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Page 1
SEVENTH EDITION
Using Multivariate
Statistics
Barbara G. Tabachnick
California State University, Northridge
Linda S. Fidell
California State University, Northridge
330 Hudson Street, NY NY 10013
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Library of Congress Cataloging-in-Publication Data
Names: Tabachnick, Barbara G., author. | Fidell, Linda S., author.
Title: Using multivariate statistics/Barbara G. Tabachnick, California State University, Northridge,
Linda S. Fidell, California State University, Northridge.
Description: Seventh edition. | Boston: Pearson, [2019] | Chapter 14,
by Jodie B. Ullman.
Identifiers: LCCN 2017040173| ISBN 9780134790541 | ISBN 0134790545
Subjects: LCSH: Multivariate analysis. | Statistics.
Classification: LCC QA278 .T3 2019 | DDC 519.5/35—dc23
LC record available at https://lccn.loc.gov/2017040173
1 18
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Contents
Preface
xiv
1 Introduction
1
1.1 Multivariate Statistics: Why?
1
1.1.1 The Domain of Multivariate Statistics:
Numbers of IVs and DVs
2
1.1.2 Experimental and Nonexperimental
Research
2
1.1.3 Computers and Multivariate Statistics
3
1.1.4 Garbage In, Roses Out?
4
1.2 Some Useful Definitions
5
1.2.1 Continuous, Discrete, and Dichotomous
Data
5
1.2.2 Samples and Populations
6
1.2.3 Descriptive and Inferential Statistics
7
1.2.4 Orthogonality: Standard and Sequential
Analyses
7
1.3 Linear Combinations of Variables
9
1.4 Number and Nature of Variables to�Include
10
1.5 Statistical Power
10
1.6 Data Appropriate for Multivariate Statistics
11
1.6.1 The Data Matrix
11
1.6.2 The Correlation Matrix
12
1.6.3 The Variance–Covariance Matrix
12
1.6.4 The Sum-of-Squares and Cross-Products
Matrix
13
1.6.5 Residuals
14
1.7 Organization of the Book
14
2 A Guide to Statistical Techniques:
Using the Book
15
2.1 Research Questions and Associated Techniques 15
2.1.1 Degree of Relationship Among Variables 15
2.1.1.1 Bivariate r
16
2.1.1.2 Multiple R
16
2.1.1.3 Sequential R
16
2.1.1.4 Canonical R
16
2.1.1.5 Multiway Frequency Analysis
17
2.1.1.6 Multilevel Modeling
17
2.1.2 Significance of Group Differences
17
2.1.2.1 One-Way ANOVA and t Test
17
2.1.2.2 One-Way ANCOVA
17
2.1.2.3 Factorial ANOVA
18
2.1.2.4 Factorial ANCOVA
18
2.1.2.5 Hotelling’s T2
18
2.1.2.6 One-Way MANOVA
18
2.1.2.7 One-Way MANCOVA
19
2.1.2.8 Factorial MANOVA
19
2.1.2.9 Factorial MANCOVA
19
2.1.2.10 Profile Analysis of Repeated Measures 19
2.1.3 Prediction of Group Membership
20
2.1.3.1 One-Way Discriminant Analysis
20
2.1.3.2 Sequential One-Way Discriminant
Analysis
20
2.1.3.3 Multiway Frequency Analysis
(Logit)
21
2.1.3.4 Logistic Regression
21
2.1.3.5 Sequential Logistic Regression
21
2.1.3.6 Factorial Discriminant Analysis
21
2.1.3.7 Sequential Factorial Discriminant
Analysis
22
2.1.4 Structure
22
2.1.4.1 Principal Components
22
2.1.4.2 Factor Analysis
22
2.1.4.3 Structural Equation Modeling
22
2.1.5 Time Course of Events
22
2.1.5.1 Survival/Failure Analysis
23
2.1.5.2 Time-Series Analysis
23
2.2 Some Further Comparisons
23
2.3 A Decision Tree
24
2.4 Technique Chapters
27
2.5 Preliminary Check of the Data
28
3 Review of Univariate and
Bivariate Statistics
29
3.1 Hypothesis Testing
29
3.1.1 One-Sample z Test as Prototype
30
3.1.2 Power
32
3.1.3 Extensions of the Model
32
3.1.4 Controversy Surrounding Significance
Testing
33
3.2 Analysis of Variance
33
3.2.1 One-Way Between-Subjects ANOVA
34
3.2.2 Factorial Between-Subjects ANOVA
36
3.2.3 Within-Subjects ANOVA
38
3.2.4 Mixed Between-Within-Subjects ANOVA 40
3.2.5 Design Complexity
41
3.2.5.1 Nesting
41
3.2.5.2 Latin-Square Designs
42
3.2.5.3 Unequal n and Nonorthogonality
42
3.2.5.4 Fixed and Random Effects
43
3.2.6 Specific Comparisons
43
3.2.6.1 Weighting Coefficients for
Comparisons
43
3.2.6.2 Orthogonality of Weighting
Coefficients
44
3.2.6.3 Obtained F for Comparisons
44
3.2.6.4 Critical F for Planned Comparisons
45
3.2.6.5 Critical F for Post Hoc Comparisons
45
3.3 Parameter Estimation
46
3.4 Effect Size
47
iii
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iv Contents
3.5 Bivariate Statistics: Correlation and Regression 48
3.5.1 Correlation
48
3.5.2 Regression
49
3.6 Chi-Square Analysis
50
4 Cleaning Up Your Act: Screening
Data Prior to Analysis
52
4.1 Important Issues in Data Screening
53
4.1.1 Accuracy of Data File
53
4.1.2 Honest Correlations
53
4.1.2.1 Inflated Correlation
53
4.1.2.2 Deflated Correlation
53
4.1.3 Missing Data
54
4.1.3.1 Deleting Cases or Variables
57
4.1.3.2 Estimating Missing Data
57
4.1.3.3 Using a Missing Data Correlation
Matrix
61
4.1.3.4 Treating Missing Data as Data
61
4.1.3.5 Repeating Analyses with and without
Missing Data
61
4.1.3.6 Choosing Among Methods for
Dealing with Missing Data
62
4.1.4 Outliers
62
4.1.4.1 Detecting Univariate and
Multivariate Outliers
63
4.1.4.2 Describing Outliers
66
4.1.4.3 Reducing the Influence
of Outliers
66
4.1.4.4 Outliers in a Solution
67
4.1.5 Normality, Linearity, and
Homoscedasticity
67
4.1.5.1 Normality
68
4.1.5.2 Linearity
72
4.1.5.3 Homoscedasticity, Homogeneity
of Variance, and Homogeneity of
Variance–Covariance Matrices
