IBM SPSS Amos 21 User Manual
IBM® SPSS® Amos™ 21
User’s Guide
James L. Arbuckle
Note: Before using this information and the product it supports, read the information in the “Notices” section
on page 631.
This edition applies to IBM® SPSS® Amos™ 21 and to all subsequent releases and modifications until
otherwise indicated in new editions.
Microsoft product screenshots reproduced with permission from Microsoft Corporation.
Licensed Materials - Property of IBM
© Copyright IBM Corp. 1983, 2012. U.S. Government Users Restricted Rights - Use, duplication or
disclosure restricted by GSA ADP Schedule Contract with IBM Corp.
© Copyright 2012 Amos Development Corporation. All Rights Reserved.
AMOS is a trademark of Amos Development Corporation.
Contents
Part I: Getting Started
1 Introduction 1
Featured Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
About the Tutorial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
About the Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
About the Documentation . . . . . . . . . . . . . . . . . . . . . . . . . . . .4
Other Sources of Information. . . . . . . . . . . . . . . . . . . . . . . . . . 4
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Tutorial: Getting Started with
Amos Graphics 7
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7
About the Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8
Launching Amos Graphics . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
Creating a New Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Specifying the Data File . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Specifying the Model and Drawing Variables . . . . . . . . . . . . . . . 11
Naming the Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
Drawing Arrows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Constraining a Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
Altering the Appearance of a Path Diagram . . . . . . . . . . . . . . . . 15
To Move an Object . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
To Reshape an Object or Double-Headed Arrow . . . . . . . . . . . 15
To Delete an Object. . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
To Undo an Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
To Redo an Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
iii
Setting Up Optional Output . . . . . . . . . . . . . . . . . . . . . . . . . . 16
Performing the Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Viewing Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
To View Text Output . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
To View Graphics Output . . . . . . . . . . . . . . . . . . . . . . . . 19
Printing the Path Diagram. . . . . . . . . . . . . . . . . . . . . . . . . . . 20
Copying the Path Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Copying Text Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Part II: Examples
1 Estimating Variances and Covariances 23
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
About the Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
Bringing In the Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
Analyzing the Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
Specifying the Model. . . . . . . . . . . . . . . . . . . . . . . . . . . 25
Naming the Variables . . . . . . . . . . . . . . . . . . . . . . . . . . 26
Changing the Font . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
Establishing Covariances . . . . . . . . . . . . . . . . . . . . . . . . 27
Performing the Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 28
Viewing Graphics Output . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
Viewing Text Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
Optional Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
Calculating Standardized Estimates . . . . . . . . . . . . . . . . . . 33
Rerunning the Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 34
Viewing Correlation Estimates as Text Output . . . . . . . . . . . . 34
Distribution Assumptions for Amos Models . . . . . . . . . . . . . . . . 35
Modeling in VB.NET . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
Generating Additional Output . . . . . . . . . . . . . . . . . . . . . . 39
Modeling in C# . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
Other Program Development Tools . . . . . . . . . . . . . . . . . . . . . 40
iv
2 Testing Hypotheses 41
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .41
About the Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .41
Parameters Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . .41
Constraining Variances . . . . . . . . . . . . . . . . . . . . . . . . . .42
Specifying Equal Parameters. . . . . . . . . . . . . . . . . . . . . . .43
Constraining Covariances . . . . . . . . . . . . . . . . . . . . . . . .44
Moving and Formatting Objects . . . . . . . . . . . . . . . . . . . . . . . .45
Data Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .46
Performing the Analysis. . . . . . . . . . . . . . . . . . . . . . . . . .47
Viewing Text Output . . . . . . . . . . . . . . . . . . . . . . . . . . . .47
Optional Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .48
Covariance Matrix Estimates. . . . . . . . . . . . . . . . . . . . . . .49
Displaying Covariance and Variance Estimates
on the Path Diagram. . . . . . . . . . . . . . . . . . . . . . . . . . . .51
Labeling Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .51
Hypothesis Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .52
Displaying Chi-Square Statistics on the Path Diagram . . . . . . . . . . .53
Modeling in VB.NET. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .55
Timing Is Everything . . . . . . . . . . . . . . . . . . . . . . . . . . . .57
3 More Hypothesis Testing 59
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .59
About the Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .59
Bringing In the Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .59
Testing a Hypothesis That Two Variables Are Uncorrelated . . . . . . .60
Specifying the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .60
Viewing Text Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .62
Viewing Graphics Output. . . . . . . . . . . . . . . . . . . . . . . . . . . .63
