SPSS 11.5 User Manual

SPSS
Advanced Models

™

11.5

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™

SPSS Advanced Models

11.5
Copyright © 2002 by SPSS Inc.
All rights reserved.
Printed in the United States of America.

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1 2 3 4 5 6 7 8 9 0 06 05 04 03 02
ISBN 1-56827-303-7

Preface

SPSS® 11.5 is a comprehensive system for analyzing data. SPSS can take data from
almost any type of fil e an d use t hem t o gener a te tabulated reports, charts, and plots of
distributions and trends, descriptive statistics, and complex statistical analyses.

The Advanced Models option is an add-on enhancement that provides additional
statistical analysis tec hniques. The pr ocedures in Advanced Mo dels must be used with
the SPSS 11.5 Base and are completely integrated into that system.

The Advanced Models option includes procedures for:

 GLM—General linear models, which can accommodate analysis of variance

(ANOVA), regression, and analysis of covariance (ANCOVA), in both univariate
and multivariate models. Rep eated measures and doubly repeated mea sures models
are also available.

 Linear mixed models.
 Variance components.
 Model selection loglinear analysis (hierarchical).
 General loglinear analysis.
 Logit loglinear analysis.
 Ordinal regression analysis.
 Survival analysis, including life tables, Kaplan-Meier survival analysis, and Cox

regression.

Installation

T o install Advanced Models, fol low the instruct ions for addin g and removing featu res
in the installation inst ructi ons suppli ed wit h the SPS S Base. (To start, double-click on
the SPSS Setup icon.)

iii

Compatibility

The SPSS system is designed to operate on many computer systems. See the
installation instructions that came with your system for specific information on
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 Technical support services
 New product announcements and upgrade announcements

Customer Service

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iv

Technical Support

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Customers may contact T echnical Support for assistance in using SPSS pr oducts or for
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Statistical introductions to procedures in the Base, Regression Models, and
Advanced Models written by Marija Norusis are pl anned to be availabl e from Prentice

v

Hall. Check with the publis her or visit the SPSS Web si te for announcements regar ding
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v

About This Manual

This manual documents the graphical user interface. Illustrations of dialog boxes are
taken from SPSS for Windows. Dialog boxes in other operating systems are similar.
The Advanced Models command syntax is included in the SPSS 11.5 Syntax Reference
Guide, available on the product CD-ROM.

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vii

Contents

1 GLM Multivariate Analysis 1

To Obtain a GLM Multivariate Analysis of Variance . . . . . . . . . . . . .3

GLM Multivariate Model . . . . . . . . . . . . . . . . . . . . . . . . . . 4

GLM Multivariate Contrasts . . . . . . . . . . . . . . . . . . . . . . . .6

GLM Multivariate Profile Plots . . . . . . . . . . . . . . . . . . . . . .8

GLM Multivariate Post Hoc Multiple Comparisons

for Observed Means . . . . . . . . . . . . . . . . . . . . . . . . . . . .9

GLM Multivariate Save. . . . . . . . . . . . . . . . . . . . . . . . . . 11

GLM Multivariate Options . . . . . . . . . . . . . . . . . . . . . . . . 12

GLM Command Additional Features . . . . . . . . . . . . . . . . . . 14

2 GLM Repeated Measures 15

GLM Repeated Measures Define Factor(s) . . . . . . . . . . . . . . . . . 18

To Obtain a GLM Repeated Measures Analysis . . . . . . . . . . . . . . 18

GLM Repeated Measures Model . . . . . . . . . . . . . . . . . . . . 21

GLM Repeated Measures Contrasts . . . . . . . . . . . . . . . . . . 23

GLM Repeated Measures Profile Plots . . . . . . . . . . . . . . . . 25

GLM Repeated Measures Post Hoc Multiple Comparisons

for Observed Means . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

GLM Repeated Measures Save. . . . . . . . . . . . . . . . . . . . . 28

GLM Repeated Measures Options . . . . . . . . . . . . . . . . . . . 29

GLM Command Additional Features . . . . . . . . . . . . . . . . . . 31

ix

3 Variance Components Analysis 33

To Obtain a Variance Components Analysis . . . . . . . . . . . . . . . . 34

Variance Components Model . . . . . . . . . . . . . . . . . . . . . . 35

Variance Components Options . . . . . . . . . . . . . . . . . . . . . 37

Sums of Squares (Variance Components) . . . . . . . . . . . . . . . 38

Variance Components Save to New File . . . . . . . . . . . . . . . . 39

VARCOMP Command Additional Features. . . . . . . . . . . . . . . 40

4 Linear Mixed Models 41

Linear Mixed Models Select Subjects/Repeated Variables . . . . . . . 42

Selecting Subjects/Repeated Variables for

Linear Mixed Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

Obtaining a Linear Mixed Models Analysis . . . . . . . . . . . . . . . . . 45

Linear Mixed Models Fixed Effects. . . . . . . . . . . . . . . . . . . 46

Linear Mixed Models Random Effects . . . . . . . . . . . . . . . . . 49

Linear Mixed Models Estimation . . . . . . . . . . . . . . . . . . . . 51

Linear Mixed Models Statistics . . . . . . . . . . . . . . . . . . . . . 52

Linear Mixed Models EM Means . . . . . . . . . . . . . . . . . . . . 54

Linear Mixed Models Save . . . . . . . . . . . . . . . . . . . . . . . 55

MIXED Command Additional Features . . . . . . . . . . . . . . . . . 56

5 Model Selection Loglinear Analysis 57

To Obtain a Model Selection Loglinear Analysis . . . . . . . . . . . . . . 58

Loglinear Analysis Define Range . . . . . . . . . . . . . . . . . . . . 59

Loglinear Analysis Model . . . . . . . . . . . . . . . . . . . . . . . . 59

Build Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

Loglinear Analysis Options . . . . . . . . . . . . . . . . . . . . . . . 60

HILOGLINEAR Command Additional Features. . . . . . . . . . . . . 61

x

6 General Loglinear Analysis 63

To Obtain a General Loglinear Analysis . . . . . . . . . . . . . . . . . . .65

General Loglinear Analysis Model . . . . . . . . . . . . . . . . . . . .66

General Loglinear Analysis Options . . . . . . . . . . . . . . . . . . .67

To Specify Options. . . . . . . . . . . . . . . . . . . . . . . . . . . . .68

General Loglinear Analysis Save . . . . . . . . . . . . . . . . . . . .68

GENLOG Command Additional Features . . . . . . . . . . . . . . . .69

7 Logit Loglinear Analysis 71

To Obtain a Logit Loglinear Analysis . . . . . . . . . . . . . . . . . . . . .73

Logit Loglinear Analysis Model . . . . . . . . . . . . . . . . . . . . .74

Build Terms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .75

Logit Loglinear Analysis Options . . . . . . . . . . . . . . . . . . . . .75

To Specify Options. . . . . . . . . . . . . . . . . . . . . . . . . . . . .76

Logit Loglinear Analysis Save . . . . . . . . . . . . . . . . . . . . . .76

GENLOG Command Additional Features . . . . . . . . . . . . . . . .77

8 Ordinal Regression 79

To Obtain an Ordinal Regression . . . . . . . . . . . . . . . . . . . . . . 80

Ordinal Regression Options. . . . . . . . . . . . . . . . . . . . . . . 81

Ordinal Regression Output . . . . . . . . . . . . . . . . . . . . . . . 82

Ordinal Regression Location Model . . . . . . . . . . . . . . . . . . 84

Ordinal Regression Scale Model. . . . . . . . . . . . . . . . . . . . 85

Ordinal Regression Command Additional Features . . . . . . . . . 86

xi

9 Life Tables 87

To Create a Life Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

Life Tables Define Event for Status Variable . . . . . . . . . . . . . 90

Life Tables Define Range . . . . . . . . . . . . . . . . . . . . . . . . 90

Life Tables Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

SURVIVAL Command Additional Features . . . . . . . . . . . . . . . 91

10 Kaplan-Meier Survival Analysis 93

To Obtain a Kaplan-Meier Survival Analysis . . . . . . . . . . . . . . . . 94

Kaplan-Meier Define Event for Status Variable . . . . . . . . . . . . 95

Kaplan-Meier Compare Factor Levels . . . . . . . . . . . . . . . . . 96

Kaplan-Meier Save New Variables. . . . . . . . . . . . . . . . . . . 96

Kaplan-Meier Options . . . . . . . . . . . . . . . . . . . . . . . . . . 97

KM Command Additional Features . . . . . . . . . . . . . . . . . . . 97

11 Cox Regression Analysis 99

Obtaining a Cox Regression Analysis . . . . . . . . . . . . . . . . . . . 100

Cox Regression Define Categorical Variables . . . . . . . . . . . 101

Cox Regression Plots. . . . . . . . . . . . . . . . . . . . . . . . . . 102

Cox Regression Save New Variables . . . . . . . . . . . . . . . . 103

Cox Regression Options . . . . . . . . . . . . . . . . . . . . . . . . 104

Cox Regression Define Event for Status Variable. . . . . . . . . . 104

COXREG Command Additional Features . . . . . . . . . . . . . . . 105

xii

12 Computing Time-Dependent Covariates 107

To Compute a Time-Dependent Covariate . . . . . . . . . . . . . . . . . 108

Cox Regression with Time-Dependent Covariates

Additional Features . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

Appendix A Categorical Variable Coding Schemes 111

Deviation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

Simple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

Helmert . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

Difference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

Polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

Repeated . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

Special . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

Indicator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

Appendix B
Covariance Structures 119

Index 123

xiii

GLM Multivariate Analysis

The GLM Multivariate procedure provides regression analysis and analysis of
variance for multiple dependent variables by one or more factor variables or
covariates. The factor variab les divid e the pop ulati on into g roups. Using t his gen eral
linear model procedure, you can test null hypotheses about the effects of factor
variables on the means of various groupings of a joint distribution of dependent
variables. You can investigate interactions between factors as well as the effects of
individual factors. In addition, the effects of covariates and covariate interactions with
factors can be included. For r egression analysis, the i ndependent (predicto r) variables
are specified as covariates.

Both balanced and unbalanced models can be t est ed. A design is balanced if each
cell in the model contains the same number of ca ses. In a multivariate model, the sums
of squares due to the effects in the model and error sums of squares are in matrix form
rather than the scalar form found in univariate analysis. These matrices are called
SSCP (sums-of-squares and cross-products) matrices. If more than one dependent
variable is specified, the multivariate an alysis of v ariance us ing Pillai ’s trace, Wilks’
lambda, Hotelling’s trace, and Roy’s largest root criterion with approximate F
statistic are provided as well as the univaria te analysis of variance for eac h dependent
variable. In addition to testing hypotheses, GLM Multivariate produces estimates of
parameters.

Commonly used a priori contrasts are available to perform hypothesis testing.
Additionally, aft er an overall F t est has shown signifi cance, you can use po st hoc tests
to evaluate differences among specific means. Estimated marginal means give
estimates of predicted mean values for the cells in the model, and profile plots

Chapter

1

1

2

Chapter 1

(interaction plots) of these means allow you to visualize some of the relationships
easily. The post hoc multiple comparison tests are performed for each dependent
variable separately.

Residuals, predicted values, Cook’s distance, and leverage values can be saved as
new variables in your data file for che cking assumpti ons. Also availabl e are a residual
SSCP matrix, which is a square matrix of sums of squares and cross-products of
residuals, a residual covariance matrix, which is the residual SSC P mat rix di vided by
the degrees of freedom of the residuals, and the residual correlation matrix, which is
the standardized form of the residual covariance matrix.

WLS Weight allows you to specify a variable used to give observations different
weights for a weighted least-squares (WLS) analysis, perhaps to compensate for
different precision of measurement.

Example. A manufacturer of plastics measures three properties of plastic film: tear

resistance, gloss, and opacity. Two rates of extrusion and two different amounts of
additive are tried, and the three properties are measured under each combination of
extrusion rate and additive amount. The manufacturer finds th at the extrusion rate and
the amount of additive individually produce significant results but that the interact ion
of the two factors is not significant.

Methods. T ype I, Type II, T ype III, and T ype IV sums of squares can be used to evaluate

different hypotheses. Type III is the default.

Statistics. Post hoc range tests and multiple comparisons: least significant difference,

Bonferroni, Sidak, Scheffé, Ryan-Einot-Gabriel-Welsch multiple F, Ryan-Einot-
Gabriel-Welsch multiple range, Student-Newman- Keuls, T ukey ’ s honestly significan t
difference, Tukey’s-b, Duncan, Hochberg’s GT2, Gabriel, Waller Duncan t test,
Dunnett (one-sided and two-sided), Tamhane’ s T2, Dunnett’s T3, Games-Howell, and
Dunnett’s C. Descriptive statistics: observed means, standard deviations, and counts
for all of the dependent variables in all cells; the Levene test for homogeneity of
variance; Box’ s M test of the homogeneity of t he covariance mat rices of the dependent
variables; and Bartlett’s test of sphericity.