73
4.1.6 Common Data Transformations
75
4.1.7 Multicollinearity and Singularity
76
4.1.8 A Checklist and Some Practical
Recommendations
79
4.2 Complete Examples of Data Screening
79
4.2.1 Screening Ungrouped Data
80
4.2.1.1 Accuracy of Input, Missing Data,
Distributions, and Univariate Outliers 81
4.2.1.2 Linearity and Homoscedasticity
84
4.2.1.3 Transformation
84
4.2.1.4 Detecting Multivariate Outliers
84
4.2.1.5 Variables Causing Cases to Be Outliers 86
4.2.1.6 Multicollinearity
88
4.2.2 Screening Grouped Data
88
4.2.2.1 Accuracy of Input, Missing Data,
Distributions, Homogeneity of Variance,
and Univariate Outliers
89
4.2.2.2 Linearity
93
4.2.2.3 Multivariate Outliers
93
4.2.2.4 Variables Causing Cases to Be Outliers 94
4.2.2.5 Multicollinearity
97
5 Multiple Regression
99
5.1 General Purpose and Description
99
5.2 Kinds of Research Questions
101
5.2.1 Degree of Relationship
101
5.2.2 Importance of IVs
102
5.2.3 Adding IVs
102
5.2.4 Changing IVs
102
5.2.5 Contingencies Among IVs
102
5.2.6 Comparing Sets of IVs
102
5.2.7 Predicting DV Scores
for Members of a New Sample
103
5.2.8 Parameter Estimates
103
5.3 Limitations to Regression Analyses
103
5.3.1 Theoretical Issues
103
5.3.2 Practical Issues
104
5.3.2.1 Ratio of Cases to IVs
105
5.3.2.2 Absence of Outliers Among
the IVs and on the DV
105
5.3.2.3 Absence of Multicollinearity and
Singularity
106
5.3.2.4 Normality, Linearity, and
Homoscedasticity of Residuals
106
5.3.2.5 Independence of Errors
108
5.3.2.6 Absence of Outliers in the Solution
109
5.4 Fundamental Equations for�Multiple�
Regression
109
5.4.1 General Linear Equations
110
5.4.2 Matrix Equations
111
5.4.3 Computer Analyses of Small-Sample
Example
113
5.5 Major Types of Multiple Regression
115
5.5.1 Standard Multiple Regression
115
5.5.2 Sequential Multiple Regression
116
5.5.3 Statistical (Stepwise) Regression
117
5.5.4 Choosing Among Regression
Strategies
121
5.6 Some Important Issues
121
5.6.1 Importance of IVs
121
5.6.1.1 Standard Multiple Regression
122
5.6.1.2 Sequential or Statistical Regression
123
5.6.1.3 Commonality Analysis
123
5.6.1.4 Relative Importance Analysis
125
5.6.2 Statistical Inference
128
5.6.2.1 Test for Multiple R
128
5.6.2.2 Test of Regression Components
129
5.6.2.3 Test of Added Subset of IVs
130
5.6.2.4 Confidence Limits
130
5.6.2.5 Comparing Two Sets of Predictors
131
5.6.3 Adjustment of R2
132
5.6.4 Suppressor Variables
133
5.6.5 Regression Approach to ANOVA
134
5.6.6 Centering When Interactions
and Powers of IVs Are Included
135
5.6.7 Mediation in Causal Sequence
137
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Contents v
5.7 Complete Examples of Regression Analysis
138
5.7.1 Evaluation of Assumptions
139
5.7.1.1 Ratio of Cases to IVs
139
5.7.1.2 Normality, Linearity,
Homoscedasticity, and
Independence of Residuals
139
5.7.1.3 Outliers
142
5.7.1.4 Multicollinearity and Singularity
144
5.7.2 Standard Multiple Regression
144
5.7.3 Sequential Regression
150
5.7.4 Example of Standard Multiple
Regression with Missing Values
Multiply Imputed
154
5.8 Comparison of Programs
162
5.8.1 IBM SPSS Package
163
5.8.2 SAS System
165
5.8.3 SYSTAT System
166
6 Analysis of Covariance
167
6.1 General Purpose and Description
167
6.2 Kinds of Research Questions
170
6.2.1 Main Effects of IVs
170
6.2.2 Interactions Among IVs
170
6.2.3 Specific Comparisons and Trend
Analysis
170
6.2.4 Effects of Covariates
170
6.2.5 Effect Size
171
6.2.6 Parameter Estimates
171
6.3 Limitations to Analysis of Covariance
171
6.3.1 Theoretical Issues
171
6.3.2 Practical Issues
172
6.3.2.1 Unequal Sample Sizes, Missing
Data, and Ratio of Cases to IVs
172
6.3.2.2 Absence of Outliers
172
6.3.2.3 Absence of Multicollinearity
and Singularity
172
6.3.2.4 Normality of Sampling Distributions 173
6.3.2.5 Homogeneity of Variance
173
6.3.2.6 Linearity
173
6.3.2.7 Homogeneity of Regression
173
6.3.2.8 Reliability of Covariates
174
6.4 Fundamental Equations for Analysis
of�Covariance
174
6.4.1 Sums of Squares and Cross-Products
175
6.4.2 Significance Test and Effect Size
177
6.4.3 Computer Analyses of Small-Sample
Example
178
6.5 Some Important Issues
179
6.5.1 Choosing Covariates
179
6.5.2 Evaluation of Covariates
180
6.5.3 Test for Homogeneity of Regression
180
6.5.4 Design Complexity
181
6.5.4.1 Within-Subjects and Mixed
Within-Between Designs
181
6.5.4.2 Unequal Sample Sizes
182
6.5.4.3 Specific Comparisons and
Trend Analysis
185
6.5.4.4 Effect Size
187
6.5.5 Alternatives to ANCOVA
187
6.6 Complete Example of Analysis of�Covariance 189
6.6.1 Evaluation of Assumptions
189
6.6.1.1 Unequal n and Missing Data
189
6.6.1.2 Normality
191
6.6.1.3 Linearity
191
6.6.1.4 Outliers
191
6.6.1.5 Multicollinearity and Singularity
192
6.6.1.6 Homogeneity of Variance
192
6.6.1.7 Homogeneity of Regression
193
6.6.1.8 Reliability of Covariates
193
6.6.2 Analysis of Covariance
193
6.6.2.1 Main Analysis
193
6.6.2.2 Evaluation of Covariates
196
6.6.2.3 Homogeneity of Regression Run
197
6.7 Comparison of Programs
200
6.7.1 IBM SPSS Package
200
6.7.2 SAS System
200
6.7.3 SYSTAT System
200
7 Multivariate Analysis of
Variance and Covariance
203
7.1 General Purpose and Description
