Spss ADVANCED MODELS 12.0 User Manual

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Preface

SPSS 12.0 is a powerful software package for microcomputer data management and analysis. The Advanced Models option is an add-on enhancement that provides

nal statistical analysis techniques. The procedures in Advanced Models must

additio be used with the SPSS 12.0 Base and are completely integrated into that system.

The Advanced Models option includes procedures for:

 Multivariate and repeated measures general linear models  Variance components analysis  Linear mixed models  Loglinear analysis, including model selection (hierarchical), general loglinear

models,

 Ordina  Surviv

Cox regression

and logit loglinear models

l regression analysis

al analysis, including life tables, Kaplan-Meier survival analysis, and

Installation

To install Advanced Models, follow the instructions for adding and removing features in the installation instructions supplied with the SPSS Base. (To start, double-click on the SPS

SSetupicon.)

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Compatibility

SPSS is designed to run on many computer systems. See the materials that came with your system for specific information on minimum and recommended requirements.

Serial Numbers

Your serial number is your identification number with SPSS Inc. You will need this serial number when you contact SPSS Inc. for information regarding support, payment, or

an upgraded system. The serial number was provided with your Base

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Additional Publications

Additional copies of SPSS product manuals may be purchased directly from SPSS Inc. Visit our Web site at http://www.spss.com/estore, or contact your local SPSS office, list

ed on the SPSS Web site at http://www.spss.com/worldwide. For telephone orders in the United States and Canada, call SPSS Inc.. at 800-543-2185. For telephone orders outside of North America, contact your local office, listed on the SPSS Web sit

The SPSS Statistical Procedures Companion, by Marija Norušis, is being prepared for publication by Prentice Hall. It contains overviews of the procedures in the SPSS Base, plus

Logistic Regression, General Linear Models, and Linear Mixed Models.

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About This Manual

This manual documents the graphical user interface for the procedures included in the Advanced Models module. Illustrations of dialog boxes are taken from SPSS for Wind

ows. Dialog boxes in other operating systems are similar. The Advanced Models command syntax is included in the SPSS 12.0 Command Syntax Reference, available on the product CD-ROM.

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Contents

1 GLM Multivar

GLMMultivariateDataConsiderations............................ .2

ObtainingGLMMultivariateTables............................... .3

iate Analysis 1

2 GLM Repeated Measures 17

GLMRepeatedMeasuresDataConsiderations..................... .18

GLMRepeatedMeasuresDefineFactors......................... .20

DefiningFactorsforGLMRepeatedMeasures..................... .21

ObtainingGLMRepeatedMeasuresTables........................ .22

3 Variance Components Analysis 35

VarianceComponentsDataConsiderations ....................... .36

ObtainingVarianceComponentsTables .......................... .36

4 Linear Mixed Models 43

LinearMixedModelsDataConsiderations........................ .44

Linear Mixed Models Select Subjects/Repeated Variables . . . . . . . . . . . . . 44

Selecting Subjects/Repeated Variables for Linear Mixed Models . . . . . . . . 46

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ObtainingaLinearMixedModelsAnalysis........................ .47

5 Model Selection Loglinear Analysis 59

Model Selection Loglinear Analysis Da ta Considerations . . . . . . . . . . . . . . 59

ObtainingaModelSelectionLoglinearAnalysis.................... .60

6 General Loglinear Analysis 65

GeneralLoglinearAnalysisDataConsiderations.................... .66

ObtainingaGeneralLoglinearAnalysis........................... .66

7 Logit Loglinear Analysis 73

LogitLoglinearAnalysisDataConsiderations...................... .74

ObtainingaLogitLoglinearAnalysis............................. .74

8 Ordinal Regression 81

OrdinalRegressionDataConsiderations.......................... .82

ObtaininganOrdinalRegression ............................... .82

9 Life Tables 91

LifeTablesDataConsiderations................................ .92

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CreatingLifeTables ......................................... .93

10 Kaplan-Meier Survival Analysis 97

Kaplan-MeierDataConsiderations.............................. .97

ObtainingaKaplan-MeierSurvivalAnalysis....................... .98

11 Cox Regression Analysis 105

CoxRegressionDataConsiderations ........................... .105

ObtainingaCoxRegressionAnalysis ........................... .106

12 Computing Time-Dependent Covariates 113

ComputingaTime-DependentCovariate......................... .114

Appendices

A Categorical Variable Coding Schemes 117

Deviation................................................ .117

Simple .................................................. .118

Helmert ................................................. .119

Difference ............................................... .120

Polynomial............................................... .120

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Repeated................................................ .121

Special ................................................. .122

Indicator ................................................ .123

B Covariance Structures 125

Index 131

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Chapter

GLM Multivar

The GLM Multivariate procedure provides regression analysis and analysis of variance fo covariates. The factor variables divide the population into groups. Using this general linear model procedure, you can test null hypotheses about the effects of factor variables 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 fact variables are specified as covariates.

Both balanced and unbalanced models can be tested. A design is balanced if each cell 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 matrice than one dependent variable is specified, the multivariate analysis of variance using Pillai's trace, Wilks' lambda, Hotelling's trace, and Roy's largest root criterion with approx for each dependent variable. In addition to testing hypotheses, GLM Multivariate produces estimates of parameters.

Common Additionally, after an overall F test has shown significance, you can use post hoc tests to evaluate differences among specific means. Estimated marginal means give estim (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 varia

r multiple dependent variables by one or more factor variables or

on the means of various groupings of a joint distribution of dependent

ors can be included. For regression analysis, the independent (predictor)

in the model contains the same number of cases. In a multivariate model,

s are called SSCP (sums-of-squares and cross-products) matrices. If more

imate F statistic are provided as well as the univariate analysis of variance

ly used aprioricontrasts are available to perform hypothesis testing.

ates of predicted mean values for the cells in the model, and profile plots

ble separately.

iate Analysis

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Chapter 1

Residuals, predicted values, Cook's distance, and leverage values can be saved as new variables in your data file for checking assumptions. Also available 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 SSCP matrix divided by the degrees of freedom of the residuals, and the residual correlation matrix, which is the standa

rdized 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

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

precision of measurement.

resistance, gloss, and opacity. Two rates of extrusion and two different amounts of additive a

re tried, and the three properties are measured under each combination of extrusion rate and additive amount. The manufacturer finds that the extrusion rate and the amount of additive individually produce significant results but that the interaction of the two

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

factorsisnotsignificant.

evaluate different hypotheses. Type III is the default.

ics.

Statist

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, Tukey's

y significant difference, Tukey's-b, Duncan, Hochberg's GT2, Gabriel, Waller

honestl 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

ions, and counts for all of the dependent variables in all cells; the Levene test

deviat for homogeneity of variance; Box's M test of the homogeneity of the covariance matrices of the dependent variables; and Bartlett's test of sphericity.

Spread-versus-level, residual, and profile (interaction).

Plots.

GLM Multivariate Data Considerations

Data. The dependent variables should be quantitative. Factors are categorical and

ve numeric values or string values of up to eight characters. Covariates are

can ha quantitative variables that are related to the dependent variable.

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Obtainin

E From the

GLM Multivaria

Assumptions. For dependent variables, the data are a random sample of vectors

te Analysis

from a multivariate normal population; in the population, the variance-covariance matrices fo

r 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 (including Box's M) and spread-versus-level plots. You can als

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

o examine residuals and residual plots.

analysis of variance. For a single dependent variable, use GLM Univariate. If you measured t

he same dependent variables on several occasions for each subject, use

GLM Repeated Measures.

g GLM Multivariate Tables

menus choose:

Analyze

General Linear Model

Multivariate...

Figure 1-

Multivar

iate dialog box

E Sele

ct at least two dependent variables.

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Chapter 1

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

GLM Multiva

Figure 1-2

Multivariate Model dialog box

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

main effects, and all factor-by-factor interactions. It does not contain covariate interactions. Select factor-by-covariate interactions. You must indicate all of the terms to be included in the model.

riate Model

Custom to specify only a subset of interactions or to specify

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

and (C) for covariate.

Model. The model depends on the nature of your data. After selecting 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.

