Hướng dẫn sử dụng AMOS toàn tập

Amos
™
6.0 User’s Guide
James L. Arbuckle
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Amos 6.0 User’s Guide
Copyright © 1995–2005 by Amos Development Corporation
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iii
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 New Features 7
Bayesian Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Data Imputation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Print Preview for Path Diagrams . . . . . . . . . . . . . . . . . . . . . . . 8
Improved Zooming and Scrolling. . . . . . . . . . . . . . . . . . . . . . . . 9
Drawing Path Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Copying Path Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Multiple Path Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Incompatibilities with Amos 5.0 . . . . . . . . . . . . . . . . . . . . . . . 11
Other Changes between Amos 5.0 and Amos 6.0 . . . . . . . . . . . . . 11
iv
3 Tutorial: Getting Started with
Amos Graphics 13
Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
About the Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
Launching Amos Graphics . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Creating a New Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
Specifying the Data File . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
Specifying the Model and Drawing Variables . . . . . . . . . . . . . . . 17
Naming the Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Drawing Arrows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
Constraining a Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
Altering the Appearance of a Path Diagram . . . . . . . . . . . . . . . . 21
Setting Up Optional Output . . . . . . . . . . . . . . . . . . . . . . . . . . 22
Performing the Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
Viewing Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
Printing the Path Diagram. . . . . . . . . . . . . . . . . . . . . . . . . . . 26
Copying the Path Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . 27
Copying Text Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
Part II: Examples
1 Estimating Variances and Covariances 29
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
About the Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
Bringing In the Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
Analyzing the Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
Viewing Graphics Output . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
Viewing Text Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
v
Optional Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .39
Distribution Assumptions for Amos Models . . . . . . . . . . . . . . . . .41
Modeling in VB.NET. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .42
Modeling in C#. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .45
Other Program Development Tools . . . . . . . . . . . . . . . . . . . . . .46
2 Testing Hypotheses 47
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .47
About the Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .47

Parameters Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . .47
Moving and Formatting Objects . . . . . . . . . . . . . . . . . . . . . . . .51
Data Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .52
Optional Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .54
Labeling Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .57
Hypothesis Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .58
Displaying Chi-Square Statistics on the Path Diagram . . . . . . . . . . .59
Modeling in VB.NET. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .61
3 More Hypothesis Testing 65
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .65
About the Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .65
Bringing In the Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .65
Testing a Hypothesis That Two Variables Are Uncorrelated . . . . . . .66
Specifying the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .66
Viewing Text Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .68
Viewing Graphics Output. . . . . . . . . . . . . . . . . . . . . . . . . . . .69
Modeling in VB.NET. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .71
vi
4 Conventional Linear Regression 73
Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
About the Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
Analysis of the Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
Specifying the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
Fixing Regression Weights . . . . . . . . . . . . . . . . . . . . . . . . . . 77
Viewing the Text Output. . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
Viewing Graphics Output . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
Viewing Additional Text Output. . . . . . . . . . . . . . . . . . . . . . . . 82
Modeling in VB.NET . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5 Unobserved Variables 87

Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
About the Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
Model A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
Measurement Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
Structural Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
Specifying the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
Results for Model A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
Model B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
Results for Model B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
Testing Model B against Model A . . . . . . . . . . . . . . . . . . . . . 102
Modeling in VB.NET . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
vii
6 Exploratory Analysis 107
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
About the Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
Model A for the Wheaton Data . . . . . . . . . . . . . . . . . . . . . . . 108
Model B for the Wheaton Data . . . . . . . . . . . . . . . . . . . . . . . 113
Model C for the Wheaton Data . . . . . . . . . . . . . . . . . . . . . . . 120
Multiple Models in a Single Analysis . . . . . . . . . . . . . . . . . . . . 122
Output from Multiple Models . . . . . . . . . . . . . . . . . . . . . . . . 125
Modeling in VB.NET. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
7 A Nonrecursive Model 135
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
About the Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
Felson and Bohrnstedt’s Model . . . . . . . . . . . . . . . . . . . . . . . 136
Model Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
Results of the Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
Modeling in VB.NET. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
8 Factor Analysis 143