203
7.2 Kinds of Research Questions
206
7.2.1 Main Effects of IVs
206
7.2.2 Interactions Among IVs
207
7.2.3 Importance of DVs
207
7.2.4 Parameter Estimates
207
7.2.5 Specific Comparisons
and Trend Analysis
207
7.2.6 Effect Size
208
7.2.7 Effects of Covariates
208
7.2.8 Repeated-Measures Analysis
of Variance
208
7.3 Limitations to Multivariate Analysis
of Variance and Covariance
208
7.3.1 Theoretical Issues
208
7.3.2 Practical Issues
209
7.3.2.1 Unequal Sample Sizes,
Missing Data, and Power
209
7.3.2.2 Multivariate Normality
210
7.3.2.3 Absence of Outliers
210
7.3.2.4 Homogeneity of Variance–
Covariance Matrices
210
7.3.2.5 Linearity
211
7.3.2.6 Homogeneity of Regression
211
7.3.2.7 Reliability of Covariates
211
7.3.2.8 Absence of Multicollinearity
and Singularity
211
7.4 Fundamental Equations for Multivariate
Analysis of Variance and Covariance
212
7.4.1 Multivariate Analysis of Variance
212
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vi Contents
7.4.2 Computer Analyses
of Small-Sample Example
218
7.4.3 Multivariate Analysis
of Covariance
221
7.5 Some Important Issues
223
7.5.1 MANOVA Versus ANOVAs
223
7.5.2 Criteria for Statistical Inference
223
7.5.3 Assessing DVs
224
7.5.3.1 Univariate F
224
7.5.3.2 Roy–Bargmann Stepdown Analysis
226
7.5.3.3 Using Discriminant Analysis
226
7.5.3.4 Choosing Among Strategies
for Assessing DVs
227
7.5.4 Specific Comparisons and Trend
Analysis
227
7.5.5 Design Complexity
228
7.5.5.1 Within-Subjects and Between-
Within Designs
228
7.5.5.2 Unequal Sample Sizes
228
7.6 Complete Examples of Multivariate
Analysis of Variance and Covariance
230
7.6.1 Evaluation of Assumptions
230
7.6.1.1 Unequal Sample Sizes
and Missing Data
230
7.6.1.2 Multivariate Normality
231
7.6.1.3 Linearity
231
7.6.1.4 Outliers
232
7.6.1.5 Homogeneity of Variance–
Covariance Matrices
233
7.6.1.6 Homogeneity of Regression
233
7.6.1.7 Reliability of Covariates
235
7.6.1.8 Multicollinearity and Singularity
235
7.6.2 Multivariate Analysis of Variance
235
7.6.3 Multivariate Analysis of Covariance
244
7.6.3.1 Assessing Covariates
244
7.6.3.2 Assessing DVs
245
7.7 Comparison of Programs
252
7.7.1 IBM SPSS Package
252
7.7.2 SAS System
254
7.7.3 SYSTAT System
255
8 Profile Analysis: The Multivariate
Approach to Repeated Measures
256
8.1 General Purpose and Description
256
8.2 Kinds of Research Questions
257
8.2.1 Parallelism of Profiles
258
8.2.2 Overall Difference Among Groups
258
8.2.3 Flatness of Profiles
258
8.2.4 Contrasts Following Profile Analysis
258
8.2.5 Parameter Estimates
258
8.2.6 Effect Size
259
8.3 Limitations to Profile Analysis
259
8.3.1 Theoretical Issues
259
8.3.2 Practical Issues
259
8.3.2.1 Sample Size, Missing Data,
and Power
259
8.3.2.2 Multivariate Normality
260
8.3.2.3 Absence of Outliers
260
8.3.2.4 Homogeneity of
Variance–Covariance Matrices
260
8.3.2.5 Linearity
260
8.3.2.6 Absence of Multicollinearity
and Singularity
260
8.4 Fundamental Equations for Profile Analysis 260
8.4.1 Differences in Levels
262
8.4.2 Parallelism
262
8.4.3 Flatness
265
8.4.4 Computer Analyses of Small-Sample
Example
266
8.5 Some Important Issues
269
8.5.1 Univariate Versus Multivariate
Approach to Repeated Measures
269
8.5.2 Contrasts in Profile Analysis
270
8.5.2.1 Parallelism and Flatness
Significant, Levels Not Significant
(Simple-Effects Analysis)
272
8.5.2.2 Parallelism and Levels Significant,
Flatness Not Significant
(Simple-Effects Analysis)
274
8.5.2.3 Parallelism, Levels, and Flatness
Significant (Interaction Contrasts)
275
8.5.2.4 Only Parallelism Significant
276
8.5.3 Doubly Multivariate Designs
277
8.5.4 Classifying Profiles
279
8.5.5 Imputation of Missing Values
279
8.6 Complete Examples of Profile Analysis
280
8.6.1 Profile Analysis of Subscales
of the WISC
280
8.6.1.1 Evaluation of Assumptions
280
8.6.1.2 Profile Analysis
283
8.6.2 Doubly Multivariate Analysis
of Reaction Time
288
8.6.2.1 Evaluation of Assumptions
289
8.6.2.2 Doubly Multivariate Analysis
of Slope and Intercept
290
8.7 Comparison of Programs
297
8.7.1 IBM SPSS Package
297
8.7.2 SAS System
298
8.7.3 SYSTAT System
298
9 Discriminant Analysis
299
9.1 General Purpose and Description
299
9.2 Kinds of Research Questions
302
9.2.1 Significance of Prediction
302
9.2.2 Number of Significant
Discriminant Functions
302
9.2.3 Dimensions of Discrimination
302
9.2.4 Classification Functions
303
9.2.5 Adequacy of Classification
303
9.2.6 Effect Size
303
9.2.7 Importance of Predictor Variables
303
9.2.8 Significance of Prediction with Covariates 304
9.2.9 Estimation of Group Means
304
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Contents vii
9.3 Limitations to Discriminant Analysis
304
9.3.1 Theoretical Issues
304
9.3.2 Practical Issues
304
9.3.2.1 Unequal Sample Sizes, Missing
Data, and Power
304
9.3.2.2 Multivariate Normality
305
9.3.2.3 Absence of Outliers
305
9.3.2.4 Homogeneity of
Variance–Covariance Matrices
305
9.3.2.5 Linearity
306
9.3.2.6 Absence of Multicollinearity
and Singularity
306
9.4 Fundamental Equations for
Discriminant Analysis
306
9.4.1 Derivation and Test of
Discriminant Functions
307
9.4.2 Classification
309
9.4.3 Computer Analyses of
Small-Sample Example
