of within-subject factors that can be specified.
The REPEATED statement is typically used for handling repeated measures designs with one repeated
response variable. Usually, the variables on the left-hand side of the equation in the MODEL statement
represent one repeated response variable. This does not mean that only one factor can be listed in the
REPEATED statement. For example, one repeated response variable (hemoglobin count) might be measured
12 times (implying variables Y1 to Y12 on the left-hand side of the equal sign in the MODEL statement),
with the associated within-subject factors treatment and time (implying two factors listed in the REPEATED
statement). See the section “Examples” on page 3492 for an example of how PROC GLM handles this case.
Designs with two or more repeated response variables can, however, be handled with the IDENTITY
transformation; see the description of this transformation in the following section, and see Example 44.9 for
an example of analyzing a doubly multivariate repeated measures design.
When a REPEATED statement appears, the GLM procedure enters a multivariate mode of handling missing
values. If any values for variables corresponding to each combination of the within-subject factors are
missing, the observation is excluded from the analysis.
If you use a CONTRAST or TEST statement with a REPEATED statement, you must enter the CONTRAST
or TEST statement before the REPEATED statement.
The simplest form of the REPEATED statement requires only a factor-name. With two repeated factors, you
must specify the factor-name and number of levels (levels) for each factor. Optionally, you can specify the
actual values for the levels (level-values), a transformation that defines single-degree-of-freedom contrasts,
and options for additional analyses and output. When you specify more than one within-subject factor, the
REPEATED Statement F 3489
factor-names (and associated level and transformation information) must be separated by a comma in the
REPEATED statement.
These terms are described in the following section, “Syntax Details.”
Syntax Details
Table 44.10 summarizes the options available in the REPEATED statement.
Table 44.10
REPEATED Statement Options
Option
Description
CANONICAL
HTYPE=
MEAN
MSTAT=
NOM
NOU
PRINTE
Performs a canonical analysis of the H and E matrices
Specifies the type of the H matrix used for analysis
Generates the overall arithmetic means of the within-subject variables
Specifies the method of evaluating the test statistics
Displays only univariate analyses results
Displays only multivariate analyses results
Displays E and partial correlation matrices for multivariate tests, and prints
sphericity tests
Displays the H matrices for multivariate tests
Displays the transformation matrices that define tested contrasts
Displays characteristic roots and vectors for multivariate tests
Produces analysis-of-variance tables for univariate tests
Specifies the F test adjustment for univariate tests
PRINTH
PRINTM
PRINTRV
SUMMARY
UEPSDEF=
You can specify the following terms in the REPEATED statement.
factor-specification
The factor-specification for the REPEATED statement can include any number of individual factor
specifications, separated by commas, of the following form:
factor-name levels < (level-values) > < transformation >
where
factor-name
names a factor to be associated with the dependent variables. The name should not
be the same as any variable name that already exists in the data set being analyzed
and should conform to the usual conventions of SAS variable names.
When specifying more than one factor, list the dependent variables in the MODEL
statement so that the within-subject factors defined in the REPEATED statement
are nested; that is, the first factor defined in the REPEATED statement should be
the one with values that change least frequently.
levels
gives the number of levels associated with the factor being defined. When there
is only one within-subject factor, the number of levels is equal to the number of
dependent variables. In this case, levels is optional. When more than one withinsubject factor is defined, however, levels is required, and the product of the number
3490 F Chapter 44: The GLM Procedure
of levels of all the factors must equal the number of dependent variables in the
MODEL statement.
(level-values)
gives values that correspond to levels of a repeated-measures factor. These values
are used to label output and as spacings for constructing orthogonal polynomial
contrasts if you specify a POLYNOMIAL transformation. The number of values
specified must correspond to the number of levels for that factor in the REPEATED
statement. Enclose the level-values in parentheses.
The following transformation keywords define single-degree-of-freedom contrasts for factors specified
in the REPEATED statement. Since the number of contrasts generated is always one less than the
number of levels of the factor, you have some control over which contrast is omitted from the analysis
by which transformation you select. The only exception is the IDENTITY transformation; this
transformation is not composed of contrasts and has the same degrees of freedom as the factor has
levels. By default, the procedure uses the CONTRAST transformation.
CONTRAST< (ordinal-reference-level) >
generates contrasts between levels of the factor and a reference level. By default, the procedure
uses the last level as the reference level; you can optionally specify a reference level in parentheses
after the keyword CONTRAST. The reference level corresponds to the ordinal value of the level
rather than the level value specified. For example, to generate contrasts between the first level of
a factor and the other levels, use
contrast(1)
HELMERT
generates contrasts between each level of the factor and the mean of subsequent levels.
IDENTITY
generates an identity transformation corresponding to the associated factor. This transformation
is not composed of contrasts; it has n degrees of freedom for an n-level factor, instead of n – 1.
This can be used for doubly multivariate repeated measures.
MEAN< (ordinal-reference-level) >
generates contrasts between levels of the factor and the mean of all other levels of the factor.
Specifying a reference level eliminates the contrast between that level and the mean. Without a reference level, the contrast involving the last level is omitted. See the CONTRAST transformation
for an example.
POLYNOMIAL
generates orthogonal polynomial contrasts. Level values, if provided, are used as spacings in the
construction of the polynomials; otherwise, equal spacing is assumed.
PROFILE
generates contrasts between adjacent levels of the factor.
You can specify the following options in the REPEATED statement after a slash (/).
REPEATED Statement F 3491
CANONICAL
performs a canonical analysis of the H and E matrices corresponding to the transformed variables
specified in the REPEATED statement.
HTYPE=n
specifies the type of the H matrix used in the multivariate tests and the type of sums of squares used
in the univariate tests. See the HTYPE= option in the specifications for the MANOVA statement for
further details.
MEAN
generates the overall arithmetic means of the within-subject variables.
MSTAT=FAPPROX | EXACT
specifies the method of evaluating the test statistics for the multivariate analysis. The default is
MSTAT=FAPPROX, which specifies that the multivariate tests are evaluated using the usual approximations based on the F distribution, as discussed in the section “Multivariate Tests” in Chapter 4,
“Introduction to Regression Procedures.” Alternatively, you can specify MSTAT=EXACT to compute
exact p-values for three of the four tests (Wilks’ lambda, the Hotelling-Lawley trace, and Roy’s
greatest root) and an improved F approximation for the fourth (Pillai’s trace). While MSTAT=EXACT
provides better control of the significance probability for the tests, especially for Roy’s greatest root,
computations for the exact p-values can be appreciably more demanding, and are in fact infeasible for
large problems (many dependent variables). Thus, although MSTAT=EXACT is more accurate for
most data, it is not the default method. For more information about the results of MSTAT=EXACT, see
the section “Multivariate Analysis of Variance” on page 3536.
NOM
displays only the results of the univariate analyses.
NOU
displays only the results of the multivariate analyses.
PRINTE
displays the E matrix for each combination of within-subject factors, as well as partial correlation
matrices for both the original dependent variables and the variables defined by the transformations
specified in the REPEATED statement. In addition, the PRINTE option provides sphericity tests for
each set of transformed variables. If the requested transformations are not orthogonal, the PRINTE
option also provides a sphericity test for a set of orthogonal contrasts.
PRINTH
displays the H (SSCP) matrix associated with each multivariate test.
PRINTM
displays the transformation matrices that define the contrasts in the analysis. PROC GLM always
displays the M matrix so that the transformed variables are defined by the rows, not the columns, of
the displayed M matrix. In other words, PROC GLM actually displays M0 .
PRINTRV
displays the characteristic roots and vectors for each multivariate test.
3492 F Chapter 44: The GLM Procedure
SUMMARY
produces analysis-of-variance tables for each contrast defined by the within-subject factors. Along
with tests for the effects of the independent variables specified in the MODEL statement, a term labeled
MEAN tests the hypothesis that the overall mean of the contrast is zero.
UEPSDEF=unbiased-epsilon-definition
specifies the type of adjustment for the univariate F test that is displayed in addition to the GreenhouseGeisser adjustment. The default is UEPSDEF=HFL, corresponding to the corrected form of the HuynhFeldt adjustment (Huynh and Feldt 1976; Lecoutre 1991). Other alternatives are UEPSDEF=HF, the
uncorrected Huynh-Feldt adjustment (the only available method in previous releases of SAS/STAT
software), and UEPSDEF=CM, the adjustment of Chi et al. (2012). See the section “Hypothesis
Testing in Repeated Measures Analysis” on page 3540 for details about these adjustments.
Examples
When specifying more than one factor, list the dependent variables in the MODEL statement so that the
within-subject factors defined in the REPEATED statement are nested; that is, the first factor defined in the
REPEATED statement should be the one with values that change least frequently. For example, assume
that three treatments are administered at each of four times, for a total of twelve dependent variables on
each experimental unit. If the variables are listed in the MODEL statement as Y1 through Y12, then the
REPEATED statement in
proc glm;
class group;
model Y1-Y12=group / nouni;
repeated trt 3, time 4;
run;
implies the following structure:
Dependent Variables
Y1 Y2 Y3 Y4 Y5 Y6 Y7 Y8 Y9 Y10 Y11 Y12
Value of trt
1
1
1
1
2
2
2
2
3
3
3
3
Value of time 1
2
3
4
1
2
3
4
1
2
3
4
The REPEATED statement always produces a table like the preceding one. For more information, see the
section “Repeated Measures Analysis of Variance” on page 3537.
STORE Statement
STORE < OUT= >item-store-name < / LABEL='label' > ;
The STORE statement requests that the procedure save the context and results of the statistical analysis. The
resulting item store has a binary file format that cannot be modified. The contents of the item store can be
processed with the PLM procedure.
For details about the syntax of the STORE statement, see the section “STORE Statement” on page 508 in
Chapter 19, “Shared Concepts and Topics.”
TEST Statement F 3493
TEST Statement
TEST < H=effects > E=effect < / options > ;
By default, for each sum of squares in the analysis an F value is computed that uses the residual MS as an
error term. Use a TEST statement to request additional F tests that use other effects as error terms. You need
a TEST statement when a nonstandard error structure (as in a split-plot design) exists. Note, however, that this
might not be appropriate if the design is unbalanced, since in most unbalanced designs with nonstandard error
structures, mean squares are not necessarily independent with equal expectations under the null hypothesis.
C AUTION : The GLM procedure does not check any of the assumptions underlying the F statistic. When
you specify a TEST statement, you assume sole responsibility for the validity of the F statistic produced. To
help validate a test, you can use the RANDOM statement and inspect the expected mean squares, or you can
use the TEST option of the RANDOM statement.
You can use as many TEST statements as you want, provided that they appear after the MODEL statement.
You can specify the following terms in the TEST statement.
H=effects
specifies which effects in the preceding model are to be used as hypothesis (numerator) effects.
E=effect
specifies one, and only one, effect to use as the error (denominator) term. The E= specification is
required.
By default, the sum of squares type for all hypothesis sum of squares and error sum of squares is the highest
type computed in the model. If the hypothesis type or error type is to be another type that was computed in
the model, you should specify one or both of the following options after a slash (/).
ETYPE=n
specifies the type of sum of squares to use for the error term. The type must be a type computed in the
model (n=1, 2, 3, or 4 ).
HTYPE=n
specifies the type of sum of squares to use for the hypothesis. The type must be a type computed in the
model (n=1, 2, 3, or 4).
This example illustrates the TEST statement with a split-plot model:
proc glm;
class a b c;
model y=a b(a) c a*c b*c(a);
test h=a e=b(a)/ htype=1 etype=1;
test h=c a*c e=b*c(a) / htype=1 etype=1;
run;
3494 F Chapter 44: The GLM Procedure
WEIGHT Statement
WEIGHT variable ;
When a WEIGHT statement is used, a weighted residual sum of squares
X
wi .yi yOi /2
i
is minimized, where wi is the value of the variable specified in the WEIGHT statement, yi is the observed
value of the response variable, and yOi is the predicted value of the response variable.
If you specify the WEIGHT statement, it must appear before the first RUN statement or it is ignored.
An observation is used in the analysis only if the value of the WEIGHT statement variable is nonmissing and
greater than zero.
The WEIGHT statement has no effect on degrees of freedom or number of observations, but it is used by the
MEANS statement when calculating means and performing multiple comparison tests (as described in the
section “MEANS Statement” on page 3474).
The normal equations used when a WEIGHT statement is present are
X0 WXˇ D X0 WY
where W is a diagonal matrix consisting of the values of the variable specified in the WEIGHT statement.
If the weights for the observations are proportional to the reciprocals of the error variances, then the weighted
least squares estimates are best linear unbiased estimators (BLUE).
Details: GLM Procedure
Statistical Assumptions for Using PROC GLM
The basic statistical assumption underlying the least squares approach to general linear modeling is that the
observed values of each dependent variable can be written as the sum of two parts: a fixed component x 0 ˇ,
which is a linear function of the independent coefficients, and a random noise, or error, component :
y D x0ˇ C The independent coefficients x are constructed from the model effects as described in the section “Parameterization of PROC GLM Models” on page 3498. Further, the errors for different observations are assumed to be
uncorrelated with identical variances. Thus, this model can be written
E.Y / D Xˇ;
Var.Y / D 2 I
where Y is the vector of dependent variable values, X is the matrix of independent coefficients, I is the
identity matrix, and 2 is the common variance for the errors. For multiple dependent variables, the model is
Specification of Effects F 3495
similar except that the errors for different dependent variables within the same observation are not assumed
to be uncorrelated. This yields a multivariate linear model of the form
E.Y / D XB;
Var.vec.Y // D † ˝ I
where Y and B are now matrices, with one column for each dependent variable, vec.Y / strings Y out by rows,
and ˝ indicates the Kronecker matrix product.
Under the assumptions thus far discussed, the least squares approach provides estimates of the linear
parameters that are unbiased and have minimum variance among linear estimators. Under the further
assumption that the errors have a normal (or Gaussian) distribution, the least squares estimates are the
maximum likelihood estimates and their distribution is known. All of the significance levels (“p values”)
and confidence limits calculated by the GLM procedure require this assumption of normality in order to be
exactly valid, although they are good approximations in many other cases.
Specification of Effects
Each term in a model, called an effect, is a variable or combination of variables. Effects are specified with a
special notation that uses variable names and operators. There are two kinds of variables: classification (or
CLASS) variables and continuous variables. There are two primary operators: crossing and nesting. A third
operator, the bar operator, is used to simplify effect specification.
In an analysis-of-variance model, independent variables must be variables that identify classification levels. In
the SAS System, these are called classification (or class) variables and are declared in the CLASS statement.
(They can also be called categorical, qualitative, discrete, or nominal variables.) Classification variables can
be either numeric or character. The values of a classification variable are called levels. For example, the
classification variable Sex has the levels “male” and “female.”
In a model, an independent variable that is not declared in the CLASS statement is assumed to be continuous.
Continuous variables, which must be numeric, are used for response variables and covariates. For example,
the heights and weights of subjects are continuous variables.
Types of Effects
There are seven different types of effects used in the GLM procedure. In the following list, assume that A, B,
C, D, and E are CLASS variables and that X1, X2, and Y are continuous variables:
• Regressor effects are specified by writing continuous variables by themselves: X1
X2.
• Polynomial effects are specified by joining two or more continuous variables with asterisks: X1*X1
X1*X2.
• Main effects are specified by writing CLASS variables by themselves: A
B
C.
• Crossed effects (interactions) are specified by joining classification variables with asterisks: A*B B*C
A*B*C.
• Nested effects are specified by following a main effect or crossed effect with a classification variable
or list of classification variables enclosed in parentheses. The main effect or crossed effect is nested
within the effects listed in parentheses: B(A) C(B*A) D*E(C*B*A). In this example, B(A) is read
“B nested within A.”
3496 F Chapter 44: The GLM Procedure
• Continuous-by-class effects are written by joining continuous variables and classification variables
with asterisks: X1*A.
• Continuous-nesting-class effects consist of continuous variables followed by a classification variable
interaction enclosed in parentheses: X1(A) X1*X2(A*B).
One example of the general form of an effect involving several variables is
X1*X2*A*B*C(D*E)
This example contains crossed continuous terms by crossed classification terms nested within more than
one classification variable. The continuous list comes first, followed by the crossed list, followed by the
nesting list in parentheses. Note that asterisks can appear within the nested list but not immediately before
the left parenthesis. For details on how the design matrix and parameters are defined with respect to the
effects specified in this section, see the section “Parameterization of PROC GLM Models” on page 3498.
The MODEL statement and several other statements use these effects. Some examples of MODEL statements
that use various kinds of effects are shown in the following table; a, b, and c represent classification variables,
and y, y1, y2, x, and z represent continuous variables.
Specification
model y=x;
Type of Model
Simple regression
model y=x z;
Multiple regression
model y=x x*x;
Polynomial regression
model y1 y2=x z;
Multivariate regression
model y=a;
One-way ANOVA
model y=a b c;
Main-effects ANOVA
model y=a b a*b;
Factorial ANOVA with interaction
model y=a b(a) c(b a);
Nested ANOVA
model y1 y2=a b;
Multivariate analysis of variance (MANOVA)
model y=a x;
Analysis of covariance
model y=a x(a);
Separate-slopes regression
model y=a x x*a;
Homogeneity-of-slopes regression
The Bar Operator
You can shorten the specification of a large factorial model by using the bar operator. For example, two ways
of writing the model for a full three-way factorial model follow:
model Y = A B C
model Y = A|B|C;
A*B A*C B*C
A*B*C;
Using PROC GLM Interactively F 3497
When the bar (|) is used, the right and left sides become effects, and the cross of them becomes an effect.
Multiple bars are permitted. The expressions are expanded from left to right, using rules 2–4 given in Searle
(1971, p. 390).
• Multiple bars are evaluated from left to right. For instance, A|B|C is evaluated as follows:
A|B|C
!
fA|Bg|C
!
f A B A*B g | C
!
A B A*B C A*C B*C A*B*C
• Crossed and nested groups of variables are combined. For example, A(B) | C(D) generates A*C(B D),
among other terms.
• Duplicate variables are removed. For example, A(C) | B(C) generates A*B(C C), among other terms,
and the extra C is removed.
• Effects are discarded if a variable occurs on both the crossed and nested parts of an effect. For instance,
A(B) | B(D E) generates A*B(B D E), but this effect is eliminated immediately.
You can also specify the maximum number of variables involved in any effect that results from bar evaluation
by specifying that maximum number, preceded by an @ sign, at the end of the bar effect. For example, the
specification A | B | [email protected] would result in only those effects that contain 2 or fewer variables: in this case,
A B A*B C A*C and B*C.
More examples of using the bar and at operators follow:
A | C(B)
is equivalent to
A C(B) A*C(B)
A(B) | C(B)
is equivalent to
A(B) C(B) A*C(B)
A(B) | B(D E)
is equivalent to
A(B) B(D E)
A | B(A) | C
is equivalent to
A B(A) C A*C B*C(A)
A | B(A) | [email protected]
is equivalent to
A B(A) C A*C
A | B | C | [email protected]
is equivalent to
A B A*B C A*C B*C D A*D B*D C*D
A*B(C*D)
is equivalent to
A*B(C D)
Using PROC GLM Interactively
You can use the GLM procedure interactively. After you specify a model with a MODEL statement and run
PROC GLM with a RUN statement, you can execute a variety of statements without reinvoking PROC GLM.
The section “Syntax: GLM Procedure” on page 3446 describes which statements can be used interactively.
These interactive statements can be executed singly or in groups by following the single statement or group
of statements with a RUN statement. Note that the MODEL statement cannot be repeated; PROC GLM
allows only one MODEL statement.
If you use PROC GLM interactively, you can end the GLM procedure with a DATA step, another PROC step,
an ENDSAS statement, or a QUIT statement.
3498 F Chapter 44: The GLM Procedure
When you are using PROC GLM interactively, additional RUN statements do not end the procedure but tell
PROC GLM to execute additional statements.
When you specify a WHERE statement with PROC GLM, it should appear before the first RUN statement.
The WHERE statement enables you to select only certain observations for analysis without using a subsetting
DATA step. For example, where group ne 5 omits observations with GROUP=5 from the analysis. See
SAS Statements: Reference for details on this statement.
When you specify a BY statement with PROC GLM, interactive processing is not possible; that is, once the
first RUN statement is encountered, processing proceeds for each BY group in the data set, and no further
statements are accepted by the procedure.
Interactivity is also disabled when there are different patterns of missing values among the dependent
variables. For details, see the section “Missing Values” on page 3549.
Parameterization of PROC GLM Models
The GLM procedure constructs a linear model according to the specifications in the MODEL statement. Each
effect generates one or more columns in a design matrix X. This section shows precisely how X is built.
Intercept
All models include a column of 1s by default to estimate an intercept parameter . You can use the NOINT
option to suppress the intercept.
Regression Effects
Regression effects (covariates) have the values of the variables copied into the design matrix directly.
Polynomial terms are multiplied out and then installed in X.
Main Effects
If a classification variable has m levels, PROC GLM generates m columns in the design matrix for its main
effect. Each column is an indicator variable for one of the levels of the classification variable. The default
order of the columns is the sort order of the values of their levels; this order can be controlled with the
ORDER= option in the PROC GLM statement, as shown in the following table.
Data
Design Matrix
A
A
B
1
1
1
2
2
2
1
2
3
1
2
3
1
1
1
1
1
1
A1
1
1
1
0
0
0
B
A2
0
0
0
1
1
1
B1
1
0
0
1
0
0
B2
0
1
0
0
1
0
B3
0
0
1
0
0
1
There are more columns for these effects than there are degrees of freedom for them; in other words, PROC
GLM is using an over-parameterized model.
Parameterization of PROC GLM Models F 3499
Crossed Effects
First, PROC GLM reorders the terms to correspond to the order of the variables in the CLASS statement;
thus, B*A becomes A*B if A precedes B in the CLASS statement. Then, PROC GLM generates columns
for all combinations of levels that occur in the data. The order of the columns is such that the rightmost
variables in the cross index faster than the leftmost variables. No columns are generated corresponding to
combinations of levels that do not occur in the data.
Data
Design Matrix
A
A B
1
1
1
2
2
2
1
2
3
1
2
3
1
1
1
1
1
1
A1
1
1
1
0
0
0
B
A2
0
0
0
1
1
1
B1
1
0
0
1
0
0
B2
0
1
0
0
1
0
B3
0
0
1
0
0
1
A* B
A1B1 A1B2 A1B3 A2B1 A2B2 A2B3
1
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
0
0
0
1
In this matrix, main-effects columns are not linearly independent of crossed-effect columns; in fact, the
column space for the crossed effects contains the space of the main effect.
Nested Effects
Nested effects are generated in the same manner as crossed effects. Hence, the design columns generated by
the following statements are the same (but the ordering of the columns is different):
model y=a b(a);
(B nested within A)
model y=a a*b;
(omitted main effect for B)
The nesting operator in PROC GLM is more a notational convenience than an operation distinct from crossing.
Nested effects are characterized by the property that the nested variables never appear as main effects. The
order of the variables within nesting parentheses is made to correspond to the order of these variables in the
CLASS statement. The order of the columns is such that variables outside the parentheses index faster than
those inside the parentheses, and the rightmost nested variables index faster than the leftmost variables.
Data
Design Matrix
B(A)
A
A
B
1
1
1
2
2
2
1
2
3
1
2
3
1
1
1
1
1
1
A1
1
1
1
0
0
0
A2
0
0
0
1
1
1
B1A1
1
0
0
0
0
0
B2A1
0
1
0
0
0
0
B3A1
0
0
1
0
0
0
B1A2
0
0
0
1
0
0
B2A2
0
0
0
0
1
0
B3A2
0
0
0
0
0
1
3500 F Chapter 44: The GLM Procedure
Continuous-Nesting-Class Effects
When a continuous variable nests with a classification variable, the design columns are constructed by
multiplying the continuous values into the design columns for the class effect.
Data
Design Matrix
X(A)
A
X
A
21
24
22
28
19
23
1
1
1
2
2
2
1
1
1
1
1
1
A1
1
1
1
0
0
0
A2
0
0
0
1
1
1
X(A1)
21
24
22
0
0
0
X(A2)
0
0
0
28
19
23
This model estimates a separate slope for X within each level of A.
Continuous-by-Class Effects
Continuous-by-class effects generate the same design columns as continuous-nesting-class effects. The
two models differ by the presence of the continuous variable as a regressor by itself, in addition to being a
contributor to X*A.
Data
Design Matrix
X*A
A
X
A
21
24
22
28
19
23
1
1
1
2
2
2
1
1
1
1
1
1
X
21
24
22
28
19
23
A1
1
1
1
0
0
0
A2
0
0
0
1
1
1
X*A1
21
24
22
0
0
0
X*A2
0
0
0
28
19
23
Continuous-by-class effects are used to test the homogeneity of slopes. If the continuous-by-class effect is
nonsignificant, the effect can be removed so that the response with respect to X is the same for all levels of
the classification variables.
General Effects
An example that combines all the effects is
X1*X2*A*B*C(D E)
The continuous list comes first, followed by the crossed list, followed by the nested list in parentheses.
The sequencing of parameters is important to learn if you use the CONTRAST or ESTIMATE statement to
compute or test some linear function of the parameter estimates.
Effects might be retitled by PROC GLM to correspond to ordering rules. For example, B*A(E D) might be
retitled A*B(D E) to satisfy the following:
Parameterization of PROC GLM Models F 3501
• Classification variables that occur outside parentheses (crossed effects) are sorted in the order in which
they appear in the CLASS statement.
• Variables within parentheses (nested effects) are sorted in the order in which they appear in a CLASS
statement.
The sequencing of the parameters generated by an effect can be described by which variables have their
levels indexed faster:
• Variables in the crossed part index faster than variables in the nested list.
• Within a crossed or nested list, variables to the right index faster than variables to the left.
For example, suppose a model includes four effects—A, B, C, and D—each having two levels, 1 and 2. If the
CLASS statement is
class A B C D;
then the order of the parameters for the effect B*A(C D), which is retitled A*B(C D), is as follows.
A1 B1 C1 D1
A1 B2 C1 D1
A2 B1 C1 D1
A2 B2 C1 D1
A1 B1 C1 D2
A1 B2 C1 D2
A2 B1 C1 D2
A2 B2 C1 D2
A1 B1 C2 D1
A1 B2 C2 D1
A2 B1 C2 D1
A2 B2 C2 D1
A1 B1 C2 D2
A1 B2 C2 D2
A2 B1 C2 D2
A2 B2 C2 D2
Note that first the crossed effects B and A are sorted in the order in which they appear in the CLASS
statement so that A precedes B in the parameter list. Then, for each combination of the nested effects in turn,
combinations of A and B appear. The B effect changes fastest because it is rightmost in the (renamed) cross
list. Then A changes next fastest. The D effect changes next fastest, and C is the slowest since it is leftmost in
the nested list.
When numeric classification variables are used, their levels are sorted by their character format, which might
not correspond to their numeric sort sequence. Therefore, it is advisable to include a format for numeric
classification variables or to use the ORDER=INTERNAL option in the PROC GLM statement to ensure that
levels are sorted by their internal values.
3502 F Chapter 44: The GLM Procedure
Degrees of Freedom
For models with classification (categorical) effects, there are more design columns constructed than there
are degrees of freedom for the effect. Thus, there are linear dependencies among the columns. In this event,
the parameters are not jointly estimable; there is an infinite number of least squares solutions. The GLM
procedure uses a generalized g2 -inverse to obtain values for the estimates; see the section “Computational
Method” on page 3552 for more details. The solution values are not produced unless the SOLUTION option
is specified in the MODEL statement. The solution has the characteristic that estimates are zero whenever the
design column for that parameter is a linear combination of previous columns. (Strictly termed, the solution
values should not be called estimates, since the parameters might not be formally estimable.) With this full
parameterization, hypothesis tests are constructed to test linear functions of the parameters that are estimable.
Other procedures (such as the CATMOD procedure) reparameterize models to full rank by using certain restrictions on the parameters. PROC GLM does not reparameterize, making the hypotheses that are commonly
tested more understandable. See Goodnight (1978a) for additional reasons for not reparameterizing.
PROC GLM does not actually construct the entire design matrix X; rather, a row xi of XPis constructed for
each observation in the data set and used to accumulate the crossproduct matrix X0 X D i xi0 xi .
Hypothesis Testing in PROC GLM
See Chapter 15, “The Four Types of Estimable Functions,” for a complete discussion of the four standard
types of hypothesis tests.
Example
To illustrate the four types of tests and the principles upon which they are based, consider a two-way design
with interaction based on the following data:
B
1
A
2
3
1
23.5
23.7
8.9
10.3
12.5
2
28.7
5.6
8.9
13.6
14.6
Hypothesis Testing in PROC GLM F 3503
Invoke PROC GLM and specify all the estimable functions options to examine what the GLM procedure can
test. The following statements produce the summary ANOVA table displayed in Figure 44.10.
data example;
input a b y @@;
datalines;
1 1 23.5 1 1 23.7
2 2 8.9 3 1 10.3
;
1 2 28.7
3 1 12.5
2 1 8.9
3 2 13.6
2 2 5.6
3 2 14.6
proc glm;
class a b;
model y=a b a*b / e e1 e2 e3 e4;
run;
Figure 44.10 Summary ANOVA Table from PROC GLM
The GLM Procedure
Dependent Variable: y
DF
Sum of
Squares
Mean Square
F Value
Pr > F
Model
5
520.4760000
104.0952000
49.66
0.0011
Error
4
8.3850000
2.0962500
Corrected Total
9
528.8610000
Source
R-Square
Coeff Var
Root MSE
y Mean
0.984145
9.633022
1.447843
15.03000
The following sections show the general form of estimable functions and discuss the four standard tests, their
properties, and abbreviated output for the two-way crossed example.
Estimability
Figure 44.11 is the general form of estimable functions for the example. In order to be testable, a hypothesis
must be able to fit within the framework displayed here.
3504 F Chapter 44: The GLM Procedure
Figure 44.11 General Form of Estimable Functions
The GLM Procedure
General Form of Estimable Functions
Effect
Coefficients
Intercept
L1
a
a
a
1
2
3
L2
L3
L1-L2-L3
b
b
1
2
L5
L1-L5
a*b
a*b
a*b
a*b
a*b
a*b
1
1
2
2
3
3
1
2
1
2
1
2
L7
L2-L7
L9
L3-L9
L5-L7-L9
L1-L2-L3-L5+L7+L9
If a hypothesis is estimable, the Ls in the preceding scheme can be set to values that match the hypothesis.
All the standard tests in PROC GLM can be shown in the preceding format, with some of the Ls zeroed and
some set to functions of other Ls.
The following sections show how many of the hypotheses can be tested by comparing the model sum-ofsquares regression from one model to a submodel. The notation used is
SS.B effectsjA effects/ D SS.B effects; A effects/
SS.A effects/
where SS(A effects) denotes the regression model sum of squares for the model consisting of A effects. This
notation is equivalent to the reduction notation defined by Searle (1971) and summarized in Chapter 15, “The
Four Types of Estimable Functions.”
Type I Tests
Type I sums of squares (SS), also called sequential sums of squares, are the incremental improvement in error
sums of squares as each effect is added to the model. They can be computed by fitting the model in steps and
recording the difference in error sum of squares at each step.
Source
A
B
AB
Type I SS
SS.A j /
SS.B j ; A/
SS.A B j; A; B/
Hypothesis Testing in PROC GLM F 3505
Type I sums of squares are displayed by default because they are easy to obtain and can be used in various
hand calculations to produce sum of squares values for a series of different models. Nelder (1994) and others
have argued that Type I and II sums are essentially the only appropriate ones for testing ANOVA effects;
however, see also the discussion of Nelder’s article, especially Rodriguez, Tobias, and Wolfinger (1995) and
Searle (1995).
The Type I hypotheses have these properties:
• Type I sum of squares for all effects add up to the model sum of squares. None of the other sum of
squares types have this property, except in special cases.
• Type I hypotheses can be derived from rows of the Forward-Dolittle transformation of X0 X (a transformation that reduces X0 X to an upper triangular matrix by row operations).
• Type I sum of squares are statistically independent of each other under the usual assumption that the
true residual errors are independent and identically normally distributed (see page 3494).
• Type I hypotheses depend on the order in which effects are specified in the MODEL statement.
• Type I hypotheses are uncontaminated by parameters corresponding to effects that precede the effect
being tested; however, the hypotheses usually involve parameters for effects following the tested effect
in the model. For example, in the model
Y=A B;
the Type I hypothesis for B does not involve A parameters, but the Type I hypothesis for A does involve
B parameters.
• Type I hypotheses are functions of the cell counts for unbalanced data; the hypotheses are not usually
the same hypotheses that are tested if the data are balanced.
• Type I sums of squares are useful for polynomial models where you want to know the contribution of a
term as though it had been made orthogonal to preceding effects. Thus, in polynomial models, Type I
sums of squares correspond to tests of the orthogonal polynomial effects.
The Type I estimable functions and associated tests for the example are shown in Figure 44.12.
3506 F Chapter 44: The GLM Procedure
Figure 44.12 Type I Estimable Functions and Tests
Type I Estimable Functions
Effect
----------------Coefficients---------------a
b
a *b
Intercept
0
0
0
a
a
a
1
2
3
L2
L3
-L2-L3
0
0
0
0
0
0
b
b
1
2
0.1667*L2-0.1667*L3
-0.1667*L2+0.1667*L3
L5
-L5
0
0
a*b
a*b
a*b
a*b
a*b
a*b
1
1
2
2
3
3
0.6667*L2
0.3333*L2
0.3333*L3
0.6667*L3
-0.5*L2-0.5*L3
-0.5*L2-0.5*L3
0.2857*L5
-0.2857*L5
0.2857*L5
-0.2857*L5
0.4286*L5
-0.4286*L5
L7
-L7
L9
-L9
-L7-L9
L7+L9
1
2
1
2
1
2
Source
a
b
a*b
DF
Type I SS
Mean Square
F Value
Pr > F
2
1
2
494.0310000
10.7142857
15.7307143
247.0155000
10.7142857
7.8653571
117.84
5.11
3.75
0.0003
0.0866
0.1209
Type II Tests
The Type II tests can also be calculated by comparing the error sums of squares (SS) for subset models. The
Type II SS are the reduction in error SS due to adding the term after all other terms have been added to the
model except terms that contain the effect being tested. An effect is contained in another effect if it can be
derived by deleting variables from the latter effect. For example, A and B are both contained in A*B. For this
model, the Type II SS are given by the reduced sums of squares as shown in the following table.
Source
A
B
AB
Type II SS
SS.A j ; B/
SS.B j ; A/
SS.A B j ; A; B/
Type II SS have these properties:
• Type II SS do not necessarily sum to the model SS.
• The hypothesis for an effect does not involve parameters of other effects except for containing effects
(which it must involve to be estimable).
• Type II SS are invariant to the ordering of effects in the model.
Hypothesis Testing in PROC GLM F 3507
• For unbalanced designs, Type II hypotheses for effects that are contained in other effects are not usually
the same hypotheses that are tested if the data are balanced. The hypotheses are generally functions of
the cell counts.
The Type II estimable functions and associated tests for the example are shown in Figure 44.13.