Plots. Spread-versus-level, residual, and profile (interaction).
Data. The dependent variables should be quantitative. Factors are categorical and can

have numeric values or string values of up to eight characters. Covariates are
quantitative variables that are related to the dependent variable.

Assumptions. For dependent variabl es, the dat a are a random sampl e of vect ors from a

multivariate normal population; in the population, the variance-covariance matrices

GLM Multivariate Analysis

for all cells are the same. Analysis of variance is robust to departures from normality,
although the data should be symmetric. To check assumptions, you can use
homogeneity of variances tests (incl uding Box’ s M) and spread-versus-level plot s. Y ou
can also examine residuals and residual plots.

Related procedures. Use the Explore procedure to examine the data before doing an

analysis of variance. For a single dependent variable, use GLM Univariate. If you
measured the same dependent variables on several occasions for each subject, use
GLM Repeated Measures.

To Obtain a GLM Multivariate Analysis of Variance

 From the menus choose:

Analyze

General Linear Model

Multivariate...

Figure 1-1

Multivariate dialog box

3

Select at least two dependent variables.



Optionally, you can specify Fixed Factor(s), Covariate(s), and WLS Weight.

4

Chapter 1

GLM Multivariate Model

Figure 1-2

Multivariate Model dialog box

Specify Model. A full factorial model contains all factor main ef fects, all covariate main

effects, and all factor-by-factor interac tions. It d oes not co ntain co variate i nteractions.
Select

Custom to specify only a subset of interact ions or to specif y factor -by-c ovariate

interactions. You must indicate all of the terms to be included in the model.

Factors and Covariates. Th e factors and covariates are listed with (F) for fixed factor and

(C) for covariate.

Model. The model d epends on the nature of y our data. Af ter selecti ng Custom, you can

select the main effects and interactions that are of interest in your analysis.

Sum of squares. The method of calculating the sums of squares. For balanced or

unbalanced models with no missing cells, the Type III sum-of-squares method is the
most commonly used type.

Include intercept in model. The intercept is usually included in the model. If you can

assume that the data pass through the origin, you can exclude the intercept.

GLM Multivariate Analysis

Build Terms

For the selected factors and covariates:

Interaction. Creates the highest-level interaction term of all selected variables. This is

the default.

Main effects. Creates a main-effects term for each variable selected.
All 2-way. Creates all possible two-way interactions of the selected variables.
All 3-way. Creates all possible three-way interactions of the selected variables.
All 4-way. Creates all possible four-way interactions of the selected variables.
All 5-way. Creates all possible five-way interactions of the selected variables.

Sums of Squares

For the model, you can choose a type of sum of squares. T ype III is the most commonly
used and is the default.

5

Type I . This method is also known as the hierarchical decomposition of the sum-of-

squares method. Each term is adjusted only for the term that precede s it in the mod el.
The Type I sum-of-squares method is commonly used for:

 A balanced ANOVA model in which any main effects are specified before any

first-order interaction effects, any first-order interaction effects are specified before
any second-order interaction effects, and so on.

 A polynomial regression model in which any lower-order terms are specified

before any higher-order terms.

 A purely nested model in whi ch the first-specified effect is nested within the

second-specified effect, the second-specified effect is nested within the third, and
so on. (This form of nesting can be specified only by using syntax.)

Type II. This method calculates the sums of squares of an effect in the model adjusted

for all other “appropriate” effects. An appropriate effect is one that corresponds to all
effects that do not contain the effect being examined. The Type II sum-of-squares
method is commonly used for:

 A balanced ANOVA model.
 Any model that has main factor effects only.

6

Chapter 1

Any regression model.



 A purely nested design. (This form of nesting can be specified by using syntax.)

Type III. This method, the default, calculates the sums of squares of an effect in the

design as the sums of squares adjusted for any other effects that do not contain it and
orthogonal to any eff ects (if any) that contai n it. The T ype III sums of squares have one
major advantage in that they are invariant with respect to the cell frequencies as long
as the general form of estimability remains constant. There fore, this type is often
considered useful for an un balance d model with no missing cell s. In a fact orial design
with no missing ce lls, this method is equivalent to the Yates’ weighted-squares-of­means technique. The Type III sum-of-squares method is commonly used for:

 Any models listed in Type I and Type II.
 Any balanced or unbalanced model with no empty cells.

Type IV. This method is designed for a situation in which there are missing cells. For any

effect F in the design, if F is not contained in any other effect, then T ype IV = Type III
= T ype II. Whe n F is contained in other effects, T ype IV distributes the contrasts being
made among the parameters in F to all higher-level effects equitably. The Type IV sum­of-squares method is commonly used for:

 Any models listed in Type I and Type II.
 Any balanced model or unbalanced model with empty cells.

GLM Multivariate Contrasts

Figure 1-3

Multivariate Contrasts dialog box

GLM Multivariate Analysis

Contrasts are used to test whet her the levels of an effect are significan tly different from
one another. You can specify a contrast for each factor in the model. Contrasts
represent linear combinations of the parameters.

Hypothesis testing is ba sed on the null hypothesis LBM = 0, where L is the contrast
coefficients matrix, M is the identity matrix, which has dimension equal to the number
of dependent variables, and B is the parameter vector. When a contrast is specified,
SPSS creates an L matrix such that the columns correspo nding to the fact or match t he
contrast. The remaining columns are adjusted so that the L matrix is estimable.

In addition to the univariate test using F statistics and the Bonferroni-type
simultaneous confidence intervals based on Student’s t distribution for the contrast
differences across all dependent variables, the multivariate tests using Pillai’s trace,
Wilks’ lambda, Hotelling’s trace, and Roy’s largest root criteria are provided.

Available contrasts are deviation, simple, difference, Helmert, repeated, and
polynomial. For deviation cont rasts and si mple contr asts, you can choose whether the
reference category is the last or first category.

Contrast Types

7

Deviation. Compares the mean of each level (except a reference category) to the mean

of all of the levels (grand mean). The levels of the factor can be in any order.

Simple. Compares the mean of each level to the mean of a specified level. This type of

contrast is useful when t here is a control gr oup. Y ou can choose the first or l ast category
as the reference.

Difference. Compares the mean of each level (except the first) t o the mean of previous

levels. (Sometimes called reverse Helmert contrasts.)

Helmert. Compares the mean of each level of the facto r (except the last) t o the mean of

subsequent levels.

Repeated. Compares the mean of each level (except the last) to the mean of the

subsequent level.

Polynomial. Compares the linear effect, quadratic effect, cubic effect, and so on. The

first degree of freedom contains the linear effect across all categories; the second
degree of freedom, the quadratic effect; and so on. These contrasts are often used to
estimate polynomial trends.

8

Chapter 1

GLM Multivariate Profile Plots

Figure 1-4

Multivariate Profile Plots dialog box

Profile plots (interaction plots) are useful for comparing marginal means in your
model. Profile plots are create d for each dependent varia ble. A profile plot is a l ine plot
in which each point indicates the estimated marginal mean of a dependent variable
(adjusted for covariates) at one level of a factor. The levels of a second factor can be
used to make separate lines. Each level in a third factor can be used to create a separate
plot. All fixed factors are available for plots.

A profile plot of one factor shows whether the estimated marginal means are
increasing or decreasing across levels. For two or more factors, parallel lines i ndi cate
that there is no interaction between factors, which means that you can investigate the
levels of only one factor. Lines that cross each other indicate an interaction.

Figure 1-5

Nonparallel plot (left) and parallel plot (right)

After a plot is specified by selecting factors for the horizontal axis, and optionally,
factors for separate lines and separate plots, the plot must be listed in the Plots list.

GLM Multivariate Post Hoc Multiple Comparisons
for Observed Means

Figure 1-6

Post Hoc Multiple Comparisons for Observed Means dialog box

9

GLM Multivariate Analysis

Post hoc multiple comparison tests. Once you have determined that differences exist

among the means, post hoc range tests and pairwise multiple comparisons can
determine which means differ. Comparisons are made on unadjusted values. These
tests are used for between-subjects factors only . The post hoc multiple compari son tests
are performed for each dependent variable separately.

The Bonferroni and Tukey’s honestly significant difference tests are commonly

used multiple comparison tests. The Bonferroni test, based on Student’s t statistic,
adjusts the observed significance level for the fact that multiple comparisons are made.
Sidak’s t test also adjusts the significance level and provides tighter bounds than the
Bonferroni test. Tukey’s honestly significant difference test uses the Studentized
range statistic to make all pairwise comparisons between groups and sets the
experimentwise error rate to the error rate for the collection for all pairwise
comparisons. When testing a large number of pairs of means, Tukey’s honestly
significant difference test is more powerful than the Bonferroni test. For a small
number of pairs, Bonferroni is more powerful.

10

Chapter 1

Hochberg’s GT2 is similar to Tukey’s honestly signi fi cant di fferen ce test, but the
Studentized maximum modulus is used. Usually, Tukey’s test is more powerful.
Gabriel’s pairwise comp arisons test also uses the Studentized maximum modulus
and is generally more powerful than Hochb erg’s GT 2 when t he cell sizes a re un equal.
Gabriel’s test may become liberal when the cell sizes vary greatly.

Dunnett’s pairwise multiple co mparison t test compares a set of treatments
against a single control mean. The last category is the default control category.
Alternatively, you can choose the first category. You can also choose a two-sided or
one-sided test. To test that the mean at any level (except the control category) of the
factor is not equal to that of the c ontrol category, use a two-si ded test . To test wh ether
the mean at any level of the factor is smaller than that of the control category, select

< Control. Likewise, to test whether the mean at any level of the factor is larger than

that of the control category, select

> Control.

Ryan, Einot, Gabriel, and Welsch (R-E-G-W) developed t wo mult ipl e step- down
range tests. Multiple step-down procedures first test whether all means are equal. If all
means are not equal, subsets of means are tested for equality. R-E-G-W F is based on an
F test and R-E-G-W Q is based on the Studentized r ange. These tests are mor e powerful
than Duncan’s multiple range test and Student-Newman-Keuls (which are also multiple
step-down procedures), but they are not recommended for unequal cell sizes.

When the variances are unequal, use Tamhane’s T2 (conservative pairwise
comparisons test based on a t test), Dunn et t ’s T3 (pai rwise comparison test based on
the Studentized maximum modulus), Games-Howell pairwise comparison test
(sometimes liberal), or Dunnett’s C (pairwise comparison test based on the
Studentized range).

Duncan’s multiple range test, Student-Newman-Keuls (S-N-K), and Tukey’s-b
are range tests that rank group means and compute a range value. These tests are not
used as frequently as the tests previously discussed.

The Waller-Duncan t test uses a Bayesian approach. This range test uses the
harmonic mean of the sample size when the sample sizes are unequal.

The significance level of the Scheffé test is designed to allow all possible linear
combinations of group means to be tested, not just pairwise comparisons available in
this feature. The result is that the Scheffé test is often more conservative than other
tests, which means that a larger difference between means is required for significance .

The least significant difference (LSD) pairwise multiple comparison test is
equivalent to multiple in dividual t tests between all pairs of gro ups. The disadvan tage
of this test is that no attempt is made to adjust the observed significa nce level for
multiple comparisons.

Tests displayed. Pairwise comparisons are provided for LSD, Sidak, Bonferroni,

Games and Howell, Tamhane’s T2 and T3, Dunnett’s C, and Dunnett’s T3.
Homogene ou s su bs et s fo r ra nge te sts ar e pr o vi de d fo r S-N-K, Tukey’s-b, Duncan,
R-E-G-W F, R-E-G-W Q, and Waller. Tukey’s honestly significant difference test,
Hochberg’s GT2, Gabriel’ s test, and Scheffé’s test are both multiple comparison tests
and range tests.

GLM Multivariate Save

Figure 1-7

Multivariate Save dialog box

11

GLM Multivariate Analysis

You can save values predicted by the model, residuals, and related measures as new
variables in the Data Editor. Many of these variables can be used for examining
assumptions about the data. To save the values for use in another SPSS session, you
must save the current data file.

Predicted Values. The values that the model predicts for each case. Unstandardized

predicted values and the standard errors of the predicted values are available. If a WLS
variable was chosen, weighted unstandardized predicted values are available.

Diagnostics. Measures to identify cases with unusual combinations of values for the

independent variables and cases that may hav e a lar ge im pact on t he model. Available
are Cook’s distance and uncentered leverage values.

Residuals. An unstandardized residual is the actual value of the dependent variable

minus the value predicted by the model. Standardized, Studentized, and deleted

12

Chapter 1

residuals are also available. If a WLS variable was chosen, weighted unstandardized
residuals are available.

Save to New File. Writes an SPSS data file containing a variance-covariance matrix of

the parameter estimates in the model. Also, for each de pendent variable, there will be
a row of parameter estimates, a row of significance values for the t statistics
corresponding to the parameter estimates, and a row of residual degrees of freedom.
For a multivariate model, there are si milar ro ws for each dependent variable. You can
use this data in other SPSS procedures.

GLM Multivariate Options

Figure 1-8

Multivariate Options dialog box

Optional statistics are available from this dialog box. Statistics are calculated using a
fixed-effects model.