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
About the Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
A Common Factor Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
Identification. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
Specifying the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
Results of the Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
Modeling in VB.NET. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
viii
9 An Alternative to Analysis of Covariance 151
Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
Analysis of Covariance and Its Alternative . . . . . . . . . . . . . . . . 151
About the Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
Analysis of Covariance . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
Model A for the Olsson Data . . . . . . . . . . . . . . . . . . . . . . . . 153
Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
Specifying Model A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
Results for Model A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
Searching for a Better Model. . . . . . . . . . . . . . . . . . . . . . . . 155
Model B for the Olsson Data . . . . . . . . . . . . . . . . . . . . . . . . 156
Results for Model B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
Model C for the Olsson Data . . . . . . . . . . . . . . . . . . . . . . . . 159
Results for Model C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
Fitting All Models At Once . . . . . . . . . . . . . . . . . . . . . . . . . 160
Modeling in VB.NET . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
10 Simultaneous Analysis of Several Groups 165
Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
Analysis of Several Groups . . . . . . . . . . . . . . . . . . . . . . . . . 165
About the Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
Model A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
Model B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

Modeling in VB.NET . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
ix
11 Felson and Bohrnstedt’s Girls and Boys 181
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
Felson and Bohrnstedt’s Model . . . . . . . . . . . . . . . . . . . . . . . 181
About the Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
Specifying Model A for Girls and Boys . . . . . . . . . . . . . . . . . . . 182
Text Output for Model A . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
Graphics Output for Model A . . . . . . . . . . . . . . . . . . . . . . . . 187
Model B for Girls and Boys . . . . . . . . . . . . . . . . . . . . . . . . . 188
Results for Model B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
Fitting Models A and B in a Single Analysis . . . . . . . . . . . . . . . . 194
Model C for Girls and Boys. . . . . . . . . . . . . . . . . . . . . . . . . . 194
Results for Model C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
Modeling in VB.NET. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
12 Simultaneous Factor Analysis for
Several Groups 201
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
About the Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
Model A for the Holzinger and Swineford Boys and Girls . . . . . . . . 202
Results for Model A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
Model B for the Holzinger and Swineford Boys and Girls . . . . . . . . 206
Results for Model B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
Modeling in VB.NET. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
x
13 Estimating and Testing Hypotheses
about Means 215
Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
Means and Intercept Modeling . . . . . . . . . . . . . . . . . . . . . . 215
About the Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

Model A for Young and Old Subjects . . . . . . . . . . . . . . . . . . . 216
Mean Structure Modeling in Amos Graphics. . . . . . . . . . . . . . . 216
Results for Model A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
Model B for Young and Old Subjects . . . . . . . . . . . . . . . . . . . 220
Results for Model B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
Comparison of Model B with Model A. . . . . . . . . . . . . . . . . . . 222
Multiple Model Input. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
Mean Structure Modeling in VB.NET . . . . . . . . . . . . . . . . . . . 223
14 Regression with an Explicit Intercept 227
Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
Assumptions Made by Amos . . . . . . . . . . . . . . . . . . . . . . . . 227
About the Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
Specifying the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
Results of the Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
Modeling in VB.NET . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
15 Factor Analysis with Structured Means 235
Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
Factor Means. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
About the Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236
xi
Model A for Boys and Girls . . . . . . . . . . . . . . . . . . . . . . . . . 236
Understanding the Cross-Group Constraints . . . . . . . . . . . . . . . 238
Results for Model A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
Model B for Boys and Girls . . . . . . . . . . . . . . . . . . . . . . . . . 241
Results for Model B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
Comparing Models A and B . . . . . . . . . . . . . . . . . . . . . . . . . 243
Modeling in VB.NET. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
16 Sörbom’s Alternative to Analysis of
Covariance 247
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247

Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
About the Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
Changing the Default Behavior . . . . . . . . . . . . . . . . . . . . . . . 249
Model A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
Results for Model A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
Model B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
Results for Model B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
Model C. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
Results for Model C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
Model D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258
Results for Model D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
Model E. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
Results for Model E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
Fitting Models A Through E in a Single Analysis . . . . . . . . . . . . . 261
Comparison of Sörbom’s Method with the Method of Example 9 . . . . 262
Model X. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262
Modeling in Amos Graphics . . . . . . . . . . . . . . . . . . . . . . . . . 262
Results for Model X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
xii
Model Y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
Results for Model Y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
Model Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266
Results for Model Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
Modeling in VB.NET . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268
17 Missing Data 275
Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275
Incomplete Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275
About the Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276
Specifying the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
Saturated and Independence Models. . . . . . . . . . . . . . . . . . . 278

