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d the product it supports, read the in
formation in the Notices section.
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iii
2Testing Hypotheses41
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .41
About the Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .41
4Conventional Linear Regression67
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
Model B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .93
Results for Model B. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .95
Testing Model B against Model A. . . . . . . . . . . . . . . . . . . . . . .97
Obtaining Tables of Indirect, Direct, and Total Effects . . . . . . . 124
Obtaining Squared Multiple Correlations . . . . . . . . . . . . . . 135
Graphics Output. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
Stability Index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
Viewing Standardized Estimates . . . . . . . . . . . . . . . . . . . 145
9An Alternative to Analysis of Covariance147
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
Analysis of Covariance and Its Alternative . . . . . . . . . . . . . . . . 147
About the Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
Analysis of Covariance. . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
Model A for the Olsson Data. . . . . . . . . . . . . . . . . . . . . . . . . 149
Identification. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
Specifying Model A. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
Results for Model A. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
Model B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
Text Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
Graphics Output. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
12Simultaneous Factor Analysis
for Several Groups197
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
About the Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
Model A for the Holzinger and Swineford Boys and Girls . . . . . . . . 198
Naming the Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
Specifying the Data . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
Results for Model A. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
Text Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
Graphics Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
Model B for the Holzinger and Swineford Boys and Girls . . . . . . . . 202
Results for Model B. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
Text Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
Graphics Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
Multiple Model Input. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
Results for Model Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
Model B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292
Output from Models A and B. . . . . . . . . . . . . . . . . . . . . . . . . 293
xvi
About the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314
Text Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317
xvii
Using BIC to Compare Models . . . . . . . . . . . . . . . . . . . . . 349
Viewing the Scree Plot . . . . . . . . . . . . . . . . . . . . . . . . . 350
Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350
23Exploratory Factor Analysis by Specification
Search351
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351
About the Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351
About the Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351
Specifying the Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352
Opening the Specification Search Window . . . . . . . . . . . . . . . . 352
Making All Regression Weights Optional . . . . . . . . . . . . . . . . . 353
Setting Options to Their Defaults . . . . . . . . . . . . . . . . . . . . . . 353
Performing the Specification Search. . . . . . . . . . . . . . . . . . . . 355
Using BCC to Compare Models . . . . . . . . . . . . . . . . . . . . . . . 356
Viewing the Scree Plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359
Viewing the Short List of Models . . . . . . . . . . . . . . . . . . . . . . 359
Heuristic Specification Search . . . . . . . . . . . . . . . . . . . . . . . 360
Performing a Stepwise Search . . . . . . . . . . . . . . . . . . . . . . . 361
Viewing the Scree Plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362
Limitations of Heuristic Specification Searches . . . . . . . . . . . . . 363
24MultipleGroup Factor Analysis365
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365
About the Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365
Model 24a: Modeling Without Means and Intercepts . . . . . . . . . . 365
Specifying the Model . . . . . . . . . . . . . . . . . . . . . . . . . . 366
Opening the MultipleGroup Analysis Dialog Box . . . . . . . . . . 366
xviii
Model 24b: Comparing Factor Means . . . . . . . . . . . . . . . . . . . 372
Specifying the Model. . . . . . . . . . . . . . . . . . . . . . . . . . 372
Removing Constraints . . . . . . . . . . . . . . . . . . . . . . . . . 373
Generating the CrossGroup Constraints . . . . . . . . . . . . . . 374
Fitting the Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . 375
Viewing the Output . . . . . . . . . . . . . . . . . . . . . . . . . . . 376
25MultipleGroup Analysis379
Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379
About the Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379
About the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379
Specifying the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380
Constraining the Latent Variable Means and Intercepts . . . . . . . . 380
Generating CrossGroup Constraints . . . . . . . . . . . . . . . . . . . 381
Fitting the Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383
Viewing the Text Output. . . . . . . . . . . . . . . . . . . . . . . . . . . 383
Examining the Modification Indices . . . . . . . . . . . . . . . . . . . . 384
Modifying the Model and Repeating the Analysis . . . . . . . . . 385
26Bayesian Estimation387
Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387
Bayesian Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387
Selecting Priors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389
Performing Bayesian Estimation Using Amos Graphics . . . . . . 390
Estimating the Covariance. . . . . . . . . . . . . . . . . . . . . . . 390
Results of Maximum Likelihood Analysis . . . . . . . . . . . . . . . . . 391
Bayesian Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392
Replicating Bayesian Analysis and Data Imputation Results. . . . . . 394
Examining the Current Seed. . . . . . . . . . . . . . . . . . . . . . 394
Changing the Current Seed . . . . . . . . . . . . . . . . . . . . . . 395
Changing the Refresh Options . . . . . . . . . . . . . . . . . . . . 397
Assessing Convergence. . . . . . . . . . . . . . . . . . . . . . . . . . . 398
Diagnostic Plots. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400
Bivariate Marginal Posterior Plots . . . . . . . . . . . . . . . . . . . . . 406
Credible Intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409
Changing the Confidence Level . . . . . . . . . . . . . . . . . . . . 409
Learning More about Bayesian Estimation . . . . . . . . . . . . . . . . 410
27Bayesian Estimation Using a
NonDiffuse Prior Distribution411
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411
About the Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411
More about Bayesian Estimation . . . . . . . . . . . . . . . . . . . . . . 411
Bayesian Analysis and Improper Solutions . . . . . . . . . . . . . . . . 412
About the Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412
Fitting a Model by Maximum Likelihood . . . . . . . . . . . . . . . . . . 413
Bayesian Estimation with a NonInformative (Diffuse) Prior. . . . . . . 414
Changing the Number of BurnIn Observations . . . . . . . . . . . 414
28Bayesian Estimation of Values
29Estimating a UserDefined Quantity
in Bayesian SEM439
Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439
About the Example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439
The Stability of Alienation Model . . . . . . . . . . . . . . . . . . . . . 439
Numeric Custom Estimands. . . . . . . . . . . . . . . . . . . . . . . . . 445
Dragging and Dropping . . . . . . . . . . . . . . . . . . . . . . . . 449
Dichotomous Custom Estimands. . . . . . . . . . . . . . . . . . . . . . 459
Defining a Dichotomous Estimand . . . . . . . . . . . . . . . . . . 459
30Data Imputation463
Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463
About the Example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463
Multiple Imputation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464
ModelBased Imputation . . . . . . . . . . . . . . . . . . . . . . . . . . 464
Performing Multiple Data Imputation Using Amos Graphics . . . . . . 464
31Analyzing Multiply
Posterior Predictive Distributions. . . . . . . . . . . . . . . . . . . . . . 483
Imputation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486
General Inequality Constraints on Data Values . . . . . . . . . . . . . . 490
33OrderedCategorical Data491
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491
About the Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491
Specifying the Data File. . . . . . . . . . . . . . . . . . . . . . . . . 493
Recoding the Data within Amos . . . . . . . . . . . . . . . . . . . . 494
Specifying the Model . . . . . . . . . . . . . . . . . . . . . . . . . . 502
Fitting the Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503
MCMC Diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 506
Posterior Predictive Distributions. . . . . . . . . . . . . . . . . . . . . . 508
Posterior Predictive Distributions for Latent Variables. . . . . . . . . . 513
Imputation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 518
34Mixture Modeling with Training Data523
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523
About the Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523
Performing the Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 526
Specifying the Data File . . . . . . . . . . . . . . . . . . . . . . . . . . . 528
Specifying the Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532
Fitting the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534
Classifying Individual Cases . . . . . . . . . . . . . . . . . . . . . . . . . 537
Latent Structure Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 539
35Mixture Modeling without
Training Data541
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 541
About the Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 541
Performing the Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 542
xxii
Specifying the Data File . . . . . . . . . . . . . . . . . . . . . . . . . . . 544
Specifying the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547
xxiii
Other Aspects of the Analysis in Addition to Model Specification . . . 590
Defining Program Variables that Correspond to Model Variables. 590
38Simple UserDefined Estimands I593
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593
The Wheaton Data Revisited . . . . . . . . . . . . . . . . . . . . . . . . 593
Estimating an Indirect Effect . . . . . . . . . . . . . . . . . . . . . . 594
Estimating the Indirect Effect without Naming Parameters . . . . 602
39Simple UserDefined Estimands II605
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605
About the Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605
A Markov Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605
ANotation613
BDiscrepancy Functions615
CMeasures of Fit619
Measures of Parsimony . . . . . . . . . . . . . . . . . . . . . . . . . . . 620
NPAR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 620
DF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 620
PRATIO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 621
Minimum Sample Discrepancy Function. . . . . . . . . . . . . . . . . . 621
CMIN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 621
P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 621
CMIN/DF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623
FMIN. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624
xxiv
Measures Based On the Population Discrepancy. . . . . . . . . . . . 624
NCP. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624
F0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625
RMSEA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625
PCLOSE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 627
xxv
DNumeric Diagnosis of NonIdentifiability641
EUsing Fit Measures to Rank Models643
FBaseline Models for
Descriptive Fit Measures647
GRescaling of AIC, BCC, and BIC649
ZeroBased Rescaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 649
Akaike Weights and Bayes Factors (Sum = 1). . . . . . . . . . . . . . . 650
Akaike Weights and Bayes Factors (Max = 1). . . . . . . . . . . . . . . 651
Notices653
Bibliography657
Index669
IBM SPSS Amos implements the general ap
proach to data analysis known as
structural equation modeling
(SEM), also known as
analysis of covariance
, or
causal modeling
. This approach includes, as special cases, many well
known conventional techniques, includ
ing the general linear model and common
factor analysis.
IBM SPSS Amos (Analysis of Moment Struct
ures) is an easytouse program for
visual SEM. With Amos, you can quickly specify, view, and modify your model
graphically using simple drawing tools. Th
en you can assess your models fit, make
any modifications, and print out a publicationquality 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
verbal
Input:
spatial
visperc
cubes
.43
lozenges
.54
wordmean
.71
paragraph
.77
sentence
.68
verbal
.70
.65
.74
.88
.83
.84
.49
Chisquare = 7.853 (8 df)
p = .448
Output:
Chapter 1
Structural equation modeling (SEM) is some
times thought of as esoteric and difficult
Introduction
models. It provides a test of univariate norm
ality for each observed variable as well as
a test of multivariate normality
Chapter 1
Example 9 and those that follow demonstrate
advanced techniques that have so far not
been used as much as they dese
rve. These techniques include:
Simultaneous analysis of data fro
m several different populations.
Estimation of means and inter
cepts in regression equations.
Maximum likelihood estimation in the presence of missing data.
Bootstrapping to obtain estimated standard
errors and confiden
ce 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 orderedcategorical data.
Mixture modeling.
Tip:
If you have questions about a particular Am
os feature, you can always refer to the
extensive online help provided by the program.
IBM SPSS Amos 23 comes with extensive do
cumentation, including online help, this
users guide, and advanced reference material
for Visual Basic or C# and the Amos API
(Application Programming Interface). If you performed a typical installation, you can
find the
IBM SPSS Amos 23 Programming Reference Guide
in the following location:
C:\Program Files\IBM\SPSS\Amos\23\Doc
umentation\Programming Reference.pdf
Although this users guide contains a good bit of expository material, it is not by any
means a complete guide to the correct and ef
fective use of structural modeling. Many
excellent SEM textbooks are available.
Structural Equation Modeling: A Multidisciplinary Journal
contains
Introduction
Carl Ferguson and Edward Rigdon estab
lished an electronic mailing list called
Chapter 2
demonstrates the menu path. For information about the toolbar buttons and keyboard
shortcuts, see the online help.
Hamilton (1990) provided several measuremen
ts on each of 21
states. Three of the
measurements will be used in this tutorial:
Average SAT score
Per capita income expressed in $1,000 units
Median education fo
r residents 25 years of age or older
You can find the data in the
directory within the Excel 8.0 workbook
in the worksheet named
Hamilton
. The data are as follows:
SATIncomeEducation
89914.34512.7
89616.3712.6
89713.53712.5
88912.55212.5
82311.44112.2
85712.75712.7
86011.79912.4
89010.68312.5
88914.11212.5
88814.57312.6
92513.14412.6
86915.28112.5
89614.12112.5
82710.75812.2
90811.58312.7
88512.34312.4
88712.72912.3
79010.07512.1
86812.63612.4
90410.68912.6