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GLM Multivaria

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. C 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 Squa

For the mode

reates all possible three-way interactions of the selected variables.

res

l, you can choose a type of sum of squares. Type III is the most

commonly used and is the default.

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

sum-of-squ

ares method. Each term is adjusted only for 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-ord

er 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.

 Apurelynestedmodelinwhichthefirst-specifiedeffectisnestedwithinthe

second-sp

ecified effect, t he second-specified effect is nested within the third, and

so on. (This form of nesting can be specified only by using syntax.)

te Analysis

Type II. T

his 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

 A balanc  Any mode

commonlyusedfor:

ed ANOVA model.

l that has main factor effects only.

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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 effects (if any) that contain it. The Type III sums of squares have one major advantage in that they are invariant with respect to the cell frequencie

saslongasthegeneralformofestimability remains constant. Therefore, this type is often considered useful for an unbalanced model with no missing cells. In a factorial design with no missing cells, this method is equivalent to the Yates' wei

ghted-squares-of-means technique. The Type III sum-of-squares method

is commonly used for:

 AnymodelslistedinTypeIandTypeII.  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 = Type II. When F is contained in other effects, Type IV distributes the contrasts being made among the parameters in F to all higher-level effects equitably. The Type I

 Any mode  Any bala

V sum-of-squares method is commonly used for:

ls listed in Type I and Type II.

nced model or unbalanced model with empty cells.

GLM Multivariate Contrasts

Figure 1-3

Multivariate Contrasts dialog box

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GLM Multivaria

Contrasts are used to test whether the levels of an effect are significantly different from one another. You can specify a contrast for each factor in the model. Contrasts represent l

inear combinations of the parameters.

Hypothesis testing is based on the null hypothesis LBM = 0,whereL is the contrast

coefficientsmatrix, M is the identity matrix, which has dimension equal to the number of dependen

tvariables,andB is the parameter vector. When a contrast is specified, SPSS creates an L matrix such that the columns corresponding to the factor match the contrast. The remaining columns are adjusted so that the L matrix is estimable.

In additio

n 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' lam

bda, Hotelling's trace, and Roy's largest root criteria are provided.

A vailable contrasts are deviation, simple, difference, Helmert, repeated, and polynomial. For deviation contrasts and simple contrasts, you can choose whether the referenc

e category is the last or first category.

Contrast Types

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.

te Analysis

Simple. C

ompares the mean of each level to the mean of a specified level. This type of contrast is useful when there is a control group. You can choose the first or last category as the reference.

nce.

Differe

Compares the mean of each level (except the first) to the mean of previous

levels. (Sometimes called reverse Helmert contrasts.)

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

of subse

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

quent levels.

subsequent level.

ial.

Polynom

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

mate polynomial trends.

to esti

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Chapter 1

GLM M ultivariate Profile Plots

Figure 1-4

Multivariat

e Profile Plots dialog box

Profile plots (interaction plots) are useful for comparing marginal means in your model. Profile plots are created for each dependent variable. A profile plot is a line 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 indicate 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)

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GLM Multivaria

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 H oc Mul

tiple Comparisons for Observed Means dialog box

te Analysis

Post ho

c 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

re used for between-subjects factors only. The post hoc multiple comparison

tests a tests are performed for each dependent variable separately.

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

used m

ultiple 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

he Bonferroni test. Tukey's honestly significant difference test uses the

than t 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

arisons. When testing a large number of pairs of means, Tukey's honestly

comp significant difference test is more powerful than the Bonferroni test. For a small number of pairs, Bonferroni is more powerful.

Page 20

Chapter 1

Hochberg's GT2 is similar to Tukey's honestly significant difference test, but the

Studentized maximum modulus is used. Usually, Tukey's test is more powerful.

Gabriel's p

airwise comparisons test also uses the Studentized maximum modulus

and is generally more powerful than Hochberg's GT2 when the cell sizes are unequal. Gabriel's test may become liberal when the cell sizes vary greatly.

Dunnett's p

airwise 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 control category, use a two-sided test. To test whether the mean at any level of the factor is smaller than that of the control category, select

Likewise, to test whether the mean at any level of the factor is larger than that

Control

of the control category, select

> Control.

Ryan, Einot, Gabriel, and Welsch (R-E-G-W) developed two multiple step-down

range tes

ts. 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 range. These tests are more

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), Dunnett's T3 (pairwise comparison test based on the Studentized maximum modulus), Games-Howell pairwise comparison

ometimes liberal), or Dunnett's C (pairwise comparison test based on the

test (s Studentized range).

Duncan's multiple range test, Student-Newman-Keuls (S-N-K), and Tukey's-b

ge tests that rank group means and compute a range value. These tests are

are ran not used as frequently as the tests previously discussed.

The Waller-Duncan t test uses a Bayesian approach. This range test uses the

nic mean of the sample size when the sample sizes are unequal.

harmo

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

eature. The result is that the Scheffé test is often more conservative than other

this f tests, which means that a larger difference between means is required for significance.

Page 21

equivalent to multiple individual t tests between all pairs of groups. The disadvantage of this test multiple comparisons.

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

Games and Ho Homogeneous subsets for range tests are provided 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' tests and range tests.

GLM Multi

Figure 1-7

Multivariate Save dialog box

GLM Multivaria

The least significant difference (LSD) pairwise multiple comparison test is

is that no attempt is made to adjust the observed significance level for

well, Tamhane's T2 and T3, Dunnett's C, and Dunnett's T3.

s GT2, Gabriel's test, and Scheffé's test are both multiple comparison

variate Save

te 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.

Page 22

Chapter 1

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

independent variables and cases that may have a large impact on the model. Ava ilable are Cook's d

 Cook's Dis

istance and uncentered leverage values.

tance.

A measure of how much the residuals of all cases would change if a particular case were excluded from the calculation of the regression coefficients. A large Cook's D indicates that excluding a case from computation of the regr

 Leverage V

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

ession statistics, changes the coefficients substantially.

alue.

The relative influence of each observation on the model's fit.

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

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

residuals are available.

 Unstandardized Residuals. The difference between an observed value and the

value pred

 Weighted

icted by the model.

Residuals.

Weighted unstandardized residuals. Available only if a

WLS variable was previously selected.

 Standardized. The residual divided by an estimate of its standard error.

Standardized residuals are also known as Pearson residuals.

 Studentized Residual. The residual divided by an estimate of its standard deviation

that vari

es from case to case, depending on the distance of each case's values on

the independent variables from the means of the independent variables.

 Deleted Residual. The residual for a case when that case is excluded from the

calculation of the regression coefficients. It is the difference between the value of the depe

ndent variable and the adjusted predicted value.

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

the parameter estimates in the model. Also, for each dependent variable, there will bearowo

f 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 similar rows for each dependent variable. You can

s data in other SPSS procedures.

use thi

Page 23

GLM Multivariate Options

Figure 1-8

Multivariat

e Options dialog box

GLM Multivaria

te Analysis

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

ted Marginal Means.

Estima

Select the factors and interactions for which you want estimates of the population marginal means in the cells. These means are adjusted for the covariates, if any. Interactions are available only if you have specified a

mmodel.

custo

 Comp

aremaineffects.

Provides uncorrected pairwise comparisons among estimated marginal means for any main effect in the model, for both between- and within-subjects factors. This item is available only if main effects are selected

the Display Means For list.

under

 Conf

idence interval adjustment.

Select least significant difference (LSD), Bonferroni, or Sidak adjustment to the confidence intervals and significance. This item is available only if

Compare main effects is selected.

Page 24

Chapter 1

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

and counts for all of the dependent variables in all cells. 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 attributable to a factor. Select

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

basedonth

e 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 matrices and the Residual

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

SSCP matri

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

each dependent variable across all level combinations of the between-subjects factors, for betwe

en-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 resi This item is disabled if there are no factors. Select

dual plots options are useful for checking assumptions about the data.