Results of the Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
Modeling in VB.NET . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
18 More about Missing Data 289
Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
Missing Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
About the Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290
Model A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
Results for Model A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293
Model B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296
Output from Models A and B . . . . . . . . . . . . . . . . . . . . . . . . 297
Modeling in VB.NET . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298
xiii
19 Bootstrapping 301
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301
The Bootstrap Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301
About the Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302
A Factor Analysis Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 302
Monitoring the Progress of the Bootstrap . . . . . . . . . . . . . . . . . 303
Results of the Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303
Modeling in VB.NET. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307
20 Bootstrapping for Model Comparison 309
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309
Bootstrap Approach to Model Comparison . . . . . . . . . . . . . . . . 309
About the Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310
Five Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316
Modeling in VB.NET. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316
21 Bootstrapping to Compare Estimation
Methods 317
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317

Estimation Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317
About the Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318
About the Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318
Modeling in VB.NET. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324
xiv
22 Specification Search 325
Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325
About the Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325
About the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325
Specification Search with Few Optional Arrows. . . . . . . . . . . . . 326
Specification Search with Many Optional Arrows . . . . . . . . . . . . 351
Limitations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355
23 Exploratory Factor Analysis by
Specification Search 357
Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357
About the Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357
About the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357
Specifying the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358
Opening the Specification Search Window . . . . . . . . . . . . . . . 358
Making All Regression Weights Optional . . . . . . . . . . . . . . . . . 359
Setting Options to Their Defaults. . . . . . . . . . . . . . . . . . . . . . 359
Performing the Specification Search . . . . . . . . . . . . . . . . . . . 361
Using BCC to Compare Models. . . . . . . . . . . . . . . . . . . . . . . 362
Viewing the Scree Plot . . . . . . . . . . . . . . . . . . . . . . . . . . . 365
Viewing the Short List of Models. . . . . . . . . . . . . . . . . . . . . . 365
Heuristic Specification Search. . . . . . . . . . . . . . . . . . . . . . . 366
Performing a Stepwise Search . . . . . . . . . . . . . . . . . . . . . . . 367
Viewing the Scree Plot . . . . . . . . . . . . . . . . . . . . . . . . . . . 368
Limitations of Heuristic Specification Searches . . . . . . . . . . . . . 369
xv

24 Multiple-Group Factor Analysis 371
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371
About the Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371
Model 24a: Modeling Without Means and Intercepts . . . . . . . . . . 371
Customizing the Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 377
Model 24b: Comparing Factor Means . . . . . . . . . . . . . . . . . . . 378
25 Multiple-Group Analysis 385
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385
About the Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385
About the Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385
Specifying the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386
Constraining the Latent Variable Means and Intercepts . . . . . . . . . 386
Generating Cross-Group Constraints . . . . . . . . . . . . . . . . . . . . 387
Fitting the Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389
Viewing the Text Output . . . . . . . . . . . . . . . . . . . . . . . . . . . 389
Examining the Modification Indices . . . . . . . . . . . . . . . . . . . . 390
26 Bayesian Estimation 393
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393
Bayesian Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393
Results of Maximum Likelihood Analysis. . . . . . . . . . . . . . . . . . 397
Bayesian Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398
Replicating Bayesian Analysis and Data Imputation Results . . . . . . 400
Assessing Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . 404
Diagnostic Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406
xvi
Bivariate Marginal Posterior Plots . . . . . . . . . . . . . . . . . . . . . 412
Credible Intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415
Learning More about Bayesian Estimation . . . . . . . . . . . . . . . . 416
27 Bayesian Estimation Using a
Non-Diffuse Prior Distribution 417

Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417
About the Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417
More about Bayesian Estimation. . . . . . . . . . . . . . . . . . . . . . 417
Bayesian Analysis and Improper Solutions . . . . . . . . . . . . . . . . 418
About the Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418
Fitting a Model by Maximum Likelihood. . . . . . . . . . . . . . . . . . 419
Bayesian Estimation with a Non-Informative (Diffuse) Prior . . . . . . 420
28 Bayesian Estimation of Values Other
Than Model Parameters 431
Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431
About the Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431
The Wheaton Data Revisited . . . . . . . . . . . . . . . . . . . . . . . . 431
Indirect Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432
Bayesian Analysis of Model C . . . . . . . . . . . . . . . . . . . . . . . 435
Additional Estimands . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436
Inferences about Indirect Effects . . . . . . . . . . . . . . . . . . . . . 439
xvii
29 Estimating a User-Defined Quantity
in Bayesian SEM 445
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445
About the Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445
The Stability of Alienation Model . . . . . . . . . . . . . . . . . . . . . . 445
Numeric Custom Estimands . . . . . . . . . . . . . . . . . . . . . . . . . 451
Dichotomous Custom Estimands . . . . . . . . . . . . . . . . . . . . . . 465
30 Data Imputation 469
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469
About the Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469
Multiple Imputation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 470
Model-Based Imputation. . . . . . . . . . . . . . . . . . . . . . . . . . . 470
Performing Multiple Data Imputation Using Amos Graphics . . . . . . 470