Estimated Marginal Means. Select the factors and i nteract ion s for which you want

estimates of the population mar ginal means in the cells. These means are adjusted for the
covariates, if any. Interactions are available only i f you have speci fied a cu stom model .

13

GLM Multivariate Analysis

 Compare main effects. Provides uncorrected pairwise comparisons among

estimated marginal means for any main effect in the model, for both between- an d
within-subjects factors. This item is available only if main effects are selected
under the Display Means For list.

 Confidence interval adjustment. Select least significant difference (LSD),

Bonferroni, or Sidak adju stment t o the conf idenc e inte rvals and si gnifi cance. Th is
item is available only if

Display. Select Descriptive statistics to produce observed means, standard deviations,

and counts for all of the d ependent vari ables in all cel ls.

Compare main effects is selected.

Estimates of ef fect si ze gi ves a

partial eta-squared value for each effect and each parameter estimate . The eta-squared
statistic describes the proportion of total variability attributable to a factor. Select

Observed power to obtain the power of the test when the alternative hypothesis is set

based on the observed value. Select

Parameter estimates to produce the parameter

estimates, standard errors, t tests, confidence intervals, and the observed power for
each test. You can display the hypothesis and error

SSCP matrix

Homogeneity tests produces the Levene test of the homogeneity of variance for each

plus Bartlett’s test of sphericity of the residual covariance matrix.

SSCP matrices and the Residual

dependent variable across all level combinations of the between-subjects factors, for
between-subjects factors only. Also, homogeneity tests include Box’s M test of the
homogeneity of the covariance matrices of the dependent variables across all level
combinations of the between-subjects factors. The spread-versus-level and residual
plots options are useful for check ing assumptions abou t the data. This item is disabled
if there are no factors. Select

Residual plots to produce an observed-by-predicted-by-

standardized residuals plot for each dependent variable. These plots are useful for
investigating the assumption of equal variance. Select

Lack of fit test to check if the

relationship between the dependent variable and the independent variables can be
adequately described by the mod el.

General estimable f unc tio n al lows you t o construct

custom hypothesis tests based on the gener al estimable function. Rows i n any contra st
coefficient matrix are linear combinations of the general estimable function.

Significance level. Y ou might want to a djust the significance level used in post hoc tests

and the confidence level used for constructing confidence intervals. The specified
value is also used to calculate the observed power for the test. When you specify a
significance level, the associated level of the confidence intervals is displayed in the
dialog box.

14

Chapter 1

GLM Command Additional Features

These features may apply to univariate, multivariate, or repeated measures analysis.
The SPSS command language also allows you to:

 Specify nested effects in the design (using the DESIGN subcommand).
 Specify tests of effects versus a linear combination of effects or a val ue (usi ng t he

TEST subcommand).

 Specify multiple contrasts (using the CONTRAST subcommand).
 Include user-missing values (using the MISSING subcommand).
 Specify EPS criteria (using the CRITERIA subcommand).
 Construct a custom L matrix, M matrix, or K matrix (using the LMATRIX,

MMATRIX, or KMATRIX subcommands).

 For deviation or simple contrasts, specify an intermediate reference category

(using the

 Specify metrics for polynomial contrasts (using the CONTRAST subcommand).
 Specify error terms for post hoc comparisons (usi ng the POSTHOC subcommand).
 Compute estimated marginal means for any fact or or fa ctor inte raction amon g the

factors in the factor list (using the

 Specify names for temporary variables (using the SAVE subcommand).
 Construct a correla tion matrix data file ( using the OUTFILE subcommand).
 Construct a matrix data file that contains statistics from the between-subjects

ANOVA table (using the

 Save the design matrix to a new data file (using the OUTFILE subcommand).

CONTRAST subcommand).

OUTFILE subcommand) .

EMMEANS subcommand).

See the SPSS Syntax Reference Guide for complete syntax information.

GLM Repeated Measures

The GLM Repeated Measures procedure provides anal ysis of variance when the same
measurement is made several times on each subject or case. If between-subjects
factors are specified, they divide th e population into grou ps. Using this general linear
model procedure, you can test null hypotheses about the effects of both the between­subjects factors and the within-subjects factors. You can investigate interactions
between factors as well as the effects of individual fact ors. I n add iti on, t he effects of
constant covariates and covariate interactions with the between-subjects factors can
be included.

In a doubly multivariate repeated measures design, the dependent variables
represent measurements of more than one variable for the different levels of the
within-subjects factors. For example, you could have measured both pulse and
respiration at three different times on each subject.

The GLM Repeated Measures procedure provides bot h univariate and multivariate
analyses for the repeated measures data. Both balanced and unbalanced models can
be tested. A design is balanced if eac h cell in the model con tains the sa me number of
cases. In a multivariate model, the sums of squa res due to the effects in the mod el and
error sums of squares are in matrix form rather than the scalar form found in
univariate analysis. These matrices are called SSCP (sums-of-squares and cross­products) matrices. In addition to testing hypotheses, GLM Repeated Measures
produces estimates of parameters.

Commonly used a priori contrasts are available to perform hypothesis testing on
between-subjects factors. Additionally, aft er an overall F test has shown significance,
you can use post hoc tests to evaluate differences among specific means. Estimated
marginal means give estimates of predicted mean values for the cells in the model,
and profile plots (inter action plots) of these means al low you to vi sualize some of t he
relationships easily.

Chapter

2

15

16

Chapter 2

Residuals, predicted values, Cook’s distance, and leverage values can be saved as
new variables in your data file for che cking assumpti ons. Also availabl e are a residual
SSCP matrix, which is a square matrix of sums of squares and cross-products of
residuals, a residual covariance matrix, which is the residual SSC P mat rix di vided by
the degrees of freedom of the residuals, and the residual correlation matrix, which is
the standardized form of the residual covariance matrix.

WLS Weight allows you to specify a variable used to give observations different
weights for a weighted least-squares (WLS) analysis, perhaps to compensate for
different precision of measurement.

Example. Twelve students are assigned to a high- or low-anxiety group based on their

scores on an anxiety-rating test. The anx iety rati ng is called a bet ween-sub jects fact or
because it divides the subjects into grou ps. The students are each given four tri als on a
learning task, and the number of errors for each trial is recorded. The errors for each
trial are recorded in separate variables, and a within-subjects factor (trial) is defined
with four levels for the four trials. The trial effect is found to be significant, while the
trial-by-anx ie ty int e ra ction is not significa n t.

Methods. T ype I, Type II, T ype III, and T ype IV sums of squares can be used to evaluate

different hypotheses. Type III is the default.

Statistics. Post hoc range tests and m ul tiple comparisons (for between -subjects

factors): least significant difference, Bonferroni, Sidak, Scheffé, Ryan-Einot-Gabriel­Welsch multiple F, Ry an-Einot-Gabriel-Welsch multiple range, Student-Newman­Keuls, Tukey’s honestly significant difference, Tukey’s-b, Duncan, Hochberg’s GT2,
Gabriel, Waller Duncan t test, Dunnett (one-sided and two-sided), Tamhane’s T2,
Dunnett’ s T3, Games-Howell, and Dunnett’s C. Descriptive statistics: observed means,
standard deviations, and counts for all of the dependent variables in all cells; the
Levene test for homogeneity of variance; Box’s M; and Mauchly’s test of sphericity.

Plots. Spread-versus-level, residual, and profile (interaction).
Data. The dependent variables should be quantitative. Between-subjects factors divide

the sample into discrete subgroups, such as male and female. These factors are
categorical and can have numeric values or string values of up to eight characters.
Within-subjects factors are defined in t he Repeated M easures Defi ne F actor(s) d ialog
box. Covariates are quantitat ive variables that are related to the dependent variable. For
a repeated measures analysis, these should remain constant at each level of a within­subjects variable.

The data file should contain a set of variables for each group of measurements on
the subjects. The set has one variable for each rep etition of the measurement within the

17

GLM Repeated Measures

group. A within-subjects fact or is defined for the group with the number of levels equ al
to the number of repetitio ns. F or example, measurements of weight could be t aken o n
different days. If measurements of the same property were taken on five days, the
within-subjects factor could be specified as day with five levels.

For multiple within-subjects factors, the number of measurement s for each su bje ct
is equal to the product of the number of levels of each factor. For example, if
measurements were taken at three different times each day for four days, the total
number of measurements is 12 for each subject. The within-subjects factors could be
specified as day(4) and time(3).

Assumptions. A repeated measures analysis ca n be appro ached in t wo ways, univari ate

and multivariate.

The univariate approach (also known as the split-plot or mixed-model approach)
considers the dependent variables as r esponses to the levels of within-s ubjects factor s.
The measurements on a subject should be a sample from a multivariate normal
distribution, and the variance-covariance mat rices are the same across the cells formed
by the between-subjects effects. Certain assumptions are made on the variance­covariance matrix of the dependent vari ables. The validity of the F statistic used in the
univariate approach can be assured if the var iance-covariance matrix is circular in form
(Huynh and Mandeville, 1979).

To test this assumption, Mau chly’s test of spherici ty can be used, whi ch performs a
test of sphericity on th e variance-covariance matr ix of an orthonormalize d transformed
dependent variable. Mauchl y’s test is aut omatically displayed for a repeated measures
analysis. For small sample sizes, this test is not very powerful. For large sample sizes,
the test may be significant ev en when the impact of the departure on the results is smal l.
If the significance of the test is large, the hypothesis of sphericity can be assumed.
However, if the significance is small and the sphericity assumption appears to be
violated, an adjustment to the numerator and denominator degrees of freedom can be
made in order to validate the univariate F statistic. Three estimates of this adjustment,
which is called epsilon, are available in the GLM Repeated Measures procedure. Both
the numerator and denominat or degrees of freedom must be mu ltiplied by epsilon, and
the significance of the F ratio must be evaluated with the new degrees of freedom.

The multivariate approach considers the measurements on a subject to be a sampl e
from a multivariate norm al distribution, and the variance-covariance matrices are the
same across the cells formed by the between-subjects effects. To test whether the
variance-covariance matrices across the cells are the same, Box’s M test can be used.

Related procedures. Use the Explore procedure to examine the data before doing an

analysis of variance. If there are not rep eated measurements on eac h subject, use GLM

18

Chapter 2

Univariate or GLM Multivaria te. If there are on ly t wo measureme nt s for eac h sub je ct
(for example, pre-test and post-test), and there are no between-subjects factors, you can
use the Paired-Samples T Test procedure.

GLM Repeated Measures Define Factor(s)

GLM Repeated Measures analyze s groups of related dependent variables that represent
different measurements of the same attribute. This dialog box lets you define one or
more within-subjects factors for use in GLM Repeated Measures. Note that the order
in which you specify within-sub je c ts fa c to rs is important. Each factor constitutes a
level within the previo us factor.

To use Repeated Measures, you must set up your data correctly. You must define
within-subjects factors in this dialog box. Notice that these factors are not existing
variables in your data but rather factors that you define here.

Example. In a wei ght-loss study, suppose the weights of several people a re measured

each week for five weeks. In the data file, each person is a subject or case. The
weights for the weeks are recorded in the vari ables weight1, weight2, and so on. The
gender of each person is recorded in another variable. The weights, measured for
each subject repeatedly, can be grouped by defining a within-subjects factor. The
factor could be called week, defined to have five levels. In the main dialog box, the
variables weight1, ..., weight5 are used to assign the five levels of week. The variable
in the data file that groups males and females (gender) can be specified as a between ­subjects factor to study the differences betwee n male s and fema les.

Measures. If subjects were tested on more than one measure at each time, click

Measure to define the measures. For example, the pulse and respiration rate could be

measured on each subject every day for a week. These measures do not exist as
variables in the data file but are defined here . A model with more than on e measure is
sometimes called a doubly multivariate repeated measures model.

To Obtain a GLM Repeated Measures Analysis

 From the menus choose:

Analyze

General Linear Model

Repeated Measures...

GLM Repeated Measures

Figure 2-1

Repeated Measures Define Factor(s) dialog box

Type a within-subject factor name and its number of levels.



 Click Add.
 Repeat these steps for each within-subjects factor.

To define measure factors for a doubly multivariate repeated measures design:

19

 Click Measure.

Figure 2-2

Expanded Repeated Measures Define Factor(s) dialog box

 Type the measure name.

 Click Add.

20

Chapter 2

 Click Define.

After defining all of your factors and measures:

Figure 2-3

Repeated Measures dialog box

Select a dependent variable that corresponds to each combination of within-subjects



factors (and optionally, measures) on the list.
To change positions of the variables, use the up and down pushbuttons.
T o make changes to the withi n-subjects factors, you c an reopen the Repeated Measure s

Define Factor(s) dialog box without closing the main dialog box.
Optionally, you can specify between-subjects factor(s) and covariates.