Residual plots to produce an

observed-by-predicted-by-standardized residuals plot for each dependent variable. These pl

Lack of fit test to check if the relationship between the dependent variable and the

ots are useful for investigating the assumption of equal variance. Select

independent variables can be adequately described by the model. General estimable

allows you to construct custom hypothesis tests based on the general

functi

estimable function. Rows in any contrast coefficient matrix are linear combinations of the general estimable function.

GLM C

Signif

icance level.

You might want to adjust 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

ecify a significance level, the associated level of the confidence intervals

you sp is displayed in the dialog box.

ommand Additional Features

e features may apply to univariate, multivariate, or repeated measures analysis.

Thes The SPSS command language also allows you to:

 Specify nested effects in the design (using the DESIGN subcommand).

Page 25

GLM Multivaria



Specify tests of effects versus a linear combination of effects or a value (using the

TEST subcom

 Specify mul  Include use  Specify EPS  Construct a

MMATRIX,orKMATRIX 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 factor interaction among the

factors in t

 Specify na  Construct a  Construct a

ANOVA table (using the

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

mand).

tiple contrasts (using the

r-missing values (using the

criteria (using the

CONTRAST subcommand).

MISSING subcommand).

CRITERIA subcommand).

custom L matrix, M matrix, or K matrix (using the

CONTRAST subcommand).

he factor list (using the

mes for temporary variables (using the

correlation matrix data file (using the

EMMEANS subcommand).

SAVE subcommand). OUTFILE subcommand).

matrix data file that contains statistics from the between-subjects

OUTFILE subcommand).

te Analysis

LMATRIX,

See the SPSS

Command Syntax Reference for complete syntax information.

Page 26

Page 27

Chapter

GLM Repeated

The GLM Repeated Measures procedure provides analysis of variance when the same measurement is made several times on each subject or case. If between-subjects factors are specified, they divide the population into groups. Using this general linear model procedure, you can test null hypotheses about the effects of both the between-subjects factors and the within-subjects factors. You can investigate interactions between factors as well as the effects of individual factors. In addition, the 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 p rovides both univariate and multivariate analyses for the repeated measures data. Both balanced and unbalanced models can be tested. A design is balanced if each cell in the model contains the same number of cases. 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 foundinunivariateanalysis.Thesematrices are called SSCP (sums-of-squares and cross-products) matrices. In addition to testing hypotheses, GLM Repeated Measures produces estimates of parameters.

Commonly used aprioricontrasts are available to perform hypothesis testing on between-subjects factors. Additionally, after 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 (interaction plots) of these means allow you to visualize some of the relationships easily.

Measures

Page 28

Chapter 2

Residuals, predicted values, Cook's distance, and leverage values can be saved as new variables in your data file for checking assumptions. Also available are a residual SSCP matrix residuals, a residual covariance matrix, which is the residual SSCP matrix divided by the degrees of freedom of the residuals, and the residual correlation matrix, which is the standa

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

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

scores on an anxiety-rating test. The anxiety rating is called a between-subjects factor because it 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 trial-by-anxiety interaction is not significant.

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

evaluat

, which is a square matrix of sums of squares and cross-products of

rdized form of the residual covariance matrix.

precision of measurement.

divides the subjects into groups. The students are each given four trials on

levels for the four trials. The trial effect is found to be significant, while the

e different hypotheses. Type III is the default.

GLM Re

Statistics. Post hoc range tests and multiple comparisons (for between-subjects

factors): least significant difference, Bonferroni, Sidak, Scheffé, Ryan-Ei Student-Newman-Keuls, Tukey's honestly significant difference, Tukey's-b,Duncan, Hochberg's GT2, Gabriel, Waller Duncan t test, Dunnett (one-sided and two-sided), Tamhan 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 o

Plots. Spread-versus-level, residual, and profile (interaction).

not-Gabriel-Welsch multiple F, Ryan-Einot-Gabriel-Welsch multiple range,

e's T2, Dunnett's T3, Games-Howell, and Dunnett's C. Descriptive statistics:

fsphericity.

peated Measures Data Considerations

The dependent variables should be quantitative. Between-subjects factors divide

Data.

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.

in-subjects factors are defined in the Repeated Measures Define Factor(s)

With

Page 29

GLM Repeated Me

asures

dialog box. Covariates are quantitative variables that are related to the dependent variable. For a repeated measures analysis, these should remain constant at each level of a within-

subjects 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 repetition of the measurement within the group. A

within-subjects factor is defined for the group with the number of levels equal to the number of repetitions. For example, measurements of weight could be taken on different days. If measurements of the same property were taken on five days, the w

ithin-subjects factor could be specified as day with five levels.

For multiple within-subjects factors, the number of measurements for each subject is equal to the product of the number of levels of each factor. For example, if measureme

nts 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).

ons.

Assumpti

A repeated measures analysis can be approached in two ways, univariate

and multivariate.

The univariate approach (also known as the split-plot or mixed-model approach)

consider

s the dependent variables as responses to the levels of within-subjects factors. The measurements on a subject should be a sample from a multivariate normal distribution, and the variance-covariance matrices are the same across the cells fo

rmed by the between-subjects effects. Certain assumptions are made on the variance-covariance matrix of the dependent variables. The validity of the F statistic used in the univariate approach can be assured i f the variance-covariance matrix is

ar in form (Huynh and Mandeville, 1979).

circul

To test this assumption, Mauchly's test of sphericity can be used, which performs

a test of sphericity on the variance-cova riance matrix of an orthonormalized

ormed dependent variable. Mauchly's test is automatically displayed for a

transf repeated measures analysis. For small sample sizes, this test is not very powerful. For large sample sizes, the test may be significant even when the impact of the departure

results is small. If the significance of the test is large, the hypothesis of

on the 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

es of freedom can be made in order to validate the univariate F statistic. Three

degre estimates of this adjustment, which is called epsilon, are available in the GLM Repeated Measures procedure. Both the numerator and denominator degrees of

dom must be multiplied by epsilon, and the significance of the F ratio must be

free evaluated with the new degrees of freedom.

Page 30

Chapter 2

The multivariate approach considers the measurements on a subject to be a sample from a multivariate normal distribution, and the variance-covariance matrices are the same across 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 GLM Univariate or GLM Multivariate. If there are only two measurements for each subject (for example, pre-test and post-test), and there are no between-subjects factors, y

the cells formed by the between-subjects effects. To test whether the

variance. If there are not repeated measurements on each subject, use

ou can use the Paired-Samples T Test procedure.

GLM Repeated Measures Define Factors

GLM Repeated Measures analyzes groups of related dependent variables that represen define one or more within-subjects factors for use in GLM Repeated Measures. Note that the order in which you specify within-subjects factors is important. Each factor constitu

within-subjects factors in this dialog box. Notice that these factors are not existing variabl

t different measurements of the same attribute. This dialog box lets you

tes a level within the previous factor.

To use Repeated Measures, you must set up your data correctly. You must define

es in your data but rather factors that you define here.

Example. In a weight-loss study, suppose the weights of several people are measured

each week for five weeks. In the data file, each person is a subject or case. The weights 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 b weight1, ..., weight5 areusedtoassignthefivelevelsofweek. The variable in the data file that groups males and females (gender) can be specified as a between-subjects facto

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

measures. For example, the pulse and respiration rate could be measured on each subje but are defined here. A model with more than one measure is sometimes called a doubly multivariate repeated measures model.

for the weeks are recorded in the variables weight1, weight2, and so on. The

e called week, defined to have five levels. In the main dialog box, the variables

r to study the differences between males and females.

ct every day for a week. These measures do not exist as variables in the data file

Page 31

Defining Fac

E From the me

Analyze

General Linear Model

Repeated Measures...

Figure 2-1

Repeated Me

tors for GLM Repeated Measures

nus choose:

asures Define Factor(s) dialog box

GLM Repeated Me

asures

E Typeaw E Click A E Repeat

ithin-subject factor name and its number of levels.

these steps for each within-subjects factor.

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

E Type the measure name. E Click Add.

After defining all of your factors and measures:

E Click Define.

Page 32

Chapter 2

Obtaining GL

Figure 2-2

Repeated Measures dialog box

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

factors (and optionally, measures) on the list.

M Repeated Measures Tables

To change positions of the variables, use the up and down arrows.