88813.06512.4
Tutorial: Getting Starte
d with Amos Graphics
The following path diagram shows a model for these data:
This is a simple regression model where one observed variable,
, is predicted as a
linear combination of the othe
r two observed variables,
Education
Income
. As with
nearly all empirical data, the predicti
on will not be perfect. The variable
represents variables other than
Education
Income
that affect
Each singleheaded arrow represents
a regression weight. The number
in the
figure specifies that
must have a weight of
1 in the prediction of
SAT
. Some such
constraint must be imposed in order to make the model
, and it is one of the
features of the model that mu
st be communicated to Amos.
You can launch Amos Graphics
in any of the following ways:
On the Start screen, click
Amos Graphics
Click
on the Windows task bar, and choose
All Programs
IBM SPSS Amos 23
Amos Graphics
Doubleclick any path diagram (
*.amw
) in Windows Explorer.
Drag a path diagram (
*.amw
) file from Windows Explorer to the Amos Graphics
window.
From within SPSS Statistics, choose
�Analyze IBM SPSS Amos
Chapter 2
From the menus, choose
File
Your work area appears. The large area on
the right is where you draw path diagrams.
The toolbar on the left provides oneclick ac
cess to the most frequently used buttons.
You can use either the toolbar or menu commands for most operations.
Tutorial: Getting Starte
d with Amos Graphics
The next step is to specify the file that cont
ains the Hamilton data. This tutorial uses a
Microsoft Excel 8.0 (
) file, but Amos supports several common database formats,
including SPSS Statistics
*.sav
files. If you launch Amos from the Addons menu in
SPSS Statistics, Amos automatically uses th
e file that is open in SPSS Statistics.
From the menus, choose
File
Data Files
In the Data Files dialog box, click
File Name
Browse to the
folder. If you performed a typi
cal installation, the path is
C:\Program Files\IBM\
SPSS\Amos\23\Tutorial\
language
In the Files of type list, select
Excel 8.0
Select
, and then click
Open
In the Data Files dialog box, click
and Drawing Variables
The next step is to draw the variables in your
model. First, youll draw three rectangles
to represent the observed variables, and then youll draw an ellipse to represent the
unobserved variable.
From the menus, choose
Diagram
Draw Observed
In the drawing area, move your mouse pointer to where you want the
Education
rectangle to appear. Click and drag to draw
the rectangle. Dont
worry about the exact
size or placement of the rectangl
e because you can
change it later.
Chapter 2
In the drawing area, move your mouse pointe
r to the right of the three rectangles and
click and drag to draw the ellipse.
The model in your drawing area should
now look similar to the following:
Naming the Variables
In the drawing area, rightclick the top left rectangle and choose
Object Properties
from
the popup menu.
Click the
tab.
In the Variable name text box, type
Education
Tutorial: Getting Starte
d with Amos Graphics
Your path diagram should now look like this:
Now you will add arrows to the path diagra
m, using the following model as your guide:
From the menus, choose
Diagram
Chapter 2
Tutorial: Getting Starte
d with Amos Graphics
You can change the appearance of your path
diagram by moving and resizing objects.
These changes are visual
only; they do not affect
the model specification.
To Move an Object
From the menus, choose
In the drawing area, cl
ick and drag the object to its new location.
To Reshape an Object or DoubleHeaded Arrow
From the menus, choose
Shape of Object
In the drawing area, click and
drag the object until you are satisfied with its size and
Chapter 2
To Undo an Action
From the menus, choose
Undo
To Redo an Action
From the menus, choose
Redo
Tutorial: Getting Starte
d with Amos Graphics
Close the Analysis Properties dialog box.
Chapter 2
The only thing left to
do is perform the calculations for
fitting the model. Note that in
order to keep the parameter estimates up to date, you must do this every time you
change the model, the data, or the options in the Analysis Properties dialog box.
From the menus, click
Analyze
Calculate Estimates
Tutorial: Getting Starte
d with Amos Graphics
To View Graphics Output
Click the
Show the output path diagram
button .
Chapter 2
Your path diagram now looks like this:
Tutorial: Getting Starte
d with Amos Graphics
Click
Example
Estimating Variances and
Covariances
This example shows you how to
estimate population variances and covariances. It also
discusses the general format of Amos input and output.
Example 1
From the menus, choose
From the menus, choose
Data Files
In the Data Files dialog box, click
File Name
Browse to the
Examples
folder. If you performed a typical installation, the path is
C:\Program Files\IBM\SPSS\A
mos\23\Examples\languag&#xlan;gu;Jg;一e
In the Files of ty
Excel 8.0 (*.xls)
, select
UserGuide.xls
, and then click
Open
In the Data Files
dialog box, click
Amos displays a list of worksheets in the
UserGuide
Estimating Variances and Covariances
Example 1
Create two more duplicate rectangles until you have four rectangles side by side.
Tip:
If you want to reposition a rectangle, choose
from the menus and drag
the rectangle to its new position.
Naming the Variables
From the menus, choose
Estimating Variances and Covariances
Rightclick a variable and choose
Properties
from the popup menu.
The Object Properties dialog box appears.
Click the
Text
tab and adjust the font attributes as desired.
If you leave the path diagram as it is, Amos
Graphics will estimate the variances of the
four variables, but it will not estimate
Example 1
From the menus, choose
Calculate Estimates
Because you have not yet saved the file, the Save As dialog box appears.
Enter a name for the file and click
Save
Viewing Graphics Output
Click the
Show the output path diagram
button.
Amos displays the output path
diagram with para
Estimating Variances and Covariances
In the output path diagram, the numbers displayed next to the boxes are estimated
variances, and the numbers displayed next to
the doubleheaded arrows are estimated
covariances. For example, the variance of
is estimated at 5.79, and that of
place1
Example 1
observation on an approximately normally
distributed random variable centered
around the population covariance with a standard deviation of about 1.16, that is, if the
assumptions in the section Distribution A
ssumptions for Amos Models on p.35 are
dividing the covariance estimate by
its standard error . This ratio
is relevant to the null hypothesis that, in the population from which Attigs 40 subjects
2.561.961.1602.562.27
2.202.561.16
Estimating Variances and Covariances
example, Runyon and Haber, 1980, p. 226) is 2.509 with 38 degrees of freedom
. In this example, both
values are less than 0.05
, so both tests agree in
rejecting the null hypothesis at the 0.05 le
vel. However, in other situations, the two
values might lie on
opposite
sides of 0.05. You might or might not regard this as
especially seriousat any rate, the two tests
can give different results. There should be
no doubt about which test is better. The
test is exact under the assumptions of
normality and independence of observations, no
matter what the sample size. In Amos,
the test based on critical ratio depends on
the same assumptions; ho
wever, with a finite
sample, the test is
only approximate.
For many interesting applications of Amos,
there is no exact test or exact standard
error or exact confidence interval available.
On the bright side, when fitting a model
for which conventional estimates exist,
maximum likelihood point estimates (for example, the numbers in the
Estimate
column) are generally identical to the conventional estimates.
Now click
Notes for Model
in the upper left pane of the Amos Output window.
The following table plays an import
ant role in every Amos analysis:
Number of distinct sample moments:10
0.016
Example 1
Number of distinct
sample moments
referred to are sample means, variances, and
covariances. In most analyses, including the
present one, Amos ignores means, so that
the sample moments are the sample
variances of the four variables,
place2
, and their sample covariances. Th
ere are four sample variances and
six sample covariances, for a
total of 10 sample moments.
Estimating Variances and Covariances
So far, we have discussed output that Amos
generates by default. You can also request
additional output.
You may be surprised to learn that Amos disp
lays estimates of covariances rather than
correlations. When the scale of
measurement is arbitrary or of no substantive interest,
correlations have more descriptive meaning
than covariances. Nevertheless, Amos and
similar programs insist on estimating covari
ances. Also, as will soon be seen, Amos
Example 1
Because you have changed the options in the Analysis Properties dialog box, you must
rerun the analysis.
From the menus, choose
Calculate Estimates
Click the
Show the output path diagram
button.
Estimating Variances and Covariances
In the tree diagram in the upper left pane of the Amos Output window, expand
and then click
Correlations
Hypothesis testing procedures, confidence intervals, and claims for efficiency in
maximum likelihood or generalized least
squares estimation depend on certain
assumptions. First, observations must be independent. For example, the 40 young
people in the Attig study have to be picked independently from the population of young
people. Second, the observed variables must
Example 1
The (conditional) expected values of the
random variables depend linearly on the
values of the fixed variables.
A typical example of a fixed variable would
be an experimental treatment, classifying
respondents into a study group and a control
group, respectively. It is all right that
treatment is nonnormally distributed, as
long as the other exogenous variables are
normally distributed for study and control ca
ses alike, and with the same conditional
variancecovariance matrix. Predictor variable
s in regression analysis (see Example 4)
are often regarded as fixed variables.
Many people are accustomed to the requ
irements for normality and independent
observations, since these are the usual requ
irements for many co
nventional procedures.
However, with Amos, you have to remember
Estimating Variances and Covariances
Example 1
Estimating Variances and Covariances
Example 1
Other Program Development Tools
The builtin program editor in Amos is used
throughout this user
s guide for writing
and executing Amos programs. However, you can use the development tool of your
choice. The
Examples
folder contains a
VisualStudio
subfolder where you can find
ExampleSWS
Testing Hypotheses
This example demonstrates how you can use Amos to test simple hypotheses about
variances and covariances. It also introduces
the chisquare test for goodness of fit and
elaborates on the concept of degrees of freedom.
We will use Attigs (1983) spatial memory data, which were described in Example 1.
We will also begin with the same path diag
ram as in Example 1. To demonstrate the
ability of Amos to use different data format
s, this example uses a data file in SPSS
Statistics format instead of an Excel file.
Example 2
Testing Hypotheses
Close the dialog box.
Example 2
require both of the variances to have the
same value without sp
ecifying ahead of time
what that value is.
Testing Hypotheses
While a horizontal layout is fine for small ex
amples, it is not practical for analyses that
are more complex. The following is a different layout of the path diagram on which
weve been working:
Example 2
You can use the following tools to rearrange your path diagram until it looks like the
one above:
To move objects, choose
Move
from the menus, and then drag the object to
its new location. You can also use the
button to drag the endpoints of arrows.
To copy formatting from one
object to another, choose
Drag Properties
from
the menus, select the proper
ties you wish to apply, and then drag from one object
to another.
For more information about
the Drag Properties feature,
refer to online help.
This example uses a data file in SPSS Sta
tistics format. If you have SPSS Statistics
installed, you can view the data as you load it. Even if you dont have SPSS Statistics
installed, Amos will still read the data.
From the menus, choose
File
Data Files
In the Data Files dialog box, click
File Name
Browse to the
Examples
folder. If you performed a typical installation, the path is
C:\Program Files\IBM\SPSS\A
mos\23\Examples\languag&#xlan;gu;Jg;一e
In the Files of ty
*.sav
, click
Attg_yng
, and then click
Open
If you have SPSS Statistic
s installed, click the
View Data
button in the Data Files dialog
box. An SPSS Statistics window
opens and displays the data.
Testing Hypotheses
Review the data and close the data view.
In the Data Files dialog box, click
From the menus, choose
Analyze
Calculate Estimates
In the Save As dialog box, enter a name for the file and click
Amos calculates the model estimates.
Viewing Text Output
From the menus, choose
Text Output
Example 2
You can see that the parameters that were sp
ecified to be equal do have equal estimates.
The standard errors here are generally smal
ler than the standard
errors obtained in
Example 1. Also, because of the constraints on the parameters, there are now positive
degrees of freedom.
in the upper left pane of the Amos Output window.
While there are still 10 sample variances an
1073
Testing Hypotheses
From the menus, choose
Analyze
Calculate Estimates
Amos recalculates the model estimates.
To see the sample variances and covariances collected into a matrix, choose
Click
Sample Moments
in the tree diagram in the upper
left corner of the Amos Output
window.
Example 2
The following is the sa
mple covariance matrix:
In the tree diagram, expand
Estimates
and then click
Matrices
The following is the matrix of implied covariances:
Testing Hypotheses
Displaying Covariance and Variance
As in Example 1, you can display the cova
riance and variance estimates on the path
diagram.
Click the
Show the output path diagram
button.
Example 2
Notice the word
in the bottom line of the figure
caption. Words that begin with
a backward slash, like
, are called
text macros
. Amos replaces text macros with
information about the currently di
splayed model. The text macro
\format
will be
replaced by the heading
Model Specification
Unstandardized estimates
Standardized estimates
, depending on which version of the path diagram is displayed.
Hypothesis Testing
The implied covariances are the best estimates of the population variances and
covariances under the null hypothesis. (The
assumptions about the population values. A
comparison of these two matrices is
Testing Hypotheses
maximum likelihood estimates, and there is no
reason to expect them to resemble the
implied covariances.
The chisquare statistic is an overall measure of how much the implied covariances
differ from the sample covariances.
In general, the more the implied covarian
ces differ from the sample covariances, the
bigger the chisquare statistic will be. If the implied covariances had been identical to
the sample covariances, as they
were in Example 1, the chisquare statistic would have
been 0. You can use the chisquare statisti
c to test the null hypothesis that the
parameters required to have
equal estimates are really equal in the population.
However, it is not simply a matter of checki
ng to see if the chisquare statistic is 0.
Since the implied covariances and the samp
le covariances are merely estimates, you
cant expect them to be identical (even if th
ey are both estimates
of the same population
covariances). Actually, you would expect them to differ enough to produce a chisquare
in the neighborhood of the degrees of freedom, even if the null hypothesis is true. In
other words, a chisquare value of 3 would not be out of the ordinary here, even with a
true null hypothesis. You can say more than
that: If the null hypothesis is true, the chi
square value (6.276) is a single observation on a random variable that has an
approximate chisquare distribution with th
ree degrees of freedom.
The probability is
about 0.099 that such an observation would