311
9.5 Types of Discriminant Analyses
315
9.5.1 Direct Discriminant Analysis
315
9.5.2 Sequential Discriminant Analysis
315
9.5.3 Stepwise (Statistical) Discriminant
Analysis
316
9.6 Some Important Issues
316
9.6.1 Statistical Inference
316
9.6.1.1 Criteria for Overall Statistical
Significance
317
9.6.1.2 Stepping Methods
317
9.6.2 Number of Discriminant Functions
317
9.6.3 Interpreting Discriminant Functions
318
9.6.3.1 Discriminant Function Plots
318
9.6.3.2 Structure Matrix of Loadings
318
9.6.4 Evaluating Predictor Variables
320
9.6.5 Effect Size
321
9.6.6 Design Complexity: Factorial Designs 321
9.6.7 Use of Classification Procedures
322
9.6.7.1 Cross-Validation and New Cases
322
9.6.7.2 Jackknifed Classification
323
9.6.7.3 Evaluating Improvement in
Classification
323
9.7 Complete Example of Discriminant Analysis 324
9.7.1 Evaluation of Assumptions
325
9.7.1.1 Unequal Sample Sizes
and Missing Data
325
9.7.1.2 Multivariate Normality
325
9.7.1.3 Linearity
325
9.7.1.4 Outliers
325
9.7.1.5 Homogeneity of Variance–
Covariance Matrices
326
9.7.1.6 Multicollinearity and Singularity
327
9.7.2 Direct Discriminant Analysis
327
9.8 Comparison of Programs
340
9.8.1 IBM SPSS Package
344
9.8.2 SAS System
344
9.8.3 SYSTAT System
345
10 Logistic Regression
346
10.1 General Purpose and Description
346
10.2 Kinds of Research Questions
348
10.2.1 Prediction of Group Membership
or Outcome
348
10.2.2 Importance of Predictors
348
10.2.3 Interactions Among Predictors
349
10.2.4 Parameter Estimates
349
10.2.5 Classification of Cases
349
10.2.6 Significance of Prediction with
Covariates
349
10.2.7 Effect Size
349
10.3 Limitations to Logistic Regression Analysis 350
10.3.1 Theoretical Issues
350
10.3.2 Practical Issues
350
10.3.2.1 Ratio of Cases to Variables
350
10.3.2.2 Adequacy of Expected
Frequencies and Power
351
10.3.2.3 Linearity in the Logit
351
10.3.2.4 Absence of Multicollinearity
351
10.3.2.5 Absence of Outliers in the Solution 351
10.3.2.6 Independence of Errors
352
10.4 Fundamental Equations for
Logistic Regression
352
10.4.1 Testing and Interpreting Coefficients 353
10.4.2 Goodness of Fit
354
10.4.3 Comparing Models
355
10.4.4 Interpretation and Analysis of
Residuals
355
10.4.5 Computer Analyses of
Small-Sample Example
356
10.5 Types of Logistic Regression
360
10.5.1 Direct Logistic Regression
360
10.5.2 Sequential Logistic Regression
360
10.5.3 Statistical (Stepwise) Logistic
Regression
362
10.5.4 Probit and Other Analyses
362
10.6 Some Important Issues
363
10.6.1 Statistical Inference
363
10.6.1.1 Assessing Goodness of Fit
of Models
363
10.6.1.2 Tests of Individual Predictors
365
10.6.2 Effect Sizes
365
10.6.2.1 Effect Size for a Model
365
10.6.2.2 Effect Sizes for Predictors
366
10.6.3 Interpretation of Coefficients
Using Odds
367
10.6.4 Coding Outcome and Predictor
Categories
368
10.6.5 Number and Type of Outcome
Categories
369
10.6.6 Classification of Cases
372
10.6.7 Hierarchical and Nonhierarchical
Analysis
372
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10.6.8 Importance of Predictors
373
10.6.9 Logistic Regression for Matched
Groups
374
10.7 Complete Examples of Logistic Regression 374
10.7.1 Evaluation of Limitations
374
10.7.1.1 Ratio of Cases to Variables
and Missing Data
374
10.7.1.2 Multicollinearity
376
10.7.1.3 Outliers in the Solution
376
10.7.2 Direct Logistic Regression with
Two-Category Outcome and
Continuous Predictors
377
10.7.2.1 Limitation: Linearity in the Logit 377
10.7.2.2 Direct Logistic Regression with
Two-Category Outcome
377
10.7.3 Sequential Logistic Regression
with Three Categories of Outcome
384
10.7.3.1 Limitations of Multinomial
Logistic Regression
384
10.7.3.2 Sequential Multinomial
Logistic Regression
387
10.8 Comparison of Programs
396
10.8.1 IBM SPSS Package
396
10.8.2 SAS System
399
10.8.3 SYSTAT System
400
11 Survival/Failure Analysis
401
11.1 General Purpose and Description
401
11.2 Kinds of Research Questions
403
11.2.1 Proportions Surviving at
Various Times
403
11.2.2 Group Differences in Survival
403
11.2.3 Survival Time with Covariates
403
11.2.3.1 Treatment Effects
403
11.2.3.2 Importance of Covariates
403
11.2.3.3 Parameter Estimates
404
11.2.3.4 Contingencies Among
Covariates
404
11.2.3.5 Effect Size and Power
404
11.3 Limitations to Survival Analysis
404
11.3.1 Theoretical Issues
404
11.3.2 Practical Issues
404
11.3.2.1 Sample Size and
Missing Data
404
11.3.2.2 Normality of Sampling
Distributions, Linearity, and
Homoscedasticity
405
11.3.2.3 Absence of Outliers
405
11.3.2.4 Differences Between
Withdrawn and Remaining
Cases
405
11.3.2.5 Change in Survival
Conditions over Time
405
11.3.2.6 Proportionality of Hazards
405
11.3.2.7 Absence of Multicollinearity
405
11.4 Fundamental Equations for
Survival Analysis
405
11.4.1 Life Tables
406
11.4.2 Standard Error of Cumulative
Proportion Surviving
408
11.4.3 Hazard and Density Functions
408
11.4.4 Plot of Life Tables
409
11.4.5 Test for Group Differences
410
11.4.6 Computer Analyses of Small-Sample
Example
411
11.5 Types of Survival Analyses
415
11.5.1 Actuarial and Product-Limit Life
Tables and Survivor Functions
415
11.5.2 Prediction of Group Survival Times