Figure 44.13 Type II Estimable Functions and Tests
Type II Estimable Functions
Effect
----------------Coefficients---------------a
b
a *b
Intercept
0
0
0
a
a
a
1
2
3
L2
L3
-L2-L3
0
0
0
0
0
0
b
b
1
2
0
0
L5
-L5
0
0
a*b
a*b
a*b
a*b
a*b
a*b
1
1
2
2
3
3
0.619*L2+0.0476*L3
0.381*L2-0.0476*L3
-0.0476*L2+0.381*L3
0.0476*L2+0.619*L3
-0.5714*L2-0.4286*L3
-0.4286*L2-0.5714*L3
0.2857*L5
-0.2857*L5
0.2857*L5
-0.2857*L5
0.4286*L5
-0.4286*L5
L7
-L7
L9
-L9
-L7-L9
L7+L9
Source
a
b
a*b
1
2
1
2
1
2
DF
Type II SS
Mean Square
F Value
Pr > F
2
1
2
499.1202857
10.7142857
15.7307143
249.5601429
10.7142857
7.8653571
119.05
5.11
3.75
0.0003
0.0866
0.1209
Type III and Type IV Tests
Type III and Type IV sums of squares (SS), sometimes referred to as partial sums of squares, are considered
by many to be the most desirable; see Searle (1987, Section 4.6). Using PROC GLM’s singular parameterization, these SS cannot, in general, be computed by comparing model SS from different models. However,
they can sometimes be computed by reduction for methods that reparameterize to full rank, when such a
reparameterization effectively imposes Type III linear constraints on the parameters. In PROC GLM, they are
computed by constructing a hypothesis matrix L and then computing the SS associated with the hypothesis
Lˇ D 0. As long as there are no missing cells in the design, Type III and Type IV SS are the same.
These are properties of Type III and Type IV SS:
• The hypothesis for an effect does not involve parameters of other effects except for containing effects
(which it must involve to be estimable).
• The hypotheses to be tested are invariant to the ordering of effects in the model.
3508 F Chapter 44: The GLM Procedure
• The hypotheses are the same hypotheses that are tested if there are no missing cells. They are not
functions of cell counts.
• The SS do not generally add up to the model SS and, in some cases, can exceed the model SS.
The SS are constructed from the general form of estimable functions. Type III and Type IV tests are different
only if the design has missing cells. In this case, the Type III tests have an orthogonality property, while the
Type IV tests have a balancing property. These properties are discussed in Chapter 15, “The Four Types of
Estimable Functions.” For this example, since the data contain observations for all pairs of levels of A and B,
Type IV tests are identical to the Type III tests that are shown in Figure 44.14. (This combines tables from
several pages of output.)
Figure 44.14 Type III Estimable Functions and Tests
Type III Estimable Functions
Source
a
b
a*b
Effect
-------------Coefficients------------a
b
a*b
Intercept
0
0
0
a
a
a
1
2
3
L2
L3
-L2-L3
0
0
0
0
0
0
b
b
1
2
0
0
L5
-L5
0
0
a*b
a*b
a*b
a*b
a*b
a*b
1
1
2
2
3
3
0.5*L2
0.5*L2
0.5*L3
0.5*L3
-0.5*L2-0.5*L3
-0.5*L2-0.5*L3
0.3333*L5
-0.3333*L5
0.3333*L5
-0.3333*L5
0.3333*L5
-0.3333*L5
L7
-L7
L9
-L9
-L7-L9
L7+L9
1
2
1
2
1
2
DF
Type III SS
Mean Square
F Value
Pr > F
2
1
2
479.1078571
9.4556250
15.7307143
239.5539286
9.4556250
7.8653571
114.28
4.51
3.75
0.0003
0.1009
0.1209
Effect Size Measures for F Tests in GLM
A significant F test in a linear model indicates that the effect of the term or contrast being tested might
be real. The next thing you want to know is, How big is the effect? Various measures have been devised
to give answers to this question that are comparable over different experimental designs. If you specify
the experimental EFFECTSIZE option in the MODEL statement, then GLM adds to each ANOVA table
estimates and confidence intervals for three different measures of effect size:
Effect Size Measures for F Tests in GLM F 3509
• the noncentrality parameter for the F test
• the proportion of total variation accounted for (also known as the semipartial correlation ratio or the
squared semipartial correlation)
• the proportion of partial variation accounted for (also known as the full partial correlation ratio or the
squared full partial correlation)
The adjectives “semipartial” and “full partial” might seem strange. They refer to how other effects are
“partialed out” of the dependent variable and the effect being tested. For “semipartial” statistics, all other
effects are partialed out of the effect in question, but not the dependent variable. This measures the (adjusted)
effect as a proportion of the total variation in the dependent variable. On the other hand, for “full partial”
statistics, all other effects are partialed out of both the dependent variable and the effect in question. This
measures the (adjusted) effect as a proportion of only the dependent variation remaining after partialing, or in
other words the partial variation. Details about the computation and interpretation of these estimates and
confidence intervals are discussed in the remainder of this section.
The noncentrality parameter is directly related to the true distribution of the F statistic when the effect being
tested has a non-null effect. The uniformly minimum variance unbiased estimate for the noncentrality is
NCUMVUE D
DF.DFE 2/FValue
DFE
DF
where FValue is the observed value of the F statistic for the test and DF and DFE are the numerator and
denominator degrees of freedom for the test, respectively. An alternative estimate that can be slightly biased
but has a somewhat lower expected mean square error is
NCminMSE D
DF.DFE 4/FValue
DFE
DF.DFE 4/
DFE 2
(See Perlman and Rasmussen (1975), cited in Johnson, Kotz, and Balakrishnan (1994).) A p 100% lower
confidence bound for the noncentrality is given by the value of NC for which probf(FValue,DF,DFE,NC) = p,
where probf() is the cumulative probability function for the non-central F distribution. This result can be
used to form a .1 ˛/ 100% confidence interval for the noncentrality.
The partial proportion of variation accounted for by the effect being tested is easiest to define by its natural
sample estimate,
O 2partial
D
SS
SS C SSE
2
where SSE is the sample error sum of squares. Note that O 2partial is actually sometimes denoted Rpartial
or
2
just R square, but in this context the R square notation is reserved for the O corresponding to the overall
model, which is just the familiar R square for the model. O 2partial is actually a biased estimate of the true
2partial ; an alternative that is approximately unbiased is given by
2
!partial
D
SS DF MSE
SS C .N DF/MSE
3510 F Chapter 44: The GLM Procedure
where MSE = SSE/DFE is the sample mean square for error and N is the number of observations. The true
2partial is related to the true noncentrality parameter NC by the formula
2partial
D
NC
NC C N
This fact can be employed to transform a confidence interval for NC into one for 2partial . Note that some
authors (Steiger and Fouladi 1997; Fidler and Thompson 2001; Smithson 2003) have published slightly
different confidence intervals for 2partial , based on a slightly different formula for the relationship between
2partial and NC, apparently due to Cohen (1988). Cohen’s formula appears to be approximately correct for
random predictor values (Maxwell 2000), but the one given previously is correct if the predictor values are
assumed fixed, as is standard for the GLM procedure.
Finally, the proportion of total variation accounted for by the effect being tested is again easiest to define by
its natural sample estimate, which is known as the (semipartial) O 2 statistic,
O 2 D
SS
SStotal
where SStotal is the total sample (corrected) sum of squares, and SS is the observed sum of squares due to
the effect being tested. As with O 2partial , O 2 is actually a biased estimate of the true 2 ; an alternative that is
approximately unbiased is the (semipartial) ! 2 statistic
!2 D
SS DF MSE
SStotal C MSE
where MSE = SSE/DFE is the sample mean square for error. Whereas 2partial depends only on the noncentrality for its associated F test, the presence of the total sum of squares in the previous formulas indicates that
2 depends on the noncentralities for all effects in the model. An exact confidence interval is not available,
but if you write the formula for O 2 as
O 2 D
SS
SS C .SStotal
SS/
then a conservative confidence interval can be constructed as for 2partial , treating SStotal SS as the SSE
and N – DF – 1 as the DFE (Smithson 2004). This confidence interval is conservative in the sense that it
implies values of the true 2 that are smaller than they should be.
Estimates and confidence intervals for effect sizes require some care in interpretation. For example, while the
true proportions of total and partial variation accounted for are nonnegative quantities, their estimates might
be less than zero. Also, confidence intervals for effect sizes are not directly related to the corresponding
estimates. In particular, it is possible for the estimate to lie outside the confidence interval.
As for interpreting the actual values of effect size measures, the approximately unbiased ! 2 estimates are
usually preferred for point estimates. Some authors have proposed certain ranges as indicating “small,”
“medium,” and “large” effects (Cohen 1988), but general benchmarks like this depend on the nature of the
data and the typical signal-to-noise ratio; they should not be expected to apply across various disciplines. For
example, while an ! 2 value of 10% might be viewed as “large” for psychometric data, it can be a relatively
small effect for industrial experimentation. Whatever the standard, confidence intervals for true effect sizes
typically span more than one category, indicating that in small experiments, it can be difficult to make firm
statements about the size of effects.
Effect Size Measures for F Tests in GLM F 3511
Example
The data for this example are similar to data analyzed in Steiger and Fouladi (1997); Fidler and Thompson
(2001); Smithson (2003). Consider the following hypothetical design, testing 28 men and 28 women on
seven different tasks.
data Test;
do Task = 1 to 7;
do Gender = 'M','F';
do i = 1 to 4;
input Response @@;
output;
end;
end;
end;
datalines;
7.1 2.8 3.9 3.7
6.5 6.5 6.5
7.1 5.5 4.8 2.6
3.6 5.4 5.6
7.2 4.6 4.9 4.6
3.3 5.4 2.8
5.6 6.2 5.4 6.5
5.6 2.7 3.8
2.2 5.4 5.6 8.4
1.2 2.0 4.3
9.1 4.5 7.6 4.9
4.3 7.7 6.5
4.5 3.8 5.9 6.1
1.7 2.5 4.3
;
6.6
4.5
1.5
2.3
4.6
7.7
2.7
This is a balanced two-way design with four replicates per cell. The following statements analyze this data.
Since this is a balanced design, you can use the SS1 option in the MODEL statement to display only the
Type I sums of squares.
proc glm data=Test;
class Gender Task;
model Response = Gender|Task / ss1;
run;
The analysis of variance results are shown in Figure 44.15.
Figure 44.15 Two-Way Analysis of Variance
The GLM Procedure
Dependent Variable: Response
Source
Gender
Task
Gender*Task
DF
Type I SS
Mean Square
F Value
Pr > F
1
6
6
14.40285714
38.15964286
35.99964286
14.40285714
6.35994048
5.99994048
6.00
2.65
2.50
0.0185
0.0285
0.0369
You can see that the two main effects as well as their interaction are all significant. Suppose you want to
compare the main effect of Gender with the interaction between Gender and Task. The sums of squares
for the interaction are more than twice as large, but it’s not clear how experimental variability might affect
this. The following statements perform the same analysis as before, but add the EFFECTSIZE option to
3512 F Chapter 44: The GLM Procedure
the MODEL statement; also, with ALPHA=0.1 option displays 90% confidence intervals, ensuring that
inferences based on the p-values at the 0.05 levels will agree with the lower confidence limit.
proc glm data=Test;
class Gender Task;
model Response = Gender|Task / ss1 effectsize alpha=0.1;
run;
The Type I analysis of variance results with added effect size information are shown in Figure 44.16.
Figure 44.16 Two-Way Analysis of Variance with Effect Sizes
The GLM Procedure
Dependent Variable: Response
Source
Gender
Task
Gender*Task
DF
Type I SS
Mean Square
F Value
Pr > F
1
6
6
14.40285714
38.15964286
35.99964286
14.40285714
6.35994048
5.99994048
6.00
2.65
2.50
0.0185
0.0285
0.0369
Noncentrality Parameter
Source
Gender
Task
Gender*Task
Min Var
Unbiased
Estimate
Low MSE
Estimate
4.72
9.14
8.29
4.48
8.69
7.87
90% Confidence Limits
0.521
0.870
0.463
17.1
27.3
25.9
Total Variation Accounted For
Source
Gender
Task
Gender*Task
Semipartial
Eta-Square
Semipartial
OmegaSquare
0.0761
0.2015
0.1901
0.0626
0.1239
0.1126
Conservative
90% Confidence Limits
0.0019
0.0000
0.0000
0.2030
0.2772
0.2639
Partial Variation Accounted For
Source
Gender
Task
Gender*Task
Partial
Eta-Square
Partial
OmegaSquare
0.1250
0.2746
0.2632
0.0820
0.1502
0.1385
90% Confidence Limits
0.0092
0.0153
0.0082
0.2342
0.3277
0.3160
The estimated effect sizes for Gender and the interaction all tell pretty much the same story: the effect of
the interaction is appreciably greater than the effect of Gender. However, the confidence intervals suggest
that this inference should be treated with some caution, since the lower confidence bound for the Gender
Absorption F 3513
effect is greater than the lower confidence bound for the interaction in all three cases. Follow-up testing is
probably in order, using the estimated effect sizes in this preliminary study to design a large enough sample
to distinguish the sizes of the effects.
Absorption
Absorption is a computational technique used to reduce computing resource needs in certain cases. The
classic use of absorption occurs when a blocking factor with a large number of levels is a term in the model.
For example, the statements
proc glm;
absorb herd;
class a b;
model y=a b a*b;
run;
are equivalent to
proc glm;
class herd a b;
model y=herd a b a*b;
run;
The exception to the previous statements is that the Type II, Type III, or Type IV SS for HERD are not
computed when HERD is absorbed.
The algorithm for absorbing variables is similar to the one used by the NESTED procedure for computing
a nested analysis of variance. As each new row of ŒXjY  (corresponding to the nonabsorbed independent
effects and the dependent variables) is constructed, it is adjusted for the absorbed effects in a Type I fashion.
The efficiency of the absorption technique is due to the fact that this adjustment can be done in one pass of
the data and without solving any linear equations, assuming that the data have been sorted by the absorbed
variables.
Several effects can be absorbed at one time. For example, these statements
proc glm;
absorb herd cow;
class a b;
model y=a b a*b;
run;
are equivalent to
proc glm;
class herd cow a b;
model y=herd cow(herd) a b a*b;
run;
When you use absorption, the size of the X0 X matrix is a function only of the effects in the MODEL statement.
The effects being absorbed do not contribute to the size of the X0 X matrix.
For the preceding example, a and b can be absorbed:
3514 F Chapter 44: The GLM Procedure
proc glm;
absorb a b;
class herd cow;
model y=herd cow(herd);
run;
Although the sources of variation in the results are listed as
a b(a) herd cow(herd)
all types of estimable functions for herd and cow(herd) are free of a, b, and a*b parameters.
To illustrate the savings in computing by using the ABSORB statement, PROC GLM is run on generated
data with 1147 degrees of freedom in the model with the following statements.
data a;
do herd=1 to 40;
do cow=1 to 30;
do treatment=1 to 3;
do rep=1 to 2;
y = herd/5 + cow/10 + treatment + rannor(1);
output;
end;
end;
end;
end;
run;
proc glm data=a;
class herd cow treatment;
model y=herd cow(herd) treatment;
run;
This analysis would have required over 6 megabytes of memory for the X0 X matrix had PROC GLM solved
it directly. However, in the following statements, the GLM procedure needs only a 4 4 matrix for the
intercept and treatment because the other effects are absorbed.
proc glm data=a;
absorb herd cow;
class treatment;
model y = treatment;
run;
These statements produce the results shown in Figure 44.17.
Figure 44.17 Absorption of Effects
The GLM Procedure
Class Level Information
Class
treatment
Levels
3
Values
1 2 3
Specification of ESTIMATE Expressions F 3515
Figure 44.17 continued
Number of Observations Read
Number of Observations Used
7200
7200
The GLM Procedure
Dependent Variable: y
DF
Sum of
Squares
Mean Square
F Value
Pr > F
Model
1201
49465.40242
41.18685
41.57
<.0001
Error
5998
5942.23647
0.99070
Corrected Total
7199
55407.63889
Source
R-Square
Coeff Var
Root MSE
y Mean
0.892754
13.04236
0.995341
7.631598
Source
herd
cow(herd)
treatment
Source
treatment
DF
Type I SS
Mean Square
F Value
Pr > F
39
1160
2
38549.18655
6320.18141
4596.03446
988.44068
5.44843
2298.01723
997.72
5.50
2319.58
<.0001
<.0001
<.0001
DF
Type III SS
Mean Square
F Value
Pr > F
2
4596.034455
2298.017228
2319.58
<.0001
Specification of ESTIMATE Expressions
Consider the model
E.Y / D ˇ0 C ˇ1 x1 C ˇ2 x2 C ˇ3 x3
The corresponding MODEL statement for PROC GLM is
model y=x1 x2 x3;
To estimate the difference between the parameters for x1 and x2 ,
ˇ1
ˇ2 D . 0 1
1 0 /ˇ; where ˇ D . ˇ0 ˇ1 ˇ2 ˇ3 /0
you can use the following ESTIMATE statement:
3516 F Chapter 44: The GLM Procedure
estimate 'B1-B2'
x1 1
x2 -1;
To predict y at x1 D 1, x2 D 0, and x3 D
ˇ0 C ˇ1
2ˇ3 D . 1 1 0
2, you can estimate
2 /ˇ
with the following ESTIMATE statement:
estimate 'B0+B1-2B3' intercept 1 x1 1 x3 -2;
Now consider models involving classification variables such as
model y=A B A*B;
with the associated parameters:
˛1 ˛2 ˛3 ˇ1 ˇ2 11 12 21 22 31 32
The LS-mean for the first level of A is Lˇ, where
L D . 1 j 1 0 0 j 0:5 0:5 j 0:5 0:5 0 0 0 0 /
You can estimate this with the following ESTIMATE statement:
estimate 'LS-mean(A1)' intercept 1 A 1 B 0.5 0.5 A*B 0.5 0.5;
Note in this statement that only one element of L is specified following the A effect, even though A has three
levels. Whenever the list of constants following an effect name is shorter than the effect’s number of levels,
zeros are used as the remaining constants. (If the list of constants is longer than the number of levels for the
effect, the extra constants are ignored, and a warning message is displayed.)
To estimate the A linear effect in the preceding model, assuming equally spaced levels for A, you can use the
following L:
LD. 0 j
1 0 1 j 0 0 j
0:5
0:5 0 0 0:5 0:5 /
The ESTIMATE statement for this L is written as
estimate 'A Linear' A -1 0 1;
If you do not specify the elements of L for an effect that contains a specified effect, then the elements of the
specified effect are equally distributed over the corresponding levels of the higher-order effect. In addition, if
you specify the intercept in an ESTIMATE or CONTRAST statement, it is distributed over all classification
effects that are not contained by any other specified effect.
The distribution of lower-order coefficients to higher-order effect coefficients follows the same general rules
as in the LSMEANS statement, and it is similar to that used to construct Type IV tests. In the previous
example, the –1 associated with ˛1 is divided by the number n1j of 1j parameters; then each 1j coefficient
is set to 1=n1j . The 1 associated with ˛3 is distributed among the 3j parameters in a similar fashion. In
the event that an unspecified effect contains several specified effects, only that specified effect with the most
factors in common with the unspecified effect is used for distribution of coefficients to the higher-order effect.
Numerous syntactical expressions for the ESTIMATE statement were considered, including many that
involved specifying the effect and level information associated with each coefficient. For models involving
Comparing Groups F 3517
higher-level effects, the requirement of specifying level information can lead to very bulky specifications.
Consequently, the simpler form of the ESTIMATE statement described earlier was implemented.
The syntax of this ESTIMATE statement puts a burden on you to know a priori the order of the parameter list
associated with each effect. You can use the ORDER= option in the PROC GLM statement to ensure that the
levels of the classification effects are sorted appropriately.
N OTE : If you use the ESTIMATE statement with unspecified effects, use the E option to make sure that the
actual L constructed by the preceding rules is the one you intended.
A Check for Estimability
Each L is checked for estimability using the relationship L D LH, where H D .X0 X/ X0 X. The L vector is
declared nonestimable, if for any i
(
if Li D 0 or
ABS.Li .LH/i / >
ABS.Li /
otherwise
where D 10 4 by default; you can change this with the SINGULAR= option. Continued fractions (like
1/3) should be specified to at least six decimal places, or the DIVISOR parameter should be used.
Comparing Groups
An important task in analyzing data with classification effects is to estimate the typical response for each level
of a given effect; often, you also want to compare these estimates to determine which levels are equivalent
in terms of the response. You can perform this task in two ways with the GLM procedure: with direct,
arithmetic group means; and with so-called least squares means (LS-means).
Means versus LS-Means
Computing and comparing arithmetic means—either simple or weighted within-group averages of the input
data—is a familiar and well-studied statistical process. This is the right approach to summarizing and
comparing groups for one-way and balanced designs. However, in unbalanced designs with more than one
effect, the arithmetic mean for a group might not accurately reflect the “typical” response for that group,
since it does not take other effects into account.
For example, the following analysis of an unbalanced two-way design produces the ANOVA, means, and
LS-means shown in Figure 44.18, Figure 44.19, and Figure 44.20.
data twoway;
input Treatment Block y @@;
datalines;
1 1 17
1 1 28
1 1 19
1 1 21
1 2 43
1 2 30
1 2 39
1 2 44
1 3 16
2 1 21
2 1 21
2 1 24
2 1 25
2 2 39
2 2 45
2 2 42
2 2 47
2 3 19
2 3 22
2 3 16
3 1 22
3 1 30
3 1 33
3 1 31
3 2 46
1 1 19
1 2 44
3518 F Chapter 44: The GLM Procedure
3 3 26
;
3 3 31
3 3 26
3 3 33
3 3 29
3 3 25
title "Unbalanced Two-way Design";
ods select ModelANOVA Means LSMeans;
proc glm data=twoway;
class Treatment Block;
model y = Treatment|Block;
means Treatment;
lsmeans Treatment;
run;
ods select all;
Figure 44.18 ANOVA Results for Unbalanced Two-Way Design
Unbalanced Two-way Design
The GLM Procedure
Dependent Variable: y
Source
Treatment
Block
Treatment*Block
Source
Treatment
Block
Treatment*Block
DF
Type I SS
Mean Square
F Value
Pr > F
2
2
4
8.060606
2621.864124
32.684361
4.030303
1310.932062
8.171090
0.24
77.95
0.49
0.7888
<.0001
0.7460
DF
Type III SS
Mean Square
F Value
Pr > F
2
2
4
266.130682
1883.729465
32.684361
133.065341
941.864732
8.171090
7.91
56.00
0.49
0.0023
<.0001
0.7460
Figure 44.19 Treatment Means for Unbalanced Two-Way Design
Unbalanced Two-way Design
The GLM Procedure
Level of
Treatment
1
2
3
N
11
11
11
--------------y-------------Mean
Std Dev
29.0909091
29.1818182
30.1818182
11.5104695
11.5569735
6.3058414
Comparing Groups F 3519
Figure 44.20 Treatment LS-means for Unbalanced Two-Way Design
Unbalanced Two-way Design
The GLM Procedure
Least Squares Means
Treatment
1
2
3
y LSMEAN
25.6000000
28.3333333
34.4444444
No matter how you look at them, these data exhibit a strong effect due to the blocks (F test p < 0:0001) and
no significant interaction between treatments and blocks (F test p > 0:7). But the lack of balance affects how
the treatment effect is interpreted: in a main-effects-only model, there are no significant differences between
the treatment means themselves (Type I F test p > 0:7), but there are highly significant differences between
the treatment means corrected for the block effects (Type III F test p < 0:01).
LS-means are, in effect, within-group means appropriately adjusted for the other effects in the model. More
precisely, they estimate the marginal means for a balanced population (as opposed to the unbalanced design).
For this reason, they are also called estimated population marginal means by Searle, Speed, and Milliken
(1980). In the same way that the Type I F test assesses differences between the arithmetic treatment means
(when the treatment effect comes first in the model), the Type III F test assesses differences between the
LS-means. Accordingly, for the unbalanced two-way design, the discrepancy between the Type I and Type
III tests is reflected in the arithmetic treatment means and treatment LS-means, as shown in Figure 44.19 and
Figure 44.20. See the section “Construction of Least Squares Means” on page 3533 for more on LS-means.
Note that, while the arithmetic means are always uncorrelated (under the usual assumptions for analysis
of variance; see page 3494), the LS-means might not be. This fact complicates the problem of multiple
comparisons for LS-means; see the following section.
Multiple Comparisons
When comparing more than two means, an ANOVA F test tells you whether the means are significantly
different from each other, but it does not tell you which means differ from which other means. Multiplecomparison procedures (MCPs), also called mean separation tests, give you more detailed information about
the differences among the means. The goal in multiple comparisons is to compare the average effects of
three or more “treatments” (for example, drugs, groups of subjects) to decide which treatments are better,
which ones are worse, and by how much, while controlling the probability of making an incorrect decision.
A variety of multiple-comparison methods are available with the MEANS and LSMEANS statement in the
GLM procedure.
The following classification is due to Hsu (1996). Multiple-comparison procedures can be categorized in two
ways: by the comparisons they make and by the strength of inference they provide. With respect to which
comparisons are made, the GLM procedure offers two types:
• comparisons between all pairs of means
• comparisons between a control and all other means
3520 F Chapter 44: The GLM Procedure
The strength of inference says what can be inferred about the structure of the means when a test is significant;
it is related to what type of error rate the MCP controls. MCPs available in the GLM procedure provide one
of the following types of inference, in order from weakest to strongest:
• Individual: differences between means, unadjusted for multiplicity
• Inhomogeneity: means are different
• Inequalities: which means are different
• Intervals: simultaneous confidence intervals for mean differences
Methods that control only individual error rates are not true MCPs at all. Methods that yield the strongest
level of inference, simultaneous confidence intervals, are usually preferred, since they enable you not only to
say which means are different but also to put confidence bounds on how much they differ, making it easier to
assess the practical significance of a difference. They are also less likely to lead nonstatisticians to the invalid
conclusion that nonsignificantly different sample means imply equal population means. Interval MCPs are
available for both arithmetic means and LS-means via the MEANS and LSMEANS statements, respectively.1
Table 44.12 and Table 44.13 display MCPs available in PROC GLM for all pairwise comparisons and
comparisons with a control, respectively, along with associated strength of inference and the syntax (when
applicable) for both the MEANS and the LSMEANS statements.
Table 44.12 Multiple-Comparison Procedures for All Pairwise Comparisons
Method
Student’s t
Duncan
Student-Newman-Keuls
REGWQ
Tukey-Kramer
Bonferroni
Sidak
Scheffé
SMM
Gabriel
Simulation
Strength of
Inference
Individual
Individual
Inhomogeneity
Inequalities
Intervals
Intervals
Intervals
Intervals
Intervals
Intervals
Intervals
MEANS
T
DUNCAN
SNK
REGWQ
TUKEY
BON
SIDAK
SCHEFFE
SMM
GABRIEL
Syntax
LSMEANS
PDIFF ADJUST=T
PDIFF ADJUST=TUKEY
PDIFF ADJUST=BON
PDIFF ADJUST=SIDAK
PDIFF ADJUST=SCHEFFE
PDIFF ADJUST=SMM
PDIFF ADJUST=SIMULATE
1 The Duncan-Waller method does not fit into the preceding scheme, since it is based on the Bayes risk rather than any particular
error rate.
Comparing Groups F 3521
Table 44.13 Multiple-Comparison Procedures for Comparisons with a Control
Method
Student’s t
Dunnett
Bonferroni
Sidak
Scheffé
SMM
Simulation
Strength of
Inference
Individual
Intervals
Intervals
Intervals
Intervals
Intervals
Intervals
MEANS
DUNNETT
Syntax
LSMEANS
PDIFF=CONTROL ADJUST=T
PDIFF=CONTROL ADJUST=DUNNETT
PDIFF=CONTROL ADJUST=BON
PDIFF=CONTROL ADJUST=SIDAK
PDIFF=CONTROL ADJUST=SCHEFFE
PDIFF=CONTROL ADJUST=SMM
PDIFF=CONTROL ADJUST=SIMULATE
N OTE : One-sided Dunnett’s tests are also available from the MEANS statement with the DUNNETTL and DUNNETTU options and from the LSMEANS statement with PDIFF=CONTROLL and
PDIFF=CONTROLU.
A note concerning the ODS tables for the results of the PDIFF or TDIFF options in the LSMEANS statement:
The p/t-values for differences are displayed in columns of the LSMeans table for PDIFF/TDIFF=CONTROL
or PDIFF/TDIFF=ANOM, and for PDIFF/TDIFF=ALL when there are only two LS-means. Otherwise (for
PDIFF/TDIFF=ALL when there are more than two LS-means), the p/t-values for differences are displayed in
a separate table called Diff.
Details of these multiple comparison methods are given in the following sections.
Pairwise Comparisons
All the methods discussed in this section depend on the standardized pairwise differences tij D .yNi
where the parts of this expression are defined as follows:
yNj /=O ij ,
• i and j are the indices of two groups
• yNi and yNj are the means or LS-means for groups i and j
• O ij is the square root of the estimated variance of yNi yNj . For simple arithmetic means, O ij2 D
s 2 .1=ni C 1=nj /, where ni and nj are the sizes of groups i and j, respectively, and s 2 is the mean
square for error, with degrees of freedom. For weighted arithmetic means, O ij2 D s 2 .1=wi C 1=wj /,
where wi and wj are the sums of the weights in groups i and j, respectively. Finally, for LS-means
defined by the linear combinations l0i b and l0j b of the parameter estimates, O ij2 D s 2 l0i .X0 X/ lj .
Furthermore, all of the methods are discussed in terms of significance tests of the form
jtij j c.˛/
where c.˛/ is some constant depending on the significance level. Such tests can be inverted to form confidence
intervals of the form
.yNi
yNj /
O ij c.˛/ i
j
.yNi
yNj / C O ij c.˛/
3522 F Chapter 44: The GLM Procedure
The simplest approach to multiple comparisons is to do a t test on every pair of means (the T option in the
MEANS statement, ADJUST=T in the LSMEANS statement). For the ith and jth means, you can reject the
null hypothesis that the population means are equal if
jtij j t .˛I /
where ˛ is the significance level, is the number of error degrees of freedom, and t .˛I / is the two-tailed
critical value from a Student’s t distribution. If the cell sizes are all equal to, say, n, the preceding formula
can be rearranged to give
r
jyNi
yNj j t .˛I /s
2
n
the value of the right-hand side being Fisher’s least significant difference (LSD).
There is a problem with repeated t tests, however. Suppose there are 10 means and each t test is performed
at the 0.05 level. There are 10.10 1/=2 D 45 pairs of means to compare, each with a 0.05 probability
of a type 1 error (a false rejection of the null hypothesis). The chance of making at least one type 1 error
is much higher than 0.05. It is difficult to calculate the exact probability, but you can derive a pessimistic
approximation by assuming that the comparisons are independent, giving an upper bound to the probability
of making at least one type 1 error (the experimentwise error rate) of
1
.1
0:05/45 D 0:90
The actual probability is somewhat less than 0.90, but as the number of means increases, the chance of
making at least one type 1 error approaches 1.
If you decide to control the individual type 1 error rates for each comparison, you are controlling the
individual or comparisonwise error rate. On the other hand, if you want to control the overall type 1 error rate
for all the comparisons, you are controlling the experimentwise error rate. It is up to you to decide whether to
control the comparisonwise error rate or the experimentwise error rate, but there are many situations in which
the experimentwise error rate should be held to a small value. Statistical methods for comparing three or more
means while controlling the probability of making at least one type 1 error are called multiple-comparison
procedures.
It has been suggested that the experimentwise error rate can be held to the ˛ level by performing the overall
ANOVA F test at the ˛ level and making further comparisons only if the F test is significant, as in Fisher’s
protected LSD. This assertion is false if there are more than three means (Einot and Gabriel 1975). Consider
again the situation with 10 means. Suppose that one population mean differs from the others by such a
sufficiently large amount that the power (probability of correctly rejecting the null hypothesis) of the F test
is near 1 but that all the other population means are equal to each other. There will be 9.9 1/=2 D 36 t
tests of true null hypotheses, with an upper limit of 0.84 on the probability of at least one type 1 error. Thus,
you must distinguish between the experimentwise error rate under the complete null hypothesis, in which all
population means are equal, and the experimentwise error rate under a partial null hypothesis, in which some
means are equal but others differ. The following abbreviations are used in the discussion:
CER
comparisonwise error rate
EERC
experimentwise error rate under the complete null hypothesis
Comparing Groups F 3523
MEER
maximum experimentwise error rate under any complete or partial null hypothesis
These error rates are associated with the different strengths of inference discussed on page 3520: individual
tests control the CER; tests for inhomogeneity of means control the EERC; tests that yield confidence
inequalities or confidence intervals control the MEER. A preliminary F test controls the EERC but not the
MEER.
You can control the MEER at the ˛ level by setting the CER to a sufficiently small value. The Bonferroni
inequality (Miller 1981) has been widely used for this purpose. If
CER D
˛
c
where c is the total number of comparisons, then the MEER is less than ˛. Bonferroni t tests (the BON option
in the MEANS statement, ADJUST=BON in the LSMEANS statement) with MEER < ˛ declare two means
to be significantly different if
jtij j t .I /
where
D
2˛
k.k 1/
for comparison of k means.
Šidák (1967) has provided a tighter bound, showing that
CER D 1
.1
˛/1=c
also ensures that MEER ˛ for any set of c comparisons. A Sidak t test (Games 1977), provided by the
SIDAK option, is thus given by
jtij j t .I /
where
D 1
.1
2
˛/ k.k
1/
for comparison of k means.
You can use the Bonferroni additive inequality and the Sidak multiplicative inequality to control the MEER
for any set of contrasts or other hypothesis tests, not just pairwise comparisons. The Bonferroni inequality
can provide simultaneous inferences in any statistical application requiring tests of more than one hypothesis.
Other methods discussed in this section for pairwise comparisons can also be adapted for general contrasts
(Miller 1981).
3524 F Chapter 44: The GLM Procedure
Scheffé (1953, 1959) proposes another method to control the MEER for any set of contrasts or other linear
hypotheses in the analysis of linear models, including pairwise comparisons, obtained with the SCHEFFE
option. Two means are declared significantly different if
jtij j p
DF F .˛I DF; /
where F .˛I DF; / is the ˛-level critical value of an F distribution with DF numerator degrees of freedom and
denominator degrees of freedom. The value of DF is k – 1 for the MEANS statement, but in other statements
the precise definition depends on context. For the LSMEANS statement, DF is the rank of the contrast matrix
L for LS-means differences. In more general contexts—for example, the ESTIMATE or LSMESTIMATE
statements in PROC GLIMMIX—DF is the rank of the contrast covariance matrix LCov.b/L0 .
Scheffé’s test is compatible with the overall ANOVA F test in that Scheffé’s method never declares a contrast
significant if the overall F test is nonsignificant. Most other multiple-comparison methods can find significant
contrasts when the overall F test is nonsignificant and, therefore, suffer a loss of power when used with a
preliminary F test.