To make changes to the within-subjects factors, you can reopen the Repeated Measures Define Factor(s) dialog box without closing the main dialog box. Optionally, you can specify between-subjects factor(s) and covariates.

Page 33

GLM Repeated Measures Model

Figure 2-3

Repeated Mea

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

sures Model dialog box

Custom to specify only a subset of interactions or to specify

GLM Repeated Me

asures

Between-Subjects. The covariates are listed with (C) for covariate. Model. The model depends on the nature of your data. After selecting Custom,you

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

Sum of squares. The method of calculating the sums of squares for the

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

Page 34

Chapter 2

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. C 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 Squa

For the mode

reates all possible three-way interactions of the selected variables.

res

l, you can choose a type of sum of squares. Type III is the most

commonly used and is the default.

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

sum-of-squ

ares 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-ord

er 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.

 Apurelynestedmodelinwhichthefirst-specifiedeffectisnestedwithinthe

second-sp

ecified effect, t he second-specified effect is nested within the third, and

so on. (This form of nesting can be specified only by using syntax.)

Type II. T

his 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

 A balanc  Any mode

commonlyusedfor:

ed ANOVA model.

l that has main factor effects only.

Page 35

GLM Repeated Me



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

asures

it and orthogonal to any effects (if any) that contain it. The Type III sums of squares have one major advantage in that they are invariant with respect to the cell frequencie

ghted-squares-of-means technique. The Type III sum-of-squares method

is commonly used for:

 AnymodelslistedinTypeIandTypeII.  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

 Any mode  Any bala

V sum-of-squares method is commonly used for:

ls listed in Type I and Type II.

nced model or unbalanced model with empty cells.

GLM Repeated Measures Contrasts

Figure 2-4

Repeated Measures Contrasts dialog box

Page 36

Chapter 2

Contrast Types

Contrasts are used to test for differences among the levels of a between-subjects factor. You can specify a contrast for each between-subjects factor in the model. Contrasts r

epresent linear combinations of the parameters.

Hypothesis testing is based on the null hypothesis LBM=0, where L is the contrast

coefficients matrix, B is the parameter vector, and M is the average matrix that correspond display t his transformation matrix by selecting

s to the average transformation for the dependent variable. You can

Transformation matrix in the Repeated

Measures Options dialog box. For example, if there are four dependent variables, a within-su

bjects 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.

A vailable contrasts are deviation, simple, difference, Helmert, repeated, and

polynomi

al. For deviation contrasts and simple contrasts, you can choose whether the

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

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

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

type of contrast is useful when there is a control group. You can choose the first or

egory as the reference.

last cat

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

levels. (Sometimes called reverse Helmert contrasts.)

Helmer

Compares the mean of each level of the factor (except the last) to the mean

of subsequent levels.

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

uent level.

subseq

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.

Page 37

GLM Repeated Measures Profile Plots

Figure 2-5

Repeated Mea

Profile plots (interaction plots) are useful for comparing marginal means in your model. A profile plot is a line plot in which each point indicates the estimated marginal mean of a dependent variable (adjusted 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.

sures Profile Plots dialog box

GLM Repeated Me

asures

Figure 2-6

Nonparallel plot (left) and parallel plot (right)

Page 38

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 2-7

Post H oc Mul

Post ho

tiple Comparisons for Observed Means dialog box

c multip le 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

s only. They are not available if there are no between-subjects factors. The

factor post hoc multiple comparison tests are performed for the average across the levels of the within-subjects factors.

The Bo

nferroni 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

Sidak's t test also adjusts the significance level and provides tighter bounds

made. than the Bonferroni test. Tukey's honestly significant difference test uses the Studentized range statistic to make all pairwise comparisons between groups and sets

xperimentwise error rate to the error rate for the collection for all pairwise

the e comparisons. When testing a large number of pairs of means, Tukey's honestly

Page 39

GLM Repeated Me

asures

significant difference test is more powerful than the Bonferroni test. For a small number of pairs, Bonferroni is more powerful.

Hochberg's G

T2 is similar to Tukey's honestly significant difference test, but the

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 gener

ally more powerful than Hochberg's GT2 when 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 s

ingle 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 n

ot equal to that of the control category, use a two-sided test. To test whether

the mean at any level of the factor is smaller than that of the control category, select

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

Control

of the con

trol category, select

> Control.

Ryan, Einot, Gabriel, and Welsch (R-E-G-W) developed two multiple step-down range tests. Multiple step-down procedures first test whether all means are equal. If all mea

ns are not equal, subsets of means are tested f or equality. R-E-G-W F is

based on an F test and R-E-G-W Q is based on the Studentized range. These tests are more powerful than Duncan's multiple range test and Student-Newman-Keuls (which a

re 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 compar

isons test based on a t test), Dunnett's T3 (pairwise 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

tized range).

Studen

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

ed as frequently as the tests previously discussed.

not us

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.

gnificance level of the Scheffé test is designed to allow all possible linear

The si 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

, which means that a larger difference between means is required for significance.

tests

Page 40

Chapter 2

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 attempt is made to adjust the observed significance level for

multiple comparisons.

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

Games and Ho

well, Tamhane's T2 and T3, Dunnett's C, and Dunnett's T3. Homogeneous subsets for range tests are provided 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 Repea

Figure 2-8

Repeated Measures Save dialog box

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

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

ted Measures Save

Page 41

GLM Repeated Me

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

asures

independent variables and cases that may have a large impact on the model. Ava ilable are Cook's d

 Cook's Dis

istance and uncentered leverage values.

tance.

 Leverage V

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

ession statistics, changes the coefficients substantially.

alue.

The relative influence of each observation on the model's fit.

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

 Unstanda

are also available.

rdized Residuals.

The difference between an observed value and the

valuepredictedbythemodel.

 Standardized. The residual divided by an estimate of its standard error.

Standardized residuals are also known as Pearson residuals.

 Studentized Residual. The residual divided by an estimate of its standard deviation

that varie

s from case to case, depending on the distance of each case's values on

the independent variables from the means of the independent variables.

 Deleted Residual. The residual for a case when that case is excluded from the

calculation of the regression coefficients. It is the difference between the value of the depen

dent variable and the adjusted predicted value.

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

the parameter estimates in the model. Also, for each dependent variable, there will bearowo

matrix file in other procedures that read an SPSS matrix file.

Page 42

Chapter 2

GLM Repeated Measures Options

Figure 2-9

Repeated Mea

sures Options dialog box

Optional statistics are available from this dialog box. Statistics are calculated u sing

-effects model.

afixed

Estimated M arginal Means. Select the factors and interactions for which you want

estimates of the population marginal means in the cells. These means are adjusted

e covariates, if any. Both within-subjects and between-subjects factors can

for th be selected.

 Compare main effects. Provides uncorrected pairwise comparisons among

ated marginal means for any main effect in the model, for both between- and

estim 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),

rroni, or Sidak adjustment to the confidence intervals and significance.

Bonfe This item is available only if

Compare main effects is selected.

Page 43

GLM Repeated Me

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

asures

and counts for all of the dependent variables in all cells. 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 attributable to a factor. Select

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

basedonth

e 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 matrices and the Residual

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

SSCP matri

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

each dependent variable across all level combinations of the between-subjects factors, for betwe

dual plots options are useful for checking assumptions about the data.

Residual plots to produce an

observed-by-predicted-by-standardized residuals plot for each dependent variable. These pl

Lack of fit test to check if the relationship between the dependent variable and the

ots are useful for investigating the assumption of equal variance. Select

independent variables can be adequately described by the model. General estimable

allows you to construct custom hypothesis tests based on the general

functi

estimable function. Rows in any contrast coefficient matrix are linear combinations of the general estimable function.

GLM C

Signif

icance level.

ecify a significance level, the associated level of the confidence intervals

you sp is displayed in the dialog box.

ommand Additional Features

e features may apply to univariate, multivariate, or repeated measures analysis.

Thes The SPSS command language also allows you to:

 Specify nested effects in the design (using the DESIGN subcommand).