from Covariates
417
11.5.2.1 Direct, Sequential,
and Statistical Analysis
417
11.5.2.2 Cox Proportional-Hazards Model 417
11.5.2.3 Accelerated Failure-Time Models 419
11.5.2.4 Choosing a Method
423
11.6 Some Important Issues
423
11.6.1 Proportionality of Hazards
423
11.6.2 Censored Data
424
11.6.2.1 Right-Censored Data
425
11.6.2.2 Other Forms of Censoring
425
11.6.3 Effect Size and Power
425
11.6.4 Statistical Criteria
426
11.6.4.1 Test Statistics for Group
Differences in Survival Functions 426
11.6.4.2 Test Statistics for Prediction
from Covariates
427
11.6.5 Predicting Survival Rate
427
11.6.5.1 Regression Coefficients
(Parameter Estimates)
427
11.6.5.2 Hazard Ratios
427
11.6.5.3 Expected Survival Rates
428
11.7 Complete Example of Survival Analysis
429
11.7.1 Evaluation of Assumptions
430
11.7.1.1 Accuracy of Input, Adequacy
of Sample Size, Missing Data,
and Distributions
430
11.7.1.2 Outliers
430
11.7.1.3 Differences Between
Withdrawn and Remaining
Cases
433
11.7.1.4 Change in Survival
Experience over Time
433
11.7.1.5 Proportionality of Hazards
433
11.7.1.6 Multicollinearity
434
11.7.2 Cox Regression Survival Analysis
436
11.7.2.1 Effect of Drug Treatment
436
11.7.2.2 Evaluation of Other
Covariates
436
11.8 Comparison of Programs
440
11.8.1 SAS System
444
11.8.2 IBM SPSS Package
445
11.8.3 SYSTAT System
445
12 Canonical Correlation
446
12.1 General Purpose and Description
446
12.2 Kinds of Research Questions
448
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Contents ix
12.2.1 Number of Canonical Variate Pairs
448
12.2.2 Interpretation of Canonical Variates 448
12.2.3 Importance of Canonical Variates
and Predictors
448
12.2.4 Canonical Variate Scores
449
12.3 Limitations
449
12.3.1 Theoretical Limitations
449
12.3.2 Practical Issues
450
12.3.2.1 Ratio of Cases to IVs
450
12.3.2.2 Normality, Linearity, and
Homoscedasticity
450
12.3.2.3 Missing Data
451
12.3.2.4 Absence of Outliers
451
12.3.2.5 Absence of Multicollinearity
and Singularity
451
12.4 Fundamental Equations for
Canonical�Correlation
451
12.4.1 Eigenvalues and Eigenvectors
452
12.4.2 Matrix Equations
454
12.4.3 Proportions of Variance Extracted
457
12.4.4 Computer Analyses of
Small-Sample Example
458
12.5 Some Important Issues
462
12.5.1 Importance of Canonical Variates
462
12.5.2 Interpretation of Canonical Variates 463
12.6 Complete Example of Canonical Correlation 463
12.6.1 Evaluation of Assumptions
463
12.6.1.1 Missing Data
463
12.6.1.2 Normality, Linearity, and
Homoscedasticity
463
12.6.1.3 Outliers
466
12.6.1.4 Multicollinearity
and Singularity
467
12.6.2 Canonical Correlation
467
12.7 Comparison of Programs
473
12.7.1 SAS System
473
12.7.2 IBM SPSS Package
474
12.7.3 SYSTAT System
475
13 Principal Components
and Factor Analysis
476
13.1 General Purpose and Description
476
13.2 Kinds of Research Questions
479
13.2.1 Number of Factors
479
13.2.2 Nature of Factors
479
13.2.3 Importance of Solutions and Factors 480
13.2.4 Testing Theory in FA
480
13.2.5 Estimating Scores on Factors
480
13.3 Limitations
480
13.3.1 Theoretical Issues
480
13.3.2 Practical Issues
481
13.3.2.1 Sample Size and Missing Data
481
13.3.2.2 Normality
482
13.3.2.3 Linearity
482
13.3.2.4 Absence of Outliers Among Cases 482
13.3.2.5 Absence of Multicollinearity
and Singularity
482
13.3.2.6 Factorability of R
482
13.3.2.7 Absence of Outliers Among
Variables
483
13.4 Fundamental Equations for Factor
Analysis
483
13.4.1 Extraction
485
13.4.2 Orthogonal Rotation
487
13.4.3 Communalities, Variance, and
Covariance
488
13.4.4 Factor Scores
489
13.4.5 Oblique Rotation
491
13.4.6 Computer Analyses of
Small-Sample Example
493
13.5 Major Types of Factor Analyses
496
13.5.1 Factor Extraction Techniques
496
13.5.1.1 PCA Versus FA
496
13.5.1.2 Principal Components
498
13.5.1.3 Principal Factors
498
13.5.1.4 Image Factor Extraction
498
13.5.1.5 Maximum Likelihood
Factor Extraction
499
13.5.1.6 Unweighted Least
Squares Factoring
499
13.5.1.7 Generalized (Weighted)
Least Squares Factoring
499
13.5.1.8 Alpha Factoring
499
13.5.2 Rotation
500
13.5.2.1 Orthogonal Rotation
500
13.5.2.2 Oblique Rotation
501
13.5.2.3 Geometric Interpretation
502
13.5.3 Some Practical Recommendations
503
13.6 Some Important Issues
504
13.6.1 Estimates of Communalities
504
13.6.2 Adequacy of Extraction and
Number of Factors
504
13.6.3 Adequacy of Rotation and
Simple Structure
507
13.6.4 Importance and Internal
Consistency of Factors
508
13.6.5 Interpretation of Factors
509
13.6.6 Factor Scores
510
13.6.7 Comparisons Among Solutions
and Groups
511
13.7 Complete Example of FA
511
13.7.1 Evaluation of Limitations
511
13.7.1.1 Sample Size and
Missing Data
512
13.7.1.2 Normality
512
13.7.1.3 Linearity
512
13.7.1.4 Outliers
513
13.7.1.5 Multicollinearity
and Singularity
514
13.7.1.6 Factorability of R
514
13.7.1.7 Outliers Among Variables
515
13.7.2 Principal Factors Extraction with
Varimax Rotation
515
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x Contents
13.8 Comparison of Programs
525
13.8.1 IBM SPSS Package
527
13.8.2 SAS System
527
13.8.3 SYSTAT System
527
14 Structural Equation Modeling
by Jodie B. Ullman
528