Scheffé’s method might be more powerful than the Bonferroni or Sidak method if the number of comparisons
is large relative to the number of means. For pairwise comparisons, Sidak t tests are generally more powerful.
Tukey (1952, 1953) proposes a test designed specifically for pairwise comparisons based on the studentized
range, sometimes called the “honestly significant difference test,” that controls the MEER when the sample
sizes are equal. Tukey (1953) and Kramer (1956) independently propose a modification for unequal cell sizes.
The Tukey or Tukey-Kramer method is provided by the TUKEY option in the MEANS statement and the
ADJUST=TUKEY option in the LSMEANS statement. This method has fared extremely well in Monte Carlo
studies (Dunnett 1980). In addition, Hayter (1984) gives a proof that the Tukey-Kramer procedure controls
the MEER for means comparisons, and Hayter (1989) describes the extent to which the Tukey-Kramer
procedure has been proven to control the MEER for LS-means comparisons. The Tukey-Kramer method
is more powerful than the Bonferroni, Sidak, or Scheffé method for pairwise comparisons. Two means are
considered significantly different by the Tukey-Kramer criterion if
jtij j q.˛I k; /
where q.˛I k; / is the ˛-level critical value of a studentized range distribution of k independent normal
random variables with degrees of freedom.
Hochberg (1974) devised a method (the GT2 or SMM option) similar to Tukey’s, but it uses the studentized
maximum modulus instead of the studentized range and employs the uncorrelated t inequality of Šidák
(1967). It is proven to hold the MEER at a level not exceeding ˛ with unequal sample sizes. It is generally
less powerful than the Tukey-Kramer method and always less powerful than Tukey’s test for equal cell sizes.
Two means are declared significantly different if
jtij j m.˛I c; /
where m.˛I c; / is the ˛-level critical value of the studentized maximum modulus distribution of c independent normal random variables with degrees of freedom and c D k.k 1/=2.
Comparing Groups F 3525
Gabriel (1978) proposes another method (the GABRIEL option) based on the studentized maximum modulus.
This method is applicable only to arithmetic means. It rejects if
jyNi
s
p1
2ni
yNj j
C
p1
2nj
m.˛I k; /
For equal cell sizes, Gabriel’s test is equivalent to Hochberg’s GT2 method. For unequal cell sizes, Gabriel’s
method is more powerful than GT2 but might become liberal with highly disparate cell sizes (see also
Dunnett (1980)). Gabriel’s test is the only method for unequal sample sizes that lends itself to a graphical
representation as intervals around the means. Assuming yNi > yNj , you can rewrite the preceding inequality as
yNi
m.˛I k; / p
s
2ni
yNj C m.˛I k; / p
s
2nj
The expression on the left does not depend on j, nor does the expression on the right depend on i. Hence,
you can form what Gabriel calls an .l; u/-interval around each sample mean and declare two means to be
significantly different if their .l; u/-intervals do not overlap. See Hsu (1996, section 5.2.1.1) for a discussion
of other methods of graphically representing all pairwise comparisons.
Comparing All Treatments to a Control
One special case of means comparison is that in which the only comparisons that need to be tested are
between a set of new treatments and a single control. In this case, you can achieve better power by using a
method that is restricted to test only comparisons to the single control mean. Dunnett (1955) proposes a test
for this situation that declares a mean significantly different from the control if
jti 0 j d.˛I k; ; 1 ; : : : ; k
1/
where yN0 is the control mean and d.˛I k; ; 1 ; : : : ; k 1 / is the critical value of the “many-to-one t statistic”
(Miller 1981; Krishnaiah and Armitage 1966) for k means to be compared to a control, with error degrees
of freedom and correlations 1 ; : : : ; k 1 , i D ni =.n0 C ni /. The correlation terms arise because each of
the treatment means is being compared to the same control. Dunnett’s test holds the MEER to a level not
exceeding the stated ˛.
Analysis of Means: Comparing Each Treatments to the Average
Analysis of means (ANOM) refers to a technique for comparing group means and displaying the comparisons
graphically so that you can easily see which ones are different. Means are judged as different if they are
significantly different from the overall average, with significance adjusted for multiplicity. The overall
average is computed as a weighted mean of the LS-means, the weights being inversely proportional to the
variances. If you use the PDIFF=ANOM option in the LSMEANS statement, the procedure will display
the p-values (adjusted for multiplicity, by default) for tests of the differences between each LS-mean and
the average LS-mean. The ANOM procedure in SAS/QC software displays both tables and graphics for
the analysis of means with a variety of response types. For one-way designs, confidence intervals for
PDIFF=ANOM comparisons are equivalent to the results of PROC ANOM. The difference is that PROC
GLM directly displays the confidence intervals for the differences, while the graphical output of PROC
ANOM displays them as decision limits around the overall mean.
3526 F Chapter 44: The GLM Procedure
If the LS-means being compared are uncorrelated, exact adjusted p-values and critical values for confidence
limits can be computed; see Nelson (1982, 1991, 1993) and Guirguis and Tobias (2004). For correlated
LS-means, an approach similar to that of Hsu (1992) is employed, using a factor-analytic approximation of
the correlation between the LS-means to derive approximate “effective sample sizes” for which exact critical
values are computed. Note that computing the exact adjusted p-values and critical values for unbalanced
designs can be computationally intensive. A simulation-based approach, as specified by the ADJUST=SIM
option, while nondeterministic, might provide inferences that are accurate enough in much less time. See the
section “Approximate and Simulation-Based Methods” on page 3526 for more details.
Approximate and Simulation-Based Methods
Tukey’s, Dunnett’s, and Nelson’s tests are all based on the same general quantile calculation:
q t .˛; ; R/ D fq 3 P .max.jt1 j; : : : ; jtn j/ > q/ D ˛g
where the ti have a joint multivariate t distribution with degrees of freedom and correlation matrix R. In
general, evaluating q t .˛; ; R/ requires repeated numerical calculation of an .n C 1/-fold integral. This is
usually intractable, but the problem reduces to a feasible 2-fold integral when R has a certain symmetry in the
case of Tukey’s test, and a factor analytic structure (Hsu 1992) in the case of Dunnett’s and Nelson’s tests.
The R matrix has the required symmetry for exact computation of Tukey’s test in the following two cases:
• The ti s are studentized differences between k.k
variances—that is, equal sample sizes.
1/=2 pairs of k uncorrelated means with equal
• The ti s are studentized differences between k.k 1/=2 pairs of k LS-means from a variance-balanced
design (for example, a balanced incomplete block design).
See Hsu (1992, 1996) for more information. The R matrix has the factor analytic structure for exact
computation of Dunnett’s and Nelson’s tests in the following two cases:
• if the ti s are studentized differences between k – 1 means and a control mean, all uncorrelated.
(Dunnett’s one-sided methods depend on a similar probability calculation, without the absolute values.)
Note that it is not required that the variances of the means (that is, the sample sizes) be equal.
• if the ti s are studentized differences between k – 1 LS-means and a control LS-mean from either a
variance-balanced design, or a design in which the other factors are orthogonal to the treatment factor
(for example, a randomized block design with proportional cell frequencies)
However, other important situations that do not result in a correlation matrix R that has the structure for exact
computation are the following:
• all pairwise differences with unequal sample sizes
• differences between LS-means in many unbalanced designs
In these situations, exact calculation of q t .˛; ; R/ is intractable in general. Most of the preceding methods
can be viewed as using various approximations for q t .˛; ; R/. When the sample sizes are unequal, the TukeyKramer test is equivalent to another approximation. For comparisons with a control when the correlation
R does not have a factor analytic structure, Hsu (1992) suggests approximating R with a matrix R that
Comparing Groups F 3527
does have such a structure and correspondingly approximating q t .˛; ; R/ with q t .˛; ; R /. When you
request Dunnett’s or Nelson’s test for LS-means (the PDIFF=CONTROL and ADJUST=DUNNETT options
or the PDIFF=ANOM and ADJUST=NELSON options, respectively), the GLM procedure automatically
uses Hsu’s approximation when appropriate.
Finally, Edwards and Berry (1987) suggest calculating q t .˛; ; R/ by simulation. Multivariate t vectors are
sampled from a distribution with the appropriate and R parameters, and Edwards and Berry (1987) suggest
estimating q t .˛; ; R/ by q,
O the ˛ percentile of the observed values of max.jt1 j; : : : ; jtn j/. Sufficient samples
are generated for the true P .max.jt1 j; : : : ; jtn j/ > q/
O to be within a certain accuracy radius of ˛ with
accuracy confidence 100.1 /. You can approximate q t .˛; ; R/ by simulation for comparisons between
LS-means by specifying ADJUST=SIM (with any PDIFF= type). By default, D 0:005 and D 0:01, so
that the tail area of qO is within 0.005 of ˛ with 99% confidence. You can use the ACC= and EPS= options
with ADJUST=SIM to reset and , or you can use the NSAMP= option to set the sample size directly. You
can also control the random number sequence with the SEED= option.
Hsu and Nelson (1998) suggest a more accurate simulation method for estimating q t .˛; ; R/, using a
control variate adjustment technique. The same independent, standardized normal variates that are used to
generate multivariate t vectors from a distribution with the appropriate and R parameters are also used
to generate multivariate t vectors from a distribution for which the exact value of q t .˛; ; R/ is known.
max.jt1 j; : : : ; jtn j/ for the second sample is used as a control variate for adjusting the quantile estimate
based on the first sample; see Hsu and Nelson (1998) for more details. The control variate adjustment has
the drawback that it takes somewhat longer than the crude technique of Edwards and Berry (1987), but it
typically yields an estimate that is many times more accurate. In most cases, if you are using ADJUST=SIM,
then you should specify ADJUST=SIM(CVADJUST). You can also specify ADJUST=SIM(CVADJUST
REPORT) to display a summary of the simulation that includes, among other things, the actual accuracy
radius , which should be substantially smaller than the target accuracy radius (0.005 by default).
Multiple-Stage Tests
You can use all of the methods discussed so far to obtain simultaneous confidence intervals (Miller 1981). By
sacrificing the facility for simultaneous estimation, you can obtain simultaneous tests with greater power by
using multiple-stage tests (MSTs). MSTs come in both step-up and step-down varieties (Welsch 1977). The
step-down methods, which have been more widely used, are available in SAS/STAT software.
Step-down MSTs first test the homogeneity of all the means at a level k . If the test results in a rejection,
then each subset of k – 1 means is tested at level k 1 ; otherwise, the procedure stops. In general, if the
hypothesis of homogeneity of a set of p means is rejected at the p level, then each subset of p 1 means is
tested at the p 1 level; otherwise, the set of p means is considered not to differ significantly and none of
its subsets are tested. The many varieties of MSTs that have been proposed differ in the levels p and the
statistics on which the subset tests are based. Clearly, the EERC of a step-down MST is not greater than k ,
and the CER is not greater than 2 , but the MEER is a complicated function of p , p D 2; : : : ; k.
With unequal cell sizes, PROC GLM uses the harmonic mean of the cell sizes as the common sample size.
However, since the resulting operating characteristics can be undesirable, MSTs are recommended only for
the balanced case. When the sample sizes are equal, using the range statistic enables you to arrange the means
in ascending or descending order and test only contiguous subsets. But if you specify the F statistic, this
shortcut cannot be taken. For this reason, only range-based MSTs are implemented. It is common practice
to report the results of an MST by writing the means in such an order and drawing lines parallel to the list
of means spanning the homogeneous subsets. This form of presentation is also convenient for pairwise
comparisons with equal cell sizes.
3528 F Chapter 44: The GLM Procedure
The best-known MSTs are the Duncan (the DUNCAN option) and Student-Newman-Keuls (the SNK option)
methods (Miller 1981). Both use the studentized range statistic and, hence, are called multiple range tests.
Duncan’s method is often called the “new” multiple range test despite the fact that it is one of the oldest
MSTs in current use.
The Duncan and SNK methods differ in the p values used. For Duncan’s method, they are
p D 1
.1
˛/p
1
whereas the SNK method uses
p D ˛
Duncan’s method controls the CER at the ˛ level. Its operating characteristics appear similar to those of
Fisher’s unprotected LSD or repeated t tests at level ˛ (Petrinovich and Hardyck 1969). Since repeated t
tests are easier to compute, easier to explain, and applicable to unequal sample sizes, Duncan’s method is
not recommended. Several published studies (for example, Carmer and Swanson (1973)) have claimed that
Duncan’s method is superior to Tukey’s because of greater power without considering that the greater power
of Duncan’s method is due to its higher type 1 error rate (Einot and Gabriel 1975).
The SNK method holds the EERC to the ˛ level but does not control the MEER (Einot and Gabriel 1975).
Consider ten population means that occur in five pairs such that means within a pair are equal, but there
are large differences between pairs. If you make the usual sampling assumptions and also assume that the
sample sizes are very large, all subset homogeneity hypotheses for three or more means are rejected. The
SNK method then comes down to five independent tests, one for each pair, each at the ˛ level. Letting ˛ be
0.05, the probability of at least one false rejection is
1
.1
0:05/5 D 0:23
As the number of means increases, the MEER approaches 1. Therefore, the SNK method cannot be
recommended.
A variety of MSTs that control the MEER have been proposed, but these methods are not as well known as
those of Duncan and SNK. One approach (Ryan 1959, 1960; Einot and Gabriel 1975; Welsch 1977) sets
(
p D
1
.1
˛/p=k
˛
for p < k
1
for p k
1
You can use range statistics, leading to what is called the REGWQ method, after the authors’ initials. If you
assume that the sample means have been arranged in descending order from yN1 through yNk , the homogeneity
of means yNi ; : : : ; yNj ; i < j , is rejected by REGWQ if
yNi
yNj
s
q.p I p; / p
n
where p D j i C 1 and the summations are over u D i; : : : ; j (Einot and Gabriel 1975). To ensure that
the MEER is controlled, the current implementation checks whether q.p I p; / is monotonically increasing
in p. If not, then a set of critical values that are increasing in p is substituted instead.
Comparing Groups F 3529
REGWQ appears to be the most powerful step-down MST in the current literature (for example, Ramsey
1978). Use of a preliminary F test decreases the power of all the other multiple-comparison methods discussed
previously except for Scheffé’s test.
Bayesian Approach
Waller and Duncan (1969) and Duncan (1975) take an approach to multiple comparisons that differs from all
the methods previously discussed in minimizing the Bayes risk under additive loss rather than controlling
ij
ij
type 1 error rates. For each pair of population means i and j , null .H0 / and alternative .Ha / hypotheses
are defined:
ij
H0 W
i
j 0
Haij W
i
j > 0
ij
ij
For any i, j pair, let d0 indicate a decision in favor of H0 and da indicate a decision in favor of Ha , and let
ı D i j . The loss function for the decision on the i, j pair is
(
L.d0 j ı/ D
0
if ı 0
ı
if ı > 0
(
L.da j ı/ D
kı
0
if ı 0
if ı > 0
where k represents a constant that you specify rather than the number of means. The loss for the joint decision
involving all pairs of means is the sum of the losses for each individual decision. The population means are
assumed to have a normal prior distribution with unknown variance, the logarithm of the variance of the
means having a uniform prior distribution. For the i, j pair, the null hypothesis is rejected if
r
yNi
yNj
tB s
2
n
where tB is the Bayesian t value (Waller and Kemp 1976) depending on k, the F statistic for the one-way
ANOVA, and the degrees of freedom for F. The value of tB is a decreasing function of F, so the Waller-Duncan
test (specified by the WALLER option) becomes more liberal as F increases.
Recommendations
In summary, if you are interested in several individual comparisons and are not concerned about the effects
of multiple inferences, you can use repeated t tests or Fisher’s unprotected LSD. If you are interested in all
pairwise comparisons or all comparisons with a control, you should use Tukey’s or Dunnett’s test, respectively,
in order to make the strongest possible inferences. If you have weaker inferential requirements and, in
particular, if you do not want confidence intervals for the mean differences, you should use the REGWQ
method. Finally, if you agree with the Bayesian approach and Waller and Duncan’s assumptions, you should
use the Waller-Duncan test.
3530 F Chapter 44: The GLM Procedure
Interpretation of Multiple Comparisons
When you interpret multiple comparisons, remember that failure to reject the hypothesis that two or more
means are equal should not lead you to conclude that the population means are, in fact, equal. Failure to
reject the null hypothesis implies only that the difference between population means, if any, is not large
enough to be detected with the given sample size. A related point is that nonsignificance is nontransitive: that
is, given three sample means, the largest and smallest might be significantly different from each other, while
neither is significantly different from the middle one. Nontransitive results of this type occur frequently in
multiple comparisons.
Multiple comparisons can also lead to counterintuitive results when the cell sizes are unequal. Consider
four cells labeled A, B, C, and D, with sample means in the order A>B>C>D. If A and D each have two
observations, and B and C each have 10,000 observations, then the difference between B and C might be
significant, while the difference between A and D is not.
Simple Effects
Suppose you use the following statements to fit a full factorial model to a two-way design:
data twoway;
input A B Y
datalines;
1 1 10.6
1 1
1 2 -0.2
1 2
1 3 0.1
1 3
2 1 19.7
2 1
2 2 -0.2
2 2
2 3 -0.9
2 3
3 1 29.7
3 1
3 2 1.5
3 2
3 3 0.2
3 3
;
@@;
11.0
1.3
0.4
19.3
0.5
-0.1
29.6
0.2
0.4
1
1
1
2
2
2
3
3
3
1
2
3
1
2
3
1
2
3
10.6
-0.2
-0.4
18.5
0.8
-0.2
29.0
-1.5
-0.4
1
1
1
2
2
2
3
3
3
1
2
3
1
2
3
1
2
3
11.3
0.2
1.0
20.4
-0.4
-1.7
30.2
1.3
-2.2
proc glm data=twoway;
class A B;
model Y = A B A*B;
run;
Partial results for the analysis of variance are shown in Figure 44.21. The Type I and Type III results are the
same because this is a balanced design.
Figure 44.21 Two-Way Design with Significant Interaction
The GLM Procedure
Dependent Variable: Y
Source
A
B
A*B
DF
Type I SS
Mean Square
F Value
Pr > F
2
2
4
219.905000
3206.101667
487.103333
109.952500
1603.050833
121.775833
165.11
2407.25
182.87
<.0001
<.0001
<.0001
Comparing Groups F 3531
Figure 44.21 continued
Source
A
B
A*B
DF
Type III SS
Mean Square
F Value
Pr > F
2
2
4
219.905000
3206.101667
487.103333
109.952500
1603.050833
121.775833
165.11
2407.25
182.87
<.0001
<.0001
<.0001
The interaction A*B is significant, indicating that the effect of A depends on the level of B. In some cases,
you might be interested in looking at the differences between predicted values across A for different levels
of B. Winer (1971) calls this the simple effects of A. You can compute simple effects with the LSMEANS
statement by specifying the SLICE= option. In this case, since the GLM procedure is interactive, you can
compute the simple effects of A by submitting the following statements after the preceding statements.
lsmeans A*B / slice=B;
run;
The results are shown Figure 44.22. Note that A has a significant effect for B=1 but not for B=2 and B=3.
Figure 44.22 Interaction LS-Means and Simple Effects
The GLM Procedure
Least Squares Means
A
B
Y LSMEAN
1
1
1
2
2
2
3
3
3
1
2
3
1
2
3
1
2
3
10.8750000
0.2750000
0.2750000
19.4750000
0.1750000
-0.7250000
29.6250000
0.3750000
-0.5000000
The GLM Procedure
Least Squares Means
A*B Effect Sliced by B for Y
B
DF
Sum of
Squares
Mean Square
F Value
Pr > F
1
2
3
2
2
2
704.726667
0.080000
2.201667
352.363333
0.040000
1.100833
529.13
0.06
1.65
<.0001
0.9418
0.2103
3532 F Chapter 44: The GLM Procedure
Homogeneity of Variance in One-Way Models
One of the usual assumptions in using the GLM procedure is that the underlying errors are all uncorrelated
with homogeneous variances (see page 3494). You can test this assumption in PROC GLM by using the
HOVTEST option in the MEANS statement, requesting a homogeneity of variance test. This section discusses
the computational details behind these tests. Note that the GLM procedure allows homogeneity of variance
testing for simple one-way models only. Homogeneity of variance testing for more complex models is a
subject of current research.
Bartlett (1937) proposes a test for equal variances that is a modification of the normal-theory likelihood ratio
test (the HOVTEST=BARTLETT option). While Bartlett’s test has accurate Type I error rates and optimal
power when the underlying distribution of the data is normal, it can be very inaccurate if that distribution is
even slightly nonnormal (Box 1953). Therefore, Bartlett’s test is not recommended for routine use.
An approach that leads to tests that are much more robust to the underlying distribution is to transform the
original values of the dependent variable to derive a dispersion variable and then to perform analysis of
variance on this variable. The significance level for the test of homogeneity of variance is the p-value for the
ANOVA F test on the dispersion variable. All of the homogeneity of variance tests available in PROC GLM
except Bartlett’s use this approach.
Levene’s test (Levene 1960) is widely considered to be the standard homogeneity of variance test (the
HOVTEST=LEVENE option). Levene’s test is of the dispersion-variable-ANOVA form discussed previously,
where the dispersion variable is either of the following:
2
zij
zij
D .yij
D jyij
yNi /2 (TYPE=SQUARE, the default)
yNi j
(TYPE=ABS)
2
O’Brien (1979) proposes a test (HOVTEST=OBRIEN) that is basically a modification of Levene’s zij
, using
the dispersion variable
W
zij
D
.W C ni
2/ni .yij yNi /2 W .ni
.ni 1/.ni 2/
1/i2
where ni is the size of the ith group and i2 is its sample variance. You can use the W= option in parentheses
W
to tune O’Brien’s zij
dispersion variable to match the suspected kurtosis of the underlying distribution.
The choice of the value of the W= option is rarely critical. By default, W=0.5, as suggested by O’Brien
(1979, 1981).
Finally, Brown and Forsythe (1974) suggest using the absolute deviations from the group medians:
BF
zij
D jyij
mi j
where mi is the median of the ith group. You can use the HOVTEST=BF option to specify this test.
Simulation results (Conover, Johnson, and Johnson 1981; Olejnik and Algina 1987) show that, while all of
these ANOVA-based tests are reasonably robust to the underlying distribution, the Brown-Forsythe test seems
best at providing power to detect variance differences while protecting the Type I error probability. However,
since the within-group medians are required for the Brown-Forsythe test, it can be resource intensive if there
are very many groups or if some groups are very large.
Comparing Groups F 3533
If one of these tests rejects the assumption of homogeneity of variance, you should use Welch’s ANOVA
instead of the usual ANOVA to test for differences between group means. However, this conclusion holds
only if you use one of the robust homogeneity of variance tests (that is, not for HOVTEST=BARTLETT);
even then, any homogeneity of variance test has too little power to be relied upon to always detect when
Welch’s ANOVA is appropriate. Unless the group variances are extremely different or the number of groups
is large, the usual ANOVA test is relatively robust when the groups are all about the same size. As Box
(1953) notes, “To make the preliminary test on variances is rather like putting to sea in a rowing boat to find
out whether conditions are sufficiently calm for an ocean liner to leave port!”
Example 44.10 illustrates the use of the HOVTEST and WELCH options in the MEANS statement in testing
for equal group variances and adjusting for unequal group variances in a one-way ANOVA.
Weighted Means
If you specify a WEIGHT statement and one or more of the multiple comparisons options, the variance of
the difference between weighted group means for group i and j is computed as
1
1
C
MSE wi
wj
where wi is the sum of the weights for the observations in group i.
Construction of Least Squares Means
To construct a least squares mean (LS-mean) for a given level of a given effect, construct a row vector L
according to the following rules and use it in an ESTIMATE statement to compute the value of the LS-mean:
1. Set all Li corresponding to covariates (continuous variables) to their mean value.
2. Consider effects contained by the given effect. Set the Li corresponding to levels associated with the
given level equal to 1. Set all other Li in these effects equal to 0. (See Chapter 15, “The Four Types of
Estimable Functions,” for a definition of containing.)
3. Consider the given effect. Set the Li corresponding to the given level equal to 1. Set the Li
corresponding to other levels equal to 0.
4. Consider the effects that contain the given effect. If these effects are not nested within the given effect,
then set the Li corresponding to the given level to 1=k, where k is the number of such columns. If
these effects are nested within the given effect, then set the Li corresponding to the given level to
1=.k1 k2 /, where k1 is the number of nested levels within this combination of nested effects, and k2 is
the number of such combinations. For Li corresponding to other levels, use 0.
5. Consider the other effects not yet considered. If there are no nested factors, then set all Li corresponding
to this effect to 1=j , where j is the number of levels in the effect. If there are nested factors, then set all
Li corresponding to this effect to 1=.j1 j2 /, where j1 is the number of nested levels within a given
combination of nested effects and j2 is the number of such combinations.
The consequence of these rules is that the sum of the Xs within any classification effect is 1. This set of Xs
forms a linear combination of the parameters that is checked for estimability before it is evaluated.
3534 F Chapter 44: The GLM Procedure
For example, consider the following model:
proc glm;
class A B C;
model Y=A B A*B C Z;
lsmeans A B A*B C;
run;
Assume A has 3 levels, B has 2 levels, and C has 2 levels, and assume that every combination of levels of A
and B exists in the data. Assume also that Z is a continuous variable with an average of 12.5. Then the least
squares means are computed by the following linear combinations of the parameter estimates:
LSM( )
1
LSM(A1)
LSM(A2)
LSM(A3)
1
1
1
LSM(B1)
LSM(B2)
1
1
LSM(AB11)
LSM(AB12)
LSM(AB21)
LSM(AB22)
LSM(AB31)
LSM(AB32)
1
1
1
1
1
1
LSM(C1)
LSM(C2)
1
1
A
B
1 2 3
1/3 1/3 1/3
1 2
1/2 1/2
A*B
11 12 21 22 31 32
1/6 1/6 1/6 1/6 1/6 1/6
1 2
1/2 1/2
12.5
1/2 1/2
1/2 1/2
1/2 1/2
1/2 1/2 0 0 0 0
0 0 1/2 1/2 0 0
0 0 0 0 1/2 1/2
1/2 1/2
1/2 1/2
1/2 1/2
12.5
12.5
12.5
1/3 0 1/3 0 1/3 0
0 1/3 0 1/3 0 1/3
1/2 1/2
1/2 1/2
12.5
12.5
1/2
1/2
1/2
1/2
1/2
1/2
1/2
1/2
1/2
1/2
1/2
1/2
12.5
12.5
12.5
12.5
12.5
12.5
1
0
0
1
12.5
12.5
1
0
0
0
1
0
0
0
1
1/3 1/3 1/3
1/3 1/3 1/3
1
1
0
0
0
0
0
0
1
1
0
0
0
0
0
0
1
1
1/3 1/3 1/3
1/3 1/3 1/3
1
0
0
1
1
0
1
0
1
0
0
1
0
1
0
1
1/2 1/2
1/2 1/2
1
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
0
0
0
1
1/6 1/6 1/6 1/6 1/6 1/6
1/6 1/6 1/6 1/6 1/6 1/6
C
Z
Setting Covariate Values
By default, all covariate effects are set equal to their mean values for computation of standard LS-means.
The AT option in the LSMEANS statement enables you to set the covariates to whatever values you consider
interesting.
If there is an effect containing two or more covariates, the AT option sets the effect equal to the product of
the individual means rather than the mean of the product (as with standard LS-means calculations). The AT
MEANS option leaves covariates equal to their mean values (as with standard LS-means) and incorporates
this adjustment to crossproducts of covariates.
As an example, the following is a model with a classification variable A and two continuous variables, x1 and
x2:
class A;
model y = A x1 x2 x1*x2;
The coefficients for the continuous effects with various AT specifications are shown in the following table.
Comparing Groups F 3535
Syntax
lsmeans
lsmeans
lsmeans
lsmeans
A;
A / at means;
A / at x1=1.2;
A / at (x1 x2)=(1.2 0.3);
x1
x2
x1*x2
x1
x1
1.2
1.2
x2
x2
x2
0.3
x1 x2
x1 x2
1:2 x2
1:2 0:3
For the first two LSMEANS statements, the A LS-mean coefficient for x1 is x1 (the mean of x1) and for
x2 is x2 (the mean of x2). However, for the first LSMEANS statement, the coefficient for x1*x2 is x1 x2 ,
but for the second LSMEANS statement the coefficient is x1 x2 . The third LSMEANS statement sets the
coefficient for x1 equal to 1.2 and leaves that for x2 at x2 , and the final LSMEANS statement sets these
values to 1.2 and 0.3, respectively.
Even if you specify a WEIGHT variable, the unweighted covariate means are used for the covariate coefficients if there is no AT specification. However, if you also use an AT specification, then weighted covariate
means are used for the covariate coefficients for which no explicit AT values are given, or if you specify
AT MEANS. Also, observations with missing dependent variables are included in computing the covariate
means, unless these observations form a missing cell. You can use the E option in conjunction with the AT
option to check that the modified LS-means coefficients are the ones you want.
The AT option is disabled if you specify the BYLEVEL option, in which case the coefficients for the
covariates are set equal to their means within each level of the LS-mean effect in question.
Changing the Weighting Scheme
The standard LS-means have equal coefficients across classification effects; however, the OM option in the
LSMEANS statement changes these coefficients to be proportional to those found in the input data set. This
adjustment is reasonable when you want your inferences to apply to a population that is not necessarily
balanced but has the margins observed in the original data set.
In computing the observed margins, PROC GLM uses all observations for which there are no missing
independent variables, including those for which there are missing dependent variables. Also, if there is
a WEIGHT variable, PROC GLM uses weighted margins to construct the LS-means coefficients. If the
analysis data set is balanced or if you specify a simple one-way model, the LS-means will be unchanged by
the OM option.
The BYLEVEL option modifies the observed-margins LS-means. Instead of computing the margins across
the entire data set, PROC GLM computes separate margins for each level of the LS-mean effect in question.
The resulting LS-means are actually equal to raw means in this case. The BYLEVEL option disables the AT
option if it is specified.
Note that the MIXED procedure implements a more versatile form of the OM option, enabling you to
specifying an alternative data set over which to compute observed margins. If you use the BYLEVEL option,
too, then this data set is effectively the “population” over which the population marginal means are computed.
See Chapter 63, “The MIXED Procedure,” for more information.
You might want to use the E option in conjunction with either the OM or BYLEVEL option to check that
the modified LS-means coefficients are the ones you want. It is possible that the modified LS-means are not
estimable when the standard ones are, or vice versa.
3536 F Chapter 44: The GLM Procedure
Estimability of LS-means
LS-means are defined as certain linear combinations of the parameters. As such, it is possible for them to be
inestimable. In fact, it is possible for a pair of LS-means to be both inestimable but their difference estimable.
When this happens, only the entries corresponding to the estimable difference are computed and displayed
in the Diffs table. If ADJUST=SIMULATE is specified when there are inestimable LS-means differences,
adjusted results for all differences are displayed as missing.
Multivariate Analysis of Variance
If you fit several dependent variables to the same effects, you might want to make joint tests involving
parameters of several dependent variables. Suppose you have p dependent variables, k parameters for each
dependent variable, and n observations. The models can be collected into one equation:
Y D Xˇ C where Y is n p, X is n k, ˇ is k p, and is n p. Each of the p models can be estimated and
tested separately. However, you might also want to consider the joint distribution and test the p models
simultaneously.
For multivariate tests, you need to make some assumptions about the errors. With p dependent variables,
there are n p errors that are independent across observations but not across dependent variables. Assume
vec./ N.0; In ˝ †/
where vec./ strings out by rows, ˝ denotes Kronecker product multiplication, and † is p p. † can be
estimated by
SD
e0 e
.Y
D
n r
Xb/0 .Y
n r
Xb/
where b D .X0 X/ X0 Y, r is the rank of the X matrix, and e is the matrix of residuals.
If S is scaled to unit diagonals, the values in S are called partial correlations of the Ys adjusting for the Xs.
This matrix can be displayed by PROC GLM if PRINTE is specified as a MANOVA option.
The multivariate general linear hypothesis is written
LˇM D 0
You can form hypotheses for linear combinations across columns, as well as across rows of ˇ.
The MANOVA statement of the GLM procedure tests special cases where L corresponds to Type I, Type II,
Type III, or Type IV tests, and M is the p p identity matrix. These tests are joint tests that the given type of
hypothesis holds for all dependent variables in the model, and they are often sufficient to test all hypotheses
of interest.
Finally, when these special cases are not appropriate, you can specify your own L and M matrices by using the
CONTRAST statement before the MANOVA statement and the M= specification in the MANOVA statement,
respectively. Another alternative is to use a REPEATED statement, which automatically generates a variety
of M matrices useful in repeated measures analysis of variance. See the section “REPEATED Statement” on
page 3488 and the section “Repeated Measures Analysis of Variance” on page 3537 for more information.
Repeated Measures Analysis of Variance F 3537
One useful way to think of a MANOVA analysis with an M matrix other than the identity is as an analysis of
a set of transformed variables defined by the columns of the M matrix. You should note, however, that PROC
GLM always displays the M matrix in such a way that the transformed variables are defined by the rows, not
the columns, of the displayed M matrix.
All multivariate tests carried out by the GLM procedure first construct the matrices H and E corresponding to
the numerator and denominator, respectively, of a univariate F test:
H D M0 .Lb/0 .L.X0 X/ L0 /
E D M0 .Y0 Y
1
.Lb/M
b0 .X0 X/b/M
The diagonal elements of H and E correspond to the hypothesis and error SS for univariate tests. When
the M matrix is the identity matrix (the default), these tests are for the original dependent variables on the
left side of the MODEL statement. When an M matrix other than the identity is specified, the tests are for
transformed variables defined by the columns of the M matrix. These tests can be studied by requesting the
SUMMARY option, which produces univariate analyses for each original or transformed variable.
Four multivariate test statistics, all functions of the eigenvalues of E
1H
(or .E C H/
1 H),
are constructed:
• Wilks’ lambda = det.E//det.H C E/
• Pillai’s trace = trace.H.H C E/
1/
• Hotelling-Lawley trace = trace.E
1 H/
• Roy’s greatest root = , largest eigenvalue of E
1H
By default, all four are reported with p-values based on F approximations, as discussed in the “Multivariate Tests” section in Chapter 4, “Introduction to Regression Procedures.” Alternatively, if you specify
MSTAT=EXACT in the associated MANOVA or REPEATED statement, p-values for three of the four tests
are computed exactly (Wilks’ lambda, the Hotelling-Lawley trace, and Roy’s greatest root), and the p-values
for the fourth (Pillai’s trace) are based on an F approximation that is more accurate than the default. See the
“Multivariate Tests” section in Chapter 4, “Introduction to Regression Procedures,” for more details on the
exact calculations.
Repeated Measures Analysis of Variance
When several measurements are taken on the same experimental unit (person, plant, machine, and so on), the
measurements tend to be correlated with each other. When the measurements represent qualitatively different
things, such as weight, length, and width, this correlation is best taken into account by use of multivariate
methods, such as multivariate analysis of variance. When the measurements can be thought of as responses
to levels of an experimental factor of interest, such as time, treatment, or dose, the correlation can be taken
into account by performing a repeated measures analysis of variance.