be as large as 6.276. Consequently, the
evidence against the null hypothesis is not significant at the 0.05 level.
Example 2
In the Figure Caption dialog box,
enter a caption that includes the
, and
text
macros, as follows:
When Amos displays the path diagram containing this caption, it appears as follows:
Testing Hypotheses
Example 2
This table gives a linebyline explanation of the program:
Program StatementExplanation
Dim Sem As New AmosEngine
Declares
as an object of type
AmosEngine
Testing Hypotheses
To perform the analysis,
Timing Is Everything
lines must appear
after
BeginGroup
; otherwise, Amos will not recognize
that the variables named in the
lines are observed variables in the
attg_yng.sav
dataset.
In general, the order of statements matter
s in an Amos program. In organizing an
Example
More Hypothesis Testing
This example demonstrates how to test th
e null hypothesis that two variables are
concept of degrees of freed
om, and demonstrates, in a
Example 3
In the Files of ty
Text (*.txt)
, select
Attg_old.txt
, and then click
In the Data Files dialog box, click
Testing a Hypothesis That Tw
o Variables Are Uncorrelated
More Hypothesis Testing
model specified by the simple
path diagram above specifies
that the covariance (and
Example 3
From the menus, choose
Calculate Estimates
The Save As dialog box appears.
Enter a name for the file and click
Save
Amos calculates the model estimates.
Viewing Text Output
From the menus, choose
Text Output
In the tree diagram in the upper left
pane of the Amos Output window, click
Estimates
Although the parameter estimates are not of pr
imary interest in this analysis, they are
as follows:
In this analysis, there is one degree of free
dom, corresponding to the single constraint
that
age
and
be uncorrelated. The degrees of freedom
can also be arrived
at by the computation shown in the follo
wing text. To display this computation:
Click
Notes for Model
in the upper left pane of the Amos Output window.
More Hypothesis Testing
The three sample moments are the variances of
age
and
and their
Example 3
statistic for testing this nu
ll hypothesis is 0.59 (,
twosided). The probability
level associated with the
statistic is exact. The probability
level of 0.555 of the chisquare statistic is
, owing to the fact that it does not have an
exact chisquare distribution
in finite samples. Even so,
the probability level of 0.555
is not bad.
Here is an interesting ques
tion: If you use the probability level displayed by Amos
to test the null hypothesis
at either the 0.05 or 0.01 level, then what is the
actual
probability of rejecting a true null hypo
thesis? In the case
of the present null
hypothesis, this question ha
s an answer, although the an
swer depends on the sample
size. The second column in the next table
le sizes, the real
probability of a Type I error when using Amos to test the null hypothesis of
at the 0.05 level. The third column
shows the real probability of a Type I
error if you use a significance level of 0.01
. The table shows that the bigger the sample
size, the closer the true significance level is
to what it is supposed
to be. Its too bad
that such a table cannot be constructed for every hypothesis that Amos can be used to
test. However, this much can be said about
any such table: Moving from top to bottom,
the numbers in the 0.05 colu
mn would approach 0.05, an
d the numbers in the 0.01
column would approach 0.01. This is what is meant when it is said that hypothesis tests
based on maximum likelihood theory are
asymptotically
correct.
The following table shows the actual probab
ility of a Type I error when using Amos
to test the hypothesis that
two variables are uncorrelated:
Sample Size
Nominal Significance Level
0.050.01
0.2500.122
0.1500.056
0.1150.038
0.0730.018
0.0600.013
0.0560.012
0.0550.012
0.0540.011
100
0.0520.011
150
0.0510.010
200
0.0510.010
500
0.0500.010
More Hypothesis Testing
Example
Conventional Linear Regression
This example demonstrates a conventional
regression analysis, predicting a single
observed variable as a linear combination of
three other observed variables. It also
introduces the concept of
identifiability
Warren, White, and Fuller (1974) studied 98 managers of farm cooperatives. We will
use the following four measurements:
A fifth measure,
past training
, was also reported, but we will not use it.
managerial role
Example 4
Here are the sample variances and covariances:
also contains the sample means. Raw da
ta are not available, but they are not
for most analyses, as long as the sample moments (that is, means,
variances, and covariances) are provided. In fact, only sample variances and
covariances are required in this example. We will not need the sample means in
for the time being, and Amos will ignore them.
Suppose you want to use scores on
knowledge
value
to predict
performance
. More specifically, suppose you think that
performance
scores can be
approximated by a li
near combination of
knowledge
, and
prediction will not be perfect, however, and the model should thus include an
variable.
Here is the initial path diagram for this relationship:
Conventional Linear Regression
The singleheaded arrows represent linear dependencies. For example, the arrow
leading from
knowledge
to
performance
indicates that performance scores depend, in
part, on knowledge. The variable
is enclosed in a circle because it is not directly
observed. Error represents much more than
random fluctuations in performance scores
due to measurement error. Error also repr
esents a composite of
age, socioeconomic
status, verbal ability, and anything else
on which performance may depend but which
was not measured in this stud
y. This variable is essential because the path diagram is
supposed to show
variables that affect performance
scores. Without the circle, the
path diagram would make the implausible claim that performance is an
exact
linear
combination of knowledge, value, and satisfaction.
The doubleheaded arrows in the path
diagram connect variables that may be
correlated with each other. The absenc
e of a doubleheaded
arrow connecting
with any other variable indicates that
is assumed to be un
correlated with every
other predictor variablea fundamental assumption in linear regression.
is also not connected to any
other variable by a doublehead
ed arrow, but this is for a
different reason. Since performance depends on the other variables, it goes without
saying that it might be
correlated with them.
Using what you learned in the firs
t three examples, do the following:
Start a new path diagram.
Specify that the dataset to be an
alyzed is in the Excel worksheet
in the file
UserGuide.xls
Draw four rectangles and label them
satisfaction
performance
Draw an ellipse for the
variable.
Draw singleheaded arro
ws that point from the
, or predictor, variables
knowledge
value
satisfaction
endogenous
, or response, variable
performance
Endogenous variables have at least one
singleheaded path pointing toward them.
Exogenous variables, in contra
st, send out only singleheade
d paths but do not receive any.
Example 4
Draw three doubleheaded arrows that connect the observed exogenous variables
knowledge
Your path diagram should look like this:
In this example, it is impossible to estimate
the regression weight for the regression of
performance
on
, and, at the same time, estimate the variance of
. It is like
Conventional Linear Regression
Setting a regression weight equal to 1 for every
variable can be tedious.
Fortunately, Amos Graphics provides a default solution that works well in most cases.
Click the
Add a unique variable to an existing variable
button.
Click an endogenous variable.
Amos automatically attaches an
error variable to it, comple
te with a fixed regression
weight of 1. Clicking the endogenous variable repeatedly changes the position of the
error variable.
Example 4
Viewing the Text Output
Here are the maximum likelihood estimates:
Amos does not display the path
performance error
because its value is fixed at the
default value of 1. You may wonder how much the other estimates would be affected
if a different constant had been chosen. It turns out that only the variance estimate for
is affected by
such a change.
The following table shows the variance esti
mate that results from various choices for
performance error
regression weight.
Suppose you fixed the path coefficient at 2
instead of 1. Then the variance estimate
would be divided by a factor of 4. You can extrapolate the rule that multiplying the path
coefficient by a fixed factor goes along with
dividing the error variance by the square
Fixed regression weightEstimated variance of error
0.50.050
0.7070.025
1.00.0125
1.4140.00625
2.00.00313
Conventional Linear Regression
of the same factor. Extending this, the product of the squared regression weight and the
error variance is always a constant. This is
what we mean when
we say the regression
Example 4
The standardized regression weights and the
correlations are inde
pendent of the units
in which all variables are measured; therefore,
they are not affected by the choice of
identification constraints.
Squared multiple correlations are also independent of units of measurement. Amos
displays a squared multiple correlation for each endogenous variable.
The squared multiple correlatio
n of a variable is the proportion of its variance that
is accounted for by its predic
tors. In the present example,
knowledge
satisfaction
account for 40% of the variance of
performance
Viewing Graphics Output
The following path diagram outp
ut shows unstandardized values:
Conventional Linear Regression
Here is the standardized solution:
Viewing Additional Text Output
In the tree diagram in the upper left pane of the Amos Output window, click
Summary
Example 4
variables are those that have single
headed arrows poin
ting to them; they
depend on other variables.
Exogenous
variables are those that do not have single
headed arrows pointing to them; they do not depend on other variables.
Inspecting the preceding list will help you catch the most common (and insidious)
errors in an input file: typing errors. If you try to type
performance
twice but
unintentionally misspell it as
preformance
one of those times, both versions will
appear on the list.
in the upper left pane of the Amos Output window.
The following output indicates that there ar
e no feedback loops
Later you will see path diagrams where you can pick a variable and, by tracing along
the singleheaded arrows, fo
llow a path that leads back to the same variable.
Path diagrams that have
feedback loops are called
nonrecursive
. Those that do
not are called
recursive
Notes for Group (Group number 1)
The model is recursive.
Conventional Linear Regression
Example 4
the specification of many models, especi
Conventional Linear Regression
Note that in the
line above, each predictor variable (on the right side of the
equation) is associated with a regression we
ight to be estimated. We could make these
regression weights explicit through th
e use of empty parentheses as follows:
Sem.AStructure("performance = ()knowledge + ()value + ()satisfaction + error(1)")
The empty parentheses are optional. By defa
ult, Amos will automatically estimate a
regression weight for each predictor.
Example
Unobserved Variables
This example demonstrates a regression
analysis with unobserved variables.
The variables in the previous example were su
rely unreliable to some degree. The fact
that the reliability of
performance
is unknown presents a minor problem when it
regression weights for perfectly reliable, hypo
Example 5
Here is a list of the input variables:
For this example, we will use the data file
to obtain the sample variances
and covariances of these subtests. The sample
means that appear in the file will not be
used in this example. Statistics on formal education (
) are present in the
file, but they also will not en
ter into the present analysis.
Variable nameDescription
performance1
12item subtest of Role Performance
performance2
12item subtest of Role Performance
13item subtest of Knowledge
13item subtest of Knowledge
15item subtest of Value Orientation
15item subtest of Value Orientation
5item subtest of Role Satisfaction
6item subtest of Role Satisfaction
degree of formal education
Unobserved Variables
The following path diagram presents a model for the eight subtests:
Four ellipses in the figure are labeled
knowledge
satisfaction
, and
performance
They represent unobserved variables that are in
directly measured by
the eight splithalf
The portion of the model that specifies ho
w the observed variables depend on the
unobserved, or latent, variables is sometimes called the
measurement model
. The
current model has four dis
tinct measurement submodels.
Example 5
Consider, for instance, the
knowledge
submodel: The scores of the two splithalf
knowledge1
knowledge2
, are hypothesized to depend on the single
underlying, but not directly observed variable,
knowledge
. According to the model,
scores on the two subtests may still disagree, owing to the influence of
and
, which represent errors of me
asurement in the two subtests.
knowledge1
and
knowledge2
are called
indicators
of the latent variable
knowledge
. The measurement
knowledge
forms a pattern that is repeated
three more times in the path
diagram shown above.
The portion of the model that specifies how the latent variables are related to each other
Unobserved Variables
The structural part of the current model is th
e same as the one in Example 4. It is only
in the measurement model th
at this example differs from the one in Example 4.
With 13 unobserved variables in this model,
it is certainly not id
entified. It will be
necessary to fix the unit of measurement of each unobserved variable by suitable
Example 5
From the menus, choose
Interface Properties
In the Interface Properties dialog box, click the
Page Layout
tab.
Unobserved Variables
Now you are ready to draw the model as shown in the path diagram on page83. There
are a number of ways to do this. One is to
start by drawing the measurement model first.
Here, we draw the measurement model
for one of the latent variables,
knowledge
then use it as a pattern for the other three.
Draw an ellipse for the unobserved variable
knowledge
From the menus, choose
Diagram
Draw Indicator Variable
Click twice inside the ellipse.
Each click creates one indicator variable for
knowledge
As you can see, with the
Draw indicator variable
button enabled, you can click multiple
times on an unobserved variable to create mu
ltiple indicators, complete with unique or
error variables. Amos Graphics maintains suitable spacing among the indicators and
inserts identification co
nstraints automatically.
Example 5
The indicators appear
by default above the
knowledge
ellipse, but you can change their
location.
From the menus, choose
Rotate
Click the
knowledge
ellipse.
Each time you click the
knowledge
ellipse, its indicators rota
te 90 clockwise. If you
click the ellipse three times, its
indicators will lo
ok like this:
The next step is to crea
te measurement models for
From the menus, choose
The measurement model turns blue.
From the menus, choose
Duplicate
Click any part of the measurement model,
and drag a copy to
beneath the original.
Repeat to create a third measur
ement model above the original.
Unobserved Variables
Your path diagram should now look like this:
Create a fourth copy for
performance
, and position it to th
e right of the original.
From the menus, choose
This repositions the two indicators of
performance
as follows:
Example 5
Entering Variable Names
Rightclick each object and select
Object Properties
from the popup menu
In the Object Properties dialog box, click the
tab, and enter a name into the
Variable Name text box.
Alternatively, you can choose
Unobserved Variables
The hypothesis that Model A is correct is accepted.
Regression Weights: (Group number 1  Default model)
EstimateS.E.C.R.PLabel
erformance
 knowledge
.337.1252.697.007
erformance
 satisfaction
.061.0541.127.260
erformance
 value
.176.0792.225.026
satisfaction2  satisfaction
.792.4381.806.071
satisfaction1  satisfaction 1.000
value2  value
.763.1854.128***
value1  value 1.000
knowledge2  knowledge
.683.1614.252***
knowledge1  knowledge 1.000
erformance1