Modeling in VB.NET. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .65
v
4 Conventional Linear Regression 67
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
About the Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
Analysis of the Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
Specifying the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
Fixing Regression Weights . . . . . . . . . . . . . . . . . . . . . . . . . . 70
Viewing the Text Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
Viewing Graphics Output . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
Viewing Additional Text Output. . . . . . . . . . . . . . . . . . . . . . . . 75
Modeling in VB.NET . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
Assumptions about Correlations among Exogenous Variables . . . 77
Equation Format for the AStructure Method . . . . . . . . . . . . . 78
5 Unobserved Variables 81
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
About the Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
Model A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
Measurement Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
Structural Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
Specifying the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
Changing the Orientation of the Drawing Area . . . . . . . . . . . . 86
Creating the Path Diagram . . . . . . . . . . . . . . . . . . . . . . . 87
Rotating Indicators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
Duplicating Measurement Models. . . . . . . . . . . . . . . . . . . 88
Entering Variable Names . . . . . . . . . . . . . . . . . . . . . . . . 90
Completing the Structural Model . . . . . . . . . . . . . . . . . . . . 90
Results for Model A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
Viewing the Graphics Output . . . . . . . . . . . . . . . . . . . . . . 93
vi
Model B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .93
Results for Model B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .94
Testing Model B against Model A. . . . . . . . . . . . . . . . . . . . . . .96
Modeling in VB.NET. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .98
Model A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .98
Model B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .99
6 Exploratory Analysis 101
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
About the Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
Model A for the Wheaton Data . . . . . . . . . . . . . . . . . . . . . . . 102
Specifying the Model . . . . . . . . . . . . . . . . . . . . . . . . . . 102
Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
Results of the Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 103
Dealing with Rejection . . . . . . . . . . . . . . . . . . . . . . . . . 104
Modification Indices. . . . . . . . . . . . . . . . . . . . . . . . . . . 105
Model B for the Wheaton Data . . . . . . . . . . . . . . . . . . . . . . . 107
Text Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
Graphics Output for Model B . . . . . . . . . . . . . . . . . . . . . . 109
Misuse of Modification Indices . . . . . . . . . . . . . . . . . . . . 110
Improving a Model by Adding New Constraints . . . . . . . . . . . 110
Model C for the Wheaton Data . . . . . . . . . . . . . . . . . . . . . . . 114
Results for Model C . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
Testing Model C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
Parameter Estimates for Model C . . . . . . . . . . . . . . . . . . . 115
Multiple Models in a Single Analysis . . . . . . . . . . . . . . . . . . . . 116
Output from Multiple Models . . . . . . . . . . . . . . . . . . . . . . . . 119
Viewing Graphics Output for Individual Models . . . . . . . . . . . 119
Viewing Fit Statistics for All Four Models. . . . . . . . . . . . . . . 119
Obtaining Optional Output . . . . . . . . . . . . . . . . . . . . . . . 121
Obtaining Tables of Indirect, Direct, and Total Effects . . . . . . . 122
vii
Modeling in VB.NET . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
Model A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
Model B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
Model C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
Fitting Multiple Models. . . . . . . . . . . . . . . . . . . . . . . . . 126
7 A Nonrecursive Model 129
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
About the Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
Felson and Bohrnstedt’s Model . . . . . . . . . . . . . . . . . . . . . . 130
Model Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
Results of the Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
Text Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
Obtaining Standardized Estimates . . . . . . . . . . . . . . . . . . 133
Obtaining Squared Multiple Correlations . . . . . . . . . . . . . . 133
Graphics Output. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
Stability Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
Modeling in VB.NET . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
8 Factor Analysis 137
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
About the Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
A Common Factor Model . . . . . . . . . . . . . . . . . . . . . . . . . . 138
Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
Specifying the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
Drawing the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
Results of the Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
Obtaining Standardized Estimates . . . . . . . . . . . . . . . . . . 142
Viewing Standardized Estimates . . . . . . . . . . . . . . . . . . . 143
Modeling in VB.NET . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
viii
9 An Alternative to Analysis of Covariance 145
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
Analysis of Covariance and Its Alternative . . . . . . . . . . . . . . . . 145
About the Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
Analysis of Covariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
Model A for the Olsson Data. . . . . . . . . . . . . . . . . . . . . . . . . 147
Identification. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