PROC GLM provides both univariate and multivariate tests for repeated measures for one response. For an
overall reference on univariate repeated measures, see Winer (1971). The multivariate approach is covered
3538 F Chapter 44: The GLM Procedure
in Cole and Grizzle (1966). For a discussion of the relative merits of the two approaches, see LaTour and
Miniard (1983).
Another approach to analysis of repeated measures is via general mixed models. This approach can handle
balanced as well as unbalanced or missing within-subject data, and it offers more options for modeling the
within-subject covariance. The main drawback of the mixed models approach is that it generally requires
iteration and, thus, might be less computationally efficient. For further details on this approach, see Chapter 63,
“The MIXED Procedure,” and Wolfinger and Chang (1995).
Organization of Data for Repeated Measure Analysis
In order to deal efficiently with the correlation of repeated measures, the GLM procedure uses the multivariate
method of specifying the model, even if only a univariate analysis is desired. In some cases, data might
already be entered in the univariate mode, with each repeated measure listed as a separate observation along
with a variable that represents the experimental unit (subject) on which measurement is taken. Consider the
following data set Old:
data Old;
input Subject Group Time y;
datalines;
1 1 1 15
1 1 2 19
1 1 3 25
2 1 1 21
2 1 2 18
2 1 3 17
1 2 1 14
1 2 2 12
1 2 3 16
2 2 1 11
2 2 2 20
2 2 3 21
... more lines ...
10 3 1 14
10 3 2 18
10 3 3 16
;
There are three observations for each subject, corresponding to measurements taken at times 1, 2, and 3.
These data could be analyzed using the following statements:
proc glm data=Old;
class Group Subject Time;
model y=Group Subject(Group) Time Group*Time;
test h=Group e=Subject(Group);
run;
However, this analysis assumes subjects’ measurements are uncorrelated across time. A repeated measures
analysis does not make this assumption. It uses the following data set New:
Repeated Measures Analysis of Variance F 3539
data New;
input Group y1 y2 y3;
datalines;
1 15 19 25
1 21 18 17
2 14 12 16
2 11 20 21
2 24 15 12
... more lines ...
3
;
14 18 16
In the data set New, the three measurements for a subject are all in one observation. For example, the
measurements for subject 1 for times 1, 2, and 3 are 15, 19, and 25, respectively. For these data, the
statements for a repeated measures analysis (assuming default options) are
proc glm data=New;
class Group;
model y1-y3 = Group / nouni;
repeated Time;
run;
To convert the univariate form of repeated measures data to the multivariate form, you can use a program like
the following:
proc sort data=Old;
by Group Subject;
run;
data New(keep=y1-y3 Group);
array yy(3) y1-y3;
do Time = 1 to 3;
set Old;
by Group Subject;
yy(Time) = y;
if last.Subject then return;
end;
run;
Alternatively, you could use PROC TRANSPOSE to achieve the same results with a program like this one:
proc sort data=Old;
by Group Subject;
run;
proc transpose out=New(rename=(_1=y1 _2=y2 _3=y3));
by Group Subject;
id Time;
run;
See the discussions in SAS Language Reference: Concepts for more information about rearrangement of data
sets.
3540 F Chapter 44: The GLM Procedure
Hypothesis Testing in Repeated Measures Analysis
In repeated measures analysis of variance, the effects of interest are as follows:
• between-subject effects (such as GROUP in the previous example)
• within-subject effects (such as TIME in the previous example)
• interactions between the two types of effects (such as GROUP*TIME in the previous example)
Repeated measures analyses are distinguished from MANOVA because of interest in testing hypotheses
about the within-subject effects and the within-subject-by-between-subject interactions.
For tests that involve only between-subjects effects, both the multivariate and univariate approaches give
rise to the same tests. These tests are provided for all effects in the MODEL statement, as well as for any
CONTRASTs specified. The ANOVA table for these tests is labeled “Tests of Hypotheses for Between
Subjects Effects” in the PROC GLM results. These tests are constructed by first adding together the dependent
variables in the model. Then an analysis of variance is performed on the sum divided by the square root of
the number of dependent variables. For example, the statements
model y1-y3=group;
repeated time;
p
give a one-way analysis of variance that uses .Y 1 C Y 2 C Y 3/= 3 as the dependent variable for performing
tests of hypothesis on the between-subject effect GROUP. Tests for between-subject effects are equivalent to
tests of the hypothesis LˇM D 0, where M is simply a vector of 1s.
For within-subject effects and for within-subject-by-between-subject interaction effects, the univariate and
multivariate approaches yield different tests. These tests are provided for the within-subject effects and for
the interactions between these effects and the other effects in the MODEL statement, as well as for any
CONTRASTs specified. The univariate tests are displayed in a table labeled “Univariate Tests of Hypotheses
for Within Subject Effects.” Results for multivariate tests are displayed in a table labeled “Repeated Measures
Analysis of Variance.”
The multivariate tests provided for within-subjects effects and interactions involving these effects are Wilks’
lambda, Pillai’s trace, Hotelling-Lawley trace, and Roy’s greatest root. For further details on these four
statistics, see the “Multivariate Tests” section in Chapter 4, “Introduction to Regression Procedures.” As an
example, the statements
model y1-y3=group;
repeated time;
produce multivariate tests for the within-subject effect TIME and the interaction TIME*GROUP.
The multivariate tests for within-subject effects are produced by testing the hypothesis LˇM D 0, where the
L matrix is the usual matrix corresponding to the Type I, Type II, Type III, or Type IV hypotheses test, and
the M matrix is one of several matrices depending on the transformation that you specify in the REPEATED
statement. These multivariate tests require that the column rank of M be less than or equal to the number of
error degrees of freedom. Besides that, the only assumption required for valid tests is that the dependent
variables in the model have a multivariate normal distribution with a common covariance matrix across the
between-subject effects.
Repeated Measures Analysis of Variance F 3541
The univariate tests for within-subject effects and interactions involving these effects require some assumptions for the probabilities provided by the ordinary F tests to be correct. Specifically, these tests require certain
patterns of covariance matrices, known as Type H covariances (Huynh and Feldt 1970). Data with these
patterns in the covariance matrices are said to satisfy the Huynh-Feldt condition. You can test this assumption
(and the Huynh-Feldt condition) by applying a sphericity test (Anderson 1958) to any set of variables defined
by an orthogonal contrast transformation. Such a set of variables is known as a set of orthogonal components.
When you use the PRINTE option in the REPEATED statement, this sphericity test is applied both to the
transformed variables defined by the REPEATED statement and to a set of orthogonal components if the
specified transformation is not orthogonal. It is the test applied to the orthogonal components that is important
in determining whether your data have a Type H covariance structure. When there are only two levels of
the within-subject effect, there is only one transformed variable, and a sphericity test is not needed. The
sphericity test is labeled “Test for Sphericity” in the output.
If your data satisfy the preceding assumptions, use the usual F tests to test univariate hypotheses for the
within-subject effects and associated interactions.
If your data do not satisfy the assumption of Type H covariance, an adjustment to numerator and denominator
degrees of freedom can be used. Several such adjustments, based on a degrees-of-freedom adjustment factor
known as (epsilon) (Box 1954), are provided in PROC GLM. All these adjustments estimate and then
multiply the numerator and denominator degrees of freedom by this estimate before determining significance
levels for the F tests. Significance levels associated with the adjusted tests are labeled “Adj Pr > F” in the
output. Two such adjustments are displayed. One is the maximum likelihood estimate of Box’s factor,
which is known to be conservative, possibly very much so. The other adjustment is intended to be unbiased
although possibly at the cost of being liberal. The first adjustment is labeled as the “Greenhouse-Geisser
Epsilon.” It has the form
OGG D
trace2 .E/=b
trace.E2 /
where E is the error matrix for the corresponding multivariate test and b is the degrees of freedom for the
hypothesis being tested. OGG was initially proposed for use in data analysis by Greenhouse and Geisser
(1959). Significance levels associated with F tests thus adjusted are labeled “G-G” in the output.
Huynh and Feldt (1976) showed that OGG tends to be biased downward (that is, conservative), especially for
small samples. Alternative estimates have been proposed to overcome this conservative bias, and there are
several options for which estimate to display along with OGG .
• Huynh and Feldt (1976) proposed an estimate of Box’s epsilon, constructed using estimators of its
numerator and denominator that are intended to be unbiased. The Huynh-Feldt epsilon has the form of
a modification of the Greenhouse-Geisser epsilon,
OHF D
nb OGG 2
b.DFE b OGG /
where n is the number of subjects and DFE is the degrees of freedom for error. The numerator
of this estimate is precisely unbiased only when there are no between-subject effects, but OHF is
still often employed even with nontrivial between-subject models; it was the only unbiased epsilon
alternative in SAS/STAT releases before SAS/STAT 9.22. The Huynh-Feldt epsilon is no longer the
default, but you can request it and its corresponding F test by using the UEPSDEF=HF option in the
3542 F Chapter 44: The GLM Procedure
REPEATED statement. The estimate is labeled “Huynh-Feldt Epsilon” in the PROC GLM output, and
the significance levels associated with adjusted F tests are labeled “H-F.”
• Lecoutre (1991) gave the unbiased form of the numerator of Box’s epsilon when there is one betweensubject effect. The correct form of Huynh and Feldt’s idea in this case is
OHFL D
.DFE C 1/b OGG 2
b.DFE b OGG /
More recently, Gribbin (2007) showed that OHFL applies to general between-subject models, and Chi
et al. (2012) showed that it extends even to situations where the number of error degrees of freedom is
less than the column rank of the within-subject contrast matrix. Thus, the Lecoutre correction of the
Huynh-Feldt epsilon is displayed by default along with the Greenhouse-Geisser epsilon; you can also
explicitly request it by using the UEPSDEF=HFL option in the REPEATED statement. The estimate
is labeled “Huynh-Feldt-Lecoutre Epsilon” in the PROC GLM output, and the significance levels
associated with adjusted F tests are labeled “H-F-L.”
• Finally, Chi et al. (2012) suggest that Box’s epsilon might be better estimated by replacing the reciprocal
of an unbiased form of the denominator with an approximately unbiased form of the reciprocal itself.
The resulting estimator can be written as a multiple of the corrected Huynh-Feldt epsilon OHFL ,
OCM D OHFL .a
2/.a
4/=a2
where a D .DFE 1/ C DFE.DFE 1/=2. Simulations indicate that OCM does a good job of
providing accurate p-values without being either too conservative or too liberal. Over a wide range
of cases, it is never much worse than any other alternative epsilon and often much better. You can
request that the Chi-Muller epsilon estimate and its corresponding F test be displayed by using the
UEPSDEF=CM option in the REPEATED statement. The estimate is labeled “Chi-Muller Epsilon” in
the PROC GLM output, and the significance levels associated with adjusted F tests are labeled “C-M.”
Although must be in the range of 0 to 1, the three approximately unbiased estimators can be outside this
range. When any of these estimators is greater than 1, a value of 1 is used in all calculations for probabilities—
in other words, the probabilities are not adjusted. Additionally, if OCM < 1=b, then the degrees of freedom
are adjusted by 1=b instead of OCM .
In summary, if your data do not meet the assumptions, use adjusted F tests. However, when you strongly
suspect that your data might not have Type H covariance, all these univariate tests should be interpreted
cautiously. In such cases, you should consider using the multivariate tests instead.
The univariate sums of squares for hypotheses involving within-subject effects can be easily calculated from
the H and E matrices corresponding to the multivariate tests described in the section “Multivariate Analysis
of Variance” on page 3536. If the M matrix is orthogonal, the univariate sums of squares is calculated as the
trace (sum of diagonal elements) of the appropriate H matrix; if it is not orthogonal, PROC GLM calculates
the trace of the H matrix that results from an orthogonal M matrix transformation. The appropriate error
term for the univariate F tests is constructed in a similar way from the error SSCP matrix and is labeled
Error(factorname), where factorname indicates the M matrix that is used in the transformation.
When the design specifies more than one repeated measures factor, PROC GLM computes the M matrix for a
given effect as the direct (Kronecker) product of the M matrices defined by the REPEATED statement if the
factor is involved in the effect or as a vector of 1s if the factor is not involved. The test for the main effect of
Repeated Measures Analysis of Variance F 3543
a repeated measures factor is constructed using an L matrix that corresponds to a test that the mean of the
observation is zero. Thus, the main effect test for repeated measures is a test that the means of the variables
defined by the M matrix are all equal to zero, while interactions involving repeated measures effects are tests
that the between-subjects factors involved in the interaction have no effect on the means of the transformed
variables defined by the M matrix. In addition, you can specify other L matrices to test hypotheses of interest
by using the CONTRAST statement, since hypotheses defined by CONTRAST statements are also tested
in the REPEATED analysis. To see which combinations of the original variables the transformed variables
represent, you can specify the PRINTM option in the REPEATED statement. This option displays the
transpose of M, which is labeled as M in the PROC GLM results. The tests produced are the same for any
choice of transformation .M/ matrix specified in the REPEATED statement; however, depending on the
nature of the repeated measurements being studied, a particular choice of transformation matrix, coupled
with the CANONICAL or SUMMARY option, can provide additional insight into the data being studied.
Transformations Used in Repeated Measures Analysis of Variance
As mentioned in the specifications of the REPEATED statement, several different M matrices can be generated
automatically, based on the transformation that you specify in the REPEATED statement. Remember that both
the univariate and multivariate tests that PROC GLM performs are unaffected by the choice of transformation;
the choice of transformation is important only when you are trying to study the nature of a repeated measures
effect, particularly with the CANONICAL and SUMMARY options. If one of these matrices does not meet
your needs for a particular analysis, you might want to use the M= option in the MANOVA statement to
perform the tests of interest.
The following sections describe the transformations available in the REPEATED statement, provide an
example of the M matrix that is produced, and give guidelines for the use of the transformation. As in the
PROC GLM output, the displayed matrix is labeled M. This is the M0 matrix.
CONTRAST Transformation
This is the default transformation used by the REPEATED statement. It is useful when one level of the
repeated measures effect can be thought of as a control level against which the others are compared. For
example, if five drugs are administered to each of several animals and the first drug is a control or placebo,
the statements
proc glm;
model d1-d5= / nouni;
repeated drug 5 contrast(1) / summary printm;
run;
produce the following M matrix:
2
3
1 1 0 0 0
6 1 0 1 0 0 7
7
MD6
4 1 0 0 1 0 5
1 0 0 0 1
When you examine the analysis of variance tables produced by the SUMMARY option, you can tell which of
the drugs differed significantly from the placebo.
3544 F Chapter 44: The GLM Procedure
POLYNOMIAL Transformation
This transformation is useful when the levels of the repeated measure represent quantitative values of a
treatment, such as dose or time. If the levels are unequally spaced, level values can be specified in parentheses
after the number of levels in the REPEATED statement. For example, if five levels of a drug corresponding
to 1, 2, 5, 10, and 20 milligrams are administered to different treatment groups, represented by the variable
group, the statements
proc glm;
class group;
model r1-r5=group / nouni;
repeated dose 5 (1 2 5 10 20) polynomial / summary printm;
run;
produce the following M matrix:
2
0:4250
0:3606
6 0:4349
0:2073
MD6
4 0:4331
0:1366
0:4926
0:7800
0:1674
0:3252
0:7253
0:3743
0:1545
0:7116
0:5108
0:0936
3
0:7984
0:3946 7
7
0:0821 5
0:0066
The SUMMARY option in this example provides univariate ANOVAs for the variables defined by the rows
of this M matrix. In this case, they represent the linear, quadratic, cubic, and quartic trends for dose and are
labeled dose_1, dose_2, dose_3, and dose_4, respectively.
HELMERT Transformation
Since the Helmert transformation compares a level of a repeated measure to the mean of subsequent levels, it
is useful when interest lies in the point at which responses cease to change. For example, if four levels of a
repeated measures factor represent responses to treatments administered over time to males and females, the
statements
proc glm;
class sex;
model resp1-resp4=sex / nouni;
repeated trtmnt 4 helmert / canon printm;
run;
produce the following M matrix:
2
1
0:33333
0:33333
1
0:50000
MD4 0
0
0
1
3
0:33333
0:50000 5
1
MEAN Transformation
This transformation can be useful in the same types of situations in which the CONTRAST transformation is
useful. If you substitute the following statement for the REPEATED statement shown in the CONTRAST
Transformation section,
Random-Effects Analysis F 3545
repeated drug 5 mean / printm;
the following M matrix is produced:
2
1
0:25
0:25
6 0:25
1
0:25
MD6
4 0:25
0:25
1
0:25
0:25
0:25
0:25
0:25
0:25
1
3
0:25
0:25 7
7
0:25 5
0:25
As with the CONTRAST transformation, if you want to omit a level other than the last, you can specify it in
parentheses after the keyword MEAN in the REPEATED statement.
PROFILE Transformation
When a repeated measure represents a series of factors administered over time, but a polynomial response
is unreasonable, a profile transformation might prove useful. As an example, consider a training program
in which four different methods are employed to teach students at several different schools. The repeated
measure is the score on tests administered after each of the methods is completed. The statements
proc glm;
class school;
model t1-t4=school / nouni;
repeated method 4 profile / summary nom printm;
run;
produce the following M matrix:
2
3
1
1
0
0
1
1
0 5
MD4 0
0
0
1
1
To determine the point at which an improvement in test scores takes place, you can examine the analyses of
variance for the transformed variables representing the differences between adjacent tests. These analyses are
requested by the SUMMARY option in the REPEATED statement, and the variables are labeled METHOD.1,
METHOD.2, and METHOD.3.
Random-Effects Analysis
When some model effects are random (that is, assumed to be sampled from a normal population of effects),
you can specify these effects in the RANDOM statement in order to compute the expected values of mean
squares for various model effects and contrasts and, optionally, to perform random-effects analysis of variance
tests.
PROC GLM versus PROC MIXED for Random-Effects Analysis
Other SAS procedures that can be used to analyze models with random effects include the MIXED and
VARCOMP procedures. Note that, for these procedures, the random-effects specification is an integral part
of the model, affecting how both random and fixed effects are fit; for PROC GLM, the random effects are
treated in a post hoc fashion after the complete fixed-effect model is fit. This distinction affects other features
3546 F Chapter 44: The GLM Procedure
in the GLM procedure, such as the results of the LSMEANS and ESTIMATE statements. These features
assume that all effects are fixed, so that all tests and estimability checks for these statements are based on a
fixed-effects model, even when you use a RANDOM statement. Standard errors for estimates and LS-means
based on the fixed-effects model might be significantly smaller than those based on a true random-effects
model; in fact, some functions that are estimable under a true random-effects model might not even be
estimable under the fixed-effects model. Therefore, you should use the MIXED procedure to compute tests
involving these features that take the random effects into account; see Chapter 63, “The MIXED Procedure,”
for more information.
Note that, for balanced data, the test statistics computed when you specify the TEST option in the RANDOM
statement have an exact F distribution only when the design is balanced; for unbalanced designs, the p values
for the F tests are approximate. For balanced data, the values obtained by PROC GLM and PROC MIXED
agree; for unbalanced data, they usually do not.
Computation of Expected Mean Squares for Random Effects
The RANDOM statement in PROC GLM declares one or more effects in the model to be random rather than
fixed. By default, PROC GLM displays the coefficients of the expected mean squares for all terms in the
model. In addition, when you specify the TEST option in the RANDOM statement, the procedure determines
what tests are appropriate and provides F ratios and probabilities for these tests.
The expected mean squares are computed as follows. Consider the model
Y D X0 ˇ0 C X1 ˇ1 C C Xk ˇk C where ˇ0 represents the fixed effects and ˇ1 ; ˇ2 ; ; represent the random effects. Random effects are assumed to be normally and independently distributed. For any L in the row space of X D
.X0 j X1 j X2 j j Xk /, the expected value of the sum of squares for Lˇ is
E.SSL / D ˇ00 C00 C0 ˇ0 C SSQ.C1 /12 C SSQ.C2 /22 C C SSQ.Ck /k2 C rank.L/2
where C is of the same dimensions as L and is partitioned as the X matrix. In other words,
C D .C0 j C1 j j Ck /
Furthermore, C D ML, where M is the inverse of the lower triangular Cholesky decomposition matrix of
L.X0 X/ L0 . SSQ(A) is defined as tr.A0 A/.
For the model in the following MODEL statement
model Y=A B(A) C A*C;
random B(A);
with B(A) declared as random, the expected mean square of each effect is displayed as
Var(Error) C constant Var.B.A// C Q.A; C; A C/
If any fixed effects appear in the expected mean square of an effect, the letter Q followed by the list of fixed
effects in the expected value is displayed. The actual numeric values of the quadratic form (Q matrix) can be
displayed using the Q option.
To determine appropriate means squares for testing the effects in the model, the TEST option in the RANDOM
statement performs the following steps:
Random-Effects Analysis F 3547
1. First, it forms a matrix of coefficients of the expected mean squares of those effects that were declared
to be random.
2. Next, for each effect in the model, it determines the combination of these expected mean squares that
produce an expectation that includes all the terms in the expected mean square of the effect of interest
except the one corresponding to the effect of interest. For example, if the expected mean square of an
effect A*B is
Var(Error) C 3 Var.A/ C Var.A B/
PROC GLM determines the combination of other expected mean squares in the model that has
expectation
Var(Error) C 3 Var.A/
3. If the preceding criterion is met by the expected mean square of a single effect in the model (as is often
the case in balanced designs), the F test is formed directly. In this case, the mean square of the effect
of interest is used as the numerator, the mean square of the single effect with an expected mean square
that satisfies the criterion is used as the denominator, and the degrees of freedom for the test are simply
the usual model degrees of freedom.
4. When more than one mean square must be combined to achieve the appropriate expectation, an
approximation is employed to determine the appropriate degrees of freedom (Satterthwaite 1946).
When effects other than the effect of interest are listed after the Q in the output, tests of hypotheses
involving the effect of interest are not valid unless all other fixed effects involved in it are assumed to
be zero. When tests such as these are performed by using the TEST option in the RANDOM statement,
a note is displayed reminding you that further assumptions are necessary for the validity of these
tests. Remember that although the tests are not valid unless these assumptions are made, this does
not provide a basis for these assumptions to be true. The particulars of a given experiment must be
examined to determine whether the assumption is reasonable.
For further theoretical discussion, see Goodnight and Speed (1978), Milliken and Johnson (1984, Chapters
22 and 23), and Hocking (1985).
Sum-to-Zero Assumptions
The formulation and parameterization of the expected mean squares for random effects in mixed models are
ongoing items of controversy in the statistical literature. Confusion arises over whether or not to assume that
terms involving fixed effects sum to zero. Cornfield and Tukey (1956); Winer (1971), and others assume that
they do sum to zero; Searle (1971); Hocking (1973), and others (including PROC GLM) do not.
Different assumptions about these sum-to-zero constraints can lead to different expected mean squares for
certain terms, and hence to different F and p values.
For arguments in favor of not assuming that terms involving fixed effects sum to zero, see Section 9.7 of
Searle (1971) and Sections 1 and 4 of McLean, Sanders, and Stroup (1991). Other references are Hartley and
Searle (1969) and Searle, Casella, and McCulloch (1992).
3548 F Chapter 44: The GLM Procedure
Computing Type I, II, and IV Expected Mean Squares
When you use the RANDOM statement, by default the GLM procedure produces the Type III expected mean
squares for model effects and for contrasts specified before the RANDOM statement. In order to obtain
expected values for other types of mean squares, you need to specify which types of mean squares are of
interest in the MODEL statement. For example, in order to obtain the Type IV expected mean squares for
effects in the RANDOM and CONTRAST statements, specify the SS4 option in the MODEL statement. If
you want both Type III and Type IV expected mean squares, specify both the SS3 and SS4 options in the
MODEL statement. Since the estimable function basis is not automatically calculated for Type I and Type II
SS, the E1 (for Type I) or E2 (for Type II) option must be specified in the MODEL statement in order for the
RANDOM statement to produce the expected mean squares for the Type I or Type II sums of squares. Note
that it is important to list the fixed effects first in the MODEL statement when requesting the Type I expected
mean squares.
For example, suppose you have a two-way design with factors A and B in which the main effect for B and
the interaction are random. In order to compute the Type III expected mean squares (in addition to the
fixed-effect analysis), you can use the following statements:
proc glm;
class A B;
model Y = A B A*B;
random B A*B;
run;
Suppose you use the SS4 option in the MODEL statement, as follows:
proc glm;
class A B;
model Y = A B A*B / ss4;
random B A*B;
run;
Then only the Type IV expected mean squares are computed (as well as the Type IV fixed-effect tests). For
the Type I expected mean squares, you can use the following statements:
proc glm;
class A B;
model Y = A B A*B / e1;
random B A*B;
run;
For each of these cases, in order to perform random-effect analysis of variance tests for each effect specified
in the model, you need to specify the TEST option in the RANDOM statement, as follows:
proc glm;
class A B;
model Y = A B A*B;
random B A*B / test;
run;
The GLM procedure automatically determines the appropriate error term for each test, based on the expected
mean squares.
Missing Values F 3549
Missing Values
For an analysis involving one dependent variable, PROC GLM uses an observation if values are nonmissing
for that dependent variable and all the classification variables.
For an analysis involving multiple dependent variables without the MANOVA or REPEATED statement, or
without the MANOVA option in the PROC GLM statement, a missing value in one dependent variable does
not eliminate the observation from the analysis of other nonmissing dependent variables. On the other hand,
for an analysis with the MANOVA or REPEATED statement, or with the MANOVA option in the PROC
GLM statement, PROC GLM uses an observation if values are nonmissing for all dependent variables and all
the variables used in independent effects.
During processing, the GLM procedure groups the dependent variables by their pattern of missing values
across observations so that sums and crossproducts can be collected in the most efficient manner.
If your data have different patterns of missing values among the dependent variables, interactivity is disabled.
This can occur when some of the variables in your data set have missing values and either of the following
conditions obtain:
• You do not use the MANOVA option in the PROC GLM statement.
• You do not use a MANOVA or REPEATED statement before the first RUN statement.
Note that the REG procedure handles missing values differently in this case; see Chapter 83, “The REG
Procedure,” for more information.
Computational Resources
Memory
For large problems, most of the memory resources are required for holding the X0 X matrix of the sums
and crossproducts. The section “Parameterization of PROC GLM Models” on page 3498 describes how
columns of the X matrix are allocated for various types of effects. For each level that occurs in the data for a
combination of classification variables in a given effect, a row and a column for X0 X are needed.
The following example illustrates the calculation. Suppose A has 20 levels, B has 4 levels, and C has 3 levels.
Then consider the model
proc glm;
class A B C;
model Y1 Y2 Y3=A B A*B C A*C B*C A*B*C X1 X2;
run;
The X0 X matrix (bordered by X0 Y and Y0 Y) can have as many as 425 rows and columns:
1
for the intercept term
20
for A
4
for B
3550 F Chapter 44: The GLM Procedure
80
for A*B
3
for C
60
for A*C
12
for B*C
240
for A*B*C
2
for X1 and X2 (continuous variables)
3
for Y1, Y2, and Y3 (dependent variables)
The matrix has 425 rows and columns only if all combinations of levels occur for each effect in the model.
For m rows and columns, 8m2 bytes are needed for crossproducts. In this case, 8 4252 D 1; 445; 000 bytes,
or about 1; 445; 000=1024 D 1411K.
The required memory grows as the square of the number of columns of X; most of the memory is for the
A*B*C interaction. Without A*B*C, you have 185 columns and need 268K for X0 X. Without either A*B*C
or A*B, you need 86K. If A is recoded to have 10 levels, then the full model has only 220 columns and
requires 378K.
The second time that a large amount of memory is needed is when Type III, Type IV, or contrast sums of
squares are being calculated. This memory requirement is a function of the number of degrees of freedom of
the model being analyzed and the maximum degrees of freedom for any single source. Let Rank equal the
sum of the model degrees of freedom, MaxDF be the maximum number of degrees of freedom for any single
source, and Ny be the number of dependent variables in the model. Then the memory requirement in bytes is
8 times
Ny Rank C .Rank .Rank C 1// =2
C MaxDF Rank
C .MaxDF .MaxDF C 1// =2
C MaxDF Ny
The first two components of this formula are for the estimable model coefficients and their variance; the rest
correspond to L, L.X0 X/ L0 , and Lb in the computation of SS.Lˇ D 0/ D .Lb/0 .L.X0 X/ L0 / 1 .Lb/. If
the operating system enables SAS to run parallel computational threads on multiple CPUs, then GLM will
attempt to allocate another 8 Rank Rank bytes in order to perform these calculations in parallel. If this
much memory is not available, then the estimability calculations are performed in a single thread.
Unfortunately, these quantities are not available when the X0 X matrix is being constructed, so PROC GLM
might occasionally request additional memory even after you have increased the memory allocation available
to the program.
If you have a large model that exceeds the memory capacity of your computer, these are your options:
• Eliminate terms, especially high-level interactions.
• Reduce the number of levels for variables with many levels.
• Use the ABSORB statement for parts of the model that are large.
Computational Resources F 3551
• Use the REPEATED statement for repeated measures variables.
• Use PROC ANOVA or PROC REG rather than PROC GLM, if your design allows.
A related limitation is that for any model effect involving classification variables (interactions as well as main
effects), the number of levels cannot exceed 32,767. This is because GLM internally indexes effect levels
with signed short (16-bit) integers, for which the maximum value is 215 1 D 32; 767.
CPU Time
Typically, if the GLM procedure requires a lot of CPU time, it will be for one of several reasons. Suppose that
the input data has n rows (observations) and the model has E effects that together produce a design matrix
X with m columns. Then if m or n is relatively large, the procedure might spend a lot of time in any of the
following areas:
• collecting the sums of squares and crossproducts
• solving the normal equations
• computing the Type III tests
The time required for collecting sums and crossproducts is difficult to calculate because it is a complicated
function of the model. The worst case occurs if all columns are continuous variables, involving nm2 =2
multiplications and additions. If the columns are levels of a classification, then only m sums might be
needed, but a significant amount of time might be spent in look-up operations. Solving the normal equations
requires time for approximately m3 =2 multiplications and additions, and the number of operations required
to compute the Type III tests is also proportional to both E and m3 .
Suppose that you know that Type IV sums of squares are appropriate for the model you are analyzing (for
example, if your design has no missing cells). You can specify the SS4 option in your MODEL statement,
which saves CPU time by requesting the Type IV sums of squares instead of the more computationally
burdensome Type III sums of squares. This proves especially useful if you have a factor in your model that
has many levels and is involved in several interactions.
If the operating system enables SAS to run parallel computational threads on multiple CPUs, then both the
solution of the normal equations and the computation of Type III tests can take advantage of this to reduce the
computational time for large models. In solving the normal equations, the fundamental row sweep operations
(Goodnight 1979) are performed in parallel. In computing the Type III tests, both the orthogonalization
for the estimable functions and the sums of squares calculation have been parallelized (if there is sufficient
memory).
The reduction in computational time due to parallel processing depends on the size of the model, the number
of processors, and the parallel architecture of the operating system. If the model is large enough that the
overwhelming proportion of CPU time for the procedure is accounted for in solving the normal equations
and/or computing the Type III tests, then you can expect a reduction in computational time approximately
inversely proportional to the number of CPUs. However, as you increase the number of processors, the
efficiency of this scaling can be reduced by several effects. One mitigating factor is a purely mathematical one
known as “Amdahl’s law,” which is related to the fact that only part of the processing time for the procedure
can be parallelized. Even taking Amdahl’s law into account, the parallelization efficiency can be reduced
by cache effects related to how fast the multiple processors can access memory. See Cohen (2002) for a
3552 F Chapter 44: The GLM Procedure
discussion of these issues. For additional information about parallel processing in SAS, see the chapter on
“Support for Parallel Processing” in SAS Language Reference: Concepts.
Computational Method
Let X represent the n p design matrix and Y the n 1 vector of dependent variables. (See the section
“Parameterization of PROC GLM Models” on page 3498 for information about how X is formed from your
model specification.)
The normal equations X0 Xˇ D X0 Y are solved using a modified sweep routine that produces a generalized
inverse .X0 X/ and a solution b D .X0 X/ X0 y. The modification is that rows and columns corresponding to
diagonal elements that are found during sweeping to be zero (or within the expected level of numerical error
of zero) are zeroed out. The .X0 X/ produced by this procedure satisfies the following two equations:
.X0 X/ .X0 X/ .X0 X/
D .X0 X/
.X0 X/ .X0 X/ .X0 X/
D .X0 X/
Pringle and Rayner (1971) call a generalized inverse with these characteristics a g2 -inverse, and this is the
term usually used in SAS documentation and output. Urquhart (1968) uses the term reflexive g-inverse to
emphasize that .X0 X/ is a generalized inverse of X0 X in the same way that X0 X is a generalized inverse of
.X0 X/ . Note that a g2 -inverse is not necessarily unique: if X0 X is singular, then sweeping the matrix in a
different order will result in a different g2 -inverse that also satisfies the two preceding equations.
For each effect in the model, a matrix L is computed such that the rows of L are estimable. Tests of the
hypothesis Lˇ D 0 are then made by first computing
SS.Lˇ D 0/ D .Lb/0 .L.X0 X/ L0 /
1
.Lb/
and then computing the associated F value by using the mean squared error.
Output Data Sets
OUT= Data Set Created by the OUTPUT Statement
The OUTPUT statement produces an output data set that contains the following:
• all original data from the SAS data set input to PROC GLM
• the new variables corresponding to the diagnostic measures specified with statistics keywords in the
OUTPUT statement (PREDICTED=, RESIDUAL=, and so on)
With multiple dependent variables, a name can be specified for any of the diagnostic measures for each of the
dependent variables in the order in which they occur in the MODEL statement.
For example, suppose that the input data set A contains the variables y1, y2, y3, x1, and x2. Then you can use
the following statements:
Output Data Sets F 3553
proc glm data=A;
model y1 y2 y3=x1;
output out=out p=y1hat y2hat y3hat
r=y1resid lclm=y1lcl uclm=y1ucl;
run;
The output data set out contains y1, y2, y3, x1, x2, y1hat, y2hat, y3hat, y1resid, y1lcl, and y1ucl. The variable
x2 is output even though it is not used by PROC GLM. Although predicted values are generated for all three
dependent variables, residuals are output for only the first dependent variable.
When any independent variable in the analysis (including all class variables) is missing for an observation,
then all new variables that correspond to diagnostic measures are missing for the observation in the output
data set.
When a dependent variable in the analysis is missing for an observation, then some new variables that
correspond to diagnostic measures are missing for the observation in the output data set, and some are
still available. Specifically, in this case, the new variables that correspond to COOKD, COVRATIO,
DFFITS, PRESS, R, RSTUDENT, STDR, and STUDENT are missing in the output data set. The variables
corresponding to H, LCL, LCLM, P, STDI, STDP, UCL, and UCLM are not missing.