erforman
ce 1.000
erformance2

erformance
.867.1167.450***
Covariances: (Group number 1  Default model)
EstimateS.E.C.R.PLabel
value&#x00;  knowledge
.037.0123.036.002
satisfaction .9;.2;耀 value 
.008.013
.610.542
satisfaction .7;�6.;耀 knowledge
.004.009.462.644
Variances: (Group number 1  Default model)
EstimateS.E.C.R.PLabel
satisfaction
.090.0521.745.081
value
.100.0323.147.002
knowledge
.046.0153.138.002
error9
.007.0032.577.010
error3
.041
.011
3.611***
error4
.035.0075.167***
error5
.080.0253.249.001
error6
.087.0184.891***
error7
.022.049.451.652
error8
.045.0321.420.156
error1
.007.0023.110.002
error2
.007.0023.871***
Example 5
Standardized estimates, on the other hand
, are not affected by the identification
constraints. To calculate standardized estimates:
From the menus, choose
Analysis Properties
In the Analysis Properties dialog box, click the
Output
tab.
Standardized estimates
Standardized Regression Weights: (Group number 1  Default model)
Estimate
performance 
knowledge
performance 
satisfaction
performance 
value
satisfaction2 
satisfaction
satisfaction1 
satisfaction
value2 
value
value1 
value
knowledge2 
knowledge
knowledge1 
knowledge
performance1
performance
performance2
performance
Correlations: (Group number 1  Default model)
Estimate
value&#x00; 
knowledge
satisfacti&#x00;on 
value 
satisfacti&#x00;on 
knowledge
Unobserved Variables
Viewing the Graphics Output
The path diagram with standardized para
meter estimates displayed is as follows:
The value above
performance
indicates that
pure
knowledge
, and
satisfaction
account for 66% of
the variance of
performance
. The values displayed above the
observed variables are reliabili
ty estimates for the eight in
dividual subtests. A formula
for the reliability of the original tests (befor
e they were split in half) can be found in
Rock et al
(1977) or any book on mental test theory.
Assuming that Model A is correct (and there
is no evidence to the
contrary), consider
the additional hypothesis that
knowledge1
and
knowledge2
are parallel tests. Under the
parallel tests hypothesis, the regression of
knowledge1
on
knowledge
same as the regression of
knowledge2
on
knowledge
. Furthermore, the
variables
associated with
knowledge1
and
knowledge2
should have identical variances. Similar
consequences flow from the assumption that
value1
value2
are parallel tests, as
Example 5
well as
performance1
and
performance2
But it is not altogether
reasonable to assume
that
and
satisfaction2
are parallel. One of the su
btests is slightly longer
than the other because the original test ha
d an odd number of items and could not be
split exactly in half. As a result,
satisfaction2
is 20% longer than
satisfaction1
Assuming that the tests diff
er only in length leads to
the following conclusions:
The regression weight for regressing
on
should be 1.2
times the weight for regressing
on
Given equal variances for
and
, the regression weight for
error8
should be times as large as the regression weight for
error7
You do not need to redraw the path diagram from scratch in order to impose these
1.21.095445
Unobserved Variables
Example 5
Standardized Regression Weights: (Group number 1  Default model)
Estimate
erformance
 knowledge .529
erformance
 satisfaction .114
erformance
 value .382
satisfaction2  error8 .578
satisfaction2  satisfaction .816
satisfaction1  satisfaction .790
value2  value .685
value1  value .685
knowledge2  knowledge .663
knowledge1  knowledge .663
erformance1