14.1 General Purpose and Description
528
14.2 Kinds of Research Questions
531
14.2.1 Adequacy of the Model
531
14.2.2 Testing Theory
531
14.2.3 Amount of Variance in the Variables
Accounted for by the Factors
532
14.2.4 Reliability of the Indicators
532
14.2.5 Parameter Estimates
532
14.2.6 Intervening Variables
532
14.2.7 Group Differences
532
14.2.8 Longitudinal Differences
532
14.2.9 Multilevel Modeling
533
14.2.10 Latent Class Analysis
533
14.3 Limitations to Structural Equation
Modeling
533
14.3.1 Theoretical Issues
533
14.3.2 Practical Issues
534
14.3.2.1 Sample Size and
Missing Data
534
14.3.2.2 Multivariate Normality
and Outliers
534
14.3.2.3 Linearity
534
14.3.2.4 Absence of Multicollinearity
and Singularity
535
14.3.2.5 Residuals
535
14.4 Fundamental Equations for Structural
Equations Modeling
535
14.4.1 Covariance Algebra
535
14.4.2 Model Hypotheses
537
14.4.3 Model Specification
538
14.4.4 Model Estimation
540
14.4.5 Model Evaluation
543
14.4.6 Computer Analysis of
Small-Sample Example
545
14.5 Some Important Issues
555
14.5.1 Model Identification
555
14.5.2 Estimation Techniques
557
14.5.2.1 Estimation Methods
and Sample Size
559
14.5.2.2 Estimation Methods
and Nonnormality
559
14.5.2.3 Estimation Methods
and Dependence
559
14.5.2.4 Some Recommendations
for Choice of Estimation
Method
560
14.5.3 Assessing the Fit of the Model
560
14.5.3.1 Comparative Fit Indices
560
14.5.3.2 Absolute Fit Index
562
14.5.3.3 Indices of Proportion
of Variance Accounted
562
14.5.3.4 Degree of Parsimony
Fit Indices
563
14.5.3.5 Residual-Based Fit Indices
563
14.5.3.6 Choosing Among Fit Indices
564
14.5.4 Model Modification
564
14.5.4.1 Chi-Square Difference Test
564
14.5.4.2 Lagrange Multiplier (LM) Test
565
14.5.4.3 Wald Test
569
14.5.4.4 Some Caveats and Hints on
Model Modification
570
14.5.5 Reliability and Proportion of Variance 570
14.5.6 Discrete and Ordinal Data
571
14.5.7 Multiple Group Models
572
14.5.8 Mean and Covariance Structure
Models
573
14.6 Complete Examples of Structural Equation
Modeling Analysis
574
14.6.1 Confirmatory Factor Analysis
of the WISC
574
14.6.1.1 Model Specification for CFA
574
14.6.1.2 Evaluation of Assumptions
for CFA
574
14.6.1.3 CFA Model Estimation and
Preliminary Evaluation
576
14.6.1.4 Model Modification
583
14.6.2 SEM of Health Data
589
14.6.2.1 SEM Model Specification
589
14.6.2.2 Evaluation of Assumptions
for SEM
591
14.6.2.3 SEM Model Estimation and
Preliminary Evaluation
593
14.6.2.4 Model Modification
596
14.7 Comparison of Programs
607
14.7.1 EQS
607
14.7.2 LISREL
607
14.7.3 AMOS
612
14.7.4 SAS System
612
15 Multilevel Linear Modeling
613
15.1 General Purpose and Description
613
15.2 Kinds of Research Questions
616
15.2.1 Group Differences in Means
616
15.2.2 Group Differences in Slopes
616
15.2.3 Cross-Level Interactions
616
15.2.4 Meta-Analysis
616
15.2.5 Relative Strength of Predictors
at Various Levels
617
15.2.6 Individual and Group Structure
617
15.2.7 Effect Size
617
15.2.8 Path Analysis at Individual
and Group Levels
617
15.2.9 Analysis of Longitudinal Data
617
15.2.10 Multilevel Logistic Regression
618
15.2.11 Multiple Response Analysis
618
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Contents xi
15.3 Limitations to Multilevel Linear Modeling 618
15.3.1 Theoretical Issues
618
15.3.2 Practical Issues
618
15.3.2.1 Sample Size, Unequal-n,
and Missing Data
619
15.3.2.2 Independence of Errors
619
15.3.2.3 Absence of Multicollinearity
and Singularity
620
15.4 Fundamental Equations
620
15.4.1 Intercepts-Only Model
623
15.4.1.1 The Intercepts-Only Model:
Level-1 Equation
623
15.4.1.2 The Intercepts-Only Model:
Level-2 Equation
623
15.4.1.3 Computer Analyses
of Intercepts-Only Model
624
15.4.2 Model with a First-Level Predictor
627
15.4.2.1 Level-1 Equation for a
Model with a Level-1
Predictor
627
15.4.2.2 Level-2 Equations for a
Model with a Level-1
Predictor
628
15.4.2.3 Computer Analysis of a
Model with a Level-1
Predictor
630
15.4.3 Model with Predictors at First
and Second Levels
633
15.4.3.1 Level-1 Equation for
Model with Predictors at
Both Levels
633
15.4.3.2 Level-2 Equations for
Model with Predictors
at Both Levels
633
15.4.3.3 Computer Analyses of
Model with Predictors at
First and Second Levels
634
15.5 Types of MLM
638
15.5.1 Repeated Measures
638
15.5.2 Higher-Order MLM
642
15.5.3 Latent Variables
642
15.5.4 Nonnormal Outcome Variables
643
15.5.5 Multiple Response Models
644
15.6 Some Important Issues
644
15.6.1 Intraclass Correlation
644
15.6.2 Centering Predictors and Changes
in Their Interpretations
646
15.6.3 Interactions
648
15.6.4 Random and Fixed Intercepts
and Slopes
648
15.6.5 Statistical Inference
651
15.6.5.1 Assessing Models
651
15.6.5.2 Tests of Individual Effects
652
15.6.6 Effect Size
653
15.6.7 Estimation Techniques and
Convergence Problems
653
15.6.8 Exploratory Model Building
654
15.7 Complete Example of MLM
655
15.7.1 Evaluation of Assumptions
656
15.7.1.1 Sample Sizes, Missing
Data, and Distributions
656
15.7.1.2 Outliers
659
15.7.1.3 Multicollinearity
and Singularity
659
15.7.1.4 Independence of Errors:
Intraclass Correlations
659
15.7.2 Multilevel Modeling
661
15.8 Comparison of Programs
