Amos
™
6.0 User’s Guide
James L. Arbuckle
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Amos 6.0 User’s Guide
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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

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

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

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

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

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

Index 535

1Chapter
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

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

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

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