erformance
.835
erformance2

erformance
.835
Correlations: (Group number 1  Default model)
Estimate
satisfaction &#x7.;耀 value .085
value &#x7.;退 knowledge .565
satisfaction &#x7.;退 knowledge .094
Squared Multiple Correlations: (Group number 1  Default model)
Estimate
erformance
erformance2
erformance1
satisfaction2
satisfaction1
value2
value1
knowledge2
knowledge1
Unobserved Variables
Here are the standardized esti
mates and squared multiple co
rrelations displayed on the
path diagram:
Testing Model B against Model A
statistic from the larger one. In this
example, the new statistic is 16.632
(that is, ). If the stronger model
(Model B) is correctly specified, this stat
istic will have an approximate chisquare
26.96710.335
Example 5
distribution with degrees of
freedom equal to the difference between the degrees of
2214
Unobserved Variables
Example 5
The following program fits Model B:
Sub Main()
Dim Sem As New AmosEngine
Try
Sem.TextOutput()
Sem.Standardized()
Sem.Smc()
Sem.BeginGroup(Sem.A
mosDir & "Examples\Warren9v.wk1")
Sem.AStructure("
performance1  performance (1)")
Sem.AStructure("
performance2  performance (1)")
Sem.AStructure("
knowledge1  knowledge (1)")
Sem.AStructure("
knowledge2  knowledge (1)")
Sem.AStructure("value1  value (1)")
Sem.AStructure("value2  value (1)")
Sem.AStructure("sati
sfaction1  sati
Sem.AStructure("satisfaction
2  satisfaction (" & CStr(1.2) & ")")
Sem.AStructure("performance  knowledge")
Sem.AStructu
re("performance  value")
Sem.AStructure("
performance  satisfaction")
Sem.AStructure("
performance  error9 (1)")
Sem.AStructure("
performance1  error1 (1)")
Sem.AStructure("
performance2  error2 (1)")
Sem.AStructure("
knowledge1  error3 (1)")
Sem.AStructure("
knowledge2  error4 (1)")
Sem.AStructure("value1  error5 (1)")
Sem.AStructure("value2  error6 (1)")
Sem.AStructure("
satisfaction1  error7 (1)")
Sem.AStructure("satisfaction
2  error8 (" & CS
tr(1.095445) & ")")
Sem.AStructure("error1 (alpha)")
Sem.AStructure("error2 (alpha)")
Sem.AStructure("error8 (delta)")
Sem.AStructure("error7 (delta)")
Sem.AStructure("error6 (gamma)")
Sem.AStructure("error5 (gamma)")
101
Example
Exploratory Analysis
This example demonstrates structural modeli
ng with timerelated latent variables, the
use of modification indices and critical ratio
s in exploratory analyses, how to compare
multiple models in a single
analysis, and computation of
implied moments, factor
score weights, total effects, and indirect effects.
Example 6
variances and covariances, as needed for the analysis. We will not use the sample
means in the analysis.
Jreskog and Srbom (1984) proposed the model shown on p. 103 for the Wheaton
data, referring to it as their Model A. The model asserts that all of the observed
variables depend on underlying, un
observed variables. For example,
anomia67
and
powles67
both depend on the unobserved variable
alienation67
, a hypothetical variable
that Jreskog and Srbom referred to
as alienation. The unobserved variables
and
appear to play the same role as the variables
and
did in Example 5.
However, their interpretation here
is different. In Example 5,
and
103
Exploratory Analysis
Model A is identified except
for the usual problem that th
e measurement scale of each
unobserved variable is inde
terminate. The measurement scale of each unobserved
variable may be fixed arbitrarily by setting a regression weight to unity (1) for one of
the paths that points away from it. The path diagram shows 11 regression weights fixed
at unity (1), that is, one constraint for eac
h unobserved variable. These constraints are
sufficient to make the model identified.
Example 6
105
Exploratory Analysis
happen to constrain any parameters to be equal to other parameters, but if such
constraints were present, you might consider removing them in hopes of getting a
Example 6
The column heading
M.I.
in this table is short for
Modification Index
. The modification
indices produced are those described by Jreskog and Srbom (1984). The first
modification index listed (
5.905
) is a conservative estimate of the decrease in
chisquare that will occur if
eps2
delta1
are allowed to be correlated. The new
chisquare statistic would have 5 degrees of freedom and would be no
greater than 65.639 (). The actual decrease of the chisquare statistic
might be much larger than 5.905. The column labeled
Par Change
gives approximate
estimates of how much each parameter would change if it were estimated rather than
fixed at 0. Amos estimates that the covariance between
and
delta1
would be
. Based on the small modification index, it does not look as though much would
be gained by allowing
eps2
to be correlated. Beside
s, it would be hard to
justify this particular modification on th
=
71.5445.905
0.424
107
Exploratory Analysis
tion Index Threshold
By default, Amos displays only modificatio
n indices that are greater than 4, but you
can change this threshold.
From the menus, choose
Analysis Properties
In the Analysis Properties
dialog box, click the
tab.
Enter a value in the
Threshold for modification indices
text box. A very small threshold
will result in the display of a lot of modifi
cation indices that are too small to be of
interest.
The largest modification index in Model A is 40.911. It indicates that allowing
eps1
eps3
to be correlated will decr
ease the chisquare statistic
by at least 40.911. This
is a modification well worth considering because it is quite plausible that these two
variables should be correlated.
represents variability in
anomia67
that is not due
to variation in
alienation67
. Similarly,
represents variability in
anomia71
that is
not due to variation in
alienation71
Anomia67
and
anomia71
are scale scores on the
same instrument (at differen
t times). If the anomia scale measures something other
Example 6
The path diagram for Model B is contained in the file
Text Output
The added covariance between
eps1
decreases the degrees of freedom by 1.
109
Exploratory Analysis
The chisquare statistic is reduced by substantially more than the promised 40.911.
Model B cannot be rejected. Since the fit of Model B is so good, we will not pursue the
possibility, mentioned earlier, of allowing
and
eps4
to be correlated. (An
Regression Weights: (Group number 1  Default model)
Estimate
S.E.
C.R.
PLabel
alienation67  ses 
.550.053
10.294***
alienation71  alienation67
.617.05012.421***
alienation71  ses 
.212.049
4.294***
 alienation71
.971.04919.650***
anomia71  alienation71
 alienation67
1.027.05319.322***
anomia67  alienation67
education  ses
SEI  ses
5.164.42112.255***
Covariances: (Group number 1  Default model)
EstimateS.E.
C.R.
Label
eps1 &#x4;&#x.600; eps3
1.886.240
Variances: (Group number 1  Default model)
EstimateS.E.
C.R.PLabel
ses
6.872.65710.458
zeta1
4.700.43310.864
zeta2
3.862.34311.257
5.059
.371
13.650
2.211.3176.968
4.806.39512.173
2.681.3298.137
delta1
2.728.5165.292
delta2
266.56718.17314.668
Example 6
The following path diagram displays the
standardized estimates and the squared
multiple correlations:
Because the error variables in
the model represent more th
an just measurement error,
the squared multiple correlati
111
Exploratory Analysis
In trying to improve upon a model, you should not be guided exclusively by
Example 6
From the menus, choose
Analysis Properties
In the Analysis Properties dialog box, click the
Output
tab.
Critical ratios for differences
When Amos calculates critical ratios for parameter differences, it generates names for
Regression Weights: (Group number 1  Default model)
EstimateS.E.
C.R.
PLabel
alienation67  ses 
.550.053
10.294***par_6
alienation71  alienation67
.617.05012.421***par_4
alienation71  ses 
.212.049
4.294***par_5
owles71
 alienation71
.971.04919.650***par_1
anomia71  alienation71
owles67
 alienation67
1.027.05319.322***par_2
anomia67  alienation67
education  ses
SEI  ses
5.164.42112.255***par_3
Covariances: (Group number 1  Default model)
Estimate
S.E.C.R.
PLabel
eps1 &#x5.;00; eps3
1.886.2407.866
***par_7
Variances: (Group number 1  Default model)
Estimate
S.E.
C.R.P
Label
.657
10.458
par_8
zeta1
.433
10.864
***par
zeta2
.343
11.257
.371
13.650
.317
6.968
.395
12.173
.329
8.137
delta1
.516
5.292
delta2 266.56718.173
14.668
113
Exploratory Analysis
Example 6
par_1
and
par_2
divided by the estimated standard error of this difference. These two
instrument on two occasions. The same goes for
powles67
and
powles71
. It is plausible
that the tests would behave the same way
on the two occasions. The critical ratios for
differences are consistent with
this hypothesis. The variances of
eps1
and
eps3
par_11
par_13
) differ with a critical ratio of 0.51. The variances of
and
eps4
par_12
par_14
) differ with a critical ratio of 1.00. The weights for the regression of
on
alienation
par_1
and
par_2
) differ with a critical
ratio of 0.88. None
of these differences, taken individually, is
significant at any conventional significance
level. This suggests that it may be worthwhile
to investigate more carefully a model in
which all three differences are constrained to be 0. We will call this new model
6.3830.7697.172
0.307
0.275
115
Exploratory Analysis
Here is the path diagram
for Model C from the file
Ex06c.amw
path_p
requires the regression
weight for predicting
powerlessness
from
alienation
to be the same in 1971 as it is in 1967. The label
is used to specify
eps1
and
eps3
have the same variance. The label
is used to specify that
eps4
have the same variance.
Example 6
Model C has three more degrees of freedom than Model B:
Testing Model C
As expected, Model C has an acceptable fit, with a higher probability level than Model B:
You can test Model C against Model B by
examining the difference in chisquare
values () and the difference in degrees of freedom ().
Achisquare value of 1.118 with 3 degrees of freedom is not significant.
Chisquare = 7.501
Degrees of freedom = 8
Probability level = 0.484
7.5016.3831.118
853
117
Exploratory Analysis
Example 6
In the following path
diagram from the file
general model (Model B) by requiring
cov1 = 0
119
Exploratory Analysis
In the
panel to the left of the path diagram, doubleclick
The Manage Models dialog box appears.
In the Model Name text box, type
Model A: No Autocorrelation
Doubleclick
in the left panel.
Notice that
cov1
Example 6
In the Manage Models
dialog box, click
In the Model Name text box, type
Model B: Most General
Model B has no constraints other than those
in the path diagram, so you can proceed
immediately to Model C.
Click
In the Model Name text box, type
Model C: TimeInvariance
121
Exploratory Analysis
Viewing Graphics Output for Individual Models
When you are fitting mu
ltiple models, use the
panel to display the diagrams
from different models. The
Models
panel is just to the left of the path diagram. To
display a model, click its name.
Viewing Fit Statistics for All Four Models
From the menus, choose
Text Output
In the tree diagram in the
upper left pane of the Amos Output window, click
The following is the portion of the outp
ut that shows the chisquare statistic:
column contains the minimum discrepa
ncy for each model. In the case of
maximum likelihood estimation (the default), the
CMIN
column contains the
chisquare statistic. The
column contains the correspondi
ng uppertail probability for
testing each model.
Example 6
This table shows, for example, that Mode
l C does not fit significantly worse than
Model B (). In other words, assuming that Model B is correct, you would
accept the hypothesis of time invariance.
On the other hand, the table shows that Mo
del A fits significantly worse than Model
B (). In other words, assu
ming that Model B is corr
ect, you would reject the
hypothesis that
are uncorrelated.
The variances and covariances among the ob
served variables can be estimated under
the assumption that Mo
del C is correct.
From the menus, choose
View� Analysis Properties
In the Analysis Properties dialog box, click the
Output
tab.
Select
Implied moments
(a check mark ap
pears next to it).
0.773
0.000
123
Exploratory Analysis
To obtain the implied variances and covariances for all the variables in the model
except error variables, select
All implied moments
For Model C, selecting
All implied moments
gives the following output:
The implied variances and covariances for the observed variables are not the same as
the sample variances and covariances. As
estimates of the corresponding population
values, the implied variances
and covariances are superior to the sample variances and
covariances (assuming that Model C is correct).
If you enable both the
Standardized estimates
and
All implied moments
check boxes
in the Analysis Properties dialog box, Amos will give you the implied
correlation
matrix of all variables as well
as the implied covariance matrix.
The matrix of implied covariances for all va
riables in the model can be used to carry
out a regression of the unobserved variables on the observed variables. The resulting
regression weight estimates can be obtained from Amos by enabling the
weights
check box. Here are the estimated factor score weights for Model C:
Implied (for all variables)
Covariances (Group number 1  Model C: TimeInvariance)
sesalienation67
alienation71
SEI
education
ses
alienation67 
3.8386.914
alienation71 
3.7204.9777.565
SEI
19.246
education
3.72035.4849.600
3.7174.9737.559
anomia71 
3.7204.9777.565
3.8356.9094.973
anomia67 
3.8386.9144.977
anomia71powles67anomia67
anomia71 7.559
12.515
4.973
anomia67 4.973
6.865
6.90911.864
Factor Score Weights (Group number 1  Model C: TimeInvariance)
SEIeducationpowles71
anomia71
owles67
anomia67
ses
.029.542
.016
.028
alienation67 
.061.134
.027.471
alienation71 
.049.491.253.134
.031
Example 6
The table of factor score weights has a sepa
rate row for each unobserved variable, and
a separate column for each observed variab
le. Suppose you wanted to estimate the
score of an individual. You would compute a weighted sum of the individuals six
observed scores using the six weights in the
row of the table.
Obtaining Tables of Indirect, Direct, and Total Effects
The coefficients associated with the singleheaded arrows in a path diagram are
sometimes called
direct effects
. In Model C, for example,
has a direct effect on
alienation71
alienation71
has a direct effect on
powles71
is then said to
have an
indirect effect
(through the intermediary of
alienation71
) on
powles71
From the menus, choose
Analysis Properties
In the Analysis Properties dialog box, click the
Output
tab.
Indirect, direct & total effects
check box.
For Model C, the output includes th
e following table of total effects:
The first row of the table indicates that
alienation67
depends, directly or indirectly, on
only. The
total effect
of
on
alienation67
is 0.56. The fact that the effect is
negative means that, all other thin
gs being equal, relatively high
scores are
associated with relatively low
alienation67
scores. Looking in the fifth row of the table,
powles71
depends, directly or indirectly, on
alienation67
alienation71
. Low
scores on
, high scores on
alienation67
, and high scores on
alienation71
associated with high scores on
. See Fox (1980) for more
help in interpreting
direct, indirect, and total effects.
Total Effects (Group number 1  Model C: TimeInvariance)
sesalienation67alienation71
alienation67 
.560.000.000
alienation71 
.542.607.000
SEI
5.174.000.000
education
1.000.000.000
.542.607.999
anomia71 
.542.6071.000
.559.999.000
anomia67 
.5601.000.000
125
Exploratory Analysis
Example 6
The following program fits Model B. It is saved as
Ex06b.vb
Main()
Dim
Sem
AmosEngine
Try
Sem.TextOutput()
Sem.Standardized()
Sem.Smc()
Sem.Crdiff()
Sem.BeginGroup(Sem
.AmosDir & "Examp
les\Wheaton.sav")
Sem.AStructure("anomia67  alienation67 (1)")
Sem.AStructure("anomia67  eps1 (1)")
Sem.AStructu
re("powles67  alienation67")
Sem.AStructu
re("powles67  eps2 (1)")
Sem.AStructure("anomia71  alienation71 (1)")
Sem.AStructure("anomia71  eps3 (1)")
Sem.AStructu
re("powles71  alienation71")
Sem.AStructu
re("powles71  eps4 (1)")
Sem.AStructure("alienation67  ses")
127
Exploratory Analysis
The following program fits Model C. It is saved as
Ex06c.vb
Main()
New
AmosEngine
Try
Sem.TextOutput()
Sem.Standardized()
Sem.Smc()
Sem.AllImpliedMoments()
Sem.FactorScoreWeights()
Sem.TotalEffects()
Sem.BeginGroup(Sem.AmosDir & "Examples\Wheaton.sav")
Sem.AStructure
("anomia67  al
Sem.AStructure("anomia67  eps1 (1)")
Sem.AStructure("powles67  alienation67 (path_p)")
Sem.AStructure("powles67  eps2 (1)")
Sem.AStructure
("anomia71  al
Sem.AStructure("anomia71  eps3 (1)")
Sem.AStructure("powles71  alienation71 (path_p)")
Sem.AStructure("powles71  eps4 (1)")
Sem.AStructu
re("alienation67  ses")
Sem.AStructure
Example 6
To fit all three models, A, B, and C in a
single analysis, start with the following
program, which assigns unique
129
Exploratory Analysis
Sem.FitModel
line, will fit the model four times, each time with a different set of
parameter constraints:
The first line defines a version of the model called
Model A: No Autocorrelation
in which
131
Example
ANonrecursive Model
This example demonstrates structural eq
uation modeling with a nonrecursive model.
Felson and Bohrnstedt (1979) studied 209 girls from sixth through eighth grade. They
made measurements on the following variables:
VariablesDescription
academic
Perceived academic ability, a soci
Example 7
Sample correlations, means, and standard
deviations for these six variables are
contained in the SPSS Statistics file,
Fels_fem.sav
. Here is the data file as it appears in
the SPSS Statistics Data Editor:
The sample means are not
used in this example.
Felson and Bohrnstedts Model
Felson and Bohrnstedt proposed the follow
ing model for six of their seven measured
variables:
A nonrecursive model
Felson and Bohrnstedt (1979)
(Female subjects)
Model Specification
133
ANonrecursive Model
Perceived
academic
performance is modeled as a function of
and perceived
attractiveness (
). Perceived attractiveness, in turn, is modeled as a function of
perceived
academic
height
weight
, and the
of attractiveness by
children from another city. Particularly noteworthy in this model is that perceived
academic ability depends on perceived attractiveness,
vice versa
these feedback loops is called
recursive
and
were defined earlier in Example 4). The cu
rrent model is nonrecursive because it is
possible to trace a path from
to
academic
and back. This path diagram is saved
in the file
We need to establish measurement unit
s for the two unobserved variables,
and
for identification purposes. The precedin
g path diagram shows two regression
weights fixed at 1. These two constraints
are enough to make the model identified.
Text Output
The model has two degrees of freedom, and th
ere is no significant evidence that the
model is wrong.
There is, however, some evidence that the model is unnecessarily complicated, as
indicated by some exceptionally small
critical ratios in the text output.
Chisquare = 2.761
Degrees of freedom = 2
Probability level = 0.251
Example 7
Judging by the critical ratios, you see that
each of these three null hypotheses would be
accepted at conventiona
l significance levels:
Perceived attractiveness does not depend
on height (critical ratio = 0.050).
Perceived academic ability does not depend on perceived attractiveness (critical
ratio = 0.039).
The residual variables
and
are uncorrelated (critical ratio =
0.382).
Strictly speaking, you cannot use the critical ra
tios to test all three hypotheses at once.
Instead, you would have to construct a mode
l that incorporates
all three constraints
simultaneously. This idea will not be pursued here.
Regression Weights: (Group number 1  Default model)
Estimate
S.E.
P Label
academic