GLM Repeated Measures Model

Figure 2-4

Repeated Measures Model dialog box

21

GLM Repeated Measures

Specify Model. A full factorial mod el contains all factor main ef fects, all covariate main

effects, and all factor-by-factor interactions. It does not contain covariate interactions.
Select

Custom to specify only a subset of interactions or to specify factor -by-covaria te

interactions. You must indicate all of the terms to be included in the model.

Between-Subjects. The covariates are listed with (C) for covariate.
Model. The model depen ds on the nat ure of y our data . After selecti ng Custom, you can

select the within-subjects effects and interactions and the between-subj ects effects and
interactions that are of interest in your analysis.

Sum of squares. The method of calculating the sums of squar es for the between-subjects

model. For balanced or unbalan ced between-subjects models wit h no missing cells, the
Type III sum-of-squares method is the most commonly used.

Build Terms

For the selected factors and covariates:

Interaction. Creates the highest-level interaction term of all selected variables. This is

the default.

22

Chapter 2

Sums of Squares

Main effects. Creates a main-effects term for each variable selected.
All 2-way. Creates all possible two-way interactions of the selected variables.
All 3-way. Creates all possible three-way interactions of the selected variables.
All 4-way. Creates all possible four-way interactions of the selected variables.
All 5-way. Creates all possible five-way interactions of the selected variables.

For the model, you can choose a type of sum of squares. Type III is the most commonl y
used and is the default.

Type I. This method is also known as the hierarchical decomposition of the sum-of-

squares method. Each term is adjusted for only the term that precedes it in the model.
The Type I sum-of-squares method is commonly used for:

 A balanced ANOVA model in which any main effects are specified before any

first-order interaction ef fects, any first-order interact ion effects are specified before
any second-order interaction effects, and so on.

 A polynomial regression model in which any lower-order terms are specified

before any higher-order terms.

 A purely nested model in which the first-specified effect is nested within the

second-specified effect, the second-specified effect is nested within the third, and
so on. (This form of nesting can be specified only by using syntax.)

Type II. This method calculat es the sum s of squar es of an effect in the m odel adj usted

for all other “appropriate” effects. An appropriate effect is one that corresponds to all
effects that do not contain the effect being examined. The Type II sum-of-squares
method is commonly used for:

 A balanced ANOVA model.
 Any model that has main factor effects only.
 Any regression model.
 A purely nested design. (This form of nesting can be specified by using syntax.)

23

GLM Repeated Measures

Type III. This method, the default, calculates the sums of squares of an effect in the

design as the sums of squares adjusted for any other effects that do not contain it and
orthogonal to any ef fects (if any) that contai n it. The T ype III sums of squares ha ve one
major advantage in that they are invariant with respect to the cell frequencies as long
as the general form of est im ability remains constan t. Th erefore, this type is ofte n
considered useful for a n u nbalanced mo del wi th no missing cells . In a f actorial desig n
with no missing cells, this method is equivalent to the Yates’ weighted-squares-of­means technique. The Type III sum-of-squares method is commonly used for:

 Any models listed in Type I and Type II.
 Any balanced or unbalanced model with no empty cells.

Type IV. This method is designed for a situation in which there are missing cells. For any

effect F in the design, if F is not contained in any other effect, then Type IV = Type III
= T ype II . When F is contained in other effects, T ype IV distributes the contrasts being
made among the parameters in F to all higher-level effects equitably. The Type IV sum­of-squares method is commonly used for:

 Any models listed in Type I and Type II.
 Any balanced model or unbalanced model with empty cells.

GLM Repeated Measures Contrasts

Figure 2-5

Repeated Measures Contrasts dialog box

Contrasts are used to test for differe nces among the levels of a between-subjec ts factor .
You can specify a contrast for each between-subjects factor in the model. Contrasts
represent linear combinations of the parameters.

24

Chapter 2

Contrast Types

Hypothesis testing is based on t he null hy pothesi s LBM=0, where L is the contrast
coefficients matrix, B is the parameter vector, and M is the average matrix that
corresponds to the average transformation for the depen dent variable. You can disp lay
this transformation matrix by selecting

Transformation matrix in the Repeated Measures

Options dialog box. For example, if there are four dependent variables, a within­subjects factor of four levels, and polynomial contrasts (the default) are used for
within-subjects factors, the M matrix will be (0.5 0.5 0.5 0.5)’. When a contrast is
specified, SPSS creates an L matrix such that the columns corresponding to the
between-subjects factor match the contrast. The remaining columns are adjusted so
that the L matrix is estimable.

Available contrasts are deviation, simple, difference, Helmert, repeated, and
polynomial. For deviatio n contrast s and simple cont rasts, you ca n choose whet her th e
reference category is the last or first category.

Deviation. Compares the mean of each level (except a reference category) to the mean

of all of the levels (grand mean). The levels of the factor can be in any order.

Simple. Compares the mean of each level to the mean of a spe cified level. Thi s type of

contrast is useful when there is a control g roup. You can choose the first or last category
as the reference.

Difference. Compares the mean of each level (except t he first) to the mean of previous

levels. (Sometimes called re verse Helmert contrasts.)

Helmert. Compares the mean of each level of the factor (excep t the last) to the mean of

subsequent levels.

Repeated. Compares the mean of each level (except the last) to the mean of the

subsequent level.

Polynomial. Compares the linear effect, quadratic effect, cubic effect, and so on. The

first degree of freedom contains the linear effect across all categories; the second
degree of freedom, the quadratic effect; and so on. These contrasts are often used to
estimate polynomial trends.

GLM Repeated Measures Profile Plots

Figure 2-6

Repeated Measures Profile Plots dialog box

Profile plots (interaction plots) are useful for comparing marginal means in your
model. A profile plot is a line pl ot in which each point i ndicates the estimated mar ginal
mean of a dependent variable (adjuste d for any covariates) at one level of a factor . The
levels of a second factor can be used to make separate lines. Each level in a third factor
can be used to create a separate plot. Both between-subjects factors and within-subjects
factors can be used in profile plots.

A profile plot of one factor shows whether the estimated marginal means are
increasing or decreasing across le vel s . For two or more f act ors, par al l el l ine s in dic ate
that there is no interaction between factors, which means that you can investigate the
levels of only one factor. Nonparallel lines indicate an interaction.

25

GLM Repeated Measures

Figure 2-7

Nonparallel plot (left) and parallel plot (right)

26

Chapter 2

After a plot is specified by selecting factors for the horizontal axis and, optionally,
factors for separate lines and separate plots, it must be added to the Plots list.

GLM Repeated Measures Post Hoc Multiple Comparisons
for Observed Means

Figure 2-8

Post Hoc Multiple Comparisons for Observed Means dialog box

Post hoc multiple comparison tests. Once you have determined that differences exist

among the means, post hoc range tests and pairwise multiple comparisons can
determine which means differ. These tests are used for fixed between-subjects factors
only. They are not available if there are no between-subjects factors. The post hoc
multiple comparison tests are performed for each dependent variable separately.

The Bonferroni and Tukey’s honestly significant difference tests are commonly

used multiple comparison tests. The Bonferroni test, based on Student’s t statistic,
adjusts the observed significance level for the fact that multiple comparisons are made.
Sidak’s t test also adjusts the significance level and provides tighter bounds than the
Bonferroni test. Tukey’s honestly significant difference test uses the Studentized
range statistic to make all pairwise comparisons between groups and sets the
experimentwise error rate to the error rate for the collection for all pairwise
comparisons. When testing a large number of pairs of means, Tukey’s honestly
significant differe nc e test is more powerful than the Bonferroni test. For a sm al l
number of pairs, Bonferroni is more powerful.

27

GLM Repeated Measures

Hochberg’s GT2 is similar to Tukey’s honestly signifi cant differ en ce test , but th e
Studentized maximum modulus is used. Usually, Tukey’s test is more powerful.
Gabriel’s pairwise comparisons test also uses the Studentized maximum modulus
and is generally more powerful than Hochberg’s GT2 whe n the cell sizes are unequal .
Gabriel’s test may become liberal when the cell sizes vary greatly.

Dunnett’s pairwise multiple comparison t test compares a set of treatments
against a single control mean. The last category is the default control category.
Alternatively, you can choose the first category. You can also choose a two-sided or
one-sided test. To test that the mean at any level (except the control category) of the
factor is not equal to that of the c ontrol category, use a two- sided t est. To test whether
the mean at any level of the factor is smaller than that of the control category, select

< Control. Likewise, to test whether the mean at any level of the factor is larger than

that of the control category, select

> Control.

Ryan, Einot, Gabriel, and Welsch (R-E-G-W) developed t wo mult iple step-down
range tests. Multiple step-down procedures first test whether all means are equal. If all
means are not equal, subsets of means are tested for equality. R-E-G-W F is based on an
F test and R-E-G-W Q i s based on the Studentized range. These tests are more power ful
than Duncan’s multiple range test and Student-Newman-Keuls (which are also multiple
step-down procedures), but they are not recommended for unequ al cel l si zes.

When the variances are unequal, use Tamhane’s T2 (conservative pairwise
comparisons test based on a t test), Dunn ett ’s T3 (pai rwise comp arison test based on
the Studentized maximum modulus), Games-Howell pairwise comparison test
(sometimes liberal), or Dunnett’s C (pairwise comparison test based on the
Studentized range).

Duncan’s multiple range test , Student-Newman-Keuls (S-N-K), and Tukey’s-b
are range tests that rank group means and compute a range value. These tests are not
used as frequently as the tests previously discussed.

The Waller-Duncan t tes t uses a Bayesian approach. This range test uses the
harmonic mean of the sample size when the sample sizes are unequal.

The significance level of the Scheffé test is designed to allow all possible linear
combinations of group means to be tested, not just pairwise comparisons available in
this feature. The result is that the Scheffé test is often mor e co nservative than other
tests, which means that a larger difference between means is required for significance.

The least significant difference (LSD) pairwise multiple comparison test is
equivalent to multiple individual t tests between all pairs of groups. The disadvantage
of this test is that no attem pt is made to adjust the observed significance level for
multiple comparisons.

28

Chapter 2

Tests displayed. Pairwise comparisons are provided for LSD, Sidak, Bonferr oni,

Games and Howell, Tamhane’s T2 and T3, Dunnett’s C, and Dunnett’s T3.
Homogeneous subsets for range tests are prov ided for S-N-K, Tukey’s-b, Duncan,
R-E-G-W F, R-E-G-W Q, and Waller. Tukey’s honestly significant difference test,
Hochberg’s GT2, Gabriel’s test, and Scheffé’s test are both multiple comparison
tests and range tests.

GLM Repeated Measures Save

Figure 2-9

Repeated Measures Save dialog box

You can save values predicted by the model, residuals, and related measures as new
variables in the Data Editor. Many of these variables can be used for examining
assumptions about the data. To save the values for use in another SPSS session, you
must save the current data file.

Predicted Values. The values that the model predicts for each case. Unstandardized

predicted values and the standard errors of the predicted values are available.

Diagnostics. Measures to identify cases with unusual combinations of values for the

independent variables and cases that may hav e a lar ge i mpact on t he model. Available
are Cook’s distance and uncentered leverage values.

Residuals. An unstandardized residual is the actual value of the dependent variable

minus the value predicted by the model. Standardized, Studentized, and deleted
residuals are also available.

Save to New File. Writes an SPSS data fil e containi ng a variance- covaria nce matrix of

the parameter estimates in the model. Also, for each dependen t varia ble, t here will be
a row of parameter estimates, a row of significance values for the t statistics
corresponding to the parameter estimates, and a row of residual degrees of freedom.
For a multivariate model, there are si milar rows for each dependent variable. You can
use this matrix file in ot her procedures that read an SPSS matrix file.

GLM Repeated Measures Options

Figure 2-10

Repeated Measures Options dialog box

29

GLM Repeated Measures

Optional statistics are available from this dialog box. Statistics are calculated using a
fixed-effects model.

Estimated Marginal Means. Select the factors and interaction s for which you want

estimates of the population marginal means in the cells. The se means are adjusted
for the covariates, if any. Both within-subjects and between-subjects factors can
be selected.

30

Chapter 2

Compare main effects. Provides uncorrected pairwise comparisons among



estimated marginal means for any main effect in the model, for both bet ween- and
within-subjects factors. This item is available only if main effects are selected
under the Display Means For list.

 Confidence interval adjustment. Select least significant difference (LSD),

Bonferroni, or Sidak adju stment t o the conf idenc e inter vals and signifi cance. T his
item is available only if

Display. Select Descriptive statistics to produce observed means, standard deviations,

and counts for all of th e depend ent vari ables in al l cell s.

Compare main effects is selected.

Estimates of effect size gives a

partial eta-squared value for each effect and each parameter estimate. The eta-squared
statistic describes the proportion of total variability attribu ta ble to a factor. Select

Observed power to obtain the power of the test when the alternative hypothesis is set

based on the observed value. Select

Parameter estimates to produce the parameter

estimates, standard errors, t tests, confidence intervals, and the observed power for
each test. You can display the hypothesis and error

SSCP matrix

Homogeneity tests produces the Levene test of the homogeneity of varianc e for each

plus Bartlett’s test of sphericity of the residual covariance matrix.