Specifying Model A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
Results for Model A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
Searching for a Better Model . . . . . . . . . . . . . . . . . . . . . . . . 149
Requesting Modification Indices . . . . . . . . . . . . . . . . . . . 149
Model B for the Olsson Data. . . . . . . . . . . . . . . . . . . . . . . . . 150
Results for Model B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
Model C for the Olsson Data . . . . . . . . . . . . . . . . . . . . . . . . . 153
Drawing a Path Diagram for Model C . . . . . . . . . . . . . . . . . 154
Results for Model C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
Fitting All Models At Once . . . . . . . . . . . . . . . . . . . . . . . . . . 154
Modeling in VB.NET. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
Model A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
Model B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
Model C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
Fitting Multiple Models . . . . . . . . . . . . . . . . . . . . . . . . . 157
10 Simultaneous Analysis of Several Groups 159
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
Analysis of Several Groups . . . . . . . . . . . . . . . . . . . . . . . . . 159
About the Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
Model A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
Conventions for Specifying Group Differences . . . . . . . . . . . 161
Specifying Model A . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
Text Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
Graphics Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
ix
Model B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
Text Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
Graphics Output. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
Modeling in VB.NET . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
Model A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
Model B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
Multiple Model Input . . . . . . . . . . . . . . . . . . . . . . . . . . 173
11 Felson and Bohrnstedt’s Girls and Boys 175
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
Felson and Bohrnstedt’s Model . . . . . . . . . . . . . . . . . . . . . . 175
About the Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
Specifying Model A for Girls and Boys . . . . . . . . . . . . . . . . . . 176
Specifying a Figure Caption . . . . . . . . . . . . . . . . . . . . . . 176
Text Output for Model A . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
Graphics Output for Model A . . . . . . . . . . . . . . . . . . . . . . . . 181
Obtaining Critical Ratios for Parameter Differences . . . . . . . . 182
Model B for Girls and Boys . . . . . . . . . . . . . . . . . . . . . . . . . 182
Results for Model B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
Text Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
Graphics Output. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
Fitting Models A and B in a Single Analysis . . . . . . . . . . . . . . . 188
Model C for Girls and Boys . . . . . . . . . . . . . . . . . . . . . . . . . 188
Results for Model C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
Modeling in VB.NET . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
Model A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
Model B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
Model C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
Fitting Multiple Models. . . . . . . . . . . . . . . . . . . . . . . . . 194
x
12 Simultaneous Factor Analysis for
Several Groups 195
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
About the Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
Model A for the Holzinger and Swineford Boys and Girls . . . . . . . . 196
Naming the Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
Specifying the Data . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
Results for Model A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
Text Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
Graphics Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
Model B for the Holzinger and Swineford Boys and Girls . . . . . . . . 200
Results for Model B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
Text Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
Graphics Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
Modeling in VB.NET. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
Model A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
Model B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
13 Estimating and Testing Hypotheses
about Means 209
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
Means and Intercept Modeling . . . . . . . . . . . . . . . . . . . . . . . 209
About the Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
Model A for Young and Old Subjects . . . . . . . . . . . . . . . . . . . . 210
Mean Structure Modeling in Amos Graphics . . . . . . . . . . . . . . . 210
Results for Model A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
Text Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
Graphics Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
Model B for Young and Old Subjects . . . . . . . . . . . . . . . . . . . . 214
Results for Model B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
Comparison of Model B with Model A . . . . . . . . . . . . . . . . . . . 216
xi
Multiple Model Input. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
Mean Structure Modeling in VB.NET . . . . . . . . . . . . . . . . . . . 217
Model A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
Model B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
Fitting Multiple Models. . . . . . . . . . . . . . . . . . . . . . . . . 219
14 Regression with an Explicit Intercept 221
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
Assumptions Made by Amos . . . . . . . . . . . . . . . . . . . . . . . . 221
About the Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
Specifying the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
Results of the Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
Text Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