OUT= Data Set Created by the LSMEANS Statement
The OUT= option in the LSMEANS statement produces an output data set that contains the following:
• the unformatted values of each classification variable specified in any effect in the LSMEANS statement
• a new variable, LSMEAN, which contains the LS-mean for the specified levels of the classification
variables
• a new variable, STDERR, which contains the standard error of the LS-mean
The variances and covariances among the LS-means are also output when the COV option is specified along
with the OUT= option. In this case, only one effect can be specified in the LSMEANS statement, and the
following variables are included in the output data set:
• new variables, COV1, COV2, . . . , COVn, where n is the number of levels of the effect specified in
the LSMEANS statement. These variables contain the covariances of each LS-mean with every other
LS-mean.
• a new variable, NUMBER, which provides an index for each observation to identify the covariances
that correspond to that observation. The covariances for the observation with NUMBER equal to n can
be found in the variable COVn.
OUTSTAT= Data Set
The OUTSTAT= option in the PROC GLM statement produces an output data set that contains the following:
• the BY variables, if any
3554 F Chapter 44: The GLM Procedure
• _TYPE_, a new character variable. _TYPE_ can take the values ‘SS1’, ‘SS2’, ‘SS3’, ‘SS4’, or ‘CONTRAST’, corresponding to the various types of sums of squares generated, or the values ‘CANCORR’,
‘STRUCTUR’, or ‘SCORE’, if a canonical analysis is performed through the MANOVA statement and
no M= matrix is specified.
• _SOURCE_, a new character variable. For each observation in the data set, _SOURCE_ contains the
name of the model effect or contrast label from which the corresponding statistics are generated.
• _NAME_, a new character variable. For each observation in the data set, _NAME_ contains the name
of one of the dependent variables in the model or, in the case of canonical statistics, the name of one of
the canonical variables (CAN1, CAN2, and so forth).
• four new numeric variables: SS, DF, F, and PROB, containing sums of squares, degrees of freedom,
F values, and probabilities, respectively, for each model or contrast sum of squares generated in the
analysis. For observations resulting from canonical analyses, these variables have missing values.
• if there is more than one dependent variable, then variables with the same names as the dependent
variables represent the following:
– for _TYPE_=SS1, SS2, SS3, SS4, or CONTRAST, the crossproducts of the hypothesis matrices
– for _TYPE_=CANCORR, canonical correlations for each variable
– for _TYPE_=STRUCTUR, coefficients of the total structure matrix
– for _TYPE_=SCORE, raw canonical score coefficients
The output data set can be used to perform special hypothesis tests (for example, with the IML procedure in
SAS/IML software), to reformat output, to produce canonical variates (through the SCORE procedure), or to
rotate structure matrices (through the FACTOR procedure).
Displayed Output
The GLM procedure produces the following output by default:
• The overall analysis-of-variance table breaks down the Total Sum of Squares for the dependent variable
into the portion attributed to the Model and the portion attributed to Error.
• The Mean Square term is the Sum of Squares divided by the degrees of freedom (DF).
• The Mean Square for Error is an estimate of 2 , the variance of the true errors.
• The F Value is the ratio produced by dividing the Mean Square for the Model by the Mean Square
for Error. It tests how well the model as a whole (adjusted for the mean) accounts for the dependent
variable’s behavior. An F test is a joint test to determine that all parameters except the intercept are
zero.
• A small significance probability, Pr > F, indicates that some linear function of the parameters is
significantly different from zero.
ODS Table Names F 3555
• R-Square, R2 , measures how much variation in the dependent variable can be accounted for by the
model. R square, which can range from 0 to 1, is the ratio of the sum of squares for the model to the
corrected total sum of squares. In general, the larger the value of R square, the better the model’s fit.
• Coeff Var, the coefficient of variation, which describes the amount of variation in the population, is 100
times the standard deviation estimate of the dependent variable, Root MSE (Mean Square for Error),
divided by the Mean. The coefficient of variation is often a preferred measure because it is unitless.
• Root MSE estimates the standard deviation of the dependent variable (or equivalently, the error term)
and equals the square root of the Mean Square for Error.
• Mean is the sample mean of the dependent variable.
These tests are used primarily in analysis-of-variance applications:
• The Type I SS (sum of squares) measures incremental sums of squares for the model as each variable
is added.
• The Type III SS is the sum of squares for a balanced test of each effect, adjusted for every other effect.
These items are used primarily in regression applications:
• The Estimates for the model Parameters (the intercept and the coefficients)
• t Value is the Student’s t value for testing the null hypothesis that the parameter (if it is estimable)
equals zero.
• The significance level, Pr > |t|, is the probability of getting a larger value of t if the parameter is truly
equal to zero. A very small value for this probability leads to the conclusion that the independent
variable contributes significantly to the model.
• The Standard Error is the square root of the estimated variance of the estimate of the true value of the
parameter.
Other portions of output are discussed in the following examples.
ODS Table Names
PROC GLM assigns a name to each table it creates. You can use these names to reference the table when
using the Output Delivery System (ODS) to select tables and create output data sets. These names are listed
in Table 44.14. For more information about ODS, see Chapter 20, “Using the Output Delivery System.”
3556 F Chapter 44: The GLM Procedure
Table 44.14 ODS Tables Produced by PROC GLM
ODS Table Name
Aliasing
Description
Type 1,2,3,4 aliasing structure
AltErrContrasts
ANOVA table for contrasts with
alternative error
ANOVA table for tests with alternative error
Bartlett’s homogeneity of variance test
Multiple comparisons of pairwise
differences
Information for multiple comparisons of pairwise differences
Multiple comparisons of means
with confidence/comparison
interval
Information for multiple comparison of means with confidence/comparison interval
Canonical analysis
AltErrTests
Bartlett
CLDiffs
CLDiffsInfo
CLMeans
CLMeansInfo
CanAnalysis
CanCoef
CanStructure
CharStruct
ClassLevels
ContrastCoef
Contrasts
DependentInfo
Diff
Epsilons
ErrorSSCP
EstFunc
Estimates
ExpectedMeanSquares
FitStatistics
GAliasing
Statement / Option
MODEL / (E1 E2 E3 or E4) and
ALIASING
CONTRAST / E=
TEST / E=
MEANS / HOVTEST=BARTLETT
MEANS / CLDIFF or DUNNETT or
(Unequal cells and not LINES)
MEANS / CLDIFF or DUNNETT or
(Unequal cells and not LINES)
MEANS / CLM
MEANS / CLM
(MANOVA or REPEATED)
/ CANONICAL
Canonical coefficients
(MANOVA or REPEATED)
/ CANONICAL
Canonical structure
(MANOVA or REPEATED)
/ CANONICAL
(MANOVA / not CANONICAL) or
Characteristic roots and vectors
(REPEATED / PRINTRV)
Classification variable levels
CLASS statement
L matrix for contrast or estimate CONTRAST / E or ESTIMATE / E
ANOVA table for contrasts
CONTRAST statement
Simultaneously analyzed depen- default when there are multiple dependent variables
dent variables with different patterns
of missing values
PDiff matrix of least squares LSMEANS / PDIFF=ALL and
means
more than two LS-means
Greenhouse-Geisser and Huynh- REPEATED statement
Feldt epsilons
Error SSCP matrix
(MANOVA or REPEATED)
/ PRINTE
Type 1,2,3,4 estimable functions MODEL / (E1 E2 E3 or E4)
Estimate statement results
ESTIMATE statement
Expected mean squares
RANDOM statement
R-Square, Coeff Var, Root MSE, default
and dependent mean
General form of aliasing
MODEL / E and ALIASING
structure
ODS Table Names F 3557
Table 44.14 continued
ODS Table Name
GEstFunc
HOVFTest
HypothesisSSCP
InvXPX
LSMeanCL
LSMeanCoef
LSMeanDiffCL
LSMeans
LSMLines
MANOVATransform
MCLines
MCLinesInfo
MCLinesRange
MatrixRepresentation
Means
ModelANOVA
MultStat
NObs
OverallANOVA
ParameterEstimates
PartialCorr
PredictedInfo
PredictedValues
QForm
RandomModelANOVA
Description
General form of estimable
functions
Homogeneity of variance
ANOVA
Hypothesis SSCP matrix
inv(X0 X) matrix
Confidence interval for LS-means
Coefficients of least squares
means
Confidence interval for LS-mean
differences
Least squares means
Least squares means comparison
lines
Multivariate transformation
matrix
Multiple comparisons LINES output
Statement / Option
MODEL / E
MEANS / HOVTEST
(MANOVA or REPEATED)
/ PRINTH
MODEL / INVERSE
LSMEANS / CL
LSMEANS / E
LSMEANS / PDIFF and CL
LSMEANS statement
LSMEANS / PDIFF=ALL LINES
MANOVA / M=
MEANS / LINES or ((DUNCAN or
WALLER or SNK or REGWQ) and
not (CLDIFF or CLM))
or (Equal cells and not CLDIFF)
Information for multiple compari- MEANS / LINES or ((DUNCAN or
son LINES output
WALLER or SNK or REGWQ) and
not (CLDIFF or CLM))
or (Equal cells and not CLDIFF)
Ranges for multiple range MC MEANS / LINES or ((DUNCAN or
tests
WALLER or SNK or REGWQ) and
not (CLDIFF or CLM))
or (Equal cells and not CLDIFF)
X matrix element
as needed for other options
representation
Group means
MEANS statement
ANOVA for model terms
default
Multivariate tests
MANOVA statement
Number of observations
default
Overall ANOVA
default
Estimated linear model
MODEL / SOLUTION
coefficients
Partial correlation matrix
(MANOVA or REPEATED)
/ PRINTE
Predicted values info
MODEL / P or CLM or CLI
Predicted values
MODEL / P or CLM or CLI
Quadratic form for expected RANDOM / Q
mean squares
Random-effect tests
RANDOM / TEST
3558 F Chapter 44: The GLM Procedure
Table 44.14 continued
ODS Table Name
RepeatedLevelInfo
RepeatedTransform
SimDetails
SimResults
SlicedANOVA
Sphericity
Tests
Tolerances
Welch
XPX
Description
Correspondence between dependents and repeated measures levels
Repeated measures transformation matrix
Details of difference quantile simulation
Evaluation of difference quantile
simulation
Sliced-effect ANOVA table
Sphericity tests
Summary ANOVA for specified
MANOVA H= effects
X0 X tolerances
Welch’s ANOVA
X0 X matrix
Statement / Option
REPEATED statement
REPEATED / PRINTM
LSMEANS
/ ADJUST=SIMULATE(REPORT)
LSMEANS
/ ADJUST=SIMULATE(REPORT)
LSMEANS / SLICE
REPEATED / PRINTE
MANOVA / H= SUMMARY
MODEL / TOLERANCE
MEANS / WELCH
MODEL / XPX
With the PDIFF or TDIFF option in the LSMEANS statement, the p/t-values for differences are displayed in columns of the LSMeans table for PDIFF/TDIFF=CONTROL or PDIFF/TDIFF=ANOM, and for
PDIFF/TDIFF=ALL when there are only two LS-means. Otherwise (for PDIFF/TDIFF=ALL when there are
more than two LS-means), the p/t-values for differences are displayed in a separate table called Diff.
ODS Graphics
Statistical procedures use ODS Graphics to create graphs as part of their output. ODS Graphics is described
in detail in Chapter 21, “Statistical Graphics Using ODS.”
Before you create graphs, ODS Graphics must be enabled (for example, by specifying the ODS GRAPHICS ON statement). For more information about enabling and disabling ODS Graphics, see the section
“Enabling and Disabling ODS Graphics” on page 606 in Chapter 21, “Statistical Graphics Using ODS.”
The overall appearance of graphs is controlled by ODS styles. Styles and other aspects of using ODS
Graphics are discussed in the section “A Primer on ODS Statistical Graphics” on page 605 in Chapter 21,
“Statistical Graphics Using ODS.”
When ODS Graphics is enabled, then for particular models the GLM procedure will produce default graphics.
• If you specify a one-way analysis of variance model, with just one CLASS variable, the GLM procedure
will produce a grouped box plot of the response values versus the CLASS levels. For an example of
the box plot, see the section “One-Way Layout with Means Comparisons” on page 946 in Chapter 26,
“The ANOVA Procedure.”
• If you specify a two-way analysis of variance model, with just two CLASS variables, the GLM
procedure will produce an interaction plot of the response values, with horizontal position representing
one CLASS variable and marker style representing the other; and with predicted response values
ODS Graphics F 3559
connected by lines representing the two-way analysis. For an example of the interaction plot, see the
section “PROC GLM for Unbalanced ANOVA” on page 3437.
• If you specify a model with a single continuous predictor, the GLM procedure will produce a fit plot of
the response values versus the covariate values, with a curve representing the fitted relationship. For
an example of the fit plot, see the section “PROC GLM for Quadratic Least Squares Regression” on
page 3440.
• If you specify a model with a two continuous predictors and no CLASS variables, the GLM procedure
will produce a panel of fit plots as in the single predictor case, with a plot of the response values versus
one of the covariates at each of several values of the other covariate.
• If you specify an analysis of covariance model, with one or two CLASS variables and one continuous
variable, the GLM procedure will produce an analysis of covariance plot of the response values versus
the covariate values, with lines representing the fitted relationship within each classification level. For
an example of the analysis of covariance plot, see Example 44.4.
• If you specify an LSMEANS statement with the PDIFF option, the GLM procedure will produce a plot
appropriate for the type of LS-means comparison. For PDIFF=ALL (which is the default if you specify
only PDIFF), the procedure produces a diffogram, which displays all pairwise LS-means differences
and their significance. The display is also known as a “mean-mean scatter plot” (Hsu 1996). For
PDIFF=CONTROL, the procedure produces a display of each noncontrol LS-mean compared to the
control LS-mean, with two-sided confidence intervals for the comparison. For PDIFF=CONTROLL
and PDIFF=CONTROLU a similar display is produced, but with one-sided confidence intervals.
Finally, for the PDIFF=ANOM option, the procedure produces an “analysis of means” plot, comparing
each LS-mean to the average LS-mean.
• If you specify a MEANS statement, the GLM procedure will produce a grouped box plot of the
response values versus the effect for which means are being calculated.
In addition to the default graphics mentioned previously, you can request plots that help you diagnose the
quality of the fitted model.
• The PLOTS=DIAGNOSTICS option in the PROC GLM statement requests that a panel of summary
diagnostics for the fit be displayed. The panel displays scatter plots of residuals, absolute residuals,
studentized residuals, and observed responses by predicted values; studentized residuals by leverage;
Cook’s D by observation; a Q-Q plot of residuals; a residual histogram; and a residual-fit spread plot.
• The PLOTS=RESIDUALS option in the PROC GLM statement requests scatter plots of the residuals
against each continuous covariate.
ODS Graph Names
PROC GLM assigns a name to each graph it creates using ODS. You can use these names to reference the
graphs when using ODS. The names are listed in Table 44.15.
ODS Graphics must be enabled before requesting plots. For more information about ODS Graphics, see
Chapter 21, “Statistical Graphics Using ODS.”
3560 F Chapter 44: The GLM Procedure
Table 44.15 Graphs Produced by PROC GLM
ODS Graph Name
ANCOVAPlot
ANOMPlot
BoxPlot
ContourFit
ControlPlot
DiagnosticsPanel
CooksDPlot
ObservedByPredicted
QQPlot
ResidualByPredicted
ResidualHistogram
RFPlot
RStudentByPredicted
RStudentByLeverage
DiffPlot
IntPlot
FitPlot
ResidualPlots
Plot Description
Analysis of covariance plot
Plot of LS-mean differences
against average LS-mean
Box plot of group means
Plot of predicted response
surface
Plot of LS-mean differences
against a control level
Panel of summary diagnostics for the fit
Cook’s D plot
Observed by predicted
Residual Q-Q plot
Residual by predicted values
Residual histogram
RF plot
Studentized residuals by predicted
RStudent by hat diagonals
Plot of LS-mean pairwise
differences
Interaction plot
Plot of predicted response by
predictor
Plots of the residuals against
each continuous covariate
Option
Analysis of covariance model
LSMEANS / PDIFF=ANOM
One-way ANOVA model or MEANS
statement
Two-predictor response surface model
LSMEANS / PDIFF=CONTROL
PLOTS=DIAGNOSTICS
PLOTS=DIAGNOSTICS(UNPACK)
PLOTS=DIAGNOSTICS(UNPACK)
PLOTS=DIAGNOSTICS(UNPACK)
PLOTS=DIAGNOSTICS(UNPACK)
PLOTS=DIAGNOSTICS(UNPACK)
PLOTS=DIAGNOSTICS(UNPACK)
PLOTS=DIAGNOSTICS(UNPACK)
PLOTS=DIAGNOSTICS(UNPACK)
LSMEANS / PDIFF
Two-way ANOVA model
Model with one continuous predictor
PLOTS=RESIDUALS
Examples: GLM Procedure
Example 44.1: Randomized Complete Blocks with Means Comparisons and
Contrasts
This example, reported by Stenstrom (1940), analyzes an experiment to investigate how snapdragons grow in
various soils. To eliminate the effect of local fertility variations, the experiment is run in blocks, with each
soil type sampled in each block. Since these data are balanced, the Type I and Type III SS are the same and
are equal to the traditional ANOVA SS.
Example 44.1: Randomized Complete Blocks with Means Comparisons and Contrasts F 3561
First, the standard analysis is shown, followed by an analysis that uses the SOLUTION option and includes
MEANS and CONTRAST statements. The ORDER=DATA option in the second PROC GLM statement
is used so that the ordering of coefficients in the CONTRAST statement can correspond to the ordering in
the input data. The SOLUTION option requests a display of the parameter estimates, which are produced
by default only if there are no CLASS variables. A MEANS statement is used to request a table of the
means with two multiple-comparison procedures requested. In experiments with focused treatment questions,
CONTRAST statements are preferable to general means comparison methods. The following statements
produce Output 44.1.1 through Output 44.1.4.
title 'Balanced Data from Randomized Complete Block';
data plants;
input Type $ @;
do Block = 1 to 3;
input StemLength @;
output;
end;
datalines;
Clarion 32.7 32.3 31.5
Clinton 32.1 29.7 29.1
Knox
35.7 35.9 33.1
O'Neill 36.0 34.2 31.2
Compost 31.8 28.0 29.2
Wabash
38.2 37.8 31.9
Webster 32.5 31.1 29.7
;
proc glm;
class Block Type;
model StemLength = Block Type;
run;
proc glm order=data;
class Block Type;
model StemLength = Block Type / solution;
/*----------------------------------clrn-cltn-knox-onel-cpst-wbsh-wstr */
contrast 'Compost vs. others' Type
-1
-1
-1
-1
6
-1
-1;
contrast 'River soils vs. non' Type
-1
-1
-1
-1
0
5
-1,
Type
-1
4
-1
-1
0
0
-1;
contrast 'Glacial vs. drift'
Type
-1
0
1
1
0
0
-1;
contrast 'Clarion vs. Webster' Type
-1
0
0
0
0
0
1;
contrast "Knox vs. O'Neill"
Type
0
0
1
-1
0
0
0;
run;
means Type / waller regwq;
run;
3562 F Chapter 44: The GLM Procedure
Output 44.1.1 Analysis of Variance for Randomized Complete Blocks
Balanced Data from Randomized Complete Block
The GLM Procedure
Class Level Information
Class
Levels
Values
Block
3
1 2 3
Type
7
Clarion Clinton Compost Knox O'Neill Wabash Webster
Number of Observations Read
Number of Observations Used
21
21
Balanced Data from Randomized Complete Block
The GLM Procedure
Dependent Variable: StemLength
DF
Sum of
Squares
Mean Square
F Value
Pr > F
Model
8
142.1885714
17.7735714
10.80
0.0002
Error
12
19.7428571
1.6452381
Corrected Total
20
161.9314286
Source
Source
Block
Type
Source
Block
Type
R-Square
Coeff Var
Root MSE
StemLength Mean
0.878079
3.939745
1.282668
32.55714
DF
Type I SS
Mean Square
F Value
Pr > F
2
6
39.0371429
103.1514286
19.5185714
17.1919048
11.86
10.45
0.0014
0.0004
DF
Type III SS
Mean Square
F Value
Pr > F
2
6
39.0371429
103.1514286
19.5185714
17.1919048
11.86
10.45
0.0014
0.0004
This analysis shows that the stem length is significantly different for the different soil types. In addition,
there are significant differences in stem length among the three blocks in the experiment.
The GLM procedure is invoked again, this time with the ORDER=DATA option. This enables you to write
accurate contrast statements more easily because you know the order SAS is using for the levels of the
Example 44.1: Randomized Complete Blocks with Means Comparisons and Contrasts F 3563
variable Type. The standard analysis is displayed again, this time including the tests for contrasts that you
specified as well as the estimated parameters. These additional results are shown in Output 44.1.2.
Output 44.1.2 Contrasts and Solutions
Balanced Data from Randomized Complete Block
The GLM Procedure
Dependent Variable: StemLength
Contrast
Compost vs. others
River soils vs. non
Glacial vs. drift
Clarion vs. Webster
Knox vs. O'Neill
Parameter
Intercept
Block
Block
Block
Type
Type
Type
Type
Type
Type
Type
DF
Contrast SS
Mean Square
F Value
Pr > F
1
2
1
1
1
29.24198413
48.24694444
22.14083333
1.70666667
1.81500000
29.24198413
24.12347222
22.14083333
1.70666667
1.81500000
17.77
14.66
13.46
1.04
1.10
0.0012
0.0006
0.0032
0.3285
0.3143
Estimate
1
2
3
Clarion
Clinton
Knox
O'Neill
Compost
Wabash
Webster
29.35714286
3.32857143
1.90000000
0.00000000
1.06666667
-0.80000000
3.80000000
2.70000000
-1.43333333
4.86666667
0.00000000
B
B
B
B
B
B
B
B
B
B
B
Standard
Error
t Value
Pr > |t|
0.83970354
0.68561507
0.68561507
.
1.04729432
1.04729432
1.04729432
1.04729432
1.04729432
1.04729432
.
34.96
4.85
2.77
.
1.02
-0.76
3.63
2.58
-1.37
4.65
.
<.0001
0.0004
0.0169
.
0.3285
0.4597
0.0035
0.0242
0.1962
0.0006
.
NOTE: The X'X matrix has been found to be singular, and a generalized inverse
was used to solve the normal equations. Terms whose estimates are
followed by the letter 'B' are not uniquely estimable.
The contrast label, degrees of freedom, sum of squares, Mean Square, F Value, and Pr > F are shown for
each contrast requested. In this example, the contrast results indicate the following inferences, at the 5%
significance level:
• The stem length of plants grown in compost soil is significantly different from the average stem length
of plants grown in other soils.
• The stem length of plants grown in river soils is significantly different from the average stem length of
those grown in nonriver soils.
• The average stem length of plants grown in glacial soils (Clarion and Webster types) is significantly
different from the average stem length of those grown in drift soils (Knox and O’Neill types).
3564 F Chapter 44: The GLM Procedure
• Stem lengths for Clarion and Webster types are not significantly different.
• Stem lengths for Knox and O’Neill types are not significantly different.
In addition to the estimates for the parameters of the model, the results of t tests about the parameters are
also displayed. The ‘B’ following the parameter estimates indicates that the estimates are biased and do not
represent a unique solution to the normal equations.
Output 44.1.3 Waller-Duncan tests
Balanced Data from Randomized Complete Block
The GLM Procedure
Waller-Duncan K-ratio t Test for StemLength
NOTE: This test minimizes the Bayes risk under additive loss and certain other
assumptions.
Kratio
100
Error Degrees of Freedom
12
Error Mean Square
1.645238
F Value
10.45
Critical Value of t
2.12034
Minimum Significant Difference
2.2206
Means with the same letter are not significantly different.
Waller Grouping
Mean
N
Type
A
A
A
A
A
35.967
3
Wabash
34.900
3
Knox
33.800
3
O'Neill
C
C
C
C
C
32.167
3
Clarion
31.100
3
Webster
30.300
3
Clinton
29.667
3
Compost
B
B
B
D
D
D
D
D
Example 44.2: Regression with Mileage Data F 3565
Output 44.1.4 Ryan-Einot-Gabriel-Welsch Multiple Range Test
Balanced Data from Randomized Complete Block
The GLM Procedure
Ryan-Einot-Gabriel-Welsch Multiple Range Test for StemLength
NOTE: This test controls the Type I experimentwise error rate.
Alpha
0.05
Error Degrees of Freedom
12
Error Mean Square
1.645238
Number of Means
2
3
4
5
6
7
Critical Range 2.9875528 3.2837322 3.4395625 3.5402383 3.5402383 3.6653133
Means with the same letter are not significantly different.
REGWQ Grouping
B
B
B
B
B
A
A
A
A
A
D
D
D
D
D
D
D
C
C
C
C
C
Mean
N
Type
35.967
3
Wabash
34.900
3
Knox
33.800
3
O'Neill
32.167
3
Clarion
31.100
3
Webster
30.300
3
Clinton
29.667
3
Compost
The final two pages of output (Output 44.1.3 and Output 44.1.4) present results of the Waller-Duncan and
REGWQ multiple-comparison procedures. For each test, notes and information pertinent to the test are given
in the output. The Type means are arranged from highest to lowest. Means with the same letter are not
significantly different. For this example, while some pairs of means are significantly different, there are no
clear equivalence classes among the different soils.
For an alternative method of analyzing and displaying mean differences, including high-resolution graphics,
see Example 44.3.
Example 44.2: Regression with Mileage Data
A car is tested for gas mileage at various speeds to determine at what speed the car achieves the highest gas
mileage. A quadratic model is fit to the experimental data. The following statements produce Output 44.2.1
through Output 44.2.4.
3566 F Chapter 44: The GLM Procedure
title 'Gasoline Mileage Experiment';
data mileage;
input mph mpg @@;
datalines;
20 15.4
30 20.2
40 25.7
50 26.2 50 26.6 50 27.4
55
.
60 24.8
;
ods graphics on;
proc glm;
model mpg=mph mph*mph / p clm;
run;
ods graphics off;
Output 44.2.1 Standard Regression Analysis
Gasoline Mileage Experiment
The GLM Procedure
Number of Observations Read
Number of Observations Used
8
7
Gasoline Mileage Experiment
The GLM Procedure
Dependent Variable: mpg
DF
Sum of
Squares
Mean Square
F Value
Pr > F
Model
2
111.8086183
55.9043091
77.96
0.0006
Error
4
2.8685246
0.7171311
Corrected Total
6
114.6771429
Source
Source
mph
mph*mph
R-Square
Coeff Var
Root MSE
mpg Mean
0.974986
3.564553
0.846836
23.75714
DF
Type I SS
Mean Square
F Value
Pr > F
1
1
85.64464286
26.16397541
85.64464286
26.16397541
119.43
36.48
0.0004
0.0038
Example 44.2: Regression with Mileage Data F 3567
Output 44.2.1 continued
Source
mph
mph*mph
DF
Type III SS
Mean Square
F Value
Pr > F
1
1
41.01171219
26.16397541
41.01171219
26.16397541
57.19
36.48
0.0016
0.0038
Parameter
Estimate
Standard
Error
t Value
Pr > |t|
Intercept
mph
mph*mph
-5.985245902
1.305245902
-0.013098361
3.18522249
0.17259876
0.00216852
-1.88
7.56
-6.04
0.1334
0.0016
0.0038
The overall F statistic is significant. The tests of mph and mph*mph in the Type I sums of squares show that
both the linear and quadratic terms in the regression model are significant. The model fits well, with an R
square of 0.97. The table of parameter estimates indicates that the estimated regression equation is
mpg D
5:9852 C 1:3052 mph
0:0131 mph2
Output 44.2.2 Results of Requesting the P and CLM Options
Observation
1
2
3
4
5
6
7 *
8
Observed
Predicted
Residual
15.40000000
20.20000000
25.70000000
26.20000000
26.60000000
27.40000000
.
24.80000000
14.88032787
21.38360656
25.26721311
26.53114754
26.53114754
26.53114754
26.18073770
25.17540984
0.51967213
-1.18360656
0.43278689
-0.33114754
0.06885246
0.86885246
.
-0.37540984
Observation
1
2
3
4
5
6
7 *
8
95% Confidence Limits for
Mean Predicted Value
12.69701317
20.01727192
23.87460041
25.44573423
25.44573423
25.44573423
24.88679308
23.05954977
17.06364257
22.74994119
26.65982582
27.61656085
27.61656085
27.61656085
27.47468233
27.29126990
The P and CLM options in the MODEL statement produce the table shown in Output 44.2.2. For each
observation, the observed, predicted, and residual values are shown. In addition, the 95% confidence limits
for a mean predicted value are shown for each observation. Note that the observation with a missing value
for mph is not used in the analysis, but predicted and confidence limit values are shown.
3568 F Chapter 44: The GLM Procedure
Output 44.2.3 Additional Results of Requesting the P and CLM Options
Sum of Residuals
Sum of Squared Residuals
Sum of Squared Residuals - Error SS
PRESS Statistic
First Order Autocorrelation
Durbin-Watson D
-0.00000000
2.86852459
-0.00000000
23.18107335
-0.54376613
2.94425592
The last portion of the output listing, shown in Output 44.2.3, gives some additional information about the
residuals. The Press statistic gives the sum of squares of predicted residual errors, as described in Chapter 4,
“Introduction to Regression Procedures.” The First Order Autocorrelation and the Durbin-Watson D statistic,
which measures first-order autocorrelation, are also given.
Output 44.2.4 Plot of Mileage Data
Finally, the ODS GRAPHICS ON command in the previous statements enables ODS Graphics, which in this
case produces the plot shown in Output 44.2.4 of the actual and predicted values for the data, as well as a
band representing the confidence limits for individual predictions. The quadratic relationship between mpg
and mph is evident.
Example 44.3: Unbalanced ANOVA for Two-Way Design with Interaction F 3569
Example 44.3: Unbalanced ANOVA for Two-Way Design with Interaction
This example uses data from Kutner (1974, p. 98) to illustrate a two-way analysis of variance. The original
data source is Afifi and Azen (1972, p. 166). These statements produce Output 44.3.1 and Output 44.3.2.
title 'Unbalanced Two-Way Analysis of Variance';
data a;
input drug disease @;
do i=1 to 6;
input y @;
output;
end;
datalines;
1 1 42 44 36 13 19 22
1 2 33 . 26 . 33 21
1 3 31 -3 . 25 25 24
2 1 28 . 23 34 42 13
2 2 . 34 33 31 . 36
2 3 3 26 28 32 4 16
3 1 . . 1 29 . 19
3 2 . 11 9 7 1 -6
3 3 21 1 . 9 3 .
4 1 24 . 9 22 -2 15
4 2 27 12 12 -5 16 15
4 3 22 7 25 5 12 .
;
proc glm;
class drug disease;
model y=drug disease drug*disease / ss1 ss2 ss3 ss4;
run;
Output 44.3.1 Classes and Levels for Unbalanced Two-Way Design
Unbalanced Two-Way Analysis of Variance
The GLM Procedure
Class Level Information
Class
Levels
Values
drug
4
1 2 3 4
disease
3
1 2 3
Number of Observations Read
Number of Observations Used
72
58
3570 F Chapter 44: The GLM Procedure
Output 44.3.2 Analysis of Variance for Unbalanced Two-Way Design
Unbalanced Two-Way Analysis of Variance
The GLM Procedure
Dependent Variable: y
Source
DF
Sum of
Squares
Mean Square
F Value
Pr > F
Model
11
4259.338506
387.212591
3.51
0.0013
Error
46
5080.816667
110.452536
Corrected Total
57
9340.155172
Source
drug
disease
drug*disease
Source
drug
disease
drug*disease
Source
drug
disease
drug*disease
Source
drug
disease
drug*disease
R-Square
Coeff Var
Root MSE
y Mean
0.456024
55.66750
10.50964
18.87931
DF
Type I SS
Mean Square
F Value
Pr > F
3
2
6
3133.238506
418.833741
707.266259
1044.412835
209.416870
117.877710
9.46
1.90
1.07
<.0001
0.1617
0.3958
DF
Type II SS
Mean Square
F Value
Pr > F
3
2
6
3063.432863
418.833741
707.266259
1021.144288
209.416870
117.877710
9.25
1.90
1.07
<.0001
0.1617
0.3958
DF
Type III SS
Mean Square
F Value
Pr > F
3
2
6
2997.471860
415.873046
707.266259
999.157287
207.936523
117.877710
9.05
1.88
1.07
<.0001
0.1637
0.3958
DF
Type IV SS
Mean Square
F Value
Pr > F
3
2
6
2997.471860
415.873046
707.266259
999.157287
207.936523
117.877710
9.05
1.88
1.07
<.0001
0.1637
0.3958
Note the differences among the four types of sums of squares. The Type I sum of squares for drug essentially
tests for differences between the expected values of the arithmetic mean response for different drugs,
unadjusted for the effect of disease. By contrast, the Type II sum of squares for drug measures the differences
between arithmetic means for each drug after adjusting for disease. The Type III sum of squares measures
the differences between predicted drug means over a balanced drugdisease population—that is, between the
LS-means for drug. Finally, the Type IV sum of squares is the same as the Type III sum of squares in this
case, since there are data for every drug-by-disease combination.
Example 44.3: Unbalanced ANOVA for Two-Way Design with Interaction F 3571
No matter which sum of squares you prefer to use, this analysis shows a significant difference among the
four drugs, while the disease effect and the drug-by-disease interaction are not significant. As the previous
discussion indicates, Type III sums of squares correspond to differences between LS-means, so you can
follow up the Type III tests with a multiple-comparison analysis of the drug LS-means. Since the GLM
procedure is interactive, you can accomplish this by submitting the following statements after the previous
ones that performed the ANOVA.
lsmeans drug / pdiff=all adjust=tukey;
run;
Both the LS-means themselves and a matrix of adjusted p-values for pairwise differences between them are
displayed; see Output 44.3.3 and Output 44.3.4.
Output 44.3.3 LS-Means for Unbalanced ANOVA
Unbalanced Two-Way Analysis of Variance
The GLM Procedure
Least Squares Means
Adjustment for Multiple Comparisons: Tukey-Kramer
drug
1
2
3
4
y LSMEAN
LSMEAN
Number
25.9944444
26.5555556
9.7444444
13.5444444
1
2
3
4
Output 44.3.4 Adjusted p-Values for Pairwise LS-Mean Differences
Least Squares Means for effect drug
Pr > |t| for H0: LSMean(i)=LSMean(j)
Dependent Variable: y
i/j
1
2
3
4
1
0.9989
0.0016
0.0107
2
3
4
0.9989
0.0016
0.0011
0.0107
0.0071
0.7870
0.0011
0.0071
0.7870
The multiple-comparison analysis shows that drugs 1 and 2 have very similar effects, and that drugs 3 and 4
are also insignificantly different from each other. Evidently, the main contribution to the significant drug
effect is the difference between the 1/2 pair and the 3/4 pair.