SSCP matrices and the Residual

dependent variable across all level combinations of the between-subjects factors, for
between-subjects factors only. Also, homogeneity tests include Box’s M test of the
homogeneity of the covariance matrices of the dependent variables across all level
combinations of the between-subjects factors. The spread-versus-level and residual
plots options are useful for checking assumptions about the data. This item is disabled
if there are no factors. Select

Residual plots to produce an observed-by-predicted-by-

standardized residuals plot for each dependent variable. These plots are useful for
investigating the assumption of equal variance. Select

Lack of fit test to check if the

relationship between the dependent variable and the independent variables can be
adequately described by the model.

General estimable f unc tio n all ows you to construct

custom hypothesis tests based on the general estimable funct ion. Rows in any contrast
coefficient matrix are linear combinations of the general estimable function.

Significance level. Y ou might want to adjust the signi ficance level used in p ost hoc tests

and the confidence level used for constructing confidence intervals. The specified
value is also used to calculate the observed power for the test. When you specify a
significance leve l, the associated level of the confidence intervals is displayed in the
dialog box.

GLM Command Additional Features

These features may apply to univariate, multivariate, or repeated measures analysis.
The SPSS command language also allows you to:

 Specify nested effects in the design (using the DESIGN subcommand).
 Specify tests of effects versus a lin ear co mbinat ion o f effects o r a val ue (using the

TEST subcommand).

 Specify multiple contrasts (using the CONTRAST subcommand).
 Include user-missing values (using the MISSING subcommand).
 Specify EPS criteria (using the CRITERIA subcommand).
 Construct a custom L matrix, M matrix, or K matrix (using the LMATRIX,

MMATRIX, and KMATRIX subcommands).

 For deviation or simple contrasts, specify an intermediate reference category

(using the

 Specify metrics for polynomial contrasts (using the CONTRAST subcommand).
 Specify error terms for post hoc comparisons (using the POSTHOC subcommand).
 Compute estimated marginal means for any factor or f actor interaction among t he

factors in the factor list (using the

 Specify names for temporary variables (using the SAVE subcommand).
 Construct a correlation matrix data file (using th e OUTFILE subcommand).
 Construct a matrix data file that contains statistics from the between-subjects

ANOVA table (using the

 Save the design matrix to a new data file (using the OUTFILE subcommand).

CONTRAST subcommand).

OUTFILE subcommand).

31

GLM Repeated Measures

EMMEANS subcommand).

See the SPSS Syntax Reference Guide for complete syntax information.

Variance Components Analysis

The Variance Components procedure, for mixed-effects models, estimates the
contribution of each random effect t o t he variance o f the depend ent var i able . Thi s
procedure is particularly interesting for ana lysis o f mixed models such as split pl ot,
univariate repeated measures, and random block designs. By calculating vari ance
components, you can determine where to focus attention in order to reduce the variance.

Four different methods are available for estimating the variance components:
minimum norm quadratic unbiased estimator (MINQUE), analysis of variance
(ANOVA), maximum likelihood (ML), and restricted maximum likelihood (REML).
Various specifications are available for the different methods.

Default output for all methods includes variance component estimates. If the ML
method or the REML method is used, an asymptotic covariance matrix table is also
displayed. Other available output includes an ANOVA table and expected mean
squares for the ANOVA method, and an iteration history for the ML and REML
methods. The Variance Components procedure is fully compatible with the GLM
Univariate procedure.

WLS Weight allows you to specify a variable used to give observations different
weights for a weighted analysis, perhaps to compensate for different precision of
measurement.

Chapter

3

Example. At an agriculture school, weight gains for pigs in six different litters are

measured after one month. The litter vari able is a random fact or with six levels . (The
six litters studied are a random sample from a large population of pig litters.) The
investigator finds out that the variance in weight gain is attribut able to the dif ference
in litters much more than to the difference in pigs within a litter.

33

34

Chapter 3

Data. The dependent variable is quantitative. Factors are categorical. They can have

numeric values or string values of up to eight characters. At least one of the factors
must be random. That is, the lev el s of t he fac tor must be a random sample of p ossib le
levels. Covariates are quantitative variables that are related to the dependent variable.

Assumptions. All methods assume that model parameters of a random ef fect have zero

means and finite constant variances and are mutually uncorrelated. Model parameters
from different random effects are also uncorrelated.

The residual term also has a zero mean and finite constant variance. It is
uncorrelated with model parameters of any random effect. Residual terms from
different observations are assumed to be uncorrelated.

Based on these assumptions, observations from the same level of a random factor
are correlated. This fact distinguishes a variance component model from a general
linear model.

ANOVA and MINQUE do not require normality assumptions. They are both robust
to moderate departures from the normality assumption.

ML and REML require the model parame te r and the residual term t o b e normally
distributed.

Related procedures. Use the Explore procedure to examine the data before doing

variance components analysis. For hypothesis testing, use GLM Univariate, GLM
Multivariate, and GLM Repeated Measures.

To Obtain a Variance Components Analysis

 From the menus choose:

Analyze

General Linear Model

Variance Components...

Figure 3-1

Variance Components dialog box

Select a dependent variable.



35

Variance Components Analysis

 Select variables for Fixed Factor(s), Random Factor(s), and Covariate(s), as

appropriate for your data. For specifying a weight variable, use WLS Weight.

Variance Components Model

Figure 3-2

Variance Components Model dialog box

36

Chapter 3

Build Terms

Specify Model. A full factorial model contains all factor main ef fects, all covariate main

effects, and all factor-by-factor interac tions. It d oes not co ntain co variate i nteractions.
Select

Custom to specify only a subset of interact ions or to specif y factor -by-c ovariate

interactions. You must indicate all of the terms to be included in the model.

Factors and Covariates. The factors and covariates are listed with (F) for a fixed f act or,

(R) for a random factor, and (C) for a covariate.

Model. The model d epends on the nature of y our data. Af ter selecti ng Custom, you can

select the main effects and interactions that are of interest in your analysis. The model
must contain a random factor.

Include intercept in model. Usually the intercept is inc luded in the model. If yo u can

assume that the data pass through the origin, you can exclude the intercept.

For the selected factors and covariates:

Interaction. Creates the highest-level interaction term of all selected variables. This is

the default.

Main effects. Creates a main-effects term for each variable selected.
All 2-way. Creates all possible two-way interactions of the selected variables.
All 3-way. Creates all possible three-way interactions of the selected variables.
All 4-way. Creates all possible four-way interactions of the selected variables.
All 5-way. Creates all possible five-way interactions of the selected variables.

Variance Components Options

Figure 3-3

Variance Components Options dialog box

Method. You can choose one of four methods used to estimate the variance

components.

37

Variance Components Analysis

 MINQUE (minimum norm quadratic unbiased estimator) produces estimates that

are invariant with respect to the fixed effects. If the data are normally distributed
and the estimates are correct, this method produces the least variance among all
unbiased estimators. You can choose a method for random-effect prior weights.

 ANOVA (analysis of variance) computes unbiased estimates using either t he Type I

or Type III sums of squares for each effect. The ANOVA method sometimes
produces negative variance estimates, which can indicate an incorrect model, an
inappropriate estimation method, or a need for more data.

 Maximum likeliho od (ML) produces estimates that would be most consistent with th e

data actually observed, using iterations. These estimates can be biased. This
method is asymptotically normal. ML and REML estimates are invariant under
translation. This method does not ta ke into account t he degrees of freedom used t o
estimate the fixed effects.

 Restricted maximum likelihood (REML) estimates reduce the ANOVA estimates for

many (if not all) cases of balanced data. Because this method is adjusted for the
fixed effects, it should have smaller standard errors than the ML method. This
method takes into account the degrees of freedom used to estimate the fix ed effects.

38

Chapter 3

Random-Effect Priors. Uniform implies that all random effects and the residual term hav e

an equal impact on the observations. The
random-effect variances. Available only for the MINQUE method.

Sum of Squares. Type I sums of squares are used for the hierarchical model, which is

often used in variance compon ent literature. If you choose
the variance estimate s can be used in GLM Univariate for hypothesis testin g with T ype
III sums of squares. Av ailable only for the ANOVA method.

Criteria. You can specify the convergence criterion and the maximum number of

iterations. Available only for the ML or REML methods.

Display. For the ANOVA method, you can choose to display sums of squares and

expected mean squares. If you selected

likelihood

, you can display a history of the iterations.

Sums of Squares (Variance Components)

For the model, you can choose a type of sum of squares. Type III is the most commonl y
used and is the default.

Zero scheme is equivalent to assuming zero

Type III, the default in GLM,

Maximum likelihood or Restricted maximum

Type I. This method is also known as the hierarchical decomposition of the sum-of-

squares method. Each term is adjusted for only the term that precedes it in the model.
The Type I sum-of-squares method is commonly used for:

 A balanced ANOVA model in which any main effects are specified before any

first-order interaction ef fects, any first-order interact ion effects are specified before
any second-order interaction effects, and so on.

 A polynomial regression model in which any lower-order terms are specified

before any higher-order terms.

 A purely nested model in which the first-specified effect is nested within the

second-specified effect, the second-specified effect is nested within the third, and
so on. (This form of nesting can be specified only by using syntax.)

Type III. This method, the default, calculates the sums of squares of an effect in the

design as the sums of squares adjusted for any other effects that do not contain it and
orthogonal to any eff ects (if any) that contai n it. The T ype III sums of squares have one
major advantage in that they are invariant with respect to the cell frequencies as long
as the general form of estimability remains constant. There fore, this type is often

considered useful for a n u nbalanced mo del wi th no missing cells . In a f actorial desig n
with no missing cells, this method is equivalent to the Yates’ weighted-squares-of­means technique. The Type III sum-of-squares method is commonly used for:

 Any models listed in Type I.
 Any balanced or unbalanced model with no empty cells.

Variance Components Save to New File

Figure 3-4

Variance Components Save to New File dialog box

39

Variance Components Analysis

You can save some results of this procedure to a new SPSS data file.

Variance component estimates. Saves estimates of the variance components and

estimate labels to a data file. These can be used in calculating more statistics or in
further analysis in the GLM procedures. For example, you can use them to calculate
confidence intervals or test hypotheses.

Component covariation. Saves a variance-covariance matrix or a correlation matrix to a

data file. A vaila ble only if

Maximum likelihood or Restricted maximum l ikelihood has been

specified.

Save to File. Allows you to specify a filename for the file containing the variance

component estimates and/or the matrix.

You can use the

MATRIX command to extract the data you need from the new file and

then compute confidence intervals or perform tests.

40

Chapter 3

VARCOMP Command Additional Features

The SPSS command language also allows you to:

 Specify nested effects in the design (using the DESIGN subcommand).
 Include user-missing values (using the MISSING subcommand).
 Specify EPS criteria (using the CRITERIA subcommand).

See the SPSS Syntax Reference Guide for complete syntax information.

Linear Mixed Models

The Linear Mixed Models pr ocedure expands the general linear model so that the data
are permitted to exhibit correlated and non-constant variability. The mixed linear
model, therefore, provides the flexibility of modeling not only the means of the data
but their variances and covariances as well.

The Linear Mixed Model s procedure is al so a flexibl e tool for f itting ot her models
that can be formulated as mixed linear models. Such models include multilevel
models, hierarchical linear models, and random coefficient models.

Chapter

4

Example. A grocery store chain is interested in the effects of various coupons on

customer spending. Taking a random sample of their regular customers, they follow
the spending of each customer for 10 weeks. In each week, a different coupon is
mailed to the customers. Linear Mixed Models is used to estimate the effect of
different coupons on spending while adjusting for correlation due to repeated
observations on each subject over the 10 weeks.

Methods. Maximum likelihood (ML) and restricted maximum likelihood (REML)

estimation.

Statistics. Descriptive statistics: sample sizes, means, and standard deviations of the

dependent variable and covariates for each distinct level combination of the factors.
Factor-level information: sorted values of the levels of each factor and their
frequencies. Also, paramet er estimates and confiden ce intervals for fixed ef fects, W al d
tests and confidence intervals for parameters of covariance matrices. Type I and Type
III sums of squares can be used to evaluate different hypotheses. Type III is the default.

41

42

Chapter 4

Data. The dependent variable should be quant itative. Factors should be cat egorical and

can have numeric values or str ing values. Covariates a nd the weight variable shou ld be
quantitative. Subjects and repeated variables may be of any type.

Assumptions. The dependent variable is assumed to be li nearly related to the fixed

factors, random factors, and covariates. The fixed effects model the mean of the
dependent variable. The random effects model the covariance structure of the
dependent variable. Multi ple random effects are considered independent of each other ,
and separate covariance matrices will be computed for each; however, model terms
specified on the same random effect can be correlated. The repeated measures model
the covariance structure of the residuals. The dependent variable is also assumed to
come from a normal distribution.

Related procedures. Use the Explore procedure to examine the data before running

an analysis. If you do not suspect there to be correlated or non-constant variability,
you can alternatively use the GLM Univariate or GLM Repeated Measures
procedure. You can alternatively use the Variance Components Analysis procedure
if the random effect s have a variance c omponen ts cov ariance st ruct ure and the re are
no repeated measures.