Graphics Output. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
Modeling in VB.NET . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
15 Factor Analysis with Structured Means 229
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
Factor Means . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
About the Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
Model A for Boys and Girls . . . . . . . . . . . . . . . . . . . . . . . . . 230
Specifying the Model. . . . . . . . . . . . . . . . . . . . . . . . . . 230
Understanding the Cross-Group Constraints . . . . . . . . . . . . . . . 232
Results for Model A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
Text Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
Graphics Output. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
Model B for Boys and Girls . . . . . . . . . . . . . . . . . . . . . . . . . 235
Results for Model B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
Comparing Models A and B. . . . . . . . . . . . . . . . . . . . . . . . . 237
xii
Modeling in VB.NET. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
Model A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
Model B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
Fitting Multiple Models . . . . . . . . . . . . . . . . . . . . . . . . . 240
16 Sörbom’s Alternative to
Analysis of Covariance 241
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
About the Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
Changing the Default Behavior . . . . . . . . . . . . . . . . . . . . . . . 243
Model A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
Specifying the Model . . . . . . . . . . . . . . . . . . . . . . . . . . 243
Results for Model A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
Text Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
Model B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
Results for Model B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
Model C. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
Results for Model C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
Model D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252
Results for Model D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
Model E. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
Results for Model E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
Fitting Models A Through E in a Single Analysis . . . . . . . . . . . . . 255
Comparison of Sörbom’s Method with the Method of Example 9 . . . . 256
Model X. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
Modeling in Amos Graphics . . . . . . . . . . . . . . . . . . . . . . . . . 256
Results for Model X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
Model Y. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
Results for Model Y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
Model Z. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260
xiii
Results for Model Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
Modeling in VB.NET . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262
Model A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262
Model B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
Model C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264
Model D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
Model E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266
Fitting Multiple Models. . . . . . . . . . . . . . . . . . . . . . . . . 267
Models X, Y, and Z . . . . . . . . . . . . . . . . . . . . . . . . . . . 268
17 Missing Data 269
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
Incomplete Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
About the Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270
Specifying the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
Saturated and Independence Models . . . . . . . . . . . . . . . . . . . 272
Results of the Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
Text Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
Graphics Output. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275
Modeling in VB.NET . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275
Fitting the Factor Model (Model A) . . . . . . . . . . . . . . . . . . 276
Fitting the Saturated Model (Model B) . . . . . . . . . . . . . . . . 277
Computing the Likelihood Ratio Chi-Square Statistic and P . . . . 281
Performing All Steps with One Program . . . . . . . . . . . . . . . 282
18 More about Missing Data 283
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
Missing Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
About the Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284
Model A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
Results for Model A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
xiv
Graphics Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
Text Output. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
Model B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290
Output from Models A and B. . . . . . . . . . . . . . . . . . . . . . . . . 291
Modeling in VB.NET. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292
Model A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292
Model B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293
19 Bootstrapping 295
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295
The Bootstrap Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295
About the Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296
A Factor Analysis Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 296
Monitoring the Progress of the Bootstrap . . . . . . . . . . . . . . . . . 297
Results of the Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297
Modeling in VB.NET. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301
20 Bootstrapping for Model Comparison 303
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303
Bootstrap Approach to Model Comparison . . . . . . . . . . . . . . . . 303
About the Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304
Five Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304
Text Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310
Modeling in VB.NET. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310