GPA .023
.004
6.241
attract 
height .000
.010
.050
.960
attract 
weight .002
.001
1.321
.186
attract 
rating .176
.027
6.444
attract 
academic
1.607
.349
4.599
academic

attract .002
.051
.039
.969
Covariances: (Group number 1  Default model)
Estimate
S.E.
C.R.
GPA  .1;16;&#x.200;
rating
.526
2.139
.032
&#x8.;က
rating
.468
2.279
.023
GPA  .1;16;&#x.200;
weight
6.710
1.435
.151
GPA  .1;16;&#x.200;
height
1.819
2.555
.011
&#x8.;က
weight
19.024
4.643
weight
&#x8.;က
rating
5.243
3.759
error1
&#x8.;က
error2
.004
.382
.702
Variances: (Group number 1  Default model)
Estimate
S.E.
C.R.
P Label
GPA
1.189
10.198
***
.826
10.198
***
weight
36.426
10.198
***
rating
.100
10.198
***
error1
.003
5.747
***
error2
.014
9.974
***
135
ANonrecursive Model
Before you perform the analysis, do the following:
From the menus, choose
Analysis Properties
In the Analysis Properties
dialog box, click the
tab.
Select
Standardized estimates
(a check mark appears next to it).
Close the dialog box.
Here it can be seen that the regression weights and the correlation that we discovered
earlier to be statistically insignificant are also, speaking descriptively, small.
The squared multiple correlations, like the st
andardized estimates, are independent of
units of measurement. To obtain squared mu
ltiple correlations, do the following before
you perform the analysis:
From the menus, choose
Analysis Properties
Standardized Regression Weights: (Group number 1 
Default model)
Estimate
academic

GPA .492
attract 
height .003
attract 
weight .078
attract 
rating .363
attract 
academic .525
academic

attract .006
Correlations: (Group number 1  Default model)
Estimate
GPA .1;.8;&#x5.6;
rating .150
height
&#x6.9;&#x000;
rating .160
GPA .1;.8;&#x5.6;
weight .100
GPA .1;.8;&#x5.6;
height .180
height
&#x6.9;&#x000;
weight .340
weight
&#x6.9;&#x000;
rating .270
error1
&#x6.9;&#x000;
error2 .076
Example 7
In the Analysis Properties dialog box, click the
Output
tab.
Select
Squared multiple correlations
(a check mark appears next to it).
Close the dialog box.
The squared multiple correlatio
ns show that the two endogenous variables in this
model are not predicted very accurately by th
e other variables in the model. This goes
to show that the chisquare test of fit
is not a measure of accuracy of prediction.
Here is the path diagram output displa
ying standardized estimates and squared
multiple correlations:
Squared Multiple Correlations: (Group number 1 
Default model)
Estimate
attract
.402
academic
.236
0
0
3
6
1
5
.
1
6
.
1
0
1
8
3
4
.
2
7
A nonrecursive model
Felson and Bohrnstedt (1979)
(Female subjects)
Standardized estimates
p = .251
137
ANonrecursive Model
The existence of feedback loops in a nonrecursive model permits certain problems to
arise that cannot occur in recursive mode
ls. In the present model, attractiveness
depends on perceived academic ability, which in turn depends on attractiveness, which
depends on perceived academic ability, and
so on. This appears to be an infinite
Example 7
139
Example
FactorAnalysis
This example demonstrates confirmatory common factor analysis.
Holzinger and Swineford (1939) administered
26 psychological tests to 301 seventh
and eighthgrade students in two Chicago schools. In the present example, we use
scores obtained by the 73 girls from a single
school (the GrantWhite school). Here is
s used in this example:
TestExplanation
visperc
Visual perception scores
cubes
Test of spatial visualization
lozenges
Test of spatial orientation
paragraph
Paragraph comprehension score
sentence
Example 8
The file
Grnt_fem.sav
contains the test scores:
Consider the following model for the six tests:
Factor analysis: Girls' sample
Holzinger and Swineford (1939)
Model Specification
141
FactorAnalysis
This model asserts that the first three test
s depend on an unobserved variable called
Spatial
Example 8
is true that the lack of a unit of measurement for unobserved variables is an
everpresent cause of nonidentif
ication. Fortunately, it is
one that is easy to cure, as
we have done repeatedly.
But other kinds of underidentification ca
n occur for which there is no simple
remedy. Conditions for identifiability have to
be established separa
tely for individual
models. Jreskog and Srbom (1984) show how to achieve identification of many
models by imposing equality constraints on
143
FactorAnalysis
Here are the unstandardized results of the an
alysis. As shown at the upper right corner
of the figure, the model fits the data quite well.
As an exercise, you may wish to confir
m the computation of degrees of freedom.
Factor analysis: Girls' sample
Holzinger and Swineford (1939)
Unstandardized estimates
p = .448
Example 8
Regression Weights: (Group number 1  Default model)
Estimate
S.E.
C.R.
visperc 
spatial
1.000
cubes 
spatial
.143
4.250
lozenges 
spatial
.272
4.405
paragrap 
verbal
1.000
sentence 
verbal
.160
8.322
wordmean