3572 F Chapter 44: The GLM Procedure
If ODS Graphics is enabled for the previous analysis, GLM also displays three additional plots by default:
• an interaction plot for the effects of disease and drug
• a mean plot of the drug LS-means
• a plot of the adjusted pairwise differences and their significance levels
The following statements reproduce the previous analysis with ODS Graphics enabled. Additionally, the
PLOTS=MEANPLOT(CL) option specifies that confidence limits for the LS-means should also be displayed
in the mean plot. The graphical results are shown in Output 44.3.5 through Output 44.3.7.
ods graphics on;
proc glm plot=meanplot(cl);
class drug disease;
model y=drug disease drug*disease;
lsmeans drug / pdiff=all adjust=tukey;
run;
ods graphics off;
Output 44.3.5 Plot of Response by Drug and Disease
Example 44.3: Unbalanced ANOVA for Two-Way Design with Interaction F 3573
Output 44.3.6 Plot of Response LS-Means for Drug
3574 F Chapter 44: The GLM Procedure
Output 44.3.7 Plot of Response LS-Mean Differences for Drug
The significance of the drug differences is difficult to discern in the original data, as displayed in Output 44.3.5,
but the plot of just the LS-means and their individual confidence limits in Output 44.3.6 makes it clearer.
Finally, Output 44.3.7 indicates conclusively that the significance of the effect of drug is due to the difference
between the two drug pairs (1, 2) and (3, 4).
Example 44.4: Analysis of Covariance
Analysis of covariance combines some of the features of both regression and analysis of variance. Typically,
a continuous variable (the covariate) is introduced into the model of an analysis-of-variance experiment.
Data in the following example are selected from a larger experiment on the use of drugs in the treatment of
leprosy (Snedecor and Cochran 1967, p. 422).
Example 44.4: Analysis of Covariance F 3575
Variables in the study are as follows:
Drug
PreTreatment
PostTreatment
two antibiotics (A and D) and a control (F)
a pretreatment score of leprosy bacilli
a posttreatment score of leprosy bacilli
Ten patients are selected for each treatment (Drug), and six sites on each patient are measured for leprosy
bacilli.
The covariate (a pretreatment score) is included in the model for increased precision in determining the effect
of drug treatments on the posttreatment count of bacilli.
The following statements create the data set, perform a parallel-slopes analysis of covariance with PROC
GLM, and compute Drug LS-means. These statements produce Output 44.4.1 and Output 44.4.2.
data DrugTest;
input Drug $ PreTreatment PostTreatment @@;
datalines;
A 11 6
A 8 0
A 5 2
A 14 8
A 19 11
A 6 4
A 10 13
A 6 1
A 11 8
A 3 0
D 6 0
D 6 2
D 7 3
D 8 1
D 18 18
D 8 4
D 19 14
D 8 9
D 5 1
D 15 9
F 16 13
F 13 10
F 11 18
F 9 5
F 21 23
F 16 12
F 12 5
F 12 16
F 7 1
F 12 20
;
proc glm data=DrugTest;
class Drug;
model PostTreatment = Drug PreTreatment / solution;
lsmeans Drug / stderr pdiff cov out=adjmeans;
run;
proc print data=adjmeans;
run;
Output 44.4.1 Classes and Levels
The GLM Procedure
Class Level Information
Class
Drug
Levels
3
Number of Observations Read
Number of Observations Used
Values
A D F
30
30
3576 F Chapter 44: The GLM Procedure
Output 44.4.2 Overall Analysis of Variance
The GLM Procedure
Dependent Variable: PostTreatment
DF
Sum of
Squares
Mean Square
F Value
Pr > F
Model
3
871.497403
290.499134
18.10
<.0001
Error
26
417.202597
16.046254
Corrected Total
29
1288.700000
Source
R-Square
Coeff Var
Root MSE
PostTreatment Mean
0.676261
50.70604
4.005778
7.900000
This model assumes that the slopes relating posttreatment scores to pretreatment scores are parallel for all
drugs. You can check this assumption by including the class-by-covariate interaction, Drug*PreTreatment, in
the model and examining the ANOVA test for the significance of this effect. This extra test is omitted in this
example, but it is insignificant, justifying the equal-slopes assumption.
In Output 44.4.3, the Type I SS for Drug (293.6) gives the between-drug sums of squares that are obtained
for the analysis-of-variance model PostTreatment=Drug. This measures the difference between arithmetic
means of posttreatment scores for different drugs, disregarding the covariate. The Type III SS for Drug
(68.5537) gives the Drug sum of squares adjusted for the covariate. This measures the differences between
Drug LS-means, controlling for the covariate. The Type I test is highly significant (p = 0.001), but the Type
III test is not. This indicates that, while there is a statistically significant difference between the arithmetic
drug means, this difference is reduced to below the level of background noise when you take the pretreatment
scores into account. From the table of parameter estimates, you can derive the least squares predictive
formula model for estimating posttreatment score based on pretreatment score and drug:
8
< . 0:435 C 3:446/ C 0:987 pre; if Drug=A
. 0:435 C 3:337/ C 0:987 pre; if Drug=D
post D
:
0:435
C 0:987 pre; if Drug=F
Output 44.4.3 Tests and Parameter Estimates
Source
Drug
PreTreatment
Source
Drug
PreTreatment
DF
Type I SS
Mean Square
F Value
Pr > F
2
1
293.6000000
577.8974030
146.8000000
577.8974030
9.15
36.01
0.0010
<.0001
DF
Type III SS
Mean Square
F Value
Pr > F
2
1
68.5537106
577.8974030
34.2768553
577.8974030
2.14
36.01
0.1384
<.0001
Example 44.4: Analysis of Covariance F 3577
Output 44.4.3 continued
Parameter
Estimate
Intercept
Drug
A
Drug
D
Drug
F
PreTreatment
-0.434671164
-3.446138280
-3.337166948
0.000000000
0.987183811
B
B
B
B
Standard
Error
t Value
Pr > |t|
2.47135356
1.88678065
1.85386642
.
0.16449757
-0.18
-1.83
-1.80
.
6.00
0.8617
0.0793
0.0835
.
<.0001
Output 44.4.4 displays the LS-means, which are, in a sense, the means adjusted for the covariate. The
STDERR option in the LSMEANS statement causes the standard error of the LS-means and the probability of
getting a larger t value under the hypothesis H0 W LS-mean D 0 to be included in this table as well. Specifying
the PDIFF option causes all probability values for the hypothesis H0 W LS-mean.i / D LS-mean.j / to be
displayed, where the indexes i and j are numbered treatment levels.
Output 44.4.4 LS-Means
The GLM Procedure
Least Squares Means
Post
Treatment
LSMEAN
Standard
Error
Pr > |t|
LSMEAN
Number
6.7149635
6.8239348
10.1611017
1.2884943
1.2724690
1.3159234
<.0001
<.0001
<.0001
1
2
3
Drug
A
D
F
Least Squares Means for effect Drug
Pr > |t| for H0: LSMean(i)=LSMean(j)
Dependent Variable: PostTreatment
i/j
1
1
2
3
0.9521
0.0793
2
3
0.9521
0.0793
0.0835
0.0835
3578 F Chapter 44: The GLM Procedure
The OUT= and COV options in the LSMEANS statement create a data set of the estimates, their standard
errors, and the variances and covariances of the LS-means, which is displayed in Output 44.4.5.
Output 44.4.5 LS-Means Output Data Set
Obs
_NAME_
1
2
3
PostTreatment
PostTreatment
PostTreatment
Drug
A
D
F
LSMEAN
STDERR
6.7150
6.8239
10.1611
1.28849
1.27247
1.31592
NUMBER
COV1
COV2
COV3
1
2
3
1.66022
0.02844
-0.08403
0.02844
1.61918
-0.04299
-0.08403
-0.04299
1.73165
The new graphical features of PROC GLM enable you to visualize the fitted analysis of covariance model.
The following statements enable ODS Graphics by specifying the ODS GRAPHICS statement and then fit an
analysis-of-covariance model with LS-means for Drug.
ods graphics on;
proc glm data=DrugTest plot=meanplot(cl);
class Drug;
model PostTreatment = Drug PreTreatment;
lsmeans Drug / pdiff;
run;
ods graphics off;
With graphics enabled, the GLM procedure output includes an analysis-of-covariance plot, as in Output 44.4.6.
The LSMEANS statement produces a plot of the LS-means; the SAS statements previously shown use
the PLOTS=MEANPLOT(CL) option to add confidence limits for the individual LS-means, shown in
Output 44.4.7. If you also specify the PDIFF option in the LSMEANS statement, the output also includes a
plot appropriate for the type of LS-mean differences computed. In this case, the default is to compare all
LS-means with each other pairwise, so the plot is a “diffogram” or “mean-mean scatter plot” (Hsu 1996), as
in Output 44.4.8. For general information about ODS Graphics, see Chapter 21, “Statistical Graphics Using
ODS.” For specific information about the graphics available in the GLM procedure, see the section “ODS
Graphics” on page 3558.
Example 44.4: Analysis of Covariance F 3579
Output 44.4.6 Analysis of Covariance Plot of PostTreatment Score by Drug and PreTreatment Score
3580 F Chapter 44: The GLM Procedure
Output 44.4.7 LS-Means for PostTreatment Score by Drug
Example 44.5: Three-Way Analysis of Variance with Contrasts F 3581
Output 44.4.8 Plot of Differences between Drug LS-Means for PostTreatment Scores
The analysis of covariance plot Output 44.4.6 makes it clear that the control (drug F) has higher posttreatment
scores across the range of pretreatment scores, while the fitted models for the two antibiotics (drugs A and D)
nearly coincide. Similarly, while the diffogram Output 44.4.7 indicates that none of the LS-mean differences
are significant at the 5% level, the difference between the LS-means for the two antibiotics is much closer to
zero than the differences between either one and the control.
Example 44.5: Three-Way Analysis of Variance with Contrasts
This example uses data from Cochran and Cox (1957, p. 176) to illustrate the analysis of a three-way factorial
design with replication, including the use of the CONTRAST statement with interactions, the OUTSTAT=
data set, and the SLICE= option in the LSMEANS statement.
The object of the study is to determine the effects of electric current on denervated muscle. The variables are
as follows:
Rep
the replicate number, 1 or 2
Time
the length of time the current is applied to the muscle, ranging from 1 to 4
3582 F Chapter 44: The GLM Procedure
Current
the level of electric current applied, ranging from 1 to 4
Number
the number of treatments per day, ranging from 1 to 3
MuscleWeight
the weight of the denervated muscle
The following statements produce Output 44.5.1 through Output 44.5.4.
data muscles;
do Rep=1 to 2;
do Time=1 to 4;
do Current=1 to 4;
do Number=1 to 3;
input MuscleWeight @@;
output;
end;
end;
end;
end;
datalines;
72 74 69 61 61 65 62 65 70 85 76 61
67 52 62 60 55 59 64 65 64 67 72 60
57 66 72 72 43 43 63 66 72 56 75 92
57 56 78 60 63 58 61 79 68 73 86 71
46 74 58 60 64 52 71 64 71 53 65 66
44 58 54 57 55 51 62 61 79 60 78 82
53 50 61 56 57 56 56 56 71 56 58 69
46 55 64 56 55 57 64 66 62 59 58 88
;
proc glm outstat=summary;
class Rep Current Time Number;
model MuscleWeight = Rep Current|Time|Number;
contrast 'Time in Current 3'
Time 1 0 0 -1 Current*Time 0 0 0 0 0 0 0 0 1 0 0 -1,
Time 0 1 0 -1 Current*Time 0 0 0 0 0 0 0 0 0 1 0 -1,
Time 0 0 1 -1 Current*Time 0 0 0 0 0 0 0 0 0 0 1 -1;
contrast 'Current 1 versus 2' Current 1 -1;
lsmeans Current*Time / slice=Current;
run;
proc print data=summary;
run;
The first CONTRAST statement examines the effects of Time within level 3 of Current. This is also called the
simple effect of Time within Current*Time. Note that, since there are three degrees of freedom, it is necessary
to specify three rows in the CONTRAST statement, separated by commas. Since the parameterization that
PROC GLM uses is determined in part by the ordering of the variables in the CLASS statement, Current
is specified before Time so that the Time parameters are nested within the Current*Time parameters; thus,
the Current*Time contrast coefficients in each row are simply the Time coefficients of that row within the
appropriate level of Current.
Example 44.5: Three-Way Analysis of Variance with Contrasts F 3583
The second CONTRAST statement isolates a single-degree-of-freedom effect corresponding to the difference
between the first two levels of Current. You can use such a contrast in a large experiment where certain
preplanned comparisons are important, but you want to take advantage of the additional error degrees of
freedom available when all levels of the factors are considered.
The LSMEANS statement with the SLICE= option is an alternative way to test for the simple effect of Time
within Current*Time. In addition to listing the LS-means for each current strength and length of time, it gives
a table of F tests for differences between the LS-means across Time within each Current level. In some cases,
this can be a way to disentangle a complex interaction.
Output 44.5.1 Overall Analysis
The GLM Procedure
Class Level Information
Class
Levels
Values
Rep
2
1 2
Current
4
1 2 3 4
Time
4
1 2 3 4
Number
3
1 2 3
Number of Observations Read
Number of Observations Used
96
96
The GLM Procedure
Dependent Variable: MuscleWeight
Source
DF
Sum of
Squares
Mean Square
F Value
Pr > F
Model
48
5782.916667
120.477431
1.77
0.0261
Error
47
3199.489583
68.074246
Corrected Total
95
8982.406250
R-Square
Coeff Var
Root MSE
MuscleWeight Mean
0.643805
13.05105
8.250712
63.21875
The output, shown in Output 44.5.2 and Output 44.5.3, indicates that the main effects for Rep, Current, and
Number are significant (with p-values of 0.0045, <0.0001, and 0.0461, respectively), but the main effect for
Time is not significant, indicating that, in general, it does not matter how long the current is applied. None
of the interaction terms are significant, nor are the contrasts significant. Notice that the row in the sliced
ANOVA table corresponding to level 3 of current matches the “Time in Current 3” contrast.
3584 F Chapter 44: The GLM Procedure
Output 44.5.2 Individual Effects and Contrasts
Source
DF
Type I SS
Mean Square
F Value
Pr > F
Rep
Current
Time
Current*Time
Number
Current*Number
Time*Number
Current*Time*Number
1
3
3
9
2
6
6
18
605.010417
2145.447917
223.114583
298.677083
447.437500
644.395833
367.979167
1050.854167
605.010417
715.149306
74.371528
33.186343
223.718750
107.399306
61.329861
58.380787
8.89
10.51
1.09
0.49
3.29
1.58
0.90
0.86
0.0045
<.0001
0.3616
0.8756
0.0461
0.1747
0.5023
0.6276
Source
DF
Type III SS
Mean Square
F Value
Pr > F
Rep
Current
Time
Current*Time
Number
Current*Number
Time*Number
Current*Time*Number
1
3
3
9
2
6
6
18
605.010417
2145.447917
223.114583
298.677083
447.437500
644.395833
367.979167
1050.854167
605.010417
715.149306
74.371528
33.186343
223.718750
107.399306
61.329861
58.380787
8.89
10.51
1.09
0.49
3.29
1.58
0.90
0.86
0.0045
<.0001
0.3616
0.8756
0.0461
0.1747
0.5023
0.6276
Contrast
DF
Contrast SS
Mean Square
F Value
Pr > F
3
1
34.83333333
99.18750000
11.61111111
99.18750000
0.17
1.46
0.9157
0.2334
Time in Current 3
Current 1 versus 2
Output 44.5.3 Simple Effects of Time
The GLM Procedure
Least Squares Means
Current*Time Effect Sliced by Current for MuscleWeight
Current
1
2
3
4
DF
Sum of
Squares
Mean Square
F Value
Pr > F
3
3
3
3
271.458333
120.666667
34.833333
94.833333
90.486111
40.222222
11.611111
31.611111
1.33
0.59
0.17
0.46
0.2761
0.6241
0.9157
0.7085
The SS, F statistics, and p-values can be stored in an OUTSTAT= data set, as shown in Output 44.5.4.
Example 44.6: Multivariate Analysis of Variance F 3585
Output 44.5.4 Contents of the OUTSTAT= Data Set
Obs
_NAME_
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
MuscleWeight
MuscleWeight
MuscleWeight
MuscleWeight
MuscleWeight
MuscleWeight
MuscleWeight
MuscleWeight
MuscleWeight
MuscleWeight
MuscleWeight
MuscleWeight
MuscleWeight
MuscleWeight
MuscleWeight
MuscleWeight
MuscleWeight
MuscleWeight
MuscleWeight
_SOURCE_
_TYPE_
DF
ERROR
Rep
Current
Time
Current*Time
Number
Current*Number
Time*Number
Current*Time*Number
Rep
Current
Time
Current*Time
Number
Current*Number
Time*Number
Current*Time*Number
Time in Current 3
Current 1 versus 2
ERROR
SS1
SS1
SS1
SS1
SS1
SS1
SS1
SS1
SS3
SS3
SS3
SS3
SS3
SS3
SS3
SS3
CONTRAST
CONTRAST
47
1
3
3
9
2
6
6
18
1
3
3
9
2
6
6
18
3
1
SS
3199.49
605.01
2145.45
223.11
298.68
447.44
644.40
367.98
1050.85
605.01
2145.45
223.11
298.68
447.44
644.40
367.98
1050.85
34.83
99.19
F
.
8.8875
10.5054
1.0925
0.4875
3.2864
1.5777
0.9009
0.8576
8.8875
10.5054
1.0925
0.4875
3.2864
1.5777
0.9009
0.8576
0.1706
1.4570
PROB
.
0.00454
0.00002
0.36159
0.87562
0.04614
0.17468
0.50231
0.62757
0.00454
0.00002
0.36159
0.87562
0.04614
0.17468
0.50231
0.62757
0.91574
0.23344
Example 44.6: Multivariate Analysis of Variance
This example employs multivariate analysis of variance (MANOVA) to measure differences in the chemical
characteristics of ancient pottery found at four kiln sites in Great Britain. The data are from Tubb, Parker,
and Nickless (1980), as reported in Hand et al. (1994).
For each of 26 samples of pottery, the percentages of oxides of five metals are measured. The following
statements create the data set and invoke the GLM procedure to perform a one-way MANOVA. Additionally,
it is of interest to know whether the pottery from one site in Wales (Llanederyn) differs from the samples
from other sites; a CONTRAST statement is used to test this hypothesis.
title "Romano-British Pottery";
data pottery;
input Site $12. Al Fe Mg Ca Na;
datalines;
Llanederyn
14.4 7.00 4.30 0.15 0.51
Llanederyn
13.8 7.08 3.43 0.12 0.17
Llanederyn
14.6 7.09 3.88 0.13 0.20
Llanederyn
11.5 6.37 5.64 0.16 0.14
Llanederyn
13.8 7.06 5.34 0.20 0.20
Llanederyn
10.9 6.26 3.47 0.17 0.22
Llanederyn
10.1 4.26 4.26 0.20 0.18
Llanederyn
11.6 5.78 5.91 0.18 0.16
Llanederyn
11.1 5.49 4.52 0.29 0.30
Llanederyn
13.4 6.92 7.23 0.28 0.20
Llanederyn
12.4 6.13 5.69 0.22 0.54
Llanederyn
13.1 6.64 5.51 0.31 0.24
Llanederyn
12.7 6.69 4.45 0.20 0.22
3586 F Chapter 44: The GLM Procedure
Llanederyn
Caldicot
Caldicot
IslandThorns
IslandThorns
IslandThorns
IslandThorns
IslandThorns
AshleyRails
AshleyRails
AshleyRails
AshleyRails
AshleyRails
;
12.5
11.8
11.6
18.3
15.8
18.0
18.0
20.8
17.7
18.3
16.7
14.8
19.1
6.44
5.44
5.39
1.28
2.39
1.50
1.88
1.51
1.12
1.14
0.92
2.74
1.64
3.94
3.94
3.77
0.67
0.63
0.67
0.68
0.72
0.56
0.67
0.53
0.67
0.60
0.22
0.30
0.29
0.03
0.01
0.01
0.01
0.07
0.06
0.06
0.01
0.03
0.10
0.23
0.04
0.06
0.03
0.04
0.06
0.04
0.10
0.06
0.05
0.05
0.05
0.03
proc glm data=pottery;
class Site;
model Al Fe Mg Ca Na = Site;
contrast 'Llanederyn vs. the rest' Site 1 1 1 -3;
manova h=_all_ / printe printh;
run;
After the summary information, displayed in Output 44.6.1, PROC GLM produces the univariate analyses
for each of the dependent variables, as shown in Output 44.6.2 through Output 44.6.6. These analyses show
that sites are significantly different for all oxides individually. You can suppress these univariate analyses by
specifying the NOUNI option in the MODEL statement.
Output 44.6.1 Summary Information about Groups
Romano-British Pottery
The GLM Procedure
Class Level Information
Class
Site
Levels
4
Values
AshleyRails Caldicot IslandThorns Llanederyn
Number of Observations Read
Number of Observations Used
26
26
Example 44.6: Multivariate Analysis of Variance F 3587
Output 44.6.2 Univariate Analysis of Variance for Aluminum Oxide
Romano-British Pottery
The GLM Procedure
Dependent Variable: Al
DF
Sum of
Squares
Mean Square
F Value
Pr > F
Model
3
175.6103187
58.5367729
26.67
<.0001
Error
22
48.2881429
2.1949156
Corrected Total
25
223.8984615
Source
R-Square
Coeff Var
Root MSE
Al Mean
0.784330
10.22284
1.481525
14.49231
Source
Site
Source
Site
DF
Type I SS
Mean Square
F Value
Pr > F
3
175.6103187
58.5367729
26.67
<.0001
DF
Type III SS
Mean Square
F Value
Pr > F
3
175.6103187
58.5367729
26.67
<.0001
Contrast
Llanederyn vs. the rest
DF
Contrast SS
Mean Square
F Value
Pr > F
1
58.58336640
58.58336640
26.69
<.0001
Output 44.6.3 Univariate Analysis of Variance for Iron Oxide
Romano-British Pottery
The GLM Procedure
Dependent Variable: Fe
DF
Sum of
Squares
Mean Square
F Value
Pr > F
Model
3
134.2216158
44.7405386
89.88
<.0001
Error
22
10.9508457
0.4977657
Corrected Total
25
145.1724615
Source
3588 F Chapter 44: The GLM Procedure
Output 44.6.3 continued
R-Square
Coeff Var
Root MSE
Fe Mean
0.924567
15.79171
0.705525
4.467692
Source
Site
Source
Site
DF
Type I SS
Mean Square
F Value
Pr > F
3
134.2216158
44.7405386
89.88
<.0001
DF
Type III SS
Mean Square
F Value
Pr > F
3
134.2216158
44.7405386
89.88
<.0001
Contrast
Llanederyn vs. the rest
DF
Contrast SS
Mean Square
F Value
Pr > F
1
71.15144132
71.15144132
142.94
<.0001
Output 44.6.4 Univariate Analysis of Variance for Calcium Oxide
Romano-British Pottery
The GLM Procedure
Dependent Variable: Ca
DF
Sum of
Squares
Mean Square
F Value
Pr > F
Model
3
0.20470275
0.06823425
29.16
<.0001
Error
22
0.05148571
0.00234026
Corrected Total
25
0.25618846
Source
R-Square
Coeff Var
Root MSE
Ca Mean
0.799032
33.01265
0.048376
0.146538
Source
Site
Source
Site
Contrast
Llanederyn vs. the rest
DF
Type I SS
Mean Square
F Value
Pr > F
3
0.20470275
0.06823425
29.16
<.0001
DF
Type III SS
Mean Square
F Value
Pr > F
3
0.20470275
0.06823425
29.16
<.0001
DF
Contrast SS
Mean Square
F Value
Pr > F
1
0.03531688
0.03531688
15.09
0.0008
Example 44.6: Multivariate Analysis of Variance F 3589
Output 44.6.5 Univariate Analysis of Variance for Magnesium Oxide
Romano-British Pottery
The GLM Procedure
Dependent Variable: Mg
DF
Sum of
Squares
Mean Square
F Value
Pr > F
Model
3
103.3505270
34.4501757
49.12
<.0001
Error
22
15.4296114
0.7013460
Corrected Total
25
118.7801385
Source
R-Square
Coeff Var
Root MSE
Mg Mean
0.870099
26.65777
0.837464
3.141538
Source
Site
Source
Site
DF
Type I SS
Mean Square
F Value
Pr > F
3
103.3505270
34.4501757
49.12
<.0001
DF
Type III SS
Mean Square
F Value
Pr > F
3
103.3505270
34.4501757
49.12
<.0001
Contrast
Llanederyn vs. the rest
DF
Contrast SS
Mean Square
F Value
Pr > F
1
56.59349339
56.59349339
80.69
<.0001
Output 44.6.6 Univariate Analysis of Variance for Sodium Oxide
Romano-British Pottery
The GLM Procedure
Dependent Variable: Na
DF
Sum of
Squares
Mean Square
F Value
Pr > F
Model
3
0.25824560
0.08608187
9.50
0.0003
Error
22
0.19929286
0.00905877
Corrected Total
25
0.45753846
Source
3590 F Chapter 44: The GLM Procedure
Output 44.6.6 continued
R-Square
Coeff Var
Root MSE
Na Mean
0.564424
60.06350
0.095178
0.158462
Source
Site
Source
Site
Contrast
Llanederyn vs. the rest
DF
Type I SS
Mean Square
F Value
Pr > F
3
0.25824560
0.08608187
9.50
0.0003
DF
Type III SS
Mean Square
F Value
Pr > F
3
0.25824560
0.08608187
9.50
0.0003
DF
Contrast SS
Mean Square
F Value
Pr > F
1
0.23344446
0.23344446
25.77
<.0001
The PRINTE option in the MANOVA statement displays the elements of the error matrix, also called the
Error Sums of Squares and Crossproducts matrix. (See Output 44.6.7.) The diagonal elements of this matrix
are the error sums of squares from the corresponding univariate analyses.
The PRINTE option also displays the partial correlation matrix associated with the E matrix. In this example,
none of the oxides are very strongly correlated; the strongest correlation (r = 0.488) is between magnesium
oxide and calcium oxide.
Output 44.6.7 Error SSCP Matrix and Partial Correlations
Romano-British Pottery
The GLM Procedure
Multivariate Analysis of Variance
E = Error SSCP Matrix
Al
Fe
Mg
Ca
Na
Al
Fe
Mg
Ca
Na
48.288142857
7.0800714286
0.6080142857
0.1064714286
0.5889571429
7.0800714286
10.950845714
0.5270571429
-0.155194286
0.0667585714
0.6080142857
0.5270571429
15.429611429
0.4353771429
0.0276157143
0.1064714286
-0.155194286
0.4353771429
0.0514857143
0.0100785714
0.5889571429
0.0667585714
0.0276157143
0.0100785714
0.1992928571
Example 44.6: Multivariate Analysis of Variance F 3591
Output 44.6.7 continued
Partial Correlation Coefficients from the Error SSCP Matrix / Prob > |r|
DF = 22
Al
Fe
Mg
Ca
Na
Al
1.000000
0.307889
0.1529
0.022275
0.9196
0.067526
0.7595
0.189853
0.3856
Fe
0.307889
0.1529
1.000000
0.040547
0.8543
-0.206685
0.3440
0.045189
0.8378
Mg
0.022275
0.9196
0.040547
0.8543
1.000000
0.488478
0.0180
0.015748
0.9431
Ca
0.067526
0.7595
-0.206685
0.3440
0.488478
0.0180
1.000000
0.099497
0.6515
Na
0.189853
0.3856
0.045189
0.8378
0.015748
0.9431
0.099497
0.6515
1.000000
The PRINTH option produces the SSCP matrix for the hypotheses being tested (Site and the contrast); see
Output 44.6.8 and Output 44.6.9. Since the Type III SS are the highest-level SS produced by PROC GLM
by default, and since the HTYPE= option is not specified, the SSCP matrix for Site gives the Type III H
matrix. The diagonal elements of this matrix are the model sums of squares from the corresponding univariate
analyses.
Four multivariate tests are computed, all based on the characteristic roots and vectors of E 1 H. These roots
and vectors are displayed along with the tests. All four tests can be transformed to variates that have F
distributions under the null hypothesis. Note that the four tests all give the same results for the contrast, since
it has only one degree of freedom. In this case, the multivariate analysis matches the univariate results: there
is an overall difference between the chemical composition of samples from different sites, and the samples
from Llanederyn are different from the average of the other sites.
Output 44.6.8 Hypothesis SSCP Matrix and Multivariate Tests for Overall Site Effect
Romano-British Pottery
The GLM Procedure
Multivariate Analysis of Variance
H = Type III SSCP Matrix for Site
Al
Fe
Mg
Ca
Na
Al
Fe
Mg
Ca
Na
175.61031868
-149.295533
-130.8097066
-5.889163736
-5.372264835
-149.295533
134.22161582
117.74503516
4.8217865934
5.3259491209
-130.8097066
117.74503516
103.35052703
4.2091613187
4.7105458242
-5.889163736
4.8217865934
4.2091613187
0.2047027473
0.154782967
-5.372264835
5.3259491209
4.7105458242
0.154782967
0.2582456044
3592 F Chapter 44: The GLM Procedure
Output 44.6.8 continued
Characteristic Roots and Vectors of: E Inverse * H, where
H = Type III SSCP Matrix for Site
E = Error SSCP Matrix
Characteristic
Root
Percent
34.1611140
1.2500994
0.0275396
0.0000000
0.0000000
96.39
3.53
0.08
0.00
0.00
Characteristic Vector
Al
0.09562211
0.02651891
0.09082220
0.03673984
0.06862324
V'EV=1
Fe
-0.26330469
-0.01239715
0.13159869
-0.15129712
0.03056912
Mg
Ca
Na
-0.05305978
0.17564390
0.03508901
0.20455529
-0.10662399
-1.87982100
-4.25929785
-0.15701602
0.54624873
2.51151978
-0.47071123
1.23727668
-1.39364544
-0.17402107
1.23668841
MANOVA Test Criteria and F Approximations for the Hypothesis of No Overall Site Effect
H = Type III SSCP Matrix for Site
E = Error SSCP Matrix
S=3
Statistic
Wilks' Lambda
Pillai's Trace
Hotelling-Lawley Trace
Roy's Greatest Root
M=0.5
N=8
Value
F Value
Num DF
Den DF
Pr > F
0.01230091
1.55393619
35.43875302
34.16111399
13.09
4.30
40.59
136.64
15
15
15
5
50.091
60
29.13
20
<.0001
<.0001
<.0001
<.0001
NOTE: F Statistic for Roy's Greatest Root is an upper bound.
Output 44.6.9 Hypothesis SSCP Matrix and Multivariate Tests for Differences between Llanederyn and the
Other Sites
H = Contrast SSCP Matrix for Llanederyn vs. the rest
Al
Fe
Mg
Ca
Na
Al
Fe
Mg
Ca
Na
58.583366402
-64.56230291
-57.57983466
-1.438395503
-3.698102513
-64.56230291
71.151441323
63.456352116
1.5851961376
4.0755256878
-57.57983466
63.456352116
56.593493386
1.4137558201
3.6347541005
-1.438395503
1.5851961376
1.4137558201
0.0353168783
0.0907993915
-3.698102513
4.0755256878
3.6347541005
0.0907993915
0.2334444577
Example 44.7: Repeated Measures Analysis of Variance F 3593
Output 44.6.9 continued
Characteristic Roots and Vectors of: E Inverse * H, where
H = Contrast SSCP Matrix for Llanederyn vs. the rest
E = Error SSCP Matrix
Characteristic
Root
Percent
16.1251646
0.0000000
0.0000000
0.0000000
0.0000000
100.00
0.00
0.00
0.00
0.00
Characteristic Vector
Al
-0.08883488
-0.00503538
0.00162771
0.04450136
0.11939206
V'EV=1
Fe
0.25458141
0.03825743
-0.08885364
-0.15722494
0.10833549
Mg
Ca
Na
0.08723574
-0.17632854
-0.01774069
0.22156791
0.00000000
0.98158668
5.16256699
-0.83096817
0.00000000
0.00000000
0.71925759
-0.01022754
2.17644566
0.00000000
0.00000000
MANOVA Test Criteria and Exact F Statistics for the Hypothesis
of No Overall Llanederyn vs. the rest Effect
H = Contrast SSCP Matrix for Llanederyn vs. the rest
E = Error SSCP Matrix
S=1
Statistic
Wilks' Lambda
Pillai's Trace
Hotelling-Lawley Trace
Roy's Greatest Root
M=1.5
N=8
Value
F Value
Num DF
Den DF
Pr > F
0.05839360
0.94160640
16.12516462
16.12516462
58.05
58.05
58.05
58.05
5
5
5
5
18
18
18
18
<.0001
<.0001
<.0001
<.0001
Example 44.7: Repeated Measures Analysis of Variance
This example uses data from Cole and Grizzle (1966) to illustrate a commonly occurring repeated measures
ANOVA design. Sixteen dogs are randomly assigned to four groups. (One animal is removed from the
analysis due to a missing value for one dependent variable.) Dogs in each group receive either morphine or
trimethaphan (variable Drug) and have either depleted or intact histamine levels (variable Depleted) before
receiving the drugs. The dependent variable is the blood concentration of histamine at 0, 1, 3, and 5 minutes
after injection of the drug. Logarithms are applied to these concentrations to minimize correlation between
the mean and the variance of the data.
The following SAS statements perform both univariate and multivariate repeated measures analyses and
produce Output 44.7.1 through Output 44.7.7.
data dogs;
input Drug $12. Depleted $ Histamine0 Histamine1
Histamine3 Histamine5;
LogHistamine0=log(Histamine0);
LogHistamine1=log(Histamine1);
LogHistamine3=log(Histamine3);
LogHistamine5=log(Histamine5);
datalines;
Morphine
N .04 .20 .10 .08
3594 F Chapter 44: The GLM Procedure
Morphine
Morphine
Morphine
Morphine
Morphine
Morphine
Morphine
Trimethaphan
Trimethaphan
Trimethaphan
Trimethaphan
Trimethaphan
Trimethaphan
Trimethaphan
Trimethaphan
;
N
N
N
Y
Y
Y
Y
N
N
N
N
Y
Y
Y
Y
.02 .06 .02 .02
.07 1.40 .48 .24
.17 .57 .35 .24
.10 .09 .13 .14
.12 .11 .10
.
.07 .07 .06 .07
.05 .07 .06 .07
.03 .62 .31 .22
.03 1.05 .73 .60
.07 .83 1.07 .80
.09 3.13 2.06 1.23
.10 .09 .09 .08
.08 .09 .09 .10
.13 .10 .12 .12
.06 .05 .05 .05
proc glm;
class Drug Depleted;
model LogHistamine0--LogHistamine5 =
Drug Depleted Drug*Depleted / nouni;
repeated Time 4 (0 1 3 5) polynomial / summary printe;
run;
The NOUNI option in the MODEL statement suppresses the individual ANOVA tables for the original
dependent variables. These analyses are usually of no interest in a repeated measures analysis. The
POLYNOMIAL option in the REPEATED statement indicates that the transformation used to implement the
repeated measures analysis is an orthogonal polynomial transformation, and the SUMMARY option requests
that the univariate analyses for the orthogonal polynomial contrast variables be displayed. The parenthetical
numbers (0 1 3 5) determine the spacing of the orthogonal polynomials used in the analysis.