Page 44

Chapter 2



Specify tests of effects versus a linear combination of effects or a value (using the

TEST subcom

 Specify mul  Include use  Specify EPS  Construct a

MMATRIX,andKMATRIX subcommands).

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

(using the

factors in t

 Specify na  Construct a  Construct a

ANOVA table (using the

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

mand).

tiple contrasts (using the

r-missing values (using the

criteria (using the

custom L matrix, M matrix, or K matrix (using the

CONTRAST subcommand).

he factor list (using the

mes for temporary variables (using the

correlation matrix data file (using the

CONTRAST subcommand).

MISSING subcommand).

CRITERIA subcommand).

LMATRIX,

EMMEANS subcommand).

SAVE subcommand). OUTFILE subcommand).

matrix data file that contains statistics from the between-subjects

OUTFILE subcommand).

See the SPSS

Command Syntax Reference for complete syntax information.

Page 45

Chapter

Variance Com

The Variance Components procedure, for mixed-effects models, estimates the contribution of each random effect to the variance of the dependent variable. This procedure is particularly interesting for analysis of mixed models such as split plot, univariate repeated measures, and random block designs. By calculating variance 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.

ponents Analysis

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

measured after one month. The litter variable is a random factor 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 attributable to the difference in litters much more than to the difference in pigs within a litter.

Page 46

Chapter 3

Variance Com

Data. The de

numeric values or string values of up to eight characters. At least one of the factors must be random. That is, the levels of the factor must be a random sample of possible levels. Cov

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

means and finite constant variances and are mutually uncorrelated. Model parameters from diffe

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

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 mo

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

ML and REML distributed.

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

variance Multivariate, and GLM Repeated Measures.

ponents Data Considerations

pendent variable is quantitative. Factors are categorical. They can have

ariates are quantitative variables that are related to the dependent variable.

rent random effects are also uncorrelated.

observations are assumed to be uncorrelated.

del.

require the model parameter and the residual term to be normally

components analysis. For hypothesis testing, use GLM Univariate, GLM

Obtaini

E From th

ng Variance Components Tables

e menus choose:

Analyze

General Linear Model

Variance Components...

Page 47

Figure 3-1

Variance Components dialog box

E Select a dependent variable.

Variance Compo

nents Analysis

E 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.

Page 48

Chapter 3

Variance Components Model

Figure 3-2

Variance Com

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

ponents Model dialog box

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-covariate 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 factor,

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

Model. The model depends on the nature of your data. After selecting 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 included in the model. If you can

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

Page 49

Build Terms

For the selected factors and covariates:

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

the default

reates all possible three-way interactions of the selected variables.

Variance Components Options

Figure 3-3

Variance Co

mponents Options dialog box

Variance Compo

nents Analysis

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

 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 e

stimates are correct, this method produces the least variance among all

unbiased estimators. You can choose a method for random-effect prior weights.

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Chapter 3

 ANOVA

IorTypeIII

(analysis of variance) computes unbiased estimates using either the Type

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 likelihood (ML) produces estimates that would be most consistent with

the data act

ually 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 take into account the degrees of freedom used to estimat

 Restrict

e the fixed effects.

ed 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 meth

od takes into account the degrees of freedom used to estimate the

fixed effects.

Random-E

ffect Priors.

Uniform implies that all random effects and the residual term

have an equal impact on the observations. The Zero scheme is equivalent to assuming zero random-effect variances. Available only for the MINQUE method.

ares.

Sum of Squ

Type I sums of squares are used for the hierarchical model, which

is often used in variance component literature. If you choose Type III, the default in GLM, the variance estimates can be used in GLM Univariate for hypothesis testing with Typ

e III sums of squares. Available 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 Maximum likelihood or Restricted maximum

likelihood

Sum of S

For the

, you can display a history of the iterations.

quares (Variance Components)

model, you can choose a type of sum of squares. Type III is the most

commonly used and is the default.

Page 51

Variance Compo

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

nents Analysis

sum-of-squares method. Each term is adjusted for only the term that precedes it in the model. The T

 A balanced

ype I sum-of-squares method is commonly used for:

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 h

 Apurelyne

igher-order terms.

sted 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 t

he sums of squares adjusted for any other effects that do not contain it and orthogonal to any effects (if any) that contain it. The Type III sums of squares have one major advantage in that they are invariant with respect to the cell frequenc

iesaslongasthegeneralformofestimability remains constant. Therefore, this type is often considered useful for an unbalanced model with no missing cells. In a factorial design with no missing cells, this method is equivalent to the Yates' w

eighted-squares-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.

Varianc

e Components Save to a File

Figure 3-4

Variance Components Save to New File dialog box

Page 52

Chapter 3

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 la

bels 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 c

ovariation.

Saves a variance-covariance matrix or a correlation matrix to a data file. Available only if Maximum likelihood or Restricted maximum likelihood has been specified.

Save to Fil

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.

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 Command Syntax Reference for complete syntax information.

Page 53

Chapter

Linear Mixed

The Linear Mixed Models procedure expands the general linear model so that the data are per model, therefore, provides the flexibility of modeling not only the means of the data but their variances and covariances as well.

The Linear that can be formulated as mixed linear models. Such models include multilevel models, h ierarchical linear models, and random coefficient models.

Example. A

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 of different coupons on spending while adjusting for correlation due to repeated observations on each subject over the 10 weeks.

Methods.

estimation.

mitted to exhibit correlated and non-constant variability. The mixed linear

Mixed Models procedure is also a flexible tool for fitting other models

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

to the customers. Linear Mixed Models is used to estimate the effect

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

Models

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

the depe factors. Factor-level information: sorted values of the levels of each factor and their frequencies. Also, parameter estimates and confidence intervals for fixed effects, Wald te and Type III sums of squares can be used to evaluate different hypotheses. Type III is the default.

ndent variable and covariates for each distinct level combination of the

sts and confidence intervals for parameters of covariance matrices. Type I

Page 54

Chapter 4

Linear Mixed

Data. The de

and can have numeric values or string values. Covariates and the weight variable should be quantitative. Subjects and repeated variables may be of any type.

Assumption

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 depend 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 is also assumed to come from a normal distribution.

Related procedures. UsetheExploreproceduretoexaminethedatabeforerunningan

analysis can alternatively use the GLM Univariate or GLM R epeated Measures procedure. You can alternatively use the Variance Components Analysis procedure if the random

have a variance components covariance structure and there are no repeated

effects measures.

Linear M

ixed Models Select Subjects/Repeated Variables

Models Data Considerations

pendent variable should be quantitative. Factors should be categorical

The dependent variable is assumed to be linearly related to the

ent variable. Multiple random effects are considered independent of

model the covariance structure of the residuals. The dependent variable

. If you do not suspect there to be correlated or nonconstant variability, you

This dia observations and to choose a covariance structure for the residuals.

Subjects. A subject is an observational unit that can be considered independent of

other s study can be considered independent of the readings from other patients. Defining subjects becomes particularly important when there are repeated measurements per su example, you might expect that blood pressure readings from a single patient during consecutive visits to the doctor are correlated.

for example, you can specify Gender and Age category as subject variables to model the belief that malesovertheageof65are similar to each other but independent of

male

log box allows you to select variables that define subjects and repeated

ubjects. For example, the blood pressure readings from a patient in a medical

bject and you want to model the correlation between these observations. For

cts can also be defined by the factor-level combination of multiple variables;

Subje

s under 65 and females.

Page 55

Linear Mixed Mo

dels

All of the variables specified in the Subjects list are used to define subjects for the residual covariance structure. You can use some or all of the variables to define subjects fo

r the random-effects covariance structure.

Repeated. The variables specified in 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 to identify daily

observations over the course of a year.

Repeated Covariance type. This specifies the covariance structure for the residuals.

The availa

 Ante-Dep  AR(1)  AR(1): He  ARMA(1,1  Compound S  Compound S  Compound S  Diagonal  Factor Ana  Factor Ana  Huynh-Fel  Scaled Ide  Toeplitz  Toeplitz:  Unstructu  Unstructu

ble structures are as follows:

endence: First Order

terogeneous

)

ymmetry ymmetry: Correlation Metric ymmetry: Heterogeneous

lytic: First Order lytic: First Order, Heterogeneous

ntity

Heterogeneous red red: Correlations

For more information, see “Covariance Structures” in Appendix B on page 125.