verbal
.263
8.482
Standardized Regression Weights: (Group number 1 
Default model)
Estimate
visperc 
spatial .703
cubes 
spatial .654
lozenges 
spatial .736
paragrap 
verbal .880
sentence 
verbal .827
wordmean

verbal .841
Covariances: (Group number 1  Default model)
Estimate
S.E.
C.R.
P Label
spatial
&#x8.;
verbal
7.315
.004
Correlations: (Group number 1  Default model)
Estimate
spatial
&#x8.;
verbal .487
Variances: (Group number 1  Default model)
Estimate
S.E.
C.R.
P Label
spatial
8.123
2.868
.004
verbal
2.159
4.485
err_v
5.986
3.988
err_c
2.584
4.490
err_l
7.892
3.583
err_p
.868
3.263
.001
err_s
1.869
4.263
err_w
4.951
4.024
145
FactorAnalysis
Squared multiple correlations
if you want squared multiple correlation for
each endogenous variable, as shown in the next graphic.
Close the dialog box.
Viewing Standardized Estimates
In the Amos Graphics window, click the
Show the output path diagram
button.
Select
Standardized estimates
in the Parameter Fo
rmats panel at the left of the path
diagram.
Here is the path diagram with
standardized estimates displayed:
Squared Multiple Correlations: (Group number 1 
Default model)
Estimate
wordmean
.708
sentence
.684
paragrap
.774
lozenges
.542
cubes
.428
visperc
.494
Factor analysis: Girls' sample
Holzinger and Swineford (1939)
Standardized estimates
p = .448
Example 8
The squared multiple correlations can be
147
Example
An Alternative to Analysis of
This example demonstrates a simple alternativ
e to an analysis of covariance that does
not require perfectly reliable covariates. A better, but more complicated, alternative
will be demonstrated in Example 16.
Analysis of covariance is a technique that
is frequently used in experimental and
quasiexperimental studies to
reduce the effect of preexisting differences among
treatment groups. Even when random assign
ment to treatment groups has eliminated
Example 9
here has been employed by Bentler and Woodward (1979) and others. Another
149
An Alternative to Analysis of Covariance
Correlations and standard deviat
ions for the five measures are contained in the Microsoft
Excel workbook
, in the
Olss_all
Example 9
Similarly, the mo
del asserts that
post_syn
and
post_opp
are imperfect measures of an
unobserved ability called
, which might be thought of
as verbal ability at
the time of the posttest.
Eps3
eps4
represent errors of
measurement and other
sources of variation not shown elsewhere in the path diagram.
The model shows two variables that may be
useful in accounting for verbal ability
at the time of the posttest. One such predictor
is verbal ability at th
Olsson (1973) test coaching study
Model Specification
151
An Alternative to Analysis of Covariance
To specify Model A, draw a path diagram similar to the one on p. 150. The path
diagram is saved as the file
Ex09a.amw
There is considerable empiri
cal evidence against Model A:
This is bad news. If we had been able to accept Model A, we could have taken the next
step of repeating the analysis with
the regression weight for regressing
post_verbal
on
fixed at 0. But there is no point in doing that now. We have to start with a
model that we believe is correct in order to use it as the basis for testing a stronger
no
treatment effect
version of the model.
Example 9
Requesting modification indices with a threshold of 4 produces the following
additional output:
According to the first mo
dification index in the
M.I.
column, the chi
will decrease by at least 13.161 if the unique variables
and
are allowed to be
correlated (the actual decrease may be gr
eater). At the same time, of course, the
number of degrees of freedom will drop by
1 because of the extra parameter that will
have to be estimated. Since 13.161 is th
e largest modification index, we should
consider it first and ask whether it is reasonable to think that
and
correlated.
represents whatever
pre_opp
measures other than verbal ability at the pretest.
represents whatever
post_opp
measures other than verbal ability at the
posttest. It is plausible that
some stable trait or ability other than verbal ability is
measured on both administrations of the Opposites test. If so, then you would expect a
Modification Indices (Group number 1  Default model)
Covariances: (Group number 1  Default model)
M.I. Par Change
eps2
&#x7.;
eps4
13.161
3.249
eps2
&#x7.;
eps3
10.813
2.822
eps1
&#x7.;
eps4
11.968
3.228
eps1
&#x7.;
eps3
9.788
2.798
153
An Alternative to Analysis of Covariance
You may find your error variables already positioned at the top of the path diagram,
with no room to draw the doubleheaded arrow. To fix the problem:
From the menus, choose
Fit to Page
Alternatively, you can:
Draw the doublehead
ed arrow and, if it is out of bounds, click the
(page with
arrows) button. Amos will shrink your path
diagram to fit within the page boundaries.
Allowing
eps2
eps4
to be correlated results in
a dramatic reduction of the
You may recall from the results of Model A that the modification index for the
covariance between
eps1
eps3
was 9.788. Clearly, freeing that covariance in
Olsson (1973) test coaching study
Model Specification
Example 9
chisquare statistic that will occur
if the corresponding constraintand
only
that
constraintis removed.
Regression Weights: (Group number 1  Default model)
Estimate S.E. C.R. P Label
post_verbal