21 Bootstrapping to Compare
Estimation Methods 311
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311
Estimation Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311
xv
About the Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312
About the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312
Text Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315
Modeling in VB.NET . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318
22 Specification Search 319
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319
About the Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319
About the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319
Specification Search with Few Optional Arrows. . . . . . . . . . . . . 320
Specifying the Model. . . . . . . . . . . . . . . . . . . . . . . . . . 320
Selecting Program Options . . . . . . . . . . . . . . . . . . . . . . 322
Performing the Specification Search . . . . . . . . . . . . . . . . 323
Viewing Generated Models . . . . . . . . . . . . . . . . . . . . . . 324
Viewing Parameter Estimates for a Model . . . . . . . . . . . . . 325
Using BCC to Compare Models . . . . . . . . . . . . . . . . . . . . 326
Viewing the Akaike Weights . . . . . . . . . . . . . . . . . . . . . 327
Using BIC to Compare Models . . . . . . . . . . . . . . . . . . . . 328
Using Bayes Factors to Compare Models . . . . . . . . . . . . . . 329
Rescaling the Bayes Factors . . . . . . . . . . . . . . . . . . . . . 331
Examining the Short List of Models. . . . . . . . . . . . . . . . . . 332
Viewing a Scatterplot of Fit and Complexity. . . . . . . . . . . . . 333
Adjusting the Line Representing Constant Fit . . . . . . . . . . . . 335
Viewing the Line Representing Constant C – df. . . . . . . . . . . 336
Adjusting the Line Representing Constant C – df . . . . . . . . . . 337
Viewing Other Lines Representing Constant Fit. . . . . . . . . . . 338
Viewing the Best-Fit Graph for C . . . . . . . . . . . . . . . . . . . 338
Viewing the Best-Fit Graph for Other Fit Measures . . . . . . . . 339
Viewing the Scree Plot for C . . . . . . . . . . . . . . . . . . . . . 340
Viewing the Scree Plot for Other Fit Measures . . . . . . . . . . . 342
Specification Search with Many Optional Arrows . . . . . . . . . . . . 344
Specifying the Model. . . . . . . . . . . . . . . . . . . . . . . . . . 345
Making Some Arrows Optional . . . . . . . . . . . . . . . . . . . . 345
Setting Options to Their Defaults . . . . . . . . . . . . . . . . . . . 345
xvi
Performing the Specification Search . . . . . . . . . . . . . . . . . 346
Using BIC to Compare Models . . . . . . . . . . . . . . . . . . . . . 347
Viewing the Scree Plot . . . . . . . . . . . . . . . . . . . . . . . . . 348
Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348
23 Exploratory Factor Analysis by
Specification Search 349
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349
About the Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349
About the Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349
Specifying the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350
Opening the Specification Search Window . . . . . . . . . . . . . . . . 350
Making All Regression Weights Optional . . . . . . . . . . . . . . . . . 351
Setting Options to Their Defaults . . . . . . . . . . . . . . . . . . . . . . 351
Performing the Specification Search . . . . . . . . . . . . . . . . . . . . 353
Using BCC to Compare Models . . . . . . . . . . . . . . . . . . . . . . . 354
Viewing the Scree Plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357
Viewing the Short List of Models . . . . . . . . . . . . . . . . . . . . . . 357
Heuristic Specification Search . . . . . . . . . . . . . . . . . . . . . . . 358
Performing a Stepwise Search . . . . . . . . . . . . . . . . . . . . . . . 359
Viewing the Scree Plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360
Limitations of Heuristic Specification Searches . . . . . . . . . . . . . 361
24 Multiple-Group Factor Analysis 363
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363
About the Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363
Model 24a: Modeling Without Means and Intercepts . . . . . . . . . . 363
Specifying the Model . . . . . . . . . . . . . . . . . . . . . . . . . . 364
Opening the Multiple-Group Analysis Dialog Box . . . . . . . . . . 364
Viewing the Parameter Subsets . . . . . . . . . . . . . . . . . . . . 366
Viewing the Generated Models . . . . . . . . . . . . . . . . . . . . 367
Fitting All the Models and Viewing the Output . . . . . . . . . . . . 368
xvii
Customizing the Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 369
Model 24b: Comparing Factor Means . . . . . . . . . . . . . . . . . . . 370
Specifying the Model. . . . . . . . . . . . . . . . . . . . . . . . . . 370
Removing Constraints . . . . . . . . . . . . . . . . . . . . . . . . . 371
Generating the Cross-Group Constraints . . . . . . . . . . . . . . 372
Fitting the Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . 373
Viewing the Output . . . . . . . . . . . . . . . . . . . . . . . . . . . 374
25 Multiple-Group Analysis 377
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377
About the Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377
About the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377
Specifying the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378
Constraining the Latent Variable Means and Intercepts . . . . . . . . 378
Generating Cross-Group Constraints . . . . . . . . . . . . . . . . . . . 379
Fitting the Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381
Viewing the Text Output . . . . . . . . . . . . . . . . . . . . . . . . . . . 381
Examining the Modification Indices . . . . . . . . . . . . . . . . . . . . 382
Modifying the Model and Repeating the Analysis . . . . . . . . . 383
26 Bayesian Estimation 385
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385
Bayesian Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385
Selecting Priors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387
Performing Bayesian Estimation Using Amos Graphics . . . . . . 388
Estimating the Covariance. . . . . . . . . . . . . . . . . . . . . . . 388
Results of Maximum Likelihood Analysis . . . . . . . . . . . . . . . . . 389