Page 56

Chapter 4

Selecting Su Models

E From the menus choose:

Analyze

Mixed Models

Linear...

Figure 4-1

Linear Mixed Mod el s Spe cify S ubj ects/Repeated Variables dialog box

bjects/Repeated Variables for Linear Mixed

E Optionally, select one or more subjects variables. E Optionally, select one or more repeated variables. E Optionally, select a residual covariance structure. E Click Continue.

Page 57

Obtaining a L

Figure 4-2

Linear Mixed Models dialog box

inear Mixed Models Analysis

Linear Mixed Mo

dels

E Select a dependent variable. E Select at least one factor or covariate. E Click Fixed or Random and specify at least a fixed-effects or random-effects model.

Optionally, select a weighting variable.

Page 58

Chapter 4

Linear Mixed Models Fixed Effects

Figure 4-3

Linear Mixed

Fixed Effects. There is no default model, so you must explicitly specify the fixed

Models Fixed Effects dialog box

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

Includ

e 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 sq uares. The method of calculating the sums of squares. For models with no

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

missin

Build Non-Nested Terms

For the selected factors and covariates:

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

s the default.

This i

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

Page 59

Linear Mixed Mo

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. Cr 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

You can build

eates all possible three-way interactions of the selected variables.

Terms

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. For example, a grocery store chain may follow the spending of

their customers at several store locations. Since each customer frequents

only one of those locations, the Customer effect can be said to be nested within the Store location effect.

Additiona

lly, you can include interaction effects or add multiple levels of nesting

to the nested term.

Limitations. Nested terms have the following restrictions:

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

specifying A*A is invalid.

 Allfactorswithinanestedeffectmustbeunique.Thus,ifA is a factor, then

specifyin

 No effect

g A(A) is invalid.

canbenestedwithinacovariate.Thus,ifA is a factor and X is a

covariate, then specifying A(X) is invalid.

dels

Sum of Squ

ares

For the mo

del, you can choose a type of sums of squares. Type III is the most

commonly used and is the default.

Page 60

Chapter 4

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 precedes it in the model. Type

 A balanced

I sums of squares are commonly used for:

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

before any h

 Apurelyne

igher-order terms.

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 orthogona

l to any effects (if any) that contain it. The Type 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. Hence, this type of sums of sq

uares is o ften considered useful for an unbalanced model with no missing cells. In a factorial design with no missing cells, this method is equivalent to the Yates' weighted-squares-of-means technique. The Type III sum-of-squares method is commo

 Any mod  Any bal

nlyusedfor:

els listed in Type I.

anced or unbalanced model with no empty cells.

Page 61

Linear Mixed Models Random Effects

Figure 4-4

Linear Mixed

Models Random Effects dialog box

Linear Mixed Mo

dels

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 term in the random-effects model.

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

Next to build the next model. Click Previous to scroll back through existing

click models. Each random-effect model is assumed to be independent of every other random-effect model; that is, separate covariance matrices will be computed for each. Terms specified in the same random-effect model can be correlated.

Page 62

Chapter 4

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

 Ante-Depe  AR(1)  AR(1): Hete  ARMA(1,1)  Compound Sy  Compound Sy  Compound Sy  Diagonal  Factor Anal  Factor Anal  Huynh-Feld  Scaled Iden  Toeplitz  Toeplitz: H  Unstructur  Unstructur  Variance Com

available structures are as follows:

ndence: First Order

rogeneous

mmetry mmetry: Correlation Metric mmetry: Heterogeneous

ytic: First Order ytic: First Order, Heterogeneous

tity

eterogeneous ed ed: Correlation Metric

ponents

For more information, see “Covariance Structures” in Appendix B on page 125.

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

in the Select Subjects/Repeated Variables dialog box. Choose some or all of these in order to defi

ne the subjects for the random-effects model.

Page 63

Linear Mixed Models Estimation

Figure 4-5

Linear Mixed

Models Estimation dialog box

Linear Mixed Mo

dels

Method. Select the maximum likelihood or restricted maximum likelihood estimation.

Iterations:

 Maximum iterations. Specify a non-negative integer.

Page 64

Chapter 4



Maximum step-halvin gs. At each iteration, the step size is reduced 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 w

ith 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.

Log-likelihood Convergence. Convergenceis assumed if the absolute change or relative

change in t

he 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 re

lative change 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 Converg enc e. For the Absolute specification, convergence is assumed if

astatist

ic 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 log-likelihood. The criterion is not used if the va

lue specified equals 0.

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

number n. Specify a positive integer.

Singula

rity tolerance.

This is the value used as tolerance in checking singularity.

Specify a positive value.

Page 65

Linear Mixed Models Statistics

Figure 4-6

Linear Mixed

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 displa

 Case P

measure variables, the repeated measure subjects, and the random-effects subjects and their frequencies.

Models Statistics dialog box

yed for each distinct level combination of the factors.

rocessing Summary.

Linear Mixed Mo

dels

Displays the sorted values of the factors, the repeated

Model Sta tistics. Produces tables for:

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

tes and their approximate standard errors.

estima

 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.

Page 66

Chapter 4



Covariances of pa ram eter estimates. Displays the asymptotic covariance matrix of

the fixed-e

 Covariance

ffects parameter estimates.

s 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 bloc

 Covarianc

k is displayed.

es of residuals.

Displays the estimated residual covariance matrix. This option i s 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 th

e fixed effects and the custom hypotheses.

Confidenc

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

Linear Mix

Figure 4-7

Linear Mixed Models EM Means dialog box

e interval.

This value is used whenever a confidence interval is constructed.

ed Models EM Means

Page 67

Linear Mixed Mo

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

dels

model-predicted estimated marginal means of the dependent variable in the cells and their st

andard errors for the specified factors. Moreover, you can request that

factor levels of main effects be compared.

 Factor(s) and Factor Interactions. This list contains factors and factor interactions

that have be

enspecifiedintheFixeddialogbox,plusan

OVERALL term. Model

terms built from covariates are excluded from this list.

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

factors and factor interactions selected to this list. If estimated

marginal means of the dependent variable are displayed, collapsing

OVERALL is selected, the

over 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 d

 Compare

ialog box.

main effects.

This option allows you to request pairwise comparisons of levels of selected main effects. The Confidence Interval Adjustment allows you to apply an adjustment to the confidence intervals and significance values to accoun

t 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 selec

ted, all pairwise comparisons will be constructed. The options for the reference category are 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

Page 68

Chapter 4

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.

MIXED Command Additional Features

The SPSS command language also allows you to:

 Specify tests of effects versus a linear combination of effects or a value (using the

TEST subcom

 Include us  Compute es

WITH keyword of the EMMEANS subcommand).

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

mand).

er-missing values (using the

timated marginal means for specified values of covariates (using the

MISSING subcommand).

See the SPSS

Command Syntax Reference for complete syntax information.

Page 69

Chapter

Model Select

The Model Selection Loglinear Analysis procedure analyzes multiway crosstabul multidimensional crosstabulations using an iterative proportional-fitting algorithm. This procedure helps you find out which categorical variables are associated. To build mode saturated models, you can request parameter estimates and tests of partial association. A saturated model adds 0.5 to all cells.

Example.

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 h brand preference.

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

interva and normal probability plots.

ations (contingency tables). It fits hierarchical loglinear models to

ls, forced entry and backward elimination methods are available. For

In a study of user preference for one of two laundry detergents, researchers

ot). They found how temperature is related to water softness and also to

ls, and tests of partial association. For custom models, plots of residuals

ion Loglinear Analysis

Model Se

Data. Fa

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

a situation where many cells have small numbers of observations, and the chi-square values may not be useful.

lection Loglinear Analysis Data Considerations

ctor variables are categorical. All variables to be analyzed must be numeric.

Avoid s

pecifying many variables with many levels. Such specifications can lead to

Page 70

Chapter 5

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 Loglinear A

nalysis 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.