31 Analyzing Multiply Imputed Data Sets 477
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477
Analyzing the Imputed Data Files Using Amos Graphics . . . . . . . . . 477
Step 2: Ten Separate Analyses . . . . . . . . . . . . . . . . . . . . . . . 478
Step 3: Combining Results of Multiply Imputed Data Files . . . . . . . . 479
Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481
xviii
Part III: Appendices
A Notation 483
B Discrepancy Functions 485
C Measures of Fit 489
Measures of Parsimony . . . . . . . . . . . . . . . . . . . . . . . . . . . 490
Minimum Sample Discrepancy Function . . . . . . . . . . . . . . . . . 491
Measures Based On the Population Discrepancy . . . . . . . . . . . . 494
Information-Theoretic Measures . . . . . . . . . . . . . . . . . . . . . 497
Comparisons to a Baseline Model . . . . . . . . . . . . . . . . . . . . . 500
Parsimony Adjusted Measures. . . . . . . . . . . . . . . . . . . . . . . 504
GFI and Related Measures . . . . . . . . . . . . . . . . . . . . . . . . . 505
Miscellaneous Measures . . . . . . . . . . . . . . . . . . . . . . . . . . 507
Selected List of Fit Measures. . . . . . . . . . . . . . . . . . . . . . . . 509
D Numeric Diagnosis of Non-Identifiability 511
E Using Fit Measures to Rank Models 513
F Baseline Models for Descriptive
Fit Measures 517
xix
G Rescaling of AIC, BCC, and BIC 519
Zero-Based Rescaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519
Akaike Weights and Bayes Factors (Sum = 1) . . . . . . . . . . . . . . . 520
Akaike Weights and Bayes Factors (Max = 1) . . . . . . . . . . . . . . . 521
Bibliography 523

Index 535

1


Chapter
1
Introduction
Amos is short for Analysis of MOment Structures. It 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.
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).
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:
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.
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
 Unweighted least squares
 Generalized least squares
 Browne’s asymptotically distribution-free criterion

 Scale-free least squares
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
3
Introduction
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.
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 Amos.

About the Examples
Many people like to learn by doing. Knowing this, we have developed 31 examples that
quickly demonstrate practical ways to use 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. Amos makes these techniques
especially easy to use, and we hope that they will become more commonplace.
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
Amos 6.0 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 Amos 6.0 Programming Reference Guide in the following location:
C:\Program Files\Amos 6\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.
 Structural Equation Modeling: A Multidisciplinary Journal contains
methodological articles as well as applications of structural modeling. It is
published by:
Lawrence Erlbaum Associates, Inc.
Journal Subscription Department
10 Industrial Avenue
Mahwah, NJ 07430-2262 USA
www.erlbaum.com
5
Introduction
 Carl Ferguson and Edward Rigdon established an electronic mailing list called
Semnet to provide a forum for discussions related to structural modeling. You can
find information about subscribing to Semnet at
www.gsu.edu/~mkteer/semnet.html.
 Edward Rigdon also maintains a list of frequently asked questions about structural
equation modeling. That FAQ is located at www.gsu.edu/~mkteer/semfaq.html.
Acknowledgements
Tor Neilands wrote the new material for this edition of the user’s guide and provided
suggestions and bug reports as he did for previous Amos versions. Joseph Schafer
reviewed portions of the manuscript and added significant passages, as well as
providing bug reports. John Raz performed testing. Pat O’Neil edited this book.
Numerous users of preliminary versions of the program provided valuable

feedback, including Stephen J. Aragon, Chris Burant, David Burns, Mark A.
Davenport, Kristen diNovi, Akihiro Inoue, Yutaka Kano, Kyle Kercher, Morton
Kleban, Sik-Yum Lee, Michelle Little, Sheela Pandey, Rachel Pruchno, and Shu Zou.
A last word of warning: While Amos Development Corporation and SPSS have
engaged in extensive program testing to ensure that Amos operates correctly, all
complicated software, Amos included, is bound to contain some undetected bugs. We
are committed to correcting any program errors. If you believe you have encountered
one, please report it to the SPSS technical support staff.
James L. Arbuckle
Ambler, Pennsylvania