Bayesian Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390
Replicating Bayesian Analysis and Data Imputation Results . . . . . . 392
Examining the Current Seed. . . . . . . . . . . . . . . . . . . . . . 392
Changing the Current Seed . . . . . . . . . . . . . . . . . . . . . . 393
Changing the Refresh Options . . . . . . . . . . . . . . . . . . . . 395
xviii
Assessing Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . 396
Diagnostic Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398
Bivariate Marginal Posterior Plots . . . . . . . . . . . . . . . . . . . . . 404
Credible Intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407
Changing the Confidence Level . . . . . . . . . . . . . . . . . . . . 407
Learning More about Bayesian Estimation . . . . . . . . . . . . . . . . 408
27 Bayesian Estimation Using a
Non-Diffuse Prior Distribution 409
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409
About the Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409
More about Bayesian Estimation . . . . . . . . . . . . . . . . . . . . . . 409
Bayesian Analysis and Improper Solutions . . . . . . . . . . . . . . . . 410
About the Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410
Fitting a Model by Maximum Likelihood . . . . . . . . . . . . . . . . . . 411
Bayesian Estimation with a Non-Informative (Diffuse) Prior. . . . . . . 412
Changing the Number of Burn-In Observations . . . . . . . . . . . 412
28 Bayesian Estimation of Values
Other Than Model Parameters 423
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423
About the Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423
The Wheaton Data Revisited . . . . . . . . . . . . . . . . . . . . . . . . 423
Indirect Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424
Estimating Indirect Effects . . . . . . . . . . . . . . . . . . . . . . . 425
Bayesian Analysis of Model C . . . . . . . . . . . . . . . . . . . . . . . . 427
Additional Estimands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428
Inferences about Indirect Effects . . . . . . . . . . . . . . . . . . . . . . 431
xix
29 Estimating a User-Defined Quantity
in Bayesian SEM 437
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437
About the Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437
The Stability of Alienation Model . . . . . . . . . . . . . . . . . . . . . 437
Numeric Custom Estimands. . . . . . . . . . . . . . . . . . . . . . . . . 443
Dragging and Dropping . . . . . . . . . . . . . . . . . . . . . . . . 447
Dichotomous Custom Estimands . . . . . . . . . . . . . . . . . . . . . . 457
Defining a Dichotomous Estimand . . . . . . . . . . . . . . . . . . 457
30 Data Imputation 461
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461
About the Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461
Multiple Imputation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462
Model-Based Imputation . . . . . . . . . . . . . . . . . . . . . . . . . . 462
Performing Multiple Data Imputation Using Amos Graphics . . . . . . 462
31 Analyzing Multiply Imputed Datasets 469
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469
Analyzing the Imputed Data Files Using SPSS Statistics . . . . . . . . 469
Step 2: Ten Separate Analyses . . . . . . . . . . . . . . . . . . . . . . . 470
Step 3: Combining Results of Multiply Imputed Data Files . . . . . . . 471
Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473
32 Censored Data 475
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475
About the Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475
Recoding the Data . . . . . . . . . . . . . . . . . . . . . . . . . . . 477
Analyzing the Data . . . . . . . . . . . . . . . . . . . . . . . . . . . 477
Performing a Regression Analysis . . . . . . . . . . . . . . . . . . 478
xx
Posterior Predictive Distributions . . . . . . . . . . . . . . . . . . . . . . 481
Imputation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484
General Inequality Constraints on Data Values . . . . . . . . . . . . . . 488
33 Ordered-Categorical Data 489
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489
About the Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489
Specifying the Data File . . . . . . . . . . . . . . . . . . . . . . . . . 491
Recoding the Data within Amos . . . . . . . . . . . . . . . . . . . . 492
Specifying the Model . . . . . . . . . . . . . . . . . . . . . . . . . . 500
Fitting the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501
MCMC Diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504
Posterior Predictive Distributions . . . . . . . . . . . . . . . . . . . . . . 506
Posterior Predictive Distributions for Latent Variables. . . . . . . . . . 511
Imputation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516
34 Mixture Modeling with Training Data 521
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521
About the Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521
Performing the Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 524
Specifying the Data File . . . . . . . . . . . . . . . . . . . . . . . . . . . 526
Specifying the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 530
Fitting the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532
Classifying Individual Cases . . . . . . . . . . . . . . . . . . . . . . . . . 535
Latent Structure Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 537
35 Mixture Modeling without Training Data 539
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539
About the Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539
Performing the Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 540
xxi
Specifying the Data File . . . . . . . . . . . . . . . . . . . . . . . . . . . 542
Specifying the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545
Constraining the Parameters . . . . . . . . . . . . . . . . . . . . . 546
Fitting the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 548
Classifying Individual Cases . . . . . . . . . . . . . . . . . . . . . . . . 551
Latent Structure Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 553
Label Switching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554
36 Mixture Regression Modeling 557
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557