Obtaining a Model Selection Loglinear Analysis

From the menus choose:

Analyze

Loglinear

Model Selection...

Figure 5-1

Model Selection Loglinear Analysis dialog box

E Select two or more numeric categorical factors. E Select one or more factor variables in the Factor(s) list, and click Define Range. E Define the range of values for each factor variable. E Select an option in the Model Building group.

Page 71

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

Loglinear A

nalysis Define Range

Figure 5-2

Loglinear Analysis Define Range dialog box

You 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, and3areu

sed. Repeat this process for each factor variable.

Loglinear Analysis Model

Model Selectio

n Loglinear Analysis

Figure 5-3

Loglinear Analysis Model dialog box

Page 72

Chapter 5

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

factor-by-factor interactions. Select Custom to specify a generating class for an unsaturate

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

dmodel.

factors appear. SPSS builds a hierarchical model containing the terms that define the generating C in the Factors list and then select

class and all lower-order relatives. Suppose you select variables A, B,and

Interaction from the Build Terms drop-down list.

The resulting model will contain the specified 3-way interaction A*B*C, the 2-way interacti

ons A*B, A*C,andB*C, and main effects for A, B,andC.Donotspecifythe

lower-order relatives in the generating class.

Build Ter

For the se

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

the default.

Main effe

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

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

lected factors and covariates:

cts.

Creates a main-effects term for each variable selected.

Creates all possible four-way interactions of the selected variables.

Page 73

Model Selection Loglinear Analysis Options

Figure 5-4

Loglinear An

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

from the model. An association table, which lists tests of partial association, is also available. This option is computationally expensive for tables with many factors.

alysis Options dialog box

. The parameter estimates may help determine which terms can be dropped

Model Selectio

n Loglinear Analysis

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

Normal probability. These will help determine how well a model 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,orDelta (a value added to all cell

frequencies for saturated models).

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).

Page 74

Chapter 5

See the SPSS Command Syntax Reference for complete syntax information.

Page 75

Chapter

General Logl

The General Loglinear Analysis procedure analyzes the frequency counts of observatio 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 (fre factors and covariates. This procedure estimates maximum likelihood parameters of hierarchical and nonhierarchical loglinear models using the Newton-Raphson method. Either a P

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 tables. Contrast variables allow computation of generalized log-odds ratios (GLOR).

SPSS automatically displays model information and goodness-of-fit statistics. You can a values in the working data file.

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

the rela nonfatal. The odds ratio indicates significant evidence of a relationship.

ns falling into each cross-classification category in a crosstabulation or

quency) in a cell of the crosstabulation, and the explanatory variables are

oisson or a multinomial distribution can be analyzed.

, fit a log-rate model, or implement the method of adjustment of marginal

lso display a variety of statistics and plots or save residuals and predicted

tionship between wearing a seat belt and whether an injury was fatal or

inear Analysis

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

design m and confidence intervals. Plots: adjusted residuals, deviance residuals, and normal probability.

atrix; parameter estimates; odds ratio; log-odds ratio; GLOR; Wald statistic;

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Chapter 6

General Logl

Data. Facto

is in the model, SPSS applies the m ean covariate value for cases in a cell to that cell. Contrast variables are continuous. They are used to compute generalized log-odds ra combination of the logs of the expected cell counts.

A cell structure variable assigns weights. For example, if some of the cells are structura a cell structure variable to weight aggregated data. Instead, choose fromtheDatamenu.

Assumptio

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

Under the multinomial distribution assumption:

 The total sample size is fixed, or the analysis is conditional on the total sample

size.

 The cell

inear Analysis Data Considerations

rs are categorical, and cell covariates are continuous. When a covariate

tios. The values of the contrast variable are the coefficients for the linear

l zeros, the cell structure variable has a value of either 0 or 1. Do not use

Weight Cases

ns.

Two distributions are available in General Loglinear Analysis: Poisson

other cells.

counts are not statistically independent.

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

the Logit Loglinear procedure when it is natural to regard one or more categorical variable

sastheresponsevariablesandtheothersastheexplanatoryvariables.

Obtaining a General Loglinear Analysis

E From the menus choose:

Analyze

Logline

General.

Page 77

General Loglin

Figure 6-1

General Loglinear Analysis dialog box

E In the General Loglinear Analysis dialog box, select up to 10 factor variables.

ear Analysis

Optionally, you can:

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

Page 78

Chapter 6

General Loglinear Analysis Model

Figure 6-2

General Logl

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

inear Analysis Model dialog box

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

acovariate.

Terms in Mod el. The model depends on the nature of your data. After selecting

Custom, you can select the main effects and interactions that are of interest in 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.

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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. Cr

eates all possible five-way interactions of the selected variables.

General Loglinear Analysis Options

Figure 6-3

General Loglinear Analysis Options dialog box

General Loglin

ear Analysis

The General Loglinear Analysis procedure displays model information and goodness-of-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 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.

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Chapter 6

Criteria. TheNewton-Raphsonmethodisusedtoobtainmaximumlikelihood

parameter estimates. You can enter new values for the maximum number of iterations

, t he convergence criterion, and delta (a constant added to all cells for initial

approximations). Delta remains in the cells for saturated models.

General Log

Figure 6-4

General Loglinear Analysis Save dialog box

Select the values you want to save as new variables in the working data file. The suffix n in the new variable names increments to make a unique name for each saved variable.

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.

The predicted values can also be saved.

 Residuals. Also called the simple or raw residual, it is the difference between the

 Standardized Residuals. The residual divided by an estimate of its standard error.

linear Analysis Save

The saved values refer to the aggregated data (cells in the contingency table), even

Four types of residuals can be saved: raw, standardized, adjusted, and deviance.

observed cell count and its expected count.

Standardized residuals are also known as Pearson residuals.

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

 Deviance Residuals. The signed square root of an individual contribution to the

GENLOG Com

The SPSS command language also allows you to:

 Calculate linear combinations of observed cell frequencies and expected cell

 Change the default threshold value for redundancy checking (using the CRITERIA

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

See the SPSS Command Syntax Reference for complete syntax information.

General Loglin

Adjusted Residuals. The standardized residual divided by its e stimated standard

error. Sinc

e the adjusted residuals are asymptotically standard normal when the

ear Analysis

selected model is correct, they are preferred over the standardized residuals for checking for normality.

likelihood

-ratio chi-square statistic (G squared), where the sign is the sign of the residual (observed count minus expected count). Deviance residuals have an asymptotic standard normal distribution.

mand Additional Features

frequenci that combination (using the

subcommand).

es, and print residuals, standardized residuals, and adjusted residuals of

GERESID subcommand).

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Page 83

Chapter

Logit Loglin

The Logit Loglinear Analysis procedure analyzes the relationship between dependent (or respons variables are always categorical, while t he independent variables can be categorical (factors). Other independent variables, cell covariates, can be continuous, but they are not app 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; procedure estimates parameters of logit loglinear models using the Newton-Raphson algorithm.

You can se 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 adjustm 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 aut You can also display a variety of statistics and plots or save residuals and predicted values in the working data file.

e) variables and independent (or explanatory) variables. The dependent

lied on a case-by-case basis. The weighted covariate mean for a cell is

these models are sometimes called multinomial logit models. This

lect from 1 to 10 dependent and factor variables combined. A cell

ent of marginal tables. Contrast variables allow computation of generalized

omatically displays model information and goodness-of-fit statistics.

ear Analysis

Exampl

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 highest in lake 3.

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

alligators; also, the odds of selecting primarily reptiles instead of fish were

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Chapter 7

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

design matrix; parameter estimates; generalized log-odds ratio; Wald statistic; and confidence probability plots.

intervals. Plots: adjusted residuals, deviance residuals, and normal

Logit Logli

Data. The d

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 variables are continuous. They are used to variable are the coefficients for the linear combination of the logs of the expected cell counts.

structural 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 distrib

 The tot

 The cell counts are not statistically independent.