Output 44.7.1 Summary Information about Groups
The GLM Procedure
Class Level Information
Class
Levels
Values
Drug
2
Morphine Trimethaphan
Depleted
2
N Y
Number of Observations Read
Number of Observations Used
16
15
The “Repeated Measures Level Information” table gives information about the repeated measures effect; it is
displayed in Output 44.7.2. In this example, the within-subject (within-dog) effect is Time, which has the
levels 0, 1, 3, and 5.
Example 44.7: Repeated Measures Analysis of Variance F 3595
Output 44.7.2 Repeated Measures Levels
The GLM Procedure
Repeated Measures Analysis of Variance
Repeated Measures Level Information
Dependent Variable
Log
Log
Log
Log
Histamine0 Histamine1 Histamine3 Histamine5
Level of Time
0
1
3
5
The multivariate analyses for within-subject effects and related interactions are displayed in Output 44.7.3.
For the example, the first table displayed shows that the TIME effect is significant. In addition, the
Time*Drug*Depleted interaction is significant, as shown in the fourth table. This means that the effect of
Time on the blood concentration of histamine is different for the four Drug*Depleted combinations studied.
Output 44.7.3 Multivariate Tests of Within-Subject Effects
MANOVA Test Criteria and Exact F Statistics for the Hypothesis of no Time Effect
H = Type III SSCP Matrix for Time
E = Error SSCP Matrix
S=1
Statistic
Wilks' Lambda
Pillai's Trace
Hotelling-Lawley Trace
Roy's Greatest Root
M=0.5
N=3.5
Value
F Value
Num DF
Den DF
Pr > F
0.11097706
0.88902294
8.01087137
8.01087137
24.03
24.03
24.03
24.03
3
3
3
3
9
9
9
9
0.0001
0.0001
0.0001
0.0001
MANOVA Test Criteria and Exact F Statistics
for the Hypothesis of no Time*Drug Effect
H = Type III SSCP Matrix for Time*Drug
E = Error SSCP Matrix
S=1
Statistic
Wilks' Lambda
Pillai's Trace
Hotelling-Lawley Trace
Roy's Greatest Root
M=0.5
N=3.5
Value
F Value
Num DF
Den DF
Pr > F
0.34155984
0.65844016
1.92774470
1.92774470
5.78
5.78
5.78
5.78
3
3
3
3
9
9
9
9
0.0175
0.0175
0.0175
0.0175
3596 F Chapter 44: The GLM Procedure
Output 44.7.3 continued
MANOVA Test Criteria and Exact F Statistics for
the Hypothesis of no Time*Depleted Effect
H = Type III SSCP Matrix for Time*Depleted
E = Error SSCP Matrix
S=1
Statistic
Wilks' Lambda
Pillai's Trace
Hotelling-Lawley Trace
Roy's Greatest Root
M=0.5
N=3.5
Value
F Value
Num DF
Den DF
Pr > F
0.12339988
0.87660012
7.10373567
7.10373567
21.31
21.31
21.31
21.31
3
3
3
3
9
9
9
9
0.0002
0.0002
0.0002
0.0002
MANOVA Test Criteria and Exact F Statistics for
the Hypothesis of no Time*Drug*Depleted Effect
H = Type III SSCP Matrix for Time*Drug*Depleted
E = Error SSCP Matrix
S=1
Statistic
Wilks' Lambda
Pillai's Trace
Hotelling-Lawley Trace
Roy's Greatest Root
M=0.5
N=3.5
Value
F Value
Num DF
Den DF
Pr > F
0.19383010
0.80616990
4.15915732
4.15915732
12.48
12.48
12.48
12.48
3
3
3
3
9
9
9
9
0.0015
0.0015
0.0015
0.0015
Output 44.7.4 displays tests of hypotheses for between-subject (between-dog) effects. This section tests
the hypotheses that the different Drugs, Depleteds, and their interactions have no effects on the dependent
variables, while ignoring the within-dog effects. From this analysis, there is a significant between-dog
effect for Depleted (p-value=0.0229). The interaction and the main effect for Drug are not significant
(p-values=0.1734 and 0.1281, respectively).
Output 44.7.4 Tests of Between-Subject Effects
The GLM Procedure
Repeated Measures Analysis of Variance
Tests of Hypotheses for Between Subjects Effects
Source
DF
Type III SS
Mean Square
F Value
Pr > F
Drug
Depleted
Drug*Depleted
Error
1
1
1
11
5.99336243
15.44840703
4.69087508
24.34683348
5.99336243
15.44840703
4.69087508
2.21334850
2.71
6.98
2.12
0.1281
0.0229
0.1734
Example 44.7: Repeated Measures Analysis of Variance F 3597
Univariate analyses for within-subject (within-dog) effects and related interactions are displayed in Output 44.7.6. The results for this example are the same as for the multivariate analyses; this is not always
the case. In addition, before the univariate analyses are used to make conclusions about the data, the result
of the sphericity test (requested with the PRINTE option in the REPEATED statement and displayed in
Output 44.7.5) should be examined. If the sphericity test is rejected, consider using the adjusted G-G or
H-F-L probabilities. See the section “Repeated Measures Analysis of Variance” on page 3537 for more
information.
Output 44.7.5 Sphericity Test
Sphericity Tests
Variables
Transformed Variates
Orthogonal Components
DF
Mauchly's
Criterion
Chi-Square
Pr > ChiSq
5
5
0.1752641
0.1752641
16.930873
16.930873
0.0046
0.0046
Output 44.7.6 Univariate Tests of Within-Subject Effects
The GLM Procedure
Repeated Measures Analysis of Variance
Univariate Tests of Hypotheses for Within Subject Effects
Source
DF
Type III SS
Mean Square
F Value
Pr > F
Time
Time*Drug
Time*Depleted
Time*Drug*Depleted
Error(Time)
3
3
3
3
33
12.05898677
1.84429514
12.08978557
2.93077939
2.48238887
4.01966226
0.61476505
4.02992852
0.97692646
0.07522391
53.44
8.17
53.57
12.99
<.0001
0.0003
<.0001
<.0001
Greenhouse-Geisser Epsilon
Huynh-Feldt-Lecoutre Epsilon
Adj Pr > F
G - G
H-F-L
<.0001
0.0039
<.0001
0.0005
<.0001
0.0023
<.0001
0.0002
0.5694
0.6636
Output 44.7.7 is produced by the SUMMARY option in the REPEATED statement. If the POLYNOMIAL
option is not used, a similar table is displayed using the default CONTRAST transformation. The linear,
quadratic, and cubic trends for Time, labeled as ‘Time_1’, ‘Time_2’, and ‘Time_3’, are displayed, and in
each case, the Source labeled ‘Mean’ gives a test for the respective trend.
3598 F Chapter 44: The GLM Procedure
Output 44.7.7 Tests of Between-Subject Effects for Transformed Variables
The GLM Procedure
Repeated Measures Analysis of Variance
Analysis of Variance of Contrast Variables
Time_N represents the nth degree polynomial contrast for Time
Contrast Variable: Time_1
Source
DF
Type III SS
Mean Square
F Value
Pr > F
Mean
Drug
Depleted
Drug*Depleted
Error
1
1
1
1
11
2.00963483
1.18069076
1.36172504
2.04346848
0.63171161
2.00963483
1.18069076
1.36172504
2.04346848
0.05742833
34.99
20.56
23.71
35.58
0.0001
0.0009
0.0005
<.0001
Source
DF
Type III SS
Mean Square
F Value
Pr > F
Mean
Drug
Depleted
Drug*Depleted
Error
1
1
1
1
11
5.40988418
0.59173192
5.94945506
0.67031587
1.04118707
5.40988418
0.59173192
5.94945506
0.67031587
0.09465337
57.15
6.25
62.86
7.08
<.0001
0.0295
<.0001
0.0221
Source
DF
Type III SS
Mean Square
F Value
Pr > F
Mean
Drug
Depleted
Drug*Depleted
Error
1
1
1
1
11
4.63946776
0.07187246
4.77860547
0.21699504
0.80949018
4.63946776
0.07187246
4.77860547
0.21699504
0.07359002
63.04
0.98
64.94
2.95
<.0001
0.3443
<.0001
0.1139
Contrast Variable: Time_2
Contrast Variable: Time_3
Example 44.8: Mixed Model Analysis of Variance with the RANDOM
Statement
Milliken and Johnson (1984) present an example of an unbalanced mixed model. Three machines, which are
considered as a fixed effect, and six employees, which are considered a random effect, are studied. Each
employee operates each machine for either one, two, or three different times. The dependent variable is an
overall rating, which takes into account the number and quality of components produced.
Example 44.8: Mixed Model Analysis of Variance with the RANDOM Statement F 3599
The following statements form the data set and perform a mixed model analysis of variance by requesting the
TEST option in the RANDOM statement. Note that the machine*person interaction is declared as a random
effect; in general, when an interaction involves a random effect, it too should be declared as random. The
results of the analysis are shown in Output 44.8.1 through Output 44.8.4.
data machine;
input machine person
datalines;
1 1 52.0 1 2 51.8 1 2
1 5 51.8 1 5 51.4 1 6
2 2 60.0 2 2 59.0 2 3
2 5 64.8 2 5 65.0 2 6
3 1 66.9 3 2 61.5 3 2
3 4 64.1 3 4 66.2 3 4
3 6 61.4 3 6 60.5
;
rating @@;
52.8
46.4
68.6
43.7
61.7
64.0
1
1
2
2
3
3
3
6
3
6
2
5
60.0
44.8
65.8
44.2
62.3
72.1
1
1
2
2
3
3
4
6
4
6
3
5
51.1
49.2
63.2
43.0
70.8
72.0
1
2
2
3
3
3
4
1
4
1
3
5
52.3
64.0
62.8
67.5
70.6
71.1
1
2
2
3
3
3
5
2
4
1
3
6
50.9
59.7
62.2
67.2
71.0
62.0
proc glm data=machine;
class machine person;
model rating=machine person machine*person;
random person machine*person / test;
run;
The TEST option in the RANDOM statement requests that PROC GLM determine the appropriate F tests
based on person and machine*person being treated as random effects. As you can see in Output 44.8.4,
this requires that a linear combination of mean squares be constructed to test both the machine and person
hypotheses; thus, F tests that use Satterthwaite approximations are needed.
Output 44.8.1 Summary Information about Groups
The GLM Procedure
Class Level Information
Class
Levels
Values
machine
3
1 2 3
person
6
1 2 3 4 5 6
Number of Observations Read
Number of Observations Used
44
44
3600 F Chapter 44: The GLM Procedure
Output 44.8.2 Fixed-Effect Model Analysis of Variance
The GLM Procedure
Dependent Variable: rating
Source
DF
Sum of
Squares
Mean Square
F Value
Pr > F
Model
17
3061.743333
180.102549
206.41
<.0001
Error
26
22.686667
0.872564
Corrected Total
43
3084.430000
R-Square
Coeff Var
Root MSE
rating Mean
0.992645
1.560754
0.934111
59.85000
Source
DF
Type I SS
Mean Square
F Value
Pr > F
machine
person
machine*person
2
5
10
1648.664722
1008.763583
404.315028
824.332361
201.752717
40.431503
944.72
231.22
46.34
<.0001
<.0001
<.0001
Source
DF
Type III SS
Mean Square
F Value
Pr > F
machine
person
machine*person
2
5
10
1238.197626
1011.053834
404.315028
619.098813
202.210767
40.431503
709.52
231.74
46.34
<.0001
<.0001
<.0001
Output 44.8.3 Expected Values of Type III Mean Squares
Source
Type III Expected Mean Square
machine
Var(Error) + 2.137 Var(machine*person) + Q(machine)
person
Var(Error) + 2.2408 Var(machine*person) + 6.7224
Var(person)
machine*person
Var(Error) + 2.3162 Var(machine*person)
Example 44.8: Mixed Model Analysis of Variance with the RANDOM Statement F 3601
Output 44.8.4 Mixed Model Analysis of Variance
The GLM Procedure
Tests of Hypotheses for Mixed Model Analysis of Variance
Dependent Variable: rating
Source
machine
DF
Type III SS
Mean Square
F Value
Pr > F
2
1238.197626
619.098813
16.57
0.0007
Error
10.036
375.057436
37.370384
Error: 0.9226*MS(machine*person) + 0.0774*MS(Error)
Source
DF
Type III SS
Mean Square
F Value
Pr > F
person
5
1011.053834
202.210767
5.17
0.0133
Error
10.015
392.005726
39.143708
Error: 0.9674*MS(machine*person) + 0.0326*MS(Error)
Source
DF
Type III SS
Mean Square
F Value
Pr > F
machine*person
10
404.315028
40.431503
46.34
<.0001
Error: MS(Error)
26
22.686667
0.872564
Note that you can also use the MIXED procedure to analyze mixed models. The following statements use
PROC MIXED to reproduce the mixed model analysis of variance; the relevant part of the PROC MIXED
results is shown in Output 44.8.5.
proc mixed data=machine method=type3;
class machine person;
model rating = machine;
random person machine*person;
run;
3602 F Chapter 44: The GLM Procedure
Output 44.8.5 PROC MIXED Mixed Model Analysis of Variance (Partial Output)
The Mixed Procedure
Type 3 Analysis of Variance
Source
DF
Sum of
Squares
Mean Square
machine
person
machine*person
Residual
2
5
10
26
1238.197626
1011.053834
404.315028
22.686667
619.098813
202.210767
40.431503
0.872564
Type 3 Analysis of Variance
Source
Expected Mean Square
machine
person
machine*person
Residual
Var(Residual) + 2.137 Var(machine*person) + Q(machine)
Var(Residual) + 2.2408 Var(machine*person) + 6.7224 Var(person)
Var(Residual) + 2.3162 Var(machine*person)
Var(Residual)
Type 3 Analysis of Variance
Source
Error Term
machine
0.9226 MS(machine*person)
+ 0.0774 MS(Residual)
0.9674 MS(machine*person)
+ 0.0326 MS(Residual)
MS(Residual)
.
person
machine*person
Residual
Error
DF
F Value
Pr > F
10.036
16.57
0.0007
10.015
5.17
0.0133
26
.
46.34
.
<.0001
.
The advantage of PROC MIXED is that it offers more versatility for mixed models; the disadvantage is that it
can be less computationally efficient for large data sets. See Chapter 63, “The MIXED Procedure,” for more
details.
Example 44.9: Analyzing a Doubly Multivariate Repeated Measures Design
This example shows how to analyze a doubly multivariate repeated measures design by using PROC GLM
with an IDENTITY factor in the REPEATED statement. Note that this differs from previous releases of
PROC GLM, in which you had to use a MANOVA statement to get a doubly repeated measures analysis.
Two responses, Y1 and Y2, are each measured three times for each subject (pretreatment, posttreatment, and
in a later follow-up). Each subject receives one of three treatments; A, B, or the control. In PROC GLM, you
use a REPEATED factor of type IDENTITY to identify the different responses and another repeated factor to
identify the different measurement times. The repeated measures analysis includes multivariate tests for time
and treatment main effects, as well as their interactions, across responses. The following statements produce
Output 44.9.1 through Output 44.9.3.
Example 44.9: Analyzing a Doubly Multivariate Repeated Measures Design F 3603
options ls=96;
data Trial;
input Treatment $ Repetition PreY1 PostY1 FollowY1
PreY2 PostY2 FollowY2;
datalines;
A
1 3 13 9 0 0 9
A
2 0 14 10 6 6 3
A
3 4
6 17 8 2 6
A
4 7
7 13 7 6 4
A
5 3 12 11 6 12 6
A
6 10 14 8 13 3 8
B
1 9 11 17 8 11 27
B
2 4 16 13 9 3 26
B
3 8 10 9 12 0 18
B
4 5
9 13 3 0 14
B
5 0 15 11 3 0 25
B
6 4 11 14 4 2 9
Control 1 10 12 15 4 3 7
Control 2 2
8 12 8 7 20
Control 3 4
9 10 2 0 10
Control 4 10
8 8 5 8 14
Control 5 11 11 11 1 0 11
Control 6 1 5 15 8 9 10
;
proc glm data=Trial;
class Treatment;
model PreY1 PostY1 FollowY1
PreY2 PostY2 FollowY2 = Treatment / nouni;
repeated Response 2 identity, Time 3;
run;
Output 44.9.1 A Doubly Multivariate Repeated Measures Design
The GLM Procedure
Class Level Information
Class
Treatment
Levels
3
Values
A B Control
Number of Observations Read
Number of Observations Used
18
18
The levels of the repeated factors are displayed in Output 44.9.2. Note that RESPONSE is 1 for all the Y1
measurements and 2 for all the Y2 measurements, while the three levels of Time identify the pretreatment,
posttreatment, and follow-up measurements within each response. The multivariate tests for within-subject
effects are displayed in Output 44.9.3.
3604 F Chapter 44: The GLM Procedure
Output 44.9.2 Repeated Factor Levels
The GLM Procedure
Repeated Measures Analysis of Variance
Repeated Measures Level Information
Dependent Variable
PreY1
Level of Response
Level of Time
1
1
PostY1 FollowY1
1
2
PreY2
1
3
2
1
PostY2 FollowY2
2
2
2
3
Output 44.9.3 Within-Subject Tests
MANOVA Test Criteria and Exact F Statistics for the Hypothesis of no Response Effect
H = Type III SSCP Matrix for Response
E = Error SSCP Matrix
S=1
Statistic
Wilks' Lambda
Pillai's Trace
Hotelling-Lawley Trace
Roy's Greatest Root
M=0
N=6
Value
F Value
Num DF
Den DF
Pr > F
0.02165587
0.97834413
45.17686368
45.17686368
316.24
316.24
316.24
316.24
2
2
2
2
14
14
14
14
<.0001
<.0001
<.0001
<.0001
MANOVA Test Criteria and F Approximations for the Hypothesis of no Response*Treatment Effect
H = Type III SSCP Matrix for Response*Treatment
E = Error SSCP Matrix
S=2
Statistic
Wilks' Lambda
Pillai's Trace
Hotelling-Lawley Trace
Roy's Greatest Root
M=-0.5
N=6
Value
F Value
Num DF
Den DF
Pr > F
0.72215797
0.27937444
0.38261660
0.37698780
1.24
1.22
1.31
2.83
4
4
4
2
28
30
15.818
15
0.3178
0.3240
0.3074
0.0908
NOTE: F Statistic for Roy's Greatest Root is an upper bound.
NOTE: F Statistic for Wilks' Lambda is exact.
Example 44.9: Analyzing a Doubly Multivariate Repeated Measures Design F 3605
Output 44.9.3 continued
MANOVA Test Criteria and Exact F Statistics for the Hypothesis of no Response*Time Effect
H = Type III SSCP Matrix for Response*Time
E = Error SSCP Matrix
S=1
Statistic
Wilks' Lambda
Pillai's Trace
Hotelling-Lawley Trace
Roy's Greatest Root
M=1
N=5
Value
F Value
Num DF
Den DF
Pr > F
0.14071380
0.85928620
6.10662362
6.10662362
18.32
18.32
18.32
18.32
4
4
4
4
12
12
12
12
<.0001
<.0001
<.0001
<.0001
MANOVA Test Criteria and F Approximations for the
Hypothesis of no Response*Time*Treatment Effect
H = Type III SSCP Matrix for Response*Time*Treatment
E = Error SSCP Matrix
S=2
Statistic
Wilks' Lambda
Pillai's Trace
Hotelling-Lawley Trace
Roy's Greatest Root
M=0.5
N=5
Value
F Value
Num DF
Den DF
Pr > F
0.22861451
0.96538785
2.52557514
2.12651905
3.27
3.03
3.64
6.91
8
8
8
4
24
26
15
13
0.0115
0.0151
0.0149
0.0033
NOTE: F Statistic for Roy's Greatest Root is an upper bound.
NOTE: F Statistic for Wilks' Lambda is exact.
The table for Response*Treatment tests for an overall treatment effect across the two responses; likewise,
the tables for Response*Time and Response*Treatment*Time test for time and the treatment-by-time
interaction, respectively. In this case, there is a strong main effect for time and possibly for the interaction,
but not for treatment.
In previous releases (before the IDENTITY transformation was introduced), in order to perform a doubly
repeated measures analysis, you had to use a MANOVA statement with a customized transformation matrix
M. You might still want to use this approach to see details of the analysis, such as the univariate ANOVA
for each transformed variate. The following statements demonstrate this approach by using the MANOVA
statement to test for the overall main effect of time and specifying the SUMMARY option.
proc glm data=Trial;
class Treatment;
model PreY1 PostY1 FollowY1
PreY2 PostY2 FollowY2 = Treatment / nouni;
manova h=intercept m=prey1 - posty1,
prey1 - followy1,
prey2 - posty2,
prey2 - followy2 / summary;
run;
3606 F Chapter 44: The GLM Procedure
The M matrix used to perform the test for time effects is displayed in Output 44.9.4, while the results of the
multivariate test are given in Output 44.9.5. Note that the test results are the same as for the Response*Time
effect in Output 44.9.3.
Output 44.9.4 M Matrix to Test for Time Effect (Repeated Measure)
The GLM Procedure
Multivariate Analysis of Variance
M Matrix Describing Transformed Variables
PreY1
PostY1
FollowY1
PreY2
PostY2
FollowY2
1
1
0
0
-1
0
0
0
0
-1
0
0
0
0
1
1
0
0
-1
0
0
0
0
-1
MVAR1
MVAR2
MVAR3
MVAR4
Output 44.9.5 Tests for Time Effect (Repeated Measure)
The GLM Procedure
Multivariate Analysis of Variance
Characteristic Roots and Vectors of: E Inverse * H, where
H = Type III SSCP Matrix for Intercept
E = Error SSCP Matrix
Variables have been transformed by the M Matrix
Characteristic
Root
Percent
6.10662362
0.00000000
0.00000000
0.00000000
100.00
0.00
0.00
0.00
Characteristic Vector
MVAR1
-0.00157729
0.00796367
-0.03534089
-0.05672137
V'EV=1
MVAR2
0.04081620
0.00493217
-0.01502146
0.04500208
MVAR3
MVAR4
-0.04210209
0.05185236
-0.00283074
0.00000000
0.03519437
0.00377940
0.04259372
0.00000000
MANOVA Test Criteria and Exact F Statistics for the Hypothesis of No Overall Intercept Effect
on the Variables Defined by the M Matrix Transformation
H = Type III SSCP Matrix for Intercept
E = Error SSCP Matrix
S=1
Statistic
Wilks' Lambda
Pillai's Trace
Hotelling-Lawley Trace
Roy's Greatest Root
M=1
N=5
Value
F Value
Num DF
Den DF
Pr > F
0.14071380
0.85928620
6.10662362
6.10662362
18.32
18.32
18.32
18.32
4
4
4
4
12
12
12
12
<.0001
<.0001
<.0001
<.0001
The SUMMARY option in the MANOVA statement creates an ANOVA table for each transformed variable
as defined by the M matrix. MVAR1 and MVAR2 contrast the pretreatment measurement for Y1 with the
Example 44.10: Testing for Equal Group Variances F 3607
posttreatment and follow-up measurements for Y1, respectively; MVAR3 and MVAR4 are the same contrasts
for Y2. Output 44.9.6 displays these univariate ANOVA tables and shows that the contrasts are all strongly
significant except for the pre-versus-post difference for Y2.
Output 44.9.6 Summary Output for the Test for Time Effect
The GLM Procedure
Multivariate Analysis of Variance
Dependent Variable: MVAR1
Source
DF
Type III SS
Mean Square
F Value
Pr > F
Intercept
Error
1
15
512.0000000
339.0000000
512.0000000
22.6000000
22.65
0.0003
The GLM Procedure
Multivariate Analysis of Variance
Dependent Variable: MVAR2
Source
DF
Type III SS
Mean Square
F Value
Pr > F
Intercept
Error
1
15
813.3888889
371.1666667
813.3888889
24.7444444
32.87
<.0001
The GLM Procedure
Multivariate Analysis of Variance
Dependent Variable: MVAR3
Source
DF
Type III SS
Mean Square
F Value
Pr > F
Intercept
Error
1
15
68.0555556
292.5000000
68.0555556
19.5000000
3.49
0.0814
The GLM Procedure
Multivariate Analysis of Variance
Dependent Variable: MVAR4
Source
DF
Type III SS
Mean Square
F Value
Pr > F
Intercept
Error
1
15
800.0000000
454.0000000
800.0000000
30.2666667
26.43
0.0001
Example 44.10: Testing for Equal Group Variances
This example demonstrates how you can test for equal group variances in a one-way design. The data come
from the University of Pennsylvania Smell Identification Test (UPSIT), reported in O’Brien and Heft (1995).
The study is undertaken to explore how age and gender are related to sense of smell. A total of 180 subjects
3608 F Chapter 44: The GLM Procedure
20 to 89 years old are exposed to 40 different odors: for each odor, subjects are asked to choose which of
four words best describes the odor. The Freeman-Tukey modified arcsine transformation (Bishop, Fienberg,
and Holland 1975) is applied to the proportion of correctly identified odors to arrive at an olfactory index.
For the following analysis, subjects are divided into five age groups:
8
1 if
ˆ
ˆ
ˆ
ˆ
< 2 if
3 if
agegroup D
ˆ
ˆ
4 if
ˆ
ˆ
:
5 if
25
40
55
70
age
age
age
age
age
<
<
<
<
25
40
55
70
The following statements create a data set named upsit, containing the age group and olfactory index for each
subject.
data upsit;
input agegroup
datalines;
1 1.381 1 1.322
1 1.492 1 1.322
1 1.275 1 1.492
1 1.234 1 1.162
1 1.322 1 1.381
1 1.381 1 1.234
2 1.234 2 1.234
2 1.381 2 1.492
2 1.322 2 1.275
2 1.322 2 1.492
2 1.098 2 1.322
2 1.196
3 1.381 3 1.381
3 1.381 3 1.234
3 1.275 3 1.230
4 1.322 4 1.381
4 1.381 4 1.322
4 1.322 4 1.129
4 1.492 4 0.810
4 1.129 4 1.492
4 1.196 4 1.234
4 0.810
5 1.322 5 1.234
5 0.909 5 0.502
5 1.381 5 1.322
5 1.013 5 0.960
5 1.098 5 1.162
5 1.162 5 1.098
;
smell @@;
1
1
1
1
1
1
2
2
2
2
2
1.162
1.381
1.322
1.381
1.275
1.105
1.381
1.492
1.275
1.196
1.381
1
1
1
1
1
1.275
1.162
1.322
1.381
1.492
1
1
1
1
1
1.381
1.013
1.492
1.381
1.275
1
1
1
1
1
1.275
1.322
1.322
1.322
1.322
1
1
1
1
1
1.322
1.322
1.381
1.381
1.275
2
2
2
2
2
1.322
1.275
1.275
1.322
1.275
2
2
2
2
2
1.492
1.492
1.322
1.275
1.492
2
2
2
2
2
1.234
1.381
1.492
1.234
1.492
2
2
2
2
2
1.381
1.492
1.381
1.322
1.381
3
3
3
4
4
4
4
4
4
1.492
1.234
1.234
1.381
1.275
0.687
1.234
1.129
0.585
3
3
3
4
4
4
4
4
4
1.492
1.129
1.234
1.322
1.275
1.322
1.381
1.098
0.785
3
3
3
4
4
4
4
4
4
1.492
1.069
1.322
1.234
1.492
1.322
1.040
1.275
1.275
3
3
3
4
4
4
4
4
4
1.098
1.234
1.322
1.234
1.234
1.234
1.381
1.322
1.322
3
3
3
4
4
4
4
4
4
1.492
1.322
1.381
1.234
1.098
1.129
1.381
1.234
0.712
5
5
5
5
5
5
1.381
1.234
1.234
0.662
1.040
0.859
5
5
5
5
5
5
1.275
1.322
1.275
1.129
0.558
1.275
5
5
5
5
5
5
1.275
1.196
1.162
0.531
0.960
1.162
5
5
5
5
5
5
1.322
0.859
1.162
1.162
1.098
0.785
5
5
5
5
5
5
1.162
1.196
0.585
0.737
0.884
0.859
Older people are more at risk for problems with their sense of smell, and this should be reflected in significant
differences in the mean of the olfactory index across the different age groups. However, many older people
also have an excellent sense of smell, which implies that the older age groups should have greater variability.
In order to test this hypothesis and to compute a one-way ANOVA for the olfactory index that is robust to
Example 44.10: Testing for Equal Group Variances F 3609
the possibility of unequal group variances, you can use the HOVTEST and WELCH options in the MEANS
statement for the GLM procedure, as shown in the following statements.
proc glm
class
model
means
run;
data=upsit;
agegroup;
smell = agegroup;
agegroup / hovtest welch;
Output 44.10.1, Output 44.10.2, and Output 44.10.3 display the usual ANOVA test for equal age group
means, Levene’s test for equal age group variances, and Welch’s test for equal age group means, respectively.
The hypotheses of age effects for mean and variance of the olfactory index are both confirmed.
Output 44.10.1 Usual ANOVA Test for Age Group Differences in Mean Olfactory Index
The GLM Procedure
Dependent Variable: smell
Source
agegroup
DF
Type III SS
Mean Square
F Value
Pr > F
4
2.13878141
0.53469535
16.65
<.0001
Output 44.10.2 Levene’s Test for Age Group Differences in Olfactory Variability
The GLM Procedure
Levene's Test for Homogeneity of smell Variance
ANOVA of Squared Deviations from Group Means
Source
agegroup
Error
DF
Sum of
Squares
Mean
Square
4
175
0.0799
0.5503
0.0200
0.00314
F Value
Pr > F
6.35
<.0001
Output 44.10.3 Welch’s Test for Age Group Differences in Mean Olfactory Index
Welch's ANOVA for smell
Source
agegroup
Error
DF
F Value
Pr > F
4.0000
78.7489
13.72
<.0001
3610 F Chapter 44: The GLM Procedure
As discussed in “Homogeneity of Variance in One-Way Models” on page 3532, Levene’s test or any other test
for homogeneity of variance should not be used as a diagnostic for the assumption of equal group variances
that underlies the usual analysis of variance. However, graphical diagnostics can be a useful informal tool for
monitoring whether your data meet the assumptions of a GLM analysis. The following statements perform a
one-way ANOVA as before, but with ODS Graphics enabled. In addition to the box plot that is produced
by default, the PLOTS=DIAGNOSTICS option requests a panel of summary diagnostics for the fit. These
additional plots are shown in Output 44.10.4 and Output 44.10.5.
ods graphics on;
proc glm data=upsit plot=diagnostics;
class agegroup;
model smell = agegroup;
run;
ods graphics off;
Output 44.10.4 Box Plot of Olfactory Index by Age Group
Example 44.10: Testing for Equal Group Variances F 3611
Output 44.10.5 Diagnostics for One-Way ANOVA of Olfactory Index by Age Group
Output 44.10.4 clearly shows different degrees of variability for olfactory index within different age groups,
with the variability generally rising with age. Likewise, several of the plots in the diagnostics panel shown in
Output 44.10.5 indicate a relationship between olfactory variability and mean olfactory index. Also, note that
the plot of Cook’s D statistic indicates that observations in the higher, more variable age groups are overly
influential on the analysis of group means. The overall inference from these plots is that an assumption of
equal group variances is probably untenable and that the analysis of the group means should thus take this
into account.
3612 F Chapter 44: The GLM Procedure
Example 44.11: Analysis of a Screening Design
Yin and Jillie (1987) describe an experiment performed on a nitride etch process for a single wafer plasma
etcher. The experiment is run using four factors: cathode power (power), gas flow (flow), reactor chamber
pressure (pressure), and electrode gap (gap). Of interest are the main effects and interaction effects of the
factors on the nitride etch rate (rate). The following statements create a SAS data set named HalfFraction,
containing the factor settings and the observed etch rate for each of eight experimental runs.
data HalfFraction;
input power flow pressure gap rate;
datalines;
0.8
4.5 125 275
550
0.8
4.5 200 325
650
0.8 550.0 125 325
642
0.8 550.0 200 275
601
1.2
4.5 125 325
749
1.2
4.5 200 275
1052
1.2 550.0 125 275
1075
1.2 550.0 200 325
729
;
Notice that each of the factors has just two values. This is a common experimental design when the intent
is to screen from the many factors that might affect the response the few that actually do. Since there are
24 D 16 different possible settings of four two-level factors, this design with only eight runs is called a “half
fraction.” The eight runs are chosen specifically to provide unambiguous information on main effects at the
cost of confounding interaction effects with each other.
One way to analyze these data is simply to use PROC GLM to compute an analysis of variance, including
both main effects and interactions in the model. The following statements demonstrate this approach.
proc glm data=HalfFraction;
class power flow pressure gap;
model rate=power|flow|pressure|[email protected];
run;
The “@2” notation in the MODEL statement includes all main effects and two-factor interactions between the
factors. The output is shown in Output 44.11.1.
Output 44.11.1 Analysis of Variance for Nitride Etch Process Half Fraction
The GLM Procedure
Class Level Information
Class
Levels
Values
power
2
0.8 1.2
flow
2
4.5 550
pressure
2
125 200
gap
2
275 325
Example 44.11: Analysis of a Screening Design F 3613
Output 44.11.1 continued
Number of Observations Read
Number of Observations Used
8
8
The GLM Procedure
Dependent Variable: rate
DF
Sum of
Squares
Mean Square
Model
7
280848.0000
40121.1429
Error
0
0.0000
Corrected Total
7
280848.0000
Source
R-Square
Coeff Var
1.000000
Source
power
flow
power*flow
pressure
power*pressure
flow*pressure
gap
power*gap
flow*gap
pressure*gap
Source
power
flow
power*flow
pressure
power*pressure
flow*pressure
gap
power*gap
flow*gap
pressure*gap
F Value
.
.
.
Root MSE
rate Mean
.
756.0000
.
Pr > F
DF
Type I SS
Mean Square
1
1
1
1
1
1
1
0
0
0
168780.5000
264.5000
200.0000
32.0000
1300.5000
78012.5000
32258.0000
0.0000
0.0000
0.0000
168780.5000
264.5000
200.0000
32.0000
1300.5000
78012.5000
32258.0000
.
.
.
DF
Type III SS
Mean Square
1
1
0
1
0
0
1
0
0
0
168780.5000
264.5000
0.0000
32.0000
0.0000
0.0000
32258.0000
0.0000
0.0000
0.0000
168780.5000
264.5000
.
32.0000
.
.
32258.0000
.
.
.
F Value
.
.
.
.
.
.
.
.
.
.
F Value
.
.
.
.
.
.
.
.
.
.
Pr > F
.
.
.
.
.
.
.
.
.
.
Pr > F
.
.
.
.
.
.
.
.
.
.
Notice that there are no error degrees of freedom. This is because there are 10 effects in the model (4
main effects plus 6 interactions) but only 8 observations in the data set. This is another cost of using a
fractional design: not only is it impossible to estimate all the main effects and interactions, but there is also
no information left to estimate the underlying error rate in order to measure the significance of the effects
that are estimable.
3614 F Chapter 44: The GLM Procedure
Another thing to notice in Output 44.11.1 is the difference between the Type I and Type III ANOVA tables.
The rows corresponding to main effects in each are the same, but no Type III interaction tests are estimable,
while some Type I interaction tests are estimable. This indicates that there is aliasing in the design: some
interactions are completely confounded with each other.