About the Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557
First Dataset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557
Second Dataset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 559
The Group Variable in the Dataset . . . . . . . . . . . . . . . . . . 560
Performing the Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 561
Specifying the Data File . . . . . . . . . . . . . . . . . . . . . . . . . . . 563
Specifying the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 566
Fitting the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 567
Classifying Individual Cases . . . . . . . . . . . . . . . . . . . . . . . . 572
Improving Parameter Estimates . . . . . . . . . . . . . . . . . . . . . . 573
Prior Distribution of Group Proportions . . . . . . . . . . . . . . . . . . 575
Label Switching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 576
37 Using Amos Graphics
without Drawing a Path Diagram 577
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 577
About the Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 578
A Common Factor Model . . . . . . . . . . . . . . . . . . . . . . . . . . 578
Creating a Plugin to Specify the Model . . . . . . . . . . . . . . . 578
Controlling Undo Capability . . . . . . . . . . . . . . . . . . . . . . 583
Compiling and Saving the Plugin . . . . . . . . . . . . . . . . . . . 585
Using the Plugin. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586
xxii
Other Aspects of the Analysis in Addition to Model Specification . . . 588
Defining Program Variables that Correspond to
Model Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 588
Part III: Appendices
A Notation 591
B Discrepancy Functions 593
C Measures of Fit 597
Measures of Parsimony . . . . . . . . . . . . . . . . . . . . . . . . . . . 598
NPAR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 598
DF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 598
PRATIO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 599
Minimum Sample Discrepancy Function . . . . . . . . . . . . . . . . . . 599
CMIN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 599
P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 599
CMIN/DF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 601
FMIN. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 602
Measures Based On the Population Discrepancy . . . . . . . . . . . . 602
NCP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 602
F0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603
RMSEA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603
PCLOSE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605
Information-Theoretic Measures . . . . . . . . . . . . . . . . . . . . . . 605
AIC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605
BCC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 606
BIC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 606
CAIC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 607
ECVI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 607
MECVI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 608
xxiii
Comparisons to a Baseline Model . . . . . . . . . . . . . . . . . . . . . 608
NFI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 609
RFI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 610
IFI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 611
TLI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 611
CFI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 612
Parsimony Adjusted Measures. . . . . . . . . . . . . . . . . . . . . . . 612
PNFI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613
PCFI. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613
GFI and Related Measures . . . . . . . . . . . . . . . . . . . . . . . . . 613
GFI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613
AGFI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614
PGFI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615
Miscellaneous Measures . . . . . . . . . . . . . . . . . . . . . . . . . . 615
HI 90 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615
HOELTER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615
LO 90 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 616
RMR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 616
Selected List of Fit Measures. . . . . . . . . . . . . . . . . . . . . . . . 617
D Numeric Diagnosis of Non-Identifiability 619
E Using Fit Measures to Rank Models 621
F Baseline Models for
Descriptive Fit Measures 625
G Rescaling of AIC, BCC, and BIC 627
Zero-Based Rescaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 627
Akaike Weights and Bayes Factors (Sum = 1) . . . . . . . . . . . . . . 628
Akaike Weights and Bayes Factors (Max = 1) . . . . . . . . . . . . . . 629
xxiv
Notices 631
Bibliography 635
Index 647
xxv
spatial
visperc
cubes
lozenges
wordmean
paragraph
sentence
e1
e2
e3
e4
e5
e6
verbal
1
1
1
1
1
1
1
1
Input:
spatial
visperc
cubes
.43
lozenges
.54
wordmean
.71
paragraph
.77
sentence
.68
e1
e2
e3
e4
e5
e6
verbal
.70
.65
.74
.88
.83
.84
.49
Chi-square = 7.853 (8 df)
p = .448
Output:
Introduction
IBM SPSS Amos implements the general approach to data analysis known as
structural equation modeling (SEM), also known as analysis of covariance
structures, or causal modeling. This approach includes, as special cases, many well-
known conventional techniques, including the general linear model and common
factor analysis.
Chapter
1
IBM SPSS Amos (Analysis of Moment Structures) is an easy-to-use program for
visual SEM. With Amos, you can quickly specify, view, and modify your model
graphically using simple drawing tools. Then you can assess your model’s fit, make
any modifications, and print out a publication-quality graphic of your final model.
Simply specify the model graphically (left). Amos quickly performs the
computations and displays the results (right).
1
2
Chapter 1
Structural equation modeling (SEM) is sometimes thought of as esoteric and difficult
to learn and use. This is incorrect. Indeed, the growing importance of SEM in data
analysis is largely due to its ease of use. SEM opens the door for nonstatisticians to
solve estimation and hypothesis testing problems that once would have required the
services of a specialist.