Linear Mixed Models Select Subjects/Repeated Variables

This dialog box allows you to select variables that define subjects and repeated
observations and to choose a covariance structure for the residuals.

Subjects. A subject is an observational unit that can be considered i ndependent of other

subjects. For example, the blood pressure readings from a patient in a medical study
can be considered independent of the readings from other patients. Defining subjects
becomes particularly impor tant when there are repeated measurements per subject and
you want to model the correlation betw een these observations. For example, you might
expect that blood pressure readings from a single patient during consecutive visits to
the doctor are correlated.

Subjects can also be defined by the factor-level combination of multipl e variables; for
example, you can specify Gender and Age category as subject variables to model the
belief that males over the age of 65 are similar to each other but independent of males
under 65 and females.

43

Linear Mixed Models

All of the variables speci f ie d in the Subjects list are used to define subjects for the
residual covariance structure. You can use some or all o f the variables to define subjects
for the random-effects covariance structure.

Repeated. The variables specified i n this list are used to identify repeated observations.

For example, a single variable Week might identify the 10 weeks of observations in a
medical study , or Month and Day might be used together t o identify d aily observati ons
over the course of a year.

Repeated Covariance Type. This specifies the covariance structure for the residuals. The

available structures are as follows:

 Ante-Dependence: First Order
 AR(1)
 AR(1): Heterogeneous
 ARMA(1,1)
 Compound Symmetry
 Compound Symme try: Correlation M etric
 Compound Symmetry: Heterogeneous
 Diagonal
 Factor Analytic: First Order
 Factor Analytic: First Order, Heterogeneous
 Huynh-Feldt
 Scaled Identity
 Toeplitz
 Toeplitz: Heterogeneous
 Unstructured
 Unstructured: Correlations

For more information, see the appendix Covariance Structures.

44

Chapter 4

Selecting Subjects/Repeated Variables for Linear Mixed Models

 From the menus choose:

Analyze

Mixed Models

Linear...

Figure 4-1

Linear Mixed Models: Specify Subjects/Repeated Variables dialog box

 Optionally, select one or more subjects variables.

 Optionally, select one or more repeated variables.

 Optionally, select a residual covariance structure.

 Click Continue.

Obtaining a Linear Mixed Models Analysis

Figure 4-2

Linear Mixed Models dialog box

45

Linear Mixed Models

 Select a dependent variable.

 Select at least one factor or covariate.

 Click Fixed or Random and specify at least a fixed-effects or random-effects model.

Optionally, select a weighting variable.

46

Chapter 4

Linear Mixed Models Fixed Effects

Figure 4-3

Linear Mixed Models: Fixed Effects dialog box

Fixed Effects. There is no defa ult model, so you must explicitly speci fy the fixed effects.

Alternatively, you can build nested or non-nested terms.

Include Intercept. The intercept is usually included in the model. If you can assume the

data pass through the origin, you can exclude the intercept.

Sum of squares. The method of calculating the sums of squares. For models with no

missing cells, the Type III method is most commonly used.

Build Non-Nested Terms

For the selected factors and covariates:

Factorial. Creates all possible interactions and main effects of the selected variables.

This is the default.

Interaction. Creates the highest-level interaction term of all selected variables.
Main Effects. Creates a main-effects term for each variable selected.

Linear Mixed Models

All 2-Way. Creates all possible two-way interactions of the selected variables.
All 3-Way. Creates all possible three-way interactions of the selected variables.
All 4-Way. Creates all possible four-way interactions of the selected variables.
All 5-Way. Creates all possible five-way interactions of the selected variables.

Build Nested Terms

You can build nested terms for your model in this procedure. Nested terms are useful
for modeling the effect of a factor or covariate whose values do not interact with the
levels of another factor . Fo r example, a grocery store chai n may follow the spending of
their customers at several store locations. Since each customer frequents only one of
those locations, the Customer ef fec t can be said to be nested within the Store location
effect.

Additionally, you can in clude interaction effect s or add mult iple levels of ne sting to

the nested term.

Limitations. Nested terms have t he following restrictions:

47

 All factors within an interaction must be unique. Thus, if A is a factor, then

specifying A*A is invalid.

 All factors within a nested effect must be unique. Thus, if A is a factor, then

specifying A(A) is inv alid.

 No effect can be nested within a covariate. Thus, if A is a factor and X is a covariate,

then specifying A(X) is invalid.

Sum of Squares

For the model, you can choose a type of sums of squares. Type III is the most
commonly used and is the default.

Ty pe I. This method is also known as t he hierarchical decomposi tion of the sum-of-squa res

method. Each term is adjusted only for the term that precedes it in the model. Type I sums
of squares are commonly used for:

 A balanced ANOVA model in which any main effects are specified before any

first-order interaction effects, any first-order interaction effects are specified before
any second-order interaction effects, and so on.

48

Chapter 4

A polynomial regression model in which any lower-order terms are specified



before any higher-order terms.

 A purely nested model in which the first-specified effect is nested within the

second-specified effect, the second-specified effect is nested within the third, and
so on. (This form of nesting can be specified only by using syntax.)

Type III. The default. This method calculates the sums of squares of an effect in the

design as the sums of squares adjusted for any other effects that do not contain it and
orthogonal to any eff ects (if any) that contai n it. The T ype III sums of squares have one
major advantage in that they are invariant with respect to the cell frequencies as long
as the general form of estimability remains constant. Hen c e, this type of sums of
squares is often considered usefu l for an unbal anced mod el with no missing cel ls. In a
factorial design with no missing cells, this method i s equivalent to the Yates’ weighted­squares-of-means technique. The Type III sum-of-squares method is commonly used
for:

 Any models listed in Type I.
 Any balanced or unbalanced model with no empty cells.

Linear Mixed Models Random Effects

Figure 4-4

Linear Mixed Models: Random Effects dialog box

49

Linear Mixed Models

Random Effects. There is no default model, so you must explicitly specify the random

effects. Alternatively, you can build nested or non-nested terms. You can also choose
to include an intercept te rm in the random-effects model.

You can specify multiple random-effects models. After building the first model, click

Next to build the next model. Click Previous to scroll back throug h existing models .

Each random-effect model is assumed to be independ ent of every other ra ndom-ef fect
model; that is, separate covariance matrices will be computed for each. T erms specified
in the same random-effect model can be correlated.

Covariance type. This allows you to specify the covariance structure for the random-

effects model. A separate covariance matrix is estimated for each random effect. The
available structures are as follows:

50

Chapter 4

Ante-Dependence: First Order



 AR(1)
 AR(1): Heterogeneous
 ARMA(1,1)
 Compound Symmetry
 Compound Symmetry: Correlation Metric
 Compound Symmetry: Heterogeneous
 Diagonal
 Factor Analytic: First Order
 Factor Analytic: First Order, Heterogeneous
 Huynh-Feldt
 Scaled Identity
 Toeplitz
 Toeplitz: Heterog eneous
 Unstructured
 Unstructured: Correlation Metric
 Variance Components

For more information, see the appendix Covariance Structures.

Subject Groupings. The variables listed are those that you selected as subj ect variables

in the Select Subjects/Repeated Variables dialog box. Choose some or all of these in
order to define the subjects for the random-effects model.

Linear Mixed Models Estimation

Figure 4-5

Linear Mixed Models: Estimation dialog box

51

Linear Mixed Models

Method. Select the maximum like lihood or restr icted maximum likeli hood estimation .
Iterations:

 Maximum iterations. Specify a non-negative integer.
 Maximum step-halvings. At each iteration, the step size is redu ced by a factor of 0.5

until the log-likelihood increases or maximum step-halving is reached. Specify a
positive integer.

 Print iteration history for every n step(s). Displays a table containing the log-

likelihood function value and parameter estimates at every n iterations beginning

th

with the 0

iteration (the initial estimates). If you choose to print the iteration

history, the last iteration is always printed regardless of the value of n.

52

Chapter 4

Log-likelihood Convergence. Converg ence is assumed if the absolute cha nge or relative

change in the log-likelihood function is less than the value specified, which must be
non-negative. The criterion is not used if the value specified equals 0.

Parameter Convergence. Convergence is assumed if the maximum absolute change or

maximum relative cha ng e in the parameter estimates is less than the value specified,
which must be non-negative. The criterion is not used if the value specified equals 0.

Hessian Convergence. For the Absolute specification, convergence is assumed if a

statistic based on the Hessian is less than the value specified. For the

Relative

specification, convergence is assumed if the statistic is less than the product of the
value specified and the absolute value of the lo g-likelihood. The cri terion is not used if
the value specified equals 0.

Maximum scoring steps. Requests to use the Fisher scoring algorithm up to iteration

number n. Specify a positive integer.

Singularity tolerance. This is the value used as tolerance in checking singularity.

Specify a positive value.

Linear Mixed Models Statistics

Figure 4-6

Linear Mixed Models: Statistics dialog box

53

Linear Mixed Models

Summary Statistics. Produces tables for:

 Descriptive statistics. Displays the sample sizes, means, and standard deviations of

the dependent variable and covariates (if specified). These statistics are displayed
for each distinct level combination of the factors.

 Case Processing Summary. Displ ays the sorted values of the factors, the repeated

measure variables, the repeated measure su bject s, and the random-effec ts subjects
and their frequencies.

Model Statistics. Produces tables for:

 Parameter estimates. Displays the fixed-effects and random-effects parameter

estimates and their approximate standard errors.

 Tests for covariance parameters. Displays the asymptotic standard errors and Wald

tests for the covariance parameters.

 Correlations of parameter estimates. Displays the asymptotic correlation matrix of

the fixed-effects parameter estimates.

 Covariances of parameter estimates. Displays the asymptotic covariance matrix of

the fixed-effects parameter estimates.

 Covariances of random effects. Displays the estimated covariance matrix of random

effects. This option is available only when at least one random effect is specified.
If a subject variable is specified for a random effect, then the common block is
displayed.

 Covariances of residuals. Displays the estimated residual covariance matrix. This

option is available only when a repeated variable has been specified. If a subject
variable is specified, the common block is displayed.

 Contrast coefficient matrix. This option displays the estimable functions used for

testing the fixed effects and the custom hypotheses.

Confidence interval. This value is used whenever a confidence interval is constructed.

Specify a value greater than or equal to 0 and less than 100. The default value is 95.

54

Chapter 4

Linear Mixed Models EM Means

Figure 4-7

Linear Mixed Models: EM Means dialog box

Estimated Marginal Means of Fitted Models. This group allows you to request model-

predicted estimated marginal means of the dependent variable in the cells and their
standard errors for the specified factors. Moreover, you can request that factor le vels of
main effects be compared.

 Factor(s) and Factor Interactions. This listbox contains factors and factor

interactions that have been specified in the Fixed dialog box, plus an

OVERALL

term. Model terms built from covariates are excluded from this listbox.

 Display Means for. The procedure will compute the estimated marginal means for

factors and factor interactio ns selected for this listbox. If

OVERALL is selected, the

estimated marginal means of t he dependent va riable are displ ayed, collapsing ov er
all factors. Note that any selected factors or factor interactions remain selected
unless an associated variable has been removed from the Factors list in the main
dialog box.

 Compare main effects. This option allows you to request pairwise comparisons of

levels of selected main effects. The Confidence Interval Adjustment allows you t o
apply an adjustment to the confidence intervals and significance values to account for
multiple comparisons. The available methods are LSD (no adjustment), Bonferroni,
and Sidak. Finally, for each factor, you can select a reference category to which

comparisons are made. If no reference category is selected, all pairwise comparisons
will be constructed. The options for the reference category are the first, last, or
custom-specified (in which case, you enter the value of the reference category).

Linear Mixed Models Save

Figure 4-8

Linear Mixed Models: Save dialog box

55

Linear Mixed Models

This dialog box allows you to save various model results to the working file.

Fixed Predicted Values. Saves variables related to the regression means without the

effects.

 Predicted values. The regression means without the random effects.
 Standard errors. The standard errors of the estimates.
 Degrees of freedom. The degrees of freedom associated with the estimates.

Predicted Values and Residuals. Saves variables related to the model fitted value.

 Predicted values. The model fitted value.
 Standard errors. The standard errors of the estimates.
 Degrees of freedom. The degrees of freedom associated with the estimates.
 Residuals. The data value minus the predicted value.

56

Chapter 4

MIXED Command Additional Features

The SPSS command language also allows you to:

 Specify tests of effects versus a linear combination of effects or a val ue (usi ng t he

TEST subcommand).

 Include user-missing values (using the MISSING subcommand).
 Compute estimated marginal means for specified values of covariates (using the

WITH keyword of the EMMEANS subcommand).

 Compare simple main effects of interactions (using the EMMEANS subcommand).

See the SPSS Syntax Reference Guide for complete syntax information.