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

near Analysis Data Considerations

ependent variables are categorical. Factors are categorical. Cell covariates

compute generalized log-odds ratios (GLOR). The values of the contrast

A cell str

size.

ucture variable assigns weights. For example, if some of the cells are

ution assumption:

al sample size is fixed, or the analysis is conditional on the total sample

procedures.

Use the Crosstabs procedure to display the contingency tables.

Obtaining a Logit Loglinear A nalysis

E From the menus choose:

Analyze

Loglinear

Logit...

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Figure 7-1

Logit Loglinear Analysis dialog box

E In the L E Select

ogit Loglinear Analysis dialog box, select one or more dependent variables.

one or more factor variables.

Logit Loglinea

rAnalysis

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 o r more contrast variables.

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Chapter 7

Logit Loglinear Analysis Model

Figure 7-2

Logit Loglin

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

ear Analysis Model dialog box

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

acovariate.

Terms in Mod el. The model depends on the nature of your data. After selecting

Custom, you can select the main effects and interactions that are of interest in your

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

Terms are added to the design by taking all possible combinations of the 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 included, the model list contains 1, M1,andM2. The resultant design includes combinations of each model term with each dependent term:

D1, D2, D1*D2

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Logit Loglinea

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

acustommod

el.

Build Terms

For the selected factors and covariates:

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

the default

reates all possible three-way interactions of the selected variables.

rAnalysis

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Chapter 7

Logit Loglinear Analysis Options

Figure 7-3

Logit Loglin

ear Analysis Options dialog box

The Logit Loglinear Analysis procedure displays model information and goodness-of-fit 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 estimates for 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. TheNewton-Raphsonmethodisusedtoobtainmaximumlikelihood

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 initial approximations). Delta remains in the cells for saturated models.

Page 89

Logit Loglinear Analysis Save

Figure 7-4

Logit Loglin

Select the values you want to save as new variables in the working data file. The suffix n in the new variable names increments to make a unique name for 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 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, standardized, adjusted, and deviance. The predicted values can also be saved.

 Residuals. Also called the simple or raw residual, it is the difference between the

 Standardized Residuals. The residual divided by an estimate of its standard error.

 Adjusted Residuals. The standardized residual divided by its estimated standard

 Deviance Residuals. The signed square root of an individual contribution to the

ear Analysis Save dialog box

observed cell count and its expected count.

Standardized residuals are also known as Pearson residuals.

error. Since the adjusted residuals are asymptotically standard normal when the selected model is correct, they are preferred over the standardized residuals for checking for normality.

likelihood-ratio chi-square statistic (G squared), where the sign is the sign of the residual (observed count minus expected count). Deviance residuals have an asymptotic standard normal distribution.

Logit Loglinea

rAnalysis

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Chapter 7

GENLOG Command Additional Features

The SPSS command language also allows you to:

 Calculate linear combinations of observed cell frequencies and expected cell

frequencies that combination (using the

 Change the default threshold value for redundancy checking (using the CRITERIA

subcommand).

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

See the SPSS Command Syntax Reference for complete syntax information.

, and print residuals, standardized residuals, and adjusted residuals of

GERESID subcommand).

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Chapter

Ordinal Regr

Ordinal Regression allows you to model the dependence of a polytomous ordinal response on Ordinal Regression is based on the methodology of McCullagh (1980, 1998), and the procedure is referred to as

Standard l differences between a response (dependent) variable and a weighted combination of predictor (independent) variables. The estimated coefficients reflect how changes in the predic sense that changes in the level of the response are equivalent throughout the range of the response. For example, the difference in height between a person who is 150 cm tall and a 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 a

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 diff quantify and is based on perception. Moreover, the difference between a mild and moderateresponsemaybegreaterorlessthanthedifferencebetweenamoderateand severe

a set of predictors, which can be factors or covariates. The design of

inear regression analysis involves minimizing the sum-of-squared

tors affect the response. The response is assumed to be numerical, in the

person who is 140 cm tall is 10 cm, which has the same meaning as the

nd number of response categories can be quite arbitrary.

erence between a mild and moderate reaction is difficult or impossible to

response.

ession

PLUM in the syntax.

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

Pearson residuals for frequencies and cumulative frequencies, observed and expected

bilities, observed and expected cumulative probabilities of each response

proba category by covariate pattern, asymptotic correlation and covariance matrices of parameter estimates, Pearson's chi-square and likelihood-ratio chi-square,

ess-of-fit statistics, iteration history, test of parallel lines assumption, parameter

goodn

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Chapter 8

estimates, standard errors, confidence intervals, and Cox and S nell's, Nagelkerke's, and McFadden's R2statistics.

Ordinal Regression Data Considerations

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 ascendi assumed to be categorical. Covariate variables must be numeric. Note that using more than one continuous covariate can easily result in the creation of a very large cell proba

Assumptions. Only one response variable is allowed, and it must be specified. Also,

for each distinct pattern of values across the independent variables, the responses are assum

Related procedures. Nominal logistic regression uses similar models for nominal

dependent variables.

ng order. The lowest value defines the first category. Factor variables are

bilities table.

ed to be independent multinomial variables.

Obtaining an Ordinal Regression

E From the menus choose:

Analyze

Regression

Ordinal...

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Figure 8-1

Ordinal Regression dialog box

E Select one dependent variable. E Click OK.

Ordinal Regres

sion

Ordinal Regression Options

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

Page 94

Chapter 8

Figure 8-2

Ordinal Regression Options dialog box

Iterations. You can customize the iterative algorithm.

 Maximum iterations. Specify a non-negative integer. If 0 is specified, the

procedure returns the i nitial estimates.

 Maximum step-halving. Specify a positive integer.  Log-likelihood convergence. The algorithm stops if the absolute or relative 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.

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 highly dependent predictors. Select a

value from the list of options.

Page 95

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

and Probit functions.

Ordinal Regression Output

The Output dialog box allows you to produce tables for display in the Viewer and save variables to the working file.

Figure 8-3

Ordinal Regression Output dialog box

Ordinal Regres

sion

Display. Produces tables for:

 Print ite ration history. The log-likelihood and parameter estimates are printed for

nt iteration frequency specified. The first and last iterations are always

the pri 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.

 Summary statistics. Cox and Snell's, Nagelkerke's, and McFadden's R  Parameter estimates. Parameter estimates, standard errors, and confidence

vals.

inter

statistics.

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Chapter 8



Asymptotic correla tion of parameter estimates. Matrix of parameter estimate

correlatio

 Asymptotic

ns.

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 pr

obabilities, and observed and expected cumulative probabilities of each response category by covariate pattern. Note that for models with many covariate patterns (for example, models with continuous covariates), this option can genera

 Test of pa

te a very large, unwieldy table.

rallel lines.

Test of the hypothesis that the location parameters are equivalent across the levels of the dependent variable. This is available 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/c

ovariate pattern into the predicted category. This probability is also the

maximum of the estimated probabilities of the factor/covariate pattern.

 Actual categ ory probability. Estimated probability of classifying a factor/covariate

pattern into the actual category.

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

constant

product

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

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

Ordinal Regression Location Model

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

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sion

Figure 8-4

Ordinal Regression Location dialog b ox

Specify

model.

A main-effects model contains the covariate and factor main effects

Ordinal Regres

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

Factor

s/covariates.

The factors and covariates are listed with (F) forfactorand(C)

for covariate.

Location model. The model depends on the main effects and interaction effects that

you sel

ect.

Build Terms

For the selected factors and covariates:

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

fects.

Main ef

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-w

Creates a main-effects term for each variable selected.

ay.

Creates all possible four-way interactions of the selected variables.

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Chapter 8

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

Ordinal Reg

The S cale di

Figure 8-5

Ordinal Regression Scale dialog box

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

for covariate.

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

Build Terms

ression Scale Model

alog box a llows you to specify the scale model for your analysis.

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.

Page 99

PLUM Command Additional Features

You can customize your Ordinal Regression if you paste your selections into a syntax window and edit the resulting PLUM command syntax. The SPSS command language al

 Create cus

See the SPSS

so allows you to:

tomized hypothesis tests by specifying null hypotheses as linear

combinations of parameters.

Command Syntax Reference for complete syntax information.

Ordinal Regres

sion

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