IBM SPSS Amos was originally designed as a tool for teaching this powerful and
fundamentally simple method. For this reason, every effort was made to see that it is
easy to use. Amos integrates an easy-to-use graphical interface with an advanced
computing engine for SEM. The publication-quality path diagrams of Amos provide a
clear representation of models for students and fellow researchers. The numeric
methods implemented in Amos are among the most effective and reliable available.
Featured Methods
Amos provides the following methods for estimating structural equation models:
Maximum likelihood
Unweighted least squares
Generalized least squares
Browne’s asymptotically distribution-free criterion
Scale-free least squares
Bayesian estimation
IBM SPSS Amos goes well beyond the usual capabilities found in other structural
equation modeling programs. When confronted with missing data, Amos performs
state-of-the-art estimation by full information maximum likelihood instead of relying
on ad-hoc methods like listwise or pairwise deletion, or mean imputation. The program
can analyze data from several populations at once. It can also estimate means for
exogenous variables and intercepts in regression equations.
The program makes bootstrapped standard errors and confidence intervals available
for all parameter estimates, effect estimates, sample means, variances, covariances,
and correlations. It also implements percentile intervals and bias-corrected percentile
intervals (Stine, 1989), as well as Bollen and Stine’s (1992) bootstrap approach to
model testing.
Multiple models can be fitted in a single analysis. Amos examines every pair of
models in which one model can be obtained by placing restrictions on the parameters
of the other. The program reports several statistics appropriate for comparing such
models. It provides a test of univariate normality for each observed variable as well as
a test of multivariate normality and attempts to detect outliers.
IBM SPSS Amos accepts a path diagram as a model specification and displays
parameter estimates graphically on a path diagram. Path diagrams used for model
specification and those that display parameter estimates are of presentation quality.
They can be printed directly or imported into other applications such as word
processors, desktop publishing programs, and general-purpose graphics programs.
About the Tutorial
The tutorial is designed to get you up and running with Amos Graphics. It covers some
of the basic functions and features and guides you through your first Amos analysis.
Once you have worked through the tutorial, you can learn about more advanced
functions using the online Help, or you can continue working through the examples to
get a more extended introduction to structural modeling with IBM SPSS Amos.
3
Introduction
About the Examples
Many people like to learn by doing. Knowing this, we have developed many examples
that quickly demonstrate practical ways to use IBM SPSS Amos. The initial examples
introduce the basic capabilities of Amos as applied to simple problems. You learn
which buttons to click, how to access the several supported data formats, and how to
maneuver through the output. Later examples tackle more advanced modeling
problems and are less concerned with program interface issues.
Examples 1 through 4 show how you can use Amos to do some conventional
analyses—analyses that could be done using a standard statistics package. These
examples show a new approach to some familiar problems while also demonstrating
all of the basic features of Amos. There are sometimes good reasons for using Amos
to do something simple, like estimating a mean or correlation or testing the hypothesis
that two means are equal. For one thing, you might want to take advantage of the ability
of Amos to handle missing data. Or maybe you want to use the bootstrapping capability
of Amos, particularly to obtain confidence intervals.
Examples 5 through 8 illustrate the basic techniques that are commonly used
nowadays in structural modeling.
4
Chapter 1
Example 9 and those that follow demonstrate advanced techniques that have so far not
been used as much as they deserve. These techniques include:
Simultaneous analysis of data from several different populations.
Estimation of means and intercepts in regression equations.
Maximum likelihood estimation in the presence of missing data.
Bootstrapping to obtain estimated standard errors and confidence intervals. Amos
makes these techniques especially easy to use, and we hope that they will become
more commonplace.
Specification searches.
Bayesian estimation.
Imputation of missing values.
Analysis of censored data.
Analysis of ordered-categorical data.
Mixture modeling.
Tip: If you have questions about a particular Amos feature, you can always refer to the
extensive online Help provided by the program.
About the Documentation
IBM SPSS Amos 21 comes with extensive documentation, including an online Help
system, this user’s guide, and advanced reference material for Amos Basic and the
Amos API (Application Programming Interface). If you performed a typical
installation, you can find the IBM SPSS Amos 21 Programming Reference Guide in
the following location: C:\Program
Files\IBM\SPSS\Amos\21\Documentation\Programming Reference.pdf.
Other Sources of Information
Although this user’s guide contains a good bit of expository material, it is not by any
means a complete guide to the correct and effective use of structural modeling. Many
excellent SEM textbooks are available.
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