668
15.8.1 SAS System
668
15.8.2 IBM SPSS Package
670
15.8.3 HLM Program
671
15.8.4 MLwiN Program
671
15.8.5 SYSTAT System
671
16 Multiway Frequency Analysis
672
16.1 General Purpose and Description
672
16.2 Kinds of Research Questions
673
16.2.1 Associations Among Variables
673
16.2.2 Effect on a Dependent Variable
674
16.2.3 Parameter Estimates
674
16.2.4 Importance of Effects
674
16.2.5 Effect Size
674
16.2.6 Specific Comparisons and
Trend Analysis
674
16.3 Limitations to Multiway Frequency Analysis 675
16.3.1 Theoretical Issues
675
16.3.2 Practical Issues
675
16.3.2.1 Independence
675
16.3.2.2 Ratio of Cases to Variables
675
16.3.2.3 Adequacy of Expected
Frequencies
675
16.3.2.4 Absence of Outliers in the
Solution
676
16.4 Fundamental Equations for Multiway
Frequency Analysis
676
16.4.1 Screening for Effects
678
16.4.1.1 Total Effect
678
16.4.1.2 First-Order Effects
679
16.4.1.3 Second-Order Effects
679
16.4.1.4 Third-Order Effect
683
16.4.2 Modeling
683
16.4.3 Evaluation and Interpretation
685
16.4.3.1 Residuals
685
16.4.3.2 Parameter Estimates
686
16.4.4 Computer Analyses of Small-Sample
Example
690
16.5 Some Important Issues
695
16.5.1 Hierarchical and Nonhierarchical
Models
695
16.5.2 Statistical Criteria
696
16.5.2.1 Tests of Models
696
16.5.2.2 Tests of Individual Effects
696
16.5.3 Strategies for Choosing a Model
696
16.5.3.1 IBM SPSS HILOGLINEAR
(Hierarchical)
697
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xii Contents
16.5.3.2 IBM SPSS GENLOG
(General Log-Linear)
697
16.5.3.3 SAS CATMOD and IBM
SPSS LOGLINEAR (General
Log-Linear)
697
16.6 Complete Example of Multiway
Frequency Analysis
698
16.6.1 Evaluation of Assumptions:
Adequacy of Expected Frequencies
698
16.6.2 Hierarchical Log-Linear Analysis
700
16.6.2.1 Preliminary Model Screening
700
16.6.2.2 Stepwise Model Selection
702
16.6.2.3 Adequacy of Fit
702
16.6.2.4 Interpretation of the
Selected Model
705
16.7 Comparison of Programs
710
16.7.1 IBM SPSS Package
710
16.7.2 SAS System
712
16.7.3 SYSTAT System
713
17 Time-Series Analysis
714
17.1 General Purpose and Description
714
17.2 Kinds of Research Questions
716
17.2.1 Pattern of Autocorrelation
717
17.2.2 Seasonal Cycles and Trends
717
17.2.3 Forecasting
717
17.2.4 Effect of an Intervention
718
17.2.5 Comparing Time Series
718
17.2.6 Time Series with Covariates
718
17.2.7 Effect Size and Power
718
17.3 Assumptions of Time-Series Analysis
718
17.3.1 Theoretical Issues
718
17.3.2 Practical Issues
718
17.3.2.1 Normality of Distributions
of Residuals
719
17.3.2.2 Homogeneity of Variance
and Zero Mean of Residuals
719
17.3.2.3 Independence of Residuals
719
17.3.2.4 Absence of Outliers
719
17.3.2.5 Sample Size and Missing Data
719
17.4 Fundamental Equations for
Time-Series ARIMA Models
720
17.4.1 Identification of ARIMA
(p, d, q) Models
720
17.4.1.1 Trend Components, d: Making
the Process Stationary
721
17.4.1.2 Auto-Regressive Components
722
17.4.1.3 Moving Average Components
724
17.4.1.4 Mixed Models
724
17.4.1.5 ACFs and PACFs
724
17.4.2 Estimating Model Parameters
729
17.4.3 Diagnosing a Model
729
17.4.4 Computer Analysis of Small-Sample
Time-Series Example
734
17.5 Types of Time-Series Analyses
737
17.5.1 Models with Seasonal Components 737
17.5.2 Models with Interventions
738
17.5.2.1 Abrupt, Permanent Effects
741
17.5.2.2 Abrupt, Temporary Effects
742
17.5.2.3 Gradual, Permanent Effects
745
17.5.2.4 Models with Multiple Interventions 746
17.5.3 Adding Continuous Variables
747
17.6 Some Important Issues
748
17.6.1 Patterns of ACFs and PACFs
748
17.6.2 Effect Size
751
17.6.3 Forecasting
752
17.6.4 Statistical Methods for Comparing
Two Models
752
17.7 Complete Examples of Time-Series
Analysis
753
17.7.1 Time-Series Analysis of
Introduction of Seat Belt Law
753
17.7.1.1 Evaluation of Assumptions
754
17.7.1.2 Baseline Model
Identification and
Estimation
755
17.7.1.3 Baseline Model Diagnosis
758
17.7.1.4 Intervention Analysis
758
17.7.2. Time-Series Analysis of
Introduction of a Dashboard to
an Educational Computer Game
762
17.7.2.1 Evaluation of Assumptions
763
17.7.2.2 Baseline Model Identification
and Diagnosis
765
17.7.2.3 Intervention Analysis
766
17.8 Comparison of Programs
771
17.8.1 IBM SPSS Package
771
17.8.2 SAS System
774
17.8.3 SYSTAT System
774
18 An Overview of the General
Linear Model
775
18.1 Linearity and the General Linear Model
775
18.2 Bivariate to Multivariate Statistics
and Overview of Techniques
775
18.2.1 Bivariate Form
775
18.2.2 Simple Multivariate Form
777
18.2.3 Full Multivariate Form
778
18.3 Alternative Research Strategies
782
Appendix A
A Skimpy Introduction to
Matrix Algebra
783
A.1 The Trace of a Matrix
784
A.2 Addition or Subtraction of a
Constant to a Matrix
784
A.3 Multiplication or Division of a
Matrix by a Constant
784
A.4 Addition and Subtraction
of Two Matrices
785
A.5 Multiplication, Transposes, and Square
Roots of Matrice
785
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Contents xiii
A.6 Matrix “Division” (Inverses and
Determinants)
786
A.7 Eigenvalues and Eigenvectors:
Procedures for Consolidating Variance
from a Matrix
788
Appendix B
Research Designs for Complete
Examples