In order to analyze this confounding, you should examine the aliasing structure of the design by using the
ALIASING option in the MODEL statement. Before doing so, however, it is advisable to code the design,
replacing low and high levels of each factor with the values –1 and +1, respectively. This puts each factor
on an equal footing in the model and makes the aliasing structure much more interpretable. The following
statements code the data, creating a new data set named Coded.
data Coded;
power
flow
pressure
gap
run;
set HalfFraction;
= -1*(power
=0.80)
= -1*(flow
=4.50)
= -1*(pressure=125 )
= -1*(gap
=275 )
+
+
+
+
1*(power
=1.20);
1*(flow
=550 );
1*(pressure=200 );
1*(gap
=325 );
The following statements use the GLM procedure to reanalyze the coded design, displaying the parameter
estimates as well as the functions of the parameters that they each estimate.
proc glm data=Coded;
model rate=power|flow|pressure|[email protected] / solution aliasing;
run;
The parameter estimates table is shown in Output 44.11.2.
Output 44.11.2 Parameter Estimates and Aliases for Nitride Etch Process Half Fraction
The GLM Procedure
Dependent Variable: rate
Parameter
Intercept
power
flow
power*flow
pressure
power*pressure
flow*pressure
gap
power*gap
flow*gap
pressure*gap
Estimate
Standard
Error
756.0000000
145.2500000
5.7500000
-5.0000000
2.0000000
-12.7500000
-98.7500000
-63.5000000
0.0000000
0.0000000
0.0000000
.
.
.
.
.
.
.
.
.
.
.
B
B
B
B
B
B
t Value
.
.
.
.
.
.
.
.
.
.
.
Pr > |t|
.
.
.
.
.
.
.
.
.
.
.
Expected Value
Intercept
power
flow
power*flow + pressure*gap
pressure
power*pressure + flow*gap
flow*pressure + power*gap
gap
In the “Expected Value” column, notice that, while each of the main effects is unambiguously estimated by
its associated term in the model, the expected values of the interaction estimates are more complicated. For
example, the relatively large effect (–98.75) corresponding to flow*pressure actually estimates the combined
effect of flow*pressure and power*gap. Without further information, it is impossible to disentangle these
Example 44.11: Analysis of a Screening Design F 3615
aliased interactions; however, since the main effects of both power and gap are large and those for flow and
pressure are small, it is reasonable to suspect that power*gap is the more “active” of the two interactions.
Fortunately, eight more runs are available for this experiment (the other half fraction). The following
statements create a data set containing these extra runs and add it to the previous eight, resulting in a full
24 D 16 run replicate. Then PROC GLM displays the analysis of variance again.
data OtherHalf;
input power flow pressure gap rate;
datalines;
0.8
4.5 125 325
669
0.8
4.5 200 275
604
0.8 550.0 125 275
633
0.8 550.0 200 325
635
1.2
4.5 125 275
1037
1.2
4.5 200 325
868
1.2 550.0 125 325
860
1.2 550.0 200 275
1063
;
data FullRep;
set HalfFraction OtherHalf;
run;
proc glm data=FullRep;
class power flow pressure gap;
model rate=power|flow|pressure|[email protected];
run;
The results are displayed in Output 44.11.3.
Output 44.11.3 Analysis of Variance for Nitride Etch Process Full Replicate
The GLM Procedure
Class Level Information
Class
Levels
power
2
0.8 1.2
flow
2
4.5 550
pressure
2
125 200
gap
2
275 325
Number of Observations Read
Number of Observations Used
Values
16
16
3616 F Chapter 44: The GLM Procedure
Output 44.11.3 continued
The GLM Procedure
Dependent Variable: rate
Source
DF
Sum of
Squares
Mean Square
F Value
Pr > F
Model
10
521234.1250
52123.4125
25.58
0.0011
Error
5
10186.8125
2037.3625
15
531420.9375
Corrected Total
Source
power
flow
power*flow
pressure
power*pressure
flow*pressure
gap
power*gap
flow*gap
pressure*gap
Source
power
flow
power*flow
pressure
power*pressure
flow*pressure
gap
power*gap
flow*gap
pressure*gap
R-Square
Coeff Var
Root MSE
rate Mean
0.980831
5.816175
45.13715
776.0625
DF
Type I SS
Mean Square
F Value
Pr > F
1
1
1
1
1
1
1
1
1
1
374850.0625
217.5625
18.0625
10.5625
1.5625
7700.0625
41310.5625
94402.5625
2475.0625
248.0625
374850.0625
217.5625
18.0625
10.5625
1.5625
7700.0625
41310.5625
94402.5625
2475.0625
248.0625
183.99
0.11
0.01
0.01
0.00
3.78
20.28
46.34
1.21
0.12
<.0001
0.7571
0.9286
0.9454
0.9790
0.1095
0.0064
0.0010
0.3206
0.7414
DF
Type III SS
Mean Square
F Value
Pr > F
1
1
1
1
1
1
1
1
1
1
374850.0625
217.5625
18.0625
10.5625
1.5625
7700.0625
41310.5625
94402.5625
2475.0625
248.0625
374850.0625
217.5625
18.0625
10.5625
1.5625
7700.0625
41310.5625
94402.5625
2475.0625
248.0625
183.99
0.11
0.01
0.01
0.00
3.78
20.28
46.34
1.21
0.12
<.0001
0.7571
0.9286
0.9454
0.9790
0.1095
0.0064
0.0010
0.3206
0.7414
With 16 runs, the analysis of variance tells the whole story: all effects are estimable and there are five degrees
of freedom left over to estimate the underlying error. The main effects of power and gap and their interaction
are all significant, and no other effects are. Notice that the Type I and Type III ANOVA tables are the same;
this is because the design is orthogonal and all effects are estimable.
This example illustrates the use of the GLM procedure for the model analysis of a screening experiment.
Typically, there is much more involved in performing an experiment of this type, from selecting the design
points to be studied to graphically assessing significant effects, optimizing the final model, and performing
References F 3617
subsequent experimentation. Specialized tools for this are available in SAS/QC software, in particular the
ADX Interface and the FACTEX and OPTEX procedures. See SAS/QC User’s Guide for more information.
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Subject Index
absorption of effects
GLM procedure, 3454, 3513
adjusted means
See least squares means, 3461
aliasing structure
GLM procedure, 3481
aliasing structure (GLM), 3614
alpha level
GLM procedure, 3464, 3476, 3481, 3486
analysis of covariance
MODEL statements (GLM), 3496
examples (GLM), 3574
analysis of means
comparing LS-means (GLM), 3525
analysis of variance
MODEL statements (GLM), 3496
mixed models (GLM), 3598
multivariate (GLM), 3470, 3536, 3585
one-way, variance-weighted, 3480
repeated measures (GLM), 3537, 3593, 3602
three-way design (GLM), 3581
unbalanced (GLM), 3437, 3517, 3569
ANOVA procedure
compared to other procedures, 3436
at sign (@) operator
GLM procedure, 3497
balanced data
example, complete block, 3560
bar (|) operator
GLM procedure, 3496
Bartlett’s test
GLM procedure, 3478, 3532
between-subject factors
repeated measures, 3488, 3540
Bonferroni adjustment
GLM procedure, 3462
Bonferroni t test, 3476, 3523
Box’s epsilon, 3541
Brown and Forsythe’s test
GLM procedure, 3478, 3532
canonical analysis
GLM procedure, 3472
CATMOD procedure
compared to other procedures, 3502
characteristic roots and vectors
GLM procedure, 3471
classification variables
GLM procedure, 3495
comparing
groups (GLM), 3517
comparisonwise error rate (GLM), 3523
complete block design
example (GLM), 3560
continuous variables, 3495
continuous-by-class effects
model parameterization (GLM), 3500
specifying (GLM), 3496
continuous-nesting-class effects
model parameterization (GLM), 3500
specifying (GLM), 3496
contrasts
GLM procedure, 3457
repeated measures (GLM), 3490
control
comparing treatments to (GLM), 3520, 3525
Cook’s D influence statistic, 3484
covariates
model parameterization (GLM), 3498
crossed effects
model parameterization (GLM), 3499
specifying (GLM), 3495
degrees of freedom
models with classification variables (GLM), 3502
DFFITS statistic
GLM procedure, 3485
discrete variables, see classification variables
Duncan’s multiple range test, 3477, 3528
Duncan-Waller test, 3479, 3529
error seriousness ratio, 3478
Dunnett’s adjustment
GLM procedure, 3462
Dunnett’s test, 3477, 3525, 3526
one-tailed lower, 3477
effect
definition, 3495
specification (GLM), 3495
effect sizes
GLM procedure, 3508
MODEL statement (GLM), 3482
Einot and Gabriel’s multiple range test
examples (GLM), 3564
GLM procedure, 3479, 3528
error seriousness ratio
Waller-Duncan test, 3478
estimability
GLM procedure, 3458
estimable functions
checking (GLM), 3458
displaying (GLM), 3482
example (GLM), 3502
general form of, 3503
GLM procedure, 3459, 3460, 3469, 3484,
3502–3504, 3506, 3507, 3517
printing (GLM), 3482
estimated population marginal means, see least squares
means
expected mean squares
computing, types (GLM), 3548
random effects, 3546
experimental design, 3617
aliasing structure (GLM), 3614
experimentwise error rate (GLM), 3523
Fisher’s LSD test, 3479
fixed effects
sum-to-zero assumptions, 3547
Forward-Dolittle transformation, 3505
G-G epsilon, 3541
Gabriel’s multiple-comparison procedure
GLM procedure, 3477, 3525
GLM procedure
absorption of effects, 3454, 3513
aliasing structure, 3481, 3614
alpha level, 3464, 3476, 3481, 3486
Bartlett’s test, 3478, 3532
Bonferroni adjustment, 3462
Brown and Forsythe’s test, 3478, 3532
canonical analysis, 3472
characteristic roots and vectors, 3471
compared to other procedures, 3436, 3487, 3545,
3601, 3617
comparing groups, 3517
computational method, 3552
computational resources, 3549
contrasts, 3457, 3490
covariate values for least squares means, 3464
diffogram, 3468
disk space, 3449
Dunnett’s adjustment, 3462
effect sizes, 3508
effect specification, 3495
error effect, 3471
estimability, 3458–3460, 3469, 3484, 3503, 3517
estimable functions, 3502
ESTIMATE specification, 3515
homogeneity of variance tests, 3478, 3532
Hsu’s adjustment, 3462
hypothesis tests, 3493, 3502
interactive use, 3497
interactivity and BY statement, 3455
interactivity and missing values, 3498, 3549
introductory example, 3437
least squares means (LS-means), 3461
Levene’s test for homogeneity of variance, 3478,
3532
means, 3474
means versus least squares means, 3517
memory requirements, reduction of, 3454
missing values, 3449, 3470, 3535, 3549
model specification, 3495
multiple comparisons, least squares means, 3462,
3466, 3519, 3522
multiple comparisons, means, 3476, 3477, 3479,
3519, 3522
multiple comparisons, procedures, 3474
multivariate analysis of variance, 3449, 3470,
3536
Nelson’s adjustment, 3462
nonstandard weights for least squares means,
3465
O’Brien’s test, 3478
observed margins for least squares means, 3465
ODS graph names, 3559
ODS table names, 3555
output data sets, 3484, 3552, 3553
parameterization, 3498
positional requirements for statements, 3446
predicted population margins, 3462
Q effects, 3546
random effects, 3487, 3545, 3546
regression, quadratic, 3440
repeated measures, 3488, 3537
Sidak’s adjustment, 3462
simple effects, 3469
simulation-based adjustment, 3463
singularity checking, 3458, 3460, 3469, 3483
sphericity tests, 3491, 3541
SSCP matrix for multivariate tests, 3471
statistical assumptions, 3494
summary of features, 3435
tests, hypothesis, 3457
transformations for MANOVA, 3471
transformations for repeated measures, 3490
Tukey’s adjustment, 3462
types of least squares means comparisons, 3466
unbalanced analysis of variance, 3437, 3517,
3569
unbalanced design, 3437, 3517, 3546, 3569, 3598
weighted analysis, 3494
weighted means, 3533
Welch’s ANOVA, 3480
WHERE statement, 3498
GLM procedure
ordering of effects, 3450
Greenhouse-Geisser epsilon, 3541
GT2 multiple-comparison method, 3479, 3524
H-F epsilon, 3542
half-fraction design, analysis, 3612
Hochberg’s GT2 multiple-comparison method, 3479,
3524
homogeneity of variance tests, 3478, 3532
Bartlett’s test (GLM), 3478, 3532
Brown and Forsythe’s test (GLM), 3478, 3532
examples, 3609
Levene’s test (GLM), 3478, 3532
O’Brien’s test (GLM), 3478
Welch’s ANOVA, 3532
honestly significant difference test, 3479, 3524, 3526
Hotelling-Lawley trace, 3471, 3540
HSD test, 3479, 3524, 3526
Hsu’s adjustment
GLM procedure, 3462
Huynh-Feldt
epsilon (GLM), 3542
structure (GLM), 3541
hypothesis tests
comparing adjusted means (GLM), 3470
contrasts (GLM), 3457
contrasts, examples (GLM), 3563, 3582, 3591
customized (GLM), 3493
for intercept (GLM), 3482
GLM procedure, 3502
MANOVA (GLM), 3536
random effects (GLM), 3487, 3546
repeated measures (GLM), 3540
Type I sum of squares (GLM), 3504
Type II sum of squares (GLM), 3506
Type III sum of squares (GLM), 3507
Type IV sum of squares (GLM), 3507
interaction effects
model parameterization (GLM), 3499
specifying (GLM), 3495
intercept
hypothesis tests for (GLM), 3482
model parameterization (GLM), 3498
least squares means
Bonferroni adjustment (GLM), 3462
coefficient adjustment, 3535
compared to means (GLM), 3517
comparison types (GLM), 3466
construction of, 3533
covariate values (GLM), 3464
Dunnett’s adjustment (GLM), 3462
examples (GLM), 3571, 3583
GLM procedure, 3461
Hsu’s adjustment (GLM), 3462
multiple comparisons adjustment (GLM), 3462,
3466
Nelson’s adjustment (GLM), 3462
nonstandard weights (GLM), 3465
observed margins (GLM), 3465
Sidak’s adjustment (GLM), 3462
simple effects (GLM), 3469, 3530
simulation-based adjustment (GLM), 3463
Tukey’s adjustment (GLM), 3462
least-significant-difference test, 3479
levels, of classification variable, 3495
Levene’s test for homogeneity of variance
GLM procedure, 3478, 3532, 3609
leverage, 3485
LS-means, see least squares means
LSD test, 3479
main effects
model parameterization (GLM), 3498
specifying (GLM), 3495
MANOVA, see multivariate analysis of variance, see
multivariate analysis of variance
mean separation tests, see multiple-comparison
procedures
means
compared to least squares means (GLM), 3517
GLM procedure, 3474
weighted (GLM), 3533
memory requirements
reduction of (GLM), 3454
missing values
and interactivity (GLM), 3498
mixed model
unbalanced (GLM), 3598
MIXED procedure
contrasted SAS procedures, 3436, 3545, 3601
model
parameterization (GLM), 3498
specification (GLM), 3495
multiple comparison procedures
GLM procedure, 3474
multiple-stage tests, 3527
pairwise (GLM), 3521
recommendations, 3529
with a control (GLM), 3525
with the average (GLM), 3525
multiple comparisons of least squares means, see also
multiple-comparison procedures
GLM procedure, 3462, 3466, 3522
interpretation, 3530
multiple comparisons of means, see also
multiple-comparison procedures
Bonferroni t test, 3476
Duncan’s multiple range test, 3477
Dunnett’s test, 3477
error mean square, 3477
examples, 3561
Fisher’s LSD test, 3479
Gabriel’s procedure, 3477
GLM procedure, 3519, 3522
GT2 method, 3479
interpretation, 3530
Ryan-Einot-Gabriel-Welsch test, 3479
Scheffé’s procedure, 3479
Sidak’s adjustment, 3479
SMM, 3479
Student-Newman-Keuls test, 3479
Tukey’s studentized range test, 3479
Waller-Duncan test, 3479
multiple-comparison procedures, 3519, see also
multiple comparisons of least squares means,
see also multiple comparisons of means
pairwise (GLM), 3520
with a control (GLM), 3520
multiple-stage tests, see multiple comparison
procedures, 3527
multivariate analysis of variance
examples (GLM), 3585
GLM procedure, 3449, 3470, 3536
hypothesis tests (GLM), 3536
partial correlations, 3536
multivariate general linear hypothesis, 3536
multivariate tests
repeated measures, 3540
Nelson’s adjustment
GLM procedure, 3462
nested effects
model parameterization (GLM), 3499
specifying (GLM), 3495
NESTED procedure
compared to other procedures, 3436
Newman-Keuls’ multiple range test, 3479, 3528
nominal variables, see also classification variables
NPAR1WAY procedure
compared to other procedures, 3436
O’Brien’s test for homogeneity of variance
GLM procedure, 3478
ODS graph names
GLM procedure, 3559
orthonormalizing transformation matrix
GLM procedure, 3472
pairwise comparisons
GLM procedure, 3520, 3521
parameterization
of models (GLM), 3498
partial correlations
multivariate analysis of variance, 3536
Pillai’s trace, 3471, 3540
polynomial effects
model parameterization (GLM), 3498
specifying (GLM), 3495
predicted population margins
GLM procedure, 3462
PRESS statistic, 3485
quadratic forms for fixed effects
displaying (GLM), 3487
quadratic regression, 3440
qualitative variables, see classification variables
R-notation, 3504
random effects
expected mean squares, 3546
GLM procedure, 3487, 3545
randomized complete block design
example, 3560
reduction notation, 3504
REG procedure
compared to other procedures, 3436
regression
MODEL statements (GLM), 3496
examples (GLM), 3565
quadratic (GLM), 3440
regression effects
model parameterization (GLM), 3498
specifying (GLM), 3495
repeated measures
contrasts (GLM), 3490
data organization (GLM), 3538
doubly multivariate design, 3602
examples (GLM), 3492, 3593
GLM procedure, 3488, 3537
hypothesis tests (GLM), 3540, 3542
more than one factor (GLM), 3542
transformations, 3543–3545
response variable, 3495
Roy’s greatest root, 3471, 3540
RSREG procedure
compared to other procedures, 3436
Ryan’s multiple range test, 3479, 3528
examples, 3564
Satterthwaite’s approximation
testing random effects, 3547
Scheffé’s multiple-comparison procedure, 3524
p, 3479
screening design, analysis, 3612
Sidak’s adjustment
GLM procedure, 3462
Sidak’s t test, 3479, 3523
simple effects
GLM procedure, 3469, 3530
simulation-based adjustment
GLM procedure, 3463
singularity checking
GLM procedure, 3458, 3460, 3469, 3483
SMM multiple-comparison method, 3479, 3524
sphericity tests, 3491, 3597
SSCP matrix
for multivariate tests (GLM), 3471, 3473
statistical
assumptions (GLM), 3494
stepdown methods
GLM procedure, 3527
Student’s multiple range test, 3479, 3528
Studentized maximum modulus
pairwise comparisons, 3479, 3524
studentized residual, 3485
sum-to-zero assumptions, 3547
sums of squares
GLM procedure, 3483
Type II (GLM), 3483
tests, hypothesis
examples (GLM), 3563
GLM procedure, 3457
transformation matrix
orthonormalizing, 3472
transformations
for multivariate ANOVA, 3471
repeated measures, 3543–3545
transformations for repeated measures
GLM procedure, 3490
TTEST procedure
compared to other procedures, 3437
Tukey’s adjustment
GLM procedure, 3462
Tukey’s studentized range test, 3479, 3524, 3526
Tukey-Kramer test, 3479, 3524, 3526
Type 1 error rate
repeated multiple comparisons, 3522
Type H covariance structure, 3541
Type I sum of squares
computing in GLM, 3548
displaying (GLM), 3483
estimable functions for, 3482
estimable functions for (GLM), 3504
examples, 3570
Type II sum of squares
computing in GLM, 3548
displaying (GLM), 3483
estimable functions for, 3482
estimable functions for (GLM), 3506
examples, 3570
Type III sum of squares
displaying (GLM), 3483
estimable functions for, 3482
estimable functions for (GLM), 3507
examples, 3570
Type IV sum of squares
computing in GLM, 3548
displaying (GLM), 3483
estimable functions for, 3482
estimable functions for (GLM), 3507
examples, 3570
unbalanced design
GLM procedure, 3437, 3517, 3546, 3569, 3598
univariate tests
repeated measures, 3540, 3541
VARCOMP procedure
compared to other procedures, 3437
Waller-Duncan test, 3479, 3529
error seriousness ratio, 3478
examples, 3564
weighted least squares
normal equations (GLM), 3494
weighted means
GLM procedure, 3533
Welch’s ANOVA, 3480
homogeneity of variance tests, 3532
using homogeneity of variance tests, 3609
Welsch’s multiple range test, 3479, 3528
examples, 3564
WHERE statement
GLM procedure, 3498
Wilks’ lambda, 3471, 3540
within-subject factors
repeated measures, 3488, 3540
Syntax Index
ABSORB statement
GLM procedure, 3454
ADJUST= option
LSMEANS statement (GLM), 3462
ALIASING option
MODEL statement (GLM), 3481, 3614
_ALL_ effect
MANOVA statement, H= option (GLM), 3471
ALPHA= option
LSMEANS statement (GLM), 3464
MEANS statement (GLM), 3476
MODEL statement (GLM), 3481
OUTPUT statement (GLM), 3486
PROC GLM statement, 3449
AT option
LSMEANS statement (GLM), 3464, 3534
BON option
MEANS statement (GLM), 3476
BY statement
GLM procedure, 3455
BYLEVEL option
LSMEANS statement (GLM), 3464, 3535
CANONICAL option
MANOVA statement (GLM), 3472
REPEATED statement (GLM), 3491
CL option
LSMEANS statement (GLM), 3464
CLASS statement
GLM procedure, 3455
CLDIFF option
MEANS statement (GLM), 3476
CLI option
MODEL statement (GLM), 3481
CLM option
MEANS statement (GLM), 3476
MODEL statement (GLM), 3481
CLPARM option
MODEL statement (GLM), 3482
CODE statement
GLM procedure, 3457
CONTRAST option
REPEATED statement (GLM), 3490, 3543
CONTRAST statement
GLM procedure, 3457
COOKD keyword
OUTPUT statement (GLM), 3484
COV option
LSMEANS statement (GLM), 3465
COVRATIO keyword
OUTPUT statement (GLM), 3485
DATA= option
PROC GLM statement, 3449
DEPONLY option
MEANS statement (GLM), 3476
DFFITS keyword
OUTPUT statement (GLM), 3485
DIVISOR= option
ESTIMATE statement (GLM), 3460
DUNCAN option
MEANS statement (GLM), 3477
DUNNETT option
MEANS statement (GLM), 3477
DUNNETTL option
MEANS statement (GLM), 3477
DUNNETTU option
MEANS statement (GLM), 3477
E option
CONTRAST statement (GLM), 3458
ESTIMATE statement (GLM), 3460
LSMEANS statement (GLM), 3465
MODEL statement (GLM), 3482
E1 option
MODEL statement (GLM), 3482
E2 option
MODEL statement (GLM), 3482
E3 option
MODEL statement (GLM), 3482
E4 option
MODEL statement (GLM), 3482
E= option
CONTRAST statement (GLM), 3458
MANOVA statement (GLM), 3471
MEANS statement (GLM), 3477
REPEATED statement (GLM), 3493
EFFECTSIZE option
MODEL statement (GLM), 3482
ESTIMATE statement
GLM procedure, 3459, 3515
ETYPE option
LSMEANS statement (GLM), 3465
ETYPE= option
CONTRAST statement (GLM), 3458
MANOVA statement (GLM), 3472
MEANS statement (GLM), 3477
TEST statement (GLM), 3493
factor specification
REPEATED statement (GLM), 3489
FREQ statement
GLM procedure, 3460
GABRIEL option
MEANS statement (GLM), 3477
GLM procedure
syntax, 3446
GLM procedure, ABSORB statement, 3454
GLM procedure, BY statement, 3455
GLM procedure, CONTRAST statement, 3457
E option, 3458
E= option, 3458
ETYPE= option, 3458
INTERCEPT effect, 3458, 3460
SINGULAR= option, 3458
GLM procedure, ESTIMATE statement, 3459
DIVISOR= option, 3460
E option, 3460
SINGULAR= option, 3460
GLM procedure, FREQ statement, 3460
GLM procedure, ID statement, 3461
GLM procedure, LSMEANS statement, 3461
ADJUST= option, 3462
ALPHA= option, 3464
AT option, 3464, 3534
BYLEVEL option, 3535
BYLEVEL options, 3464
CL option, 3464
COV option, 3465
E option, 3465
ETYPE option, 3465
LINES option, 3465
NOPRINT option, 3465
OBSMARGINS option, 3465, 3535
OM option, 3465, 3535
OUT= option, 3466
PDIFF option, 3466
PLOTS= option, 3467
SINGULAR option, 3469
SLICE= option, 3469
STDERR option, 3470
TDIFF option, 3470
GLM procedure, MANOVA statement, 3470
_ALL_ effect (H= option), 3471
CANONICAL option, 3472
E= option, 3471
ETYPE= option, 3472
H= option, 3471
HTYPE= option, 3472
INTERCEPT effect (H= option), 3471
M= option, 3471
MNAMES= option, 3472
MSTAT= option, 3472
ORTH option, 3472
PREFIX= option, 3472
PRINTE option, 3473
PRINTH option, 3473
SUMMARY option, 3473
GLM procedure, MEANS statement, 3474, 3533
ALPHA= option, 3476
BON option, 3476
CLDIFF option, 3476
CLM option, 3476
DEPONLY option, 3476
DUNCAN option, 3477
DUNNETT option, 3477
DUNNETTL option, 3477
DUNNETTU option, 3477
E= option, 3477
ETYPE= option, 3477
GABRIEL option, 3477
GT2 option, 3477
HOVTEST option, 3478, 3532
HTYPE= option, 3478
KRATIO= option, 3478
LINES option, 3478
LSD option, 3479
NOSORT option, 3479
REGWQ option, 3479
SCHEFFE option, 3479
SIDAK option, 3479
SMM option, 3479
SNK option, 3479
T option, 3479
TUKEY option, 3479
WALLER option, 3479
WELCH option, 3480
GLM procedure, MODEL statement, 3480
ALIASING option, 3481, 3614
ALPHA= option, 3481
CLI option, 3481
CLM option, 3481
CLPARM option, 3482
E option, 3482
E1 option, 3482
E2 option, 3482
E3 option, 3482
E4 option, 3482
EFFECTSIZE option, 3482
I option, 3482
INTERCEPT option, 3482
INVERSE option, 3482
NOINT option, 3483
NOUNI option, 3483
P option, 3483
SINGULAR= option, 3483
SOLUTION option, 3483
SS1 option, 3483
SS2 option, 3483
SS3 option, 3483
SS4 option, 3483
TOLERANCE option, 3483
XPX option, 3484
ZETA= option, 3484
GLM procedure, OUTPUT statement, 3484
ALPHA= option, 3486
COOKD keyword, 3484
COVRATIO keyword, 3485
DFFITS keyword, 3485
H keyword, 3485
keyword= option, 3484
LCL keyword, 3485
LCLM keyword, 3485
OUT= option, 3486
PREDICTED keyword, 3485
PRESS keyword, 3485
RESIDUAL keyword, 3485
RSTUDENT keyword, 3485
STDI keyword, 3485
STDP keyword, 3485
STDR keyword, 3485
STUDENT keyword, 3485
UCL keyword, 3486
UCLM keyword, 3486
GLM procedure, PROC GLM statement, 3448
ALPHA= option, 3449
DATA= option, 3449
MANOVA option, 3449
MULTIPASS option, 3449
NAMELEN= option, 3449
NOPRINT option, 3449
ORDER= option, 3517
OUTSTAT= option, 3450
PLOTS= option, 3450
GLM procedure, RANDOM statement, 3487
Q option, 3487, 3546
TEST option, 3487
GLM procedure, REPEATED statement, 3488
CANONICAL option, 3491
CONTRAST option, 3490, 3543
E= option, 3493
factor specification, 3489
H= option, 3493
HELMERT option, 3490, 3544
HTYPE= option, 3491
IDENTITY option, 3490, 3602
MEAN option, 3490, 3491, 3544
MSTAT= option, 3491
NOM option, 3491
NOU option, 3491
POLYNOMIAL option, 3490, 3544, 3594
PRINTE option, 3491, 3541
PRINTH option, 3491
PRINTM option, 3491
PRINTRV option, 3491
PROFILE option, 3490, 3545
SUMMARY option, 3492
UEPSDEF option, 3492
GLM procedure, TEST statement, 3493
ETYPE= option, 3493
HTYPE= option, 3493
GLM procedure, WEIGHT statement, 3494
GLM procedure, CLASS statement, 3455
REF= option, 3456
REF= variable option, 3456
TRUNCATE option, 3456
GLM procedure, CODE statement, 3457
GLM procedure, PROC GLM statement
ORDER= option, 3450
GLM procedure, STORE statement, 3492
GT2 option
MEANS statement (GLM), 3477
H keyword
OUTPUT statement (GLM), 3485
H= option
MANOVA statement (GLM), 3471
REPEATED statement (GLM), 3493
HELMERT option
REPEATED statement (GLM), 3490, 3544
HOVTEST option
MEANS statement (GLM), 3478, 3532
HTYPE= option
MANOVA statement (GLM), 3472
MEANS statement (GLM), 3478
REPEATED statement (GLM), 3491
TEST statement (GLM), 3493
I option
MODEL statement (GLM), 3482
ID statement
GLM procedure, 3461
IDENTITY option
REPEATED statement (GLM), 3490, 3602
INTERCEPT effect
CONTRAST statement (GLM), 3458, 3460
MANOVA statement, H= option (GLM), 3471
INTERCEPT option
MODEL statement (GLM), 3482
INVERSE option
MODEL statement (GLM), 3482
keyword= option
OUTPUT statement (GLM), 3484
KRATIO= option
MEANS statement (GLM), 3478
LCL keyword
OUTPUT statement (GLM), 3485
LCLM keyword
OUTPUT statement (GLM), 3485
LINES option
LSMEANS statement (GLM), 3465
MEANS statement (GLM), 3478
LSD option
MEANS statement (GLM), 3479
LSMEANS statement
GLM procedure, 3461
M= option
MANOVA statement (GLM), 3471
MANOVA option
PROC GLM statement, 3449
MANOVA statement
GLM procedure, 3470
MEAN option
REPEATED statement (GLM), 3490, 3491, 3544
MEANS statement
GLM procedure, 3474
MNAMES= option
MANOVA statement (GLM), 3472
MODEL statement
GLM procedure, 3480
MSTAT= option
MANOVA statement (GLM), 3472
REPEATED statement (GLM), 3491
MULTIPASS option
PROC GLM statement, 3449
NAMELEN= option
PROC GLM statement, 3449
NOINT option
MODEL statement (GLM), 3483
NOM option
REPEATED statement (GLM), 3491
NOPRINT option
LSMEANS statement (GLM), 3465
PROC GLM statement, 3449
NOSORT option
MEANS statement (GLM), 3479
NOU option
REPEATED statement (GLM), 3491
NOUNI option
MODEL statement (GLM), 3483
OBSMARGINS option
LSMEANS statement (GLM), 3465, 3535
OM option
LSMEANS statement (GLM), 3465, 3535
ORDER= option
PROC GLM statement, 3517
PROC GLM statement, 3450
ORTH option
MANOVA statement (GLM), 3472
OUT= option
LSMEANS statement (GLM), 3466
OUTPUT statement (GLM), 3486
OUTPUT statement
GLM procedure, 3484
OUTSTAT= option
PROC GLM statement, 3450
P option
MODEL statement (GLM), 3483
PDIFF option
LSMEANS statement (GLM), 3466
PLOTS= option
LSMEANS statement (GLM), 3467
PROC GLM statement, 3450
POLYNOMIAL option
REPEATED statement (GLM), 3490, 3544, 3594
PREDICTED keyword
OUTPUT statement (GLM), 3485
PREFIX= option
MANOVA statement (GLM), 3472
PRESS keyword
OUTPUT statement (GLM), 3485
PRINTE option
MANOVA statement (GLM), 3473
REPEATED statement (GLM), 3491, 3541
PRINTH option
MANOVA statement (GLM), 3473
REPEATED statement (GLM), 3491
PRINTM option
REPEATED statement (GLM), 3491
PRINTRV option
REPEATED statement (GLM), 3491
PROC GLM statement, see GLM procedure
PROFILE option
REPEATED statement (GLM), 3490, 3545
Q option
RANDOM statement (GLM), 3487, 3546
RANDOM statement
GLM procedure, 3487
REF= option
CLASS statement (GLM), 3456
REGWQ option
MEANS statement (GLM), 3479
REPEATED statement
GLM procedure, 3488
RESIDUAL keyword
OUTPUT statement (GLM), 3485
RSTUDENT keyword
OUTPUT statement (GLM), 3485
SCHEFFE option
MEANS statement (GLM), 3479
SIDAK option
MEANS statement (GLM), 3479
SINGULAR option
CONTRAST statement (GLM), 3458
LSMEANS statement (GLM), 3469
SINGULAR= option
ESTIMATE statement (GLM), 3460
MODEL statement (GLM), 3483
SLICE= option
LSMEANS statement (GLM), 3469
SMM option
MEANS statement (GLM), 3479
SNK option
MEANS statement (GLM), 3479
SOLUTION option
MODEL statement (GLM), 3483
SS1 option
MODEL statement (GLM), 3483
SS2 option
MODEL statement (GLM), 3483
SS3 option
MODEL statement (GLM), 3483
SS4 option
MODEL statement (GLM), 3483
STDERR option
LSMEANS statement (GLM), 3470
STDI keyword
OUTPUT statement (GLM), 3485
STDP keyword
OUTPUT statement (GLM), 3485
STDR keyword
OUTPUT statement (GLM), 3485
STORE statement
GLM procedure, 3492
STUDENT keyword
OUTPUT statement (GLM), 3485
SUMMARY option
MANOVA statement (GLM), 3473
REPEATED statement (GLM), 3492
T option
MEANS statement (GLM), 3479
TDIFF option
LSMEANS statement (GLM), 3470
TEST option
RANDOM statement (GLM), 3487
TEST statement
GLM procedure, 3493
TOLERANCE option
MODEL statement (GLM), 3483
TRUNCATE option
CLASS statement (GLM), 3456
TUKEY option
MEANS statement (GLM), 3479
UCL keyword
OUTPUT statement (GLM), 3486
UCLM keyword
OUTPUT statement (GLM), 3486
UEPSDEF option
REPEATED statement (GLM), 3492
WALLER option
MEANS statement (GLM), 3479
WEIGHT statement
GLM procedure, 3494
WELCH option
MEANS statement (GLM), 3480
WHERE statement
GLM procedure, 3498
XPX option
MODEL statement (GLM), 3484
ZETA= option
MODEL statement (GLM), 3484
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