Model Selection Loglinear Analysis

The Model Selection Loglinear Analysis procedure analyzes multiway
crosstabulations (contingency tables). It fits hierarchical loglinear models to
multidimension al crosstabulations us ing an iterative proportional-fitting algorithm.
This procedure helps y ou find out whi ch categorical va riables are associ ated. T o buil d
models, forced entry and backward elimination methods are available. For saturated
models, you can request parameter estimates and tests of partial association. A
saturated model add s 0.5 to all cells.

Chapter

5

Example. In a study of u ser preferen ce for one of two laundry detergents, researchers

counted people in each group, combining various categories of water softness (soft,
medium, or hard), previous use of one of the brands, and washing temperature (cold
or hot). They found how temperature is related to water softness and also to brand
preference.

Statistics. Frequencies, residuals, parameter estimates, standard errors, confidence

intervals, and tests of partial association. For custom models, plots of residuals and
normal probability plots.

Data. Factor variables are categorical. All variables to be analyzed must be numeric.

Categorical string variables can be recoded to numeric variables before starting the
model selection analysis.

Avoid specifying many variables with man y levels. Such specification s can lead to
a situation where many cells have small n umbers of obse rvati ons, and the ch i-squ are
values may not be useful.

Related procedures. The Model Selection procedure can help identify the terms

needed in the model. Then you can continue to evaluate the model using General

57

58

Chapter 5

Loglinear Analysis or Logit Loglinear Analysis. You can use Autorecode to recode
string variables. If a numeric variable has empty categories, use Recode to create
consecutive integer values.

To Obtain a Model Selection Loglinear Analysis

From the menus choose:

Analyze

Loglinear

Model Selection...

Figure 5-1

Model Selection Loglinear Analysis dialog box



Select two or more numeric categorical factors.

 Select one or more factor variables in the Factor(s) list, and click Define Range.

 Define the range of values for each factor variable.

 Select an option in the Model Building group.

Optionally, you can select a cell weight variable to specify structural zeros.

Loglinear Analysis Define Range

Figure 5-2

Loglinear Analysis Define Range dialog box

Y ou must indicate the range of categories for each factor variable. Values for Minimum
and Maximum correspond to the lowest and highest categories of the factor variable.
Both values must be integers, and the minimum value must be less than the maximum
value. Cases with values outside of the bounds are excluded. For example, if you
specify a minimum value of 1 and a maximum value of 3, only the values 1, 2, and 3
are used. Repeat this process for each factor variable.

Loglinear Analysis Model

59

Model Selection Loglinear Analysis

Figure 5-3

Loglinear Analysis Model dialog box

Specify Model. A saturated model contains all factor main effects and all factor-by-factor

interactions. Select

Generating Class. A generating class is a list of the highest-order terms i n which factors

Custom to specify a generating class for an unsaturated model.

appear. SPSS builds a hierarchical model containing the terms that define the
generating class and all l ower-order relat ives. Suppose you select va riables A, B, and C

60

Chapter 5

in the Factors list and then select Interaction from the Build Terms drop-down list. The
resulting model will contain the specified 3-way interaction A*B*C, the 2-way
interactions A*B, A*C, and B*C, and main effects for A, B, and C. Do not specify the
lower-order relatives in the generating class.

Build Terms

For the selected factors and covariates:

Interaction. Creates the highest-level interaction term of all selected variables. This is

the default.

Main effects. Creates a main-effects term for each variable selected.
All 2-way. Creates all possible two-way interactions of the selected variables.
All 3-way. Creates all possible three-way interactions of the selected variables.
All 4-way. Creates all possible four-way interactions of the selected variables.
All 5-way. Creates all possible five-way interactions of the selected variables.

Loglinear Analysis Options

Figure 5-4

Loglinear Analysis Options dialog box

Display. You can choose Frequencies, Residuals, or both. In a saturated model, the

observed and expected frequencies are equal, and the residuals are equal to 0.

Display for Saturated Model. For a saturated model, you can choose Parameter

estimates

. The parameter estimates may help determine which terms can be dropped
from the model. An associa tion table, which lists tests of partial association, is also
available. This option is computationally expensive for tables with many factors.

Plot. For custom models, you can choose one or both types of plots, Residuals and

Normal probability. These will help determine how well a mod el fits the data.

Model Criteria. SPSS uses an iterative proportional-fitting algorithm to obtain

parameter estimates. You can override one or more of the estimation criteria by
specifying

Maximum iterations, Convergence, or Delta (a value added to all cell

frequencies for saturated m odels).

HILOGLINEAR Command Additional Features

The SPSS command language also allows you to:

 Specify cell weights in matrix form (using the CWEIGHT subcommand).
 Generate analyses of several models with a single command (using the DESIGN

subcommand).

61

Model Selection Loglinear Analysis

See the SPSS Syntax Reference Guide for complete syntax information.

General Loglinear Analysis

The General Loglinear Analysis procedure analyzes the frequency counts of
observations falling into each cross-classification category in a crosstabulation or a
contingency table. Each cross-classification in the table constitutes a cell, and each
categorical variable is called a factor. The dependent variable is the number of cases
(frequency) in a cell of the crosstabulation, and the explanatory variables are factors
and covariates. This procedure estimates maximum likelihood parameters of
hierarchical and nonhierarchical loglinear models using the Newton-Raphson
method. Either a Poisson or a multinomial distribution can be analyzed.

You can select up to 10 factors to define the cells of a table. A cell structure variable
allows you to define structural zeros for incomplete tables, include an offset term in the
model, fit a log-rate model, or implement the method of adjustment of marginal tables.
Contrast variables allow computation of generalized log-odds ratios (GLOR).

SPSS automatically display s model information and goodness-of -fit statistics. You
can also display a variety of statistics and plots or save residuals and predi cted values
in the working data file.

Chapter

6

Example. Data from a report of automobile accidents in Florida are used to determine

the relationship between w earing a seat belt and whether an injury wa s fatal or
nonfatal. The odds ratio indicates significant evidence of a relationship.

Statistics. Observed and expected frequencies; raw , adjusted, and dev iance residuals;

design matrix; parameter estimates; odds ratio; log-odds ratio; GLOR; W ald statistic;
and confidence intervals. Plots: adjusted residuals, deviance residuals, and normal
probability.

Data. Factors are categori cal, and cell covariates ar e c ont i nuou s. Wh en a covariate is

in the model, SPSS applies the mean covariate value for cases in a cell to that cell.

63

64

Chapter 6

Contrast variables are continuous. They are used to compute generalized log-odds
ratios. The values of the contrast variable are the coef ficients for the linear combinat ion
of the logs of the expected cell counts.

A cell structure variable assigns weights. For example, if some of the cells are
structural zeros, the cell structure variable has a value of either 0 or 1. Do not use a cell
structure variable to weight aggregated data. Instead, choose

Weight Cases from the

Data menu.

Assumptions. Two distributions are available in General Loglinear Analysis: Poisson

and multinomial.
Under the Poisson distribution assumption:

 The total sample size is not fixed before the study, or the analysis is not conditional

on the total sample size .

 The event of an observation being in a cell is statistically independent of the cell

counts of other cells.

Under the multinomial distribution assumption:

 The total sample size is fixed, or the analysis is conditional on the total sample size.
 The cell counts are not statistically independent.

Related procedures. Use the Crosstabs procedure to examine the crosstabulations. Use

the Logit Loglinear procedure when it is natural to regard one or more categorical
variables as the response variables and the others as the explanatory variables.

To Obtain a General Loglinear Analysis

 From the menus choose:

Analyze

Loglinear

General...

Figure 6-1

General Loglinear Analysis dialog box

65

General Loglinear Analysis



In the General Loglinear Analysis dialog box, select up to 10 factor variables.
Optionally, you can:

 Select cell covariates.
 Select a cell structure variable to define structural zeros or include an offset term.
 Select a contrast variable.

66

Chapter 6

General Loglinear Analysis Model

Figure 6-2

General Loglinear Analysis Model dialog box

Specify Model. A saturated model contains all main effects and interactions involving

factor variables. It does not contain covariate terms. Select

Custom to specify only a

subset of interactions or to specify factor-by-covariate interactions.

Factors and Covariates. The factors and covariates are listed, with (Cov) indicating a

covariate.

Terms in Model. The model dep ends on the natu re of your data. Af ter selecting Custom,

you can select the main ef fects and interactions that are of interest i n your analysis. You
must indicate all of the terms to be included in the model.

Build Terms

For the selected factors and covariates:

Interaction. Creates the highest-level interaction term of all selected variables. This is

the default.

Main effects. Creates a main-effects term for each variable selected.
All 2-way. Creates all possible two-way interactions of the selected variables.
All 3-way. Creates all possible three-way interactions of the selected variables.

All 4-way. Creates all possible four-way interactions of the selected variables.
All 5-way. Creates all possible five-way interactions of the selected variables.

General Loglinear Analysis Options

Figure 6-3

General Loglinear Analysis Options dialog box

67

General Loglinear Analysis

The General Loglinear Analysi s procedure displays model informa ti on a nd go odness­of-fit statistics. In addition, you can choose one or more of the following:

Display. Several statistics are available for display—observed and expected cell

frequencies; raw, adjusted, and deviance residuals; a design matrix of the model; and
parameter estimates for the model.

Plot. Plots, available for custom models only , include t wo scatterplot matrices (adjusted

residuals or deviance residuals against observed and expected cell counts). You can
also display normal probability and detrended normal plots of adjusted residuals or
deviance residuals.

Confidence Interval. The confidence interval for parameter estimates can be adjusted.
Criteria. The Newton-Raphson method is used to obtain maximum likelihood

parameter estimates. You can enter new values for the maximum number of iterations,
the convergence criterion, and delta (a constant added to all cells for initia l
approximations). Delta remains in the cells for saturated models.

68

Chapter 6

To Specify Options

 From the menus choose:

Analyze

Loglinear

General...

In the General Loglinear Analysis or Logit Loglinear Analysis dialog box, click



Options.

General Loglinear Analysis Save

Figure 6-4

General Loglinear Analysis Save dialog box

Select the values you want to save as new variabl es in the workin g data file. The suf fix
n in the new variable names increments to make a unique na me for each saved variable.

The saved values refer to the aggregated data (cells in the contingency table), even
if the data are recorded in individual observations in the Data Editor. If you save
residuals or predicted values for unaggregated data, the saved value for a cell in the
contingency table is entered in the Data Editor for each case in that cell . To make sense
of the saved values, you should aggregate the data to obtain the cell counts.

Four types of residuals can be saved: raw, st andardized, adjusted, and devian ce. The
predicted values can also be saved.

GENLOG Command Additional Features

The SPSS command language also allows you to:

 Calculate linear combinations of observed cell frequencies and expected cell

frequencies, and print residuals, standardized residuals, and adjusted residuals of
that combination ( using the

 Change the default threshold val ue for re dun danc y checki ng (u sing the CRITERIA

subcommand).

 Display the standardized residuals (using the PRINT subcommand).

See the SPSS Syntax Reference Guide for complete syntax information.

GERESID subcommand).

69

General Loglinear Analysis

Logit Loglinear Analysis

The Logit Loglinear Analysis procedure analyzes the relationship between dependent
(or response) variables and independent (or explanatory) variables. The dependent
variables are always categorical, while the independent variables can be categorical
(factors). Other independent variables, cell covariates, can be continuous, but they are
not applied on a case-by-case basis. The weighted covariate mean for a cell is applied
to that cell. The logarithm of the odds of the dependent variables is expressed as a
linear combination of parameters. A multinomial distribution is automatically
assumed; these models are sometimes called multinomial logit models. This procedure
estimates parameters of logit loglinear models using the Newton-Raphson algorithm.

You can select from 1 to 10 dependent and factor variables combined. A cell
structure variable all ows you t o define stru ctural zeros for i ncomplete ta bles, incl ude
an offset term in the model, fit a log-rate mode l, or implement the method of
adjustment of marginal tables. Contrast variables allow computation of generalized
log-odds ratios (GLOR). The values of the contrast variable are the coefficients for
the linear combination of the logs of the expected cell counts.

SPSS automatically display s model information and goodness-of -fit statistics. You
can also display a variety of statistics and plots or save residuals and predi cted values
in the working data file.

Chapter

7

Example. A study in Florida included 219 alligators. How does the alligators’ food

type vary with their size and the four lakes in which they live? The study found that
the odds of a smaller alligator preferring reptiles to fish is 0.70 times lower than for
larger alligators; also, the odds of selecting primarily reptiles instead of fish were
highest in lake 3.

71

72

Chapter 7

Statistics. Observed and expected frequencies; raw, adjusted, and deviance residuals;

design matrix; parameter estimates; generalized log odds ratio; Wald statistic; and
confidence intervals. Plots: adjusted residuals, deviance residuals, and normal
probability plots.

Data. The dependent variables are categorical. Factors are categorical. Cell covariates can

be continuous, but when a covariate is in the model, SPSS applies the mean covariate
value for cases in a cell to that cell. Contrast v ariabl es are conti nuous. T hey are used to
compute generalized log odds ratios (GLOR). The values of the contrast variable are the
coefficients for the linear combination of the logs of the expected cell counts.