791
B.1 Women’s Health and Drug Study
791
B.2 Sexual Attraction Study
793
B.3 Learning Disabilities Data Bank
794
B.4 Reaction Time to Identify Figures
794
B.5 Field Studies of Noise-Induced Sleep
Disturbance
795
B.6 Clinical Trial for Primary Biliary
Cirrhosis
795
B.7 Impact of Seat Belt Law
795
B.8 The Selene Online Educational Game
796
Appendix C
Statistical Tables
797
C.1 Normal Curve Areas
798
C.2 Critical Values of the t Distribution
for a = .05 and .01, Two-Tailed Test
799
C.3 Critical Values of the F Distribution
800
C.4 Critical Values of Chi Square (x2)
804
C.5 Critical Values for Squares Multiple
Correlation (R2) in Forward Stepwise
Selection: a = .05
805
C.6 Critical Values for FMAX (S2
MAX/S2
MIN)
Distribution for a = .05 and .01
807
References
808
Index
815
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Page 14
Preface
Some good things seem to go on forever: friendship and updating this book. It is diffi-
cult to believe that the first edition manuscript was typewritten, with real cutting and
pasting. The publisher required a paper manuscript with numbered pages—that was
almost our downfall. We could write a book on multivariate statistics, but we couldn’t get the
same number of pages (about 1200, double-spaced) twice in a row. SPSS was in release 9.0,
and the other program we demonstrated was BMDP. There were a mere 11 chapters, of which
6 of them were describing techniques. Multilevel and structural equation modeling were not
yet ready for prime time. Logistic regression and survival analysis were not yet popular.
Material new to this edition includes a redo of all SAS examples, with a pretty new output
format and replacement of interactive analyses that are no longer available. We’ve also re-run
the IBM SPSS examples to show the new output format. We’ve tried to update the references in
all chapters, including only classic citations if they date prior to 2000. New work on relative im-
portance has been incorporated in multiple regression, canonical correlation, and logistic regres-
sion analysis—complete with demonstrations. Multiple imputation procedures for dealing with
missing data have been updated, and we’ve added a new time-series example, taking advantage
of an IBM SPSS expert modeler that replaces previous tea-leaf reading aspects of the analysis.
Our goals in writing the book remain the same as in all previous editions—to present com-
plex statistical procedures in a way that is maximally useful and accessible to researchers who
are not necessarily statisticians. We strive to be short on theory but long on conceptual under-
standing. The statistical packages have become increasingly easy to use, making it all the more
critical to make sure that they are applied with a good understanding of what they can and
cannot do. But above all else—what does it all mean?
We have not changed the basic format underlying all of the technique chapters, now 14 of
them. We start with an overview of the technique, followed by the types of research questions
the techniques are designed to answer. We then provide the cautionary tale—what you need to
worry about and how to deal with those worries. Then come the fundamental equations underly-
ing the technique, which some readers truly enjoy working through (we know because they help-
fully point out any errors and/or inconsistencies they find); but other readers discover they can
skim (or skip) the section without any loss to their ability to conduct meaningful analysis of their
research. The fundamental equations are in the context of a small, made-up, usually silly data set
for which computer analyses are provided—usually IBM SPSS and SAS. Next, we delve into is-
sues surrounding the technique (such as different types of the analysis, follow-up procedures to
the main analysis, and effect size, if it is not amply covered elsewhere). Finally, we provide one or
two full-bore analyses of an actual real-life data set together with a Results section appropriate for
a journal. Data sets for these examples are available at www.pearsonhighered.com in IBM SPSS,
SAS, and ASCII formats. We end each technique chapter with a comparison of features available
in IBM SPSS, SAS, SYSTAT and sometimes other specialized programs. SYSTAT is a statistical
package that we reluctantly had to drop a few editions ago for lack of space.
We apologize in advance for the heft of the book; it is not our intention to line the cof-
fers of chiropractors, physical therapists, acupuncturists, and the like, but there’s really just so
much to say. As to our friendship, it’s still going strong despite living in different cities. Art has
taken the place of creating belly dance costumes for both of us, but we remain silly in outlook,
although serious in our analysis of research.
The lineup of people to thank grows with each edition, far too extensive to list: students,
reviewers, editors, and readers who send us corrections and point out areas of confusion. As
always, we take full responsibility for remaining errors and lack of clarity.
Barbara G. Tabachnick
Linda S. Fidell
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