A cell structure variab le assigns weights. For examp le, if some of the cells are structu ral
zeros, the cell structure variable has a value of either 0 or 1. Do not use a cell structure
variable to weight aggregate data. Instead, use Weight Cases on the Data menu.

Assumptions. The counts within each combination of categories of explanatory

variables are assumed to have a multinomial distribution. Under the multinomial
distribution assumption:

 The total sample size is fixed, or the analysis is conditional on the total sample size.
 The cell counts are not statistically independent.

Related procedures. Use the Crosstabs pro cedure to display th e contingency tables. Use

the General Loglinear Analysis procedure when you want to analyze the relationship
between an observed count and a set of explanatory variables.

To Obtain a Logit Loglinear Analysis

 From the menus choose:

Analyze

Loglinear

Logit...

Figure 7-1

Logit Loglinear Analysis dialog box

73

Logit Loglinear Analysis

In the Logit Loglinear Analysis dialog box, select one or more dependent variables.



 Select one or more factor variables.

The total number of dependent and factor variables must be less than or equal to 10.
Optionally, you can:

 Select cell covariates.
 Select a cell structure variable to define structural zeros or include an offset term.
 Select one or more contrast variables.

74

Chapter 7

Logit Loglinear Analysis Model

Figure 7-2

Logit Loglinear Analysis Model dialog box

Specify Model. A saturated model contains all main effects and interactions involving

factor variables. It does not contain covariate terms. Select

Custom to specify only a

subset of interactions or to specify factor-by-covariate interactions.

Factors and Covariates. The factors and covariates are listed, with (Cov) indicating

a covariate.

Terms in Model. The model dep ends on the natu re of your data. Af ter selecting Custom,

you can select the main ef fects and interactions that are of interest i n your analysis. You
must indicate all of the terms to be included in the model.

Terms are added to the desi gn by taki ng all possib le combinat ions of th e dependent
terms and matching each combination with each term in the model list. If

constant for dependent

is selected, there is also a unit term (1) added to the model list.

Include

For example, suppose variables D1 and D2 are the dependent variables. A
dependent terms list is created by the Logit Loglinear Analysis procedure (D1, D2,
D1*D2). If the Terms in Model list contains M1 and M2 and a constant is incl uded, the
model list contains 1, M1, and M2. The result ant design includes combinations of each
model term with each de pendent term:

D1, D2, D1*D2
M1*D1, M1*D2, M1*D1*D2
M2*D1, M2*D2, M2*D1*D2.

Include constant for dependent. Includes a constant for the dependent variable in a

custom model.

Build Terms

For the selected factors and covariates:

Interaction. Creates the highest-level interaction term of all selected variables. This is

the default.

Main effects. Creates a main-effects term for each variable selected.
All 2-way. Creates all possible two-way interactions of the selected variables.
All 3-way. Creates all possible three-way interactions of the selected variables.
All 4-way. Creates all possible four-way interactions of the selected variables.

75

Logit Loglinear Analysis

All 5-way. Creates all possible five-way interactions of the selected variables.

Logit Loglinear Analysis Options

Figure 7-3

Logit Loglinear Analysis Options dialog box

76

Chapter 7

The Logit Loglinear Analysis proced ure displa ys model informati on and goodness-of­fit statistics. In addition, you can choose one or more of the following options:

Display. Several statistics are available for display: observed and expected cell

frequencies; raw, adjusted, and deviance residuals; a design matrix of the model; and
parameter estimate s f or the model.

Plot. Plots available for custom models include two scatterplot matrices (adjusted

residuals or deviance residuals against observed and expected cell counts). You can
also display normal probability and detrended normal plots of adjusted residuals or
deviance residuals.

Confidence Interval. The confidence interval for parameter estimates can be adjusted.
Criteria. The Newton-Raphson method is used to obtain maximum likelihood

parameter estimates. You can enter new v alues for the maximum number of iterations,
the convergence criterion, and delta (a constant added to all cells for initial
approximations). Delta rem a ins in the cells for saturated mod els.

To Specify Options

 From the menus choose:

Analyze

Loglinear

Logit...

In the Logit Loglinear Analysis dialog box, click Options.



Logit Loglinear Analysis Save

Figure 7-4

Logit Loglinear Analysis Save dialog box

Select the values you want to save as new variabl es in the workin g data file. Th e suffix
n in the new variable names increments to make a unique name fo r each saved variable.

The saved values refer to the aggregated data (to cells in the contingency table),
even if the data are recorded in individual observations i n the Data Editor. If you save
residuals or predicted values for unaggregated data, the saved value for a cell in the
contingency table is entered in the Data Editor for each case in that cell. To mak e sense
of the saved values, you should aggregate the data to obtain the cell counts.

Four types of residuals can be saved: raw, standar dized, adjusted, and deviance. The
predicted values can also be saved.

GENLOG Command Additional Features

The SPSS command language also allows you to:

 Calculate linear combinations of observed cell frequencies and expected cell

frequencies, and print residuals, standardized residuals, and adjusted residuals of
that combination ( using the

 Change the default threshold val ue for re dun danc y checki ng (u sing the CRITERIA

subcommand).

 Display the standardized residuals (using the PRINT subcommand).

GERESID subcommand).

77

Logit Loglinear Analysis

See the SPSS Syntax Reference Guide for complete syntax information.

Ordinal Regression

Ordinal Regression allows you to model the dependence of a polytomous ordinal
response on a set of predictors, which can be factors or covariates. The design of
Ordinal Regression is based on the methodology of McCullagh (1980, 1998), and t he
procedure is referred to as

Standard linear regression analysis involves minimizing the sum-of-squared
differences between a response (dependent) variable and a weighted combination of
predictor (independent) variables. The estimated coefficients reflect how changes in
the predictors affect the response. The response is assumed to be numerical, in the
sense that changes in the l evel of t he resp onse are equi valent throu ghout t he range o f
the response. For example, the difference in height between a person who is 150 cm
tall and a person who is 140 cm tall is 10 cm, which has the same meaning as the
difference in height between a person who is 210 cm tall and a person who is 200 cm
tall. These relationships do not necessarily hold for ordinal variables, in which the
choice and number of response categories can be quite arbitrary.

PLUM in the syntax.

Chapter

8

Example. Ordinal Regression could be used to study patient reaction to drug dosage.

The possible reactions may be classified as “none,” “mild,” “ moderate,” or “severe.”
The difference between a mild and moderate reaction is difficult or impossible to
quantify and is based on perception. Moreover, the difference between a mild and
moderate response may be greater or less than the difference between a moderate and
severe response.

Statistics and plots. Observed and expected frequencies and cumulative frequencies,

Pearson residuals for frequencies and cumulative frequencies, observed and expected
probabilities, observed and expected cumulative probabilities of each response
category by covariate pattern, asymptotic correlation and covariance matrices of
parameter estimates, Pearson’s chi-square and likelihood-ratio chi-square, goodness­of-fit statistics, iteration history, test of parallel lines assumption, parameter estimates,

79

80

Chapter 8

standard errors, confidence intervals, and Cox and Snell’s, Nagelkerke’s, and
McFadden’s statistics.

2

R

Data. The dependent variable is assumed to be ordinal and can be numeric or string.

The ordering is determined by sorting the values of the dependent variable in
ascending order. The lowest value defines the first categor y. Factor variables are
assumed to be categorical. Covariate variables must be numeric. Note that using more
than one continuous covariate c an easily result in the creation of a very large c ell
probabilities table.

Assumptions. Only one response variable is allowed, and it must be speci fied. Also, for

each distinct pattern of values across the independent variables, the responses are
assumed to be independent multinomial variables.

Related procedures. Nominal logistic regression uses similar models for nominal

dependent variables.

To Obtain an Ordinal Regression

 From the menus choose:

Analyze

Regression

Ordinal...

Figure 8-1

Ordinal Regression dialog box

 Select one dependent variable.

 Click OK.

Ordinal Regression Options

The Options dialog box allows you to adjust parameters used in the iter ative
estimation algorithm, choose a level of confidence for your parameter estimates, and
select a link function.

Figure 8-2

Ordinal Regression Options dialog box

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Iterations. You can customize the iterative algorithm.

 Maximum iterations. Specify a non-negative integer. If 0 is specifie d, the procedure

returns the initial estima te s .

 Maximum step-halving. Specify a positive integer.
 Log-likelihood convergence. The algorithm stops if the absolut e or relati ve

change in the log-likelihood is less than this value. The criterion is not used if 0
is specified.

 Parameter convergence. The algorithm stops if the absolute or relative change in

each of the parameter estimates is less than this value. The criterion is not used if
0 is specified.

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Confidence interval. Specify a value greater than or equal to 0 and less than 100.
Delta. The value added to zero cell frequencies. Specify a non-negative value less than 1.
Singularity tolerance. Used for checking for hi ghly dependent predictors. S elect a value

from the lis t of options.

Link. Choose among the Cauchit, Complementary Log-log, Logit, Negative Log-log,

and Probit functions.

Ordinal Regression Output

The Output dialog box allows you t o produce tables for disp lay in the V iewer and sav e
variables to the workin g file.

Figure 8-3

Ordinal Regression Output dialog box

Display. Produces tables for:

 Iteration history. The log-likelihood and parameter estimates are printed for the

print iteration frequency specified. The first and last iterations are always printed.

 Goodness-of-fit statistics. The Pearson and likelihood-ratio chi-square statistics.

They are computed based on the classification specified in the variable list.

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Ordinal Regression

 Summary statistics. Cox and Snell’s, Nagelkerke’s, and Mc Fadd en’ s statistics.
 Parameter estimates. Parameter estimates, standard errors, and confidence intervals.
 Asymptotic correlation of parameter estimates. Matrix of parameter estimat e

2

R

correlations.

 Asymptotic covariance of parameter estimates. Matrix of parameter estimate

covariances.

 Cell information. Observed and expected frequencies and cumulative frequencies,

Pearson residuals for frequencies and cumulative frequencies, observed and
expected probabilities, and observ ed and expected cumulative probab ilities of each
response category by covariate pattern. Note that for models with many covariate
patterns (for example, models wi th continuous covariates), this option can generate
a very large, unwieldy table.

 Test of parallel lines. Test of the hypothesis that the location parameters are

equivalent across the levels of the depe ndent variable. This is availabl e only for the
location-only model.

Saved variables. Saves the following variables to the working file:

 Estimated response probabilities. Model-estimated probabilities of classifying a

factor/covariate pattern into the response categories. There are as many
probabilities as the number of response categories.

 Predicted category. The response category that has the maximum estimated

probability for a factor/covariate pattern.

 Predicted category probability. Estimated probability of classifying a

factor/covariate pattern into the predicted category. This probability is also the
maximum of the esti m ated probabilities of the factor/covariate pattern.

 Actual category probability. Estimated probability of classifying a factor/covariate

pattern into the actual category.

Print log-likelihood. Controls the display of the log-likelihood. Including multinomial

constant

gives you the full value of the likelihood. To compare your results across

products that do not include the constant, you can choose to exclude it.

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Ordinal Regression Location Model

The Location dialog box allows you to specify the location model for your analysis.

Figure 8-4

Ordinal Regression Location dialog box

Specify model. A main-effects model contains the covariate and fact or main effects but

no interaction effects. You can create a custom model to specify subsets of factor
interactions or covariate interactions.

Factors/covariates. The factors and covariates are listed with (F) for factor and (C)

for covariate.

Location model. The model depends on the main effects and intera ction e ffects that

you select.

Build Terms

For the selected factors and covariates:

Interaction. Creates the highest-level interaction term of all selected variables.
Main effects. Creates a main-effects term for each variable selected.
All 2-way. Creates all possible two-way interactions of the selected variables.
All 3-way. Creates all possible three-way interactions of the selected variables.

All 4-way. Creates all possible four-way interactions of the selected variables.
All 5-way. Creates all possible five-way interactions of the selected variables.

Ordinal Regression Scale Model

The Scale dialog box allows you to specify the scale model for your analysis.

Figure 8-5

Ordinal Regression Scale dialog box

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Ordinal Regression

Factors/covariates. The factors and covariates are listed with (F) for factor and (C)

for covariate.

Scale model. The model depends on the main and interaction effects that you select.

Build Terms

For the selected factors and covariates:

Interaction. Creates the highest-level interaction term of all selected variables.
Main effects. Creates a main-effects term for each variable selected.
All 2-way. Creates all possible two-way interactions of the selected variables.
All 3-way. Creates all possible three-way interactions of the selected variables.
All 4-way. Creates all possible four-way interactions of the selected variables.
All 5-way. Creates all possible five-way interactions of the selected variables.

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Ordinal Regression Command Additional Features

You can customize your Ordinal Regression i f you paste your selection s i nt o a synt ax
window and edit the resulting
also allows you to:

 Create customized hypothesis tests by specifying null hypotheses as linear

combinations of parameters.

See the SPSS Syntax Reference Guide for complete syntax information.

PLUM command syntax. The SPSS command la nguage