pre_verbal
.053
16.900
***
post_verbal

treatment 3.640
.477
7.625
***
pre_syn 
pre_verbal
1.000
pre_opp 
pre_verbal
.053
16.606
***
post_syn 
post_verbal
1.000
post_opp 
post_verbal
.053
16.948
***
Covariances: (Group number 1  Default model)
Estimate S.E. C.R. P Label
pre_verbal
&#x70;
treatment
.467
.226
2.066
.039
eps2 &#x5.3;&#x.5;&#x7.7;
eps4 6.797
1.344
5.059
***
Variances: (Group number 1  Default model)
Estimate S.E. C.R. P Label
pre_verbal
38.491
4.501
8.552
treatment
.249
10.296
zeta
4.824
1.331
3.625
eps1
6.013
1.502
4.004
eps2
12.255
1.603
7.646
eps3
6.546
1.501
4.360
eps4
14.685
1.812
8.102
155
An Alternative to Analysis of Covariance
In this example, we are primarily concerne
d with testing a particular hypothesis and
not so much with parameter estimation. Ho
wever, even when the parameter estimates
themselves are not of primary interest, it is a good idea to look at them anyway to see
if they are reasonable. Here,
for instance, you may not care exactly what the correlation
model fits the data.
Now that we have a model (Model B) that we
can reasonably believe is correct, lets
see how it fares if we add the constraint that
does not depend on
In other words, we will test a new model (call it
) that is just like Model B
except that Model C specifies that
has a regression weight of 0 on
2
8
Olsson (1973) test coaching study
Standardized estimates
Example 9
To draw the path diagram for Model C:
Start with the path diagram for Model B.
Rightclick the arrow that points from
treatment
to
and choose
Properties
from the popup menu.
In the Object Properties dialog box, click the
55.3962.684
157
An Alternative to Analysis of Covariance
Example 9
This program fits Model C.
It is saved in the file
Ex09c.vb
Main()
Dim
Sem
AmosEngine
Try
Sem.TextOutput()
Sem.Mods(4)
Sem.Standardized()
Sem.Smc()
Sem.BeginGroup(Sem.AmosDir
& "Examples\UserGuide.xls", "Olss_all")
Sem.AStructure("pre_syn
= (1) pre_verbal + (1) eps1")
Sem.AStructure("pre_opp = pre_verbal + (1) eps2")
Sem.AStructure("post_syn = (1) post_verbal + (1) eps3")
Sem.AStructure("post_opp = post_verbal + (1) eps4")
Sem.AStructure("post_verbal = pre_verbal + (0) treatment + (1) zeta")
Sem.AStructure("eps2 &#x8.;0; eps4")
Sem.FitModel()
Finally
Sem.Dispose()
Try
159
An Alternative to Analysis of Covariance
(Ex09all.vb
) fits all three mode
ls (A through C).
Main()
New
AmosEngine
Try
Sem.TextOutput()
Sem.Mods(4)
Sem.Standardized()
Sem.Smc()
Sem.BeginGroup(Sem.AmosDir & "Examples\UserGuide.xls", "Olss_all")
Sem.AStructure("pre_syn = (1) pre_verbal + (1) eps1")
Sem.AStructure("pre_opp = pre_verbal + (1) eps2")
Sem.AStructure("post_syn = (1) post_verbal + (1) eps3")
Sem.AStructure("post_opp = post_verbal + (1) eps4")
Sem.AStructure("post_verbal =
pre_verbal + (effect) treatment + (1) zeta")
Sem.AStructure
("&#x0;eps2  ep
Sem.Model("Model_A", "cov2_4 = 0")
Sem.Model("Model_B")
Sem.Model("Model_C", "effect = 0")
Sem.FitAllModels()
Finally
Sem.Dispose()
Try
End Sub
161
ExampleS
Simultaneous Analysis of Several
Example 10
We will use Attigs memory data from both young and old subjects. Following is a
partial listing of the old subjec
163
Simultaneous Analysis of Several Groups
The main purpose of a multigroup analysis is
to find out the extent to which groups
differ. Do the groups all have the same path diagram with the same parameter values?
Example 10
Click
File Name
, select the Excel workbook
UserGuide.xls
that is in the Amos
Examples
directory, and click
In the Select a Data Table
dialog box, select the
Attg_yng
worksheet.
Click
to close the Select a Data Table dialog box.
Click
to close the Data Files dialog box.
From the menus, choose
Variables in Dataset
Drag observed variables
recall1
and
to the diagram.
165
Simultaneous Analysis of Several Groups
Connect
recall1
and
with a double
headed arrow.
To add a caption to the path di
agram, from the
menus, choose
Diagram
Figure Caption
and then click the path diagram at the sp
ot where you want the caption to appear.
In the Figure Caption dialog box, ente
r a title that contains the text macros
\group
and
Example 10
Click
to complete the model speci
fication for the young group.
To add a second group, from the menus, choose
Manage Groups
In the Manage Groups dialog box, change the name in the Group Name text box from
Group number 1
to
young subjects
Click
to create a second group.
Change the name in the Group Name text box from
Group number 2
old subjects
167
Simultaneous Analysis of Several Groups
Click
Close
From the menus, choose
File
Data Files
The Data Files dialog box shows that there are two groups labeled
young subjects
and
old subjects
To specify the dataset for the old subjects, in the Data Files dialog box, select
subjects
Click
File Name
, select the Excel workbook
UserGuide.xls
that is in the Amos
Examples
directory, and click
Open
In the Select a Data Tabl
e dialog box, select the
Example 10
Click
Text Output
Model A has zero degrees of freedom.
Amos computed the number of distinct sample moments this way: The young subjects
have two sample variances and one sample covariance, which makes three sample
moments. The old subjects also have thre
e sample moments, making a total of six
sample moments. The parameters to be es
timated are the population moments, and
there are six of them as well.
Since there are zero degrees
of freedom, this model is
untestable.
Chisquare = 0.000
Degrees of freedom = 0
Probability level cannot be computed
Computation of degrees of freedom (Default model)
Number of distinct sample moments: 6
Number of distinct parameters to be estimated: 6
Degrees of freedom (6  6): 0
169
Simultaneous Analysis of Several Groups
Covariances: (young subjects  Default model)
Estimate
S.E.
C.R.
Label
recall1
&#x70;
cued1
3.225
.944
3.416
***
Variances: (young subjects  Default model)
Estimate
S.E.
C.R.
Label
recall1
5.787
1.311
4.416
***
cued1
4.210
.953
4.416
***
Covariances: (old subjects  Default model)
Estimate
S.E.
C.R.
Label
recall1
&#x7.5;&#x000;
cued1
4.887
1.252
3.902
***
Variances: (old subjects  Default model)
Estimate
S.E.
C.R.
Label
recall1
5.569
1.261
4.416
***
cued1
6.694
1.516
4.416
***
5.79
recall1
4.21
cued1
3.22
Example 10: Model A
Simultaneous analysis of several groups
Attig (1983) young subjects
Unstandardized estimates
5.57
recall1
6.69
cued1
4.89
Example 10: Model A
Simultaneous analysis of several groups
Attig (1983) old subjects
Unstandardized estimates
Example 10
Click either the
View Input
View Output
button to see an input or output path
Select either
young subjects
or
old subjects
in the Groups panel.
Select either
Unstandardized estimates
or
Standardized estimates
in the Parameter
Formats panel.
It is easy to see that the parameter estimates are different for the two groups. But are
the differences significant? One way to find ou
t is to repeat the analysis, but this time
171
Simultaneous Analysis of Several Groups
In the Variance text box, enter a name for the variance of
; for example, type
var_rec
Select
All groups
(a check mark will appear next to it).
The effect of the check mark is to assign the name
to the variance of
recall1
in
all groups. Without the check mark,
would be the name of the variance for
for the young group only.
While the Object Properties
dialog box is open, click
cued1
and type the name
var_cue
for its variance.
Click the doubleheaded arrow and type the name
cov_rc
for the covariance. Always
make sure that you select
All groups
The path diagram for each group should now look like this
var_rec
recall1
var_cue
cued1
cov_rc
Example 10: Model B
Homogenous covariance structures
in two groups, Attig (1983) data.
Model Specification
Example 10
Text Output
Because of the constraints im
posed in Model B, only three distinct parameters are
estimated instead of six. As a result, the
number of degrees of freedom has increased
from 0 to 3.
Model B is acceptable at any conventional significance level.
The following are the parameter estimate
s obtained under Model B for the young
the young subjects, 0.780, 0.909, and 0.873) than the corresponding estimates obtained
under Model A (0.944, 1.311, and 0.953). The Model B estimates are to be preferred
over the ones from Model A as long as
you believe that Model B is correct.
Chisquare = 4.588
Degrees of freedom = 3
Probability level = 0.205
Computation of degrees of freedom (Default model)
Number of distinct sample moments: 6
Number of distinct parameters to be estimated: 3
Degrees of freedom (6  3): 3
Covariances: (young subjects  Default model)
Estimate
S.E.
C.R.
Label
recall1
&#x7.5;&#x000;
cued1
4.056
.780
5.202
***
cov_rc
Variances: (young subjec
ts  Default model)
Estimate
S.E.
C.R.
Label
recall1
5.678
.909
6.245
var_rec
cued1
5.452
.873
6.245
var_cue
173
Simultaneous Analysis of Several Groups
For Model B, the output path diag
ram is the same for both groups.
p = .205
Example 10
BeginGroup
method is used tw
ice in this twogroup analysis. The first
BeginGroup
line specifies the
Attg_yng
175
Simultaneous Analysis of Several Groups
Multiple Model Input
Here is a program (
) for fitting both Models A and B.
statements should appe
ar immediately after the
specifications for the last group. It does not matter which
statement goes first.
1In Example 6 (
Ex06all.vb
), multiple model constraint
s were written in a single string, within which individual
constraints were separated by semicolo
ns. In the present exampl
e, each constraint is in its own string, and the
individual strings are separated by
commas. Either syntax is acceptable.
Main()
New
AmosEngine
Try
Sem.Standardized()
Sem.TextOutput()
Sem.BeginGroup(Sem.AmosDir & "Examples\UserGuide.xls", "Attg_yng")
Sem.GroupName("young subjects")
Sem.AStructure("recall1 (yng_rec)")
Sem.AStructure("cued1 (yng_cue)")
Sem.AStructure("recall1 .70; cued1 (yng_rc)")
Sem.BeginGroup(Sem.AmosDir & "Examples\UserGuide.xls", "Attg_old")
Sem.GroupName("old subjects")
Sem.AStructure("recall1 (old_rec)")
Sem.AStructure("cued1 (old_cue)")
Sem.AStructure("recall1 .10; cued1 (old_rc)")
Sem.Model("Model A")
Sem.Model("Model B", "yng_rec=old_rec", "yng_cue=old_cue", _
"yng_rc=old_rc")
Sem.FitAllModels()
Finally
Sem.Dispose()
Try
End Sub
177
Example
Felson and Bohrnstedts Girls and
This example demonstrates how to fit a simu
Example 11
Notice that there are eight variables in the boys data file but only seven in the girls
data file. The extra variable
skills
is not used in any model of this example, so its
presence in the data file is ignored.
Consider extending the Felson and Bohrnstedt model of perceived attractiveness and
academic ability to boys as well as girls.
To do this, we will star
t with the girlsonly
model specification from Example 7 and modify it to accommodate two groups. If you
have already drawn the path diagram for Example 7, you can use it as a starting point
for this example. No additional drawing is needed.
179
Felson and Bohrnstedt
s Girls and Boys
In the Figure Caption dialog box, enter a title that contains the text macro
example:
In Example 7, where there
was only one group, the groups name didnt matter.
Accepting the default name
Group number 1
was good enough. Now that there are two
groups to keep track of, the groups should be given meaningful names.
From the menus, choose
Analyze
Manage Groups
In the Manage Groups dialog box, type
girls
for Group Name.
While the Manage Groups dialog box is open, create a second group by clicking
New
boys
in the Group Name text box.
Example 11
Click
to close the Manage Groups dialog box.
From the menus, choose
File
Data Files
In the Data Files dialog box, doubleclick
and select the data file
Fels_fem.sav
Then, doubleclick
boys
and select the data file
Fels_mal.sav
Click
to close the Data Files dialog box.
A nonrecursive, twogroup model
Felson and Bohrnstedt (1979) boys' data
Model Specification
181
Felson and Bohrnstedt
s Girls and Boys
Text Output for Model A
With two groups instead of one (as in Ex
ample 7), there are twice as many sample
moments and twice as many parameters to estimate. Therefore, you have twice as many
degrees of freedom as there were in Example 7.
The model fits the data from both groups quite well.
We accept the hypothesis that the Felson and Bohrnstedt model is correct for both boys
Computation of degrees of freedom (Default model)
Number of distinct sample moments: 42
Number of distinct parameters to be estimated: 38
Degrees of freedom (42  38): 4
Regression Weights: (girls  Default model)
Estimate S.E. C.R. P Label
academic

GPA .023
attract 
height .000
attract 
weight .002
attract 
rating .176
attract 
academic
1.607
academic

attract .002
Covariances: (girls  Default model)
Estimate S.E. C.R. P Label
GPA  .1;16;&#x.200;
rating
.526
.246
.032
&#x8.;က
rating
.468
.205
.023
GPA  .1;16;&#x.200;
weight
6.710
4.676
.151
GPA  .1;16;&#x.200;
height
1.819
.712
.011
&#x8.;က
weight
19.024
4.098
weight
&#x8.;က
rating
5.243
1.395
error1
&#x8.;က
error2
.004
.010
.702
Variances: (girls  Default model)
Estimate S.E. C.R. P Label
GPA
12.122
1.189
10.198
***
8.428
.826
10.198
***
weight
371.476
36.427
10.198
***
rating
1.015
.100
10.198
***
error1
.019
.003
5.747
***
error2
.143
.014
9.974
***
Example 11
These parameter estimates are the same as
in Example 7. Standard errors, critical
values are also the same. The followi
ng are the unstandardized estimates
for the boys:
Regression Weights: (boys  Default model)
Estimate S.E. C.R. P Label
academic

GPA .021
attract 
height .019
attract 
weight .003
attract 
rating .095
attract 
academic
1.386
academic

attract .063
Covariances: (boys  Default model)
Estimate S.E. C.R. P Label
GPA  .1;16;&#x.200;
rating
.507
.274
.064
&#x8.;က
rating
.198
.230
.390
GPA  .1;16;&#x.200;
weight
15.645
6.899
.023
GPA  .1;16;&#x.200;
height
1.508
.961
.117
&#x8.;က
weight
42.091
6.455
weight
&#x8.;က
rating
4.226
1.662
.011
error1
&#x8.;က
error2
.010
.011
.369
Variances: (boys  Default model)
Estimate S.E. C.R. P Label
GPA
16.243
1.600
10.149
***
11.572
1.140
10.149
***
weight
588.605
57.996
10.149
***
rating
.936
.092
10.149
***
error1
.015
.002
7.571
***
error2
.164
.016
10.149
***
183
Felson and Bohrnstedt
s Girls and Boys
For girls, this is the path diagram wi
th unstandardized estimates displayed:
The following is the path diagram
with the estimates for the boys:
You can visually inspect the girls and bo
ys estimates in Model A, looking for sex
differences. To find out if girls and boys di
ffer significantly with respect to any single
parameter, you could examine the table of cr
itical ratios of differences among all pairs
0
0
1
8
5
3
.
4
7
6
.
7
1
.
8
2
9
.
0
2
5
.
2
4
A nonrecursive, twogroup model
Felson and Bohrnstedt (1979) girls' data
Unstandardized estimates
1
0
5
1
2
0
1
5
.
6
4
1
.
5
1
2
.
0
9
4
.
2
3
A nonrecursive, twogroup model
Felson and Bohrnstedt (1979) boys' data
Unstandardized estimates
Example 11
185
Felson and Bohrnstedt
s Girls and Boys
variables to be groupinvariant. For Model B, you need to constrain six regression
weights in each group.
First, display the girls path diagram by clicking
girls
in the Groups panel at the left of
the path diagram.
Rightclick one of the singleheaded arrows and choose
Object Properties
from the pop
In the Object Properties
dialog box, click the
Example 11
Repeat this until you have named every regression weight. Always make sure to select
(put a check mark next to)
All groups
After you have named all of the regression
weights, the path diagram for each sample
should look something like this:
Text Output
Model B fits the data very well.
Chisquare = 9.493
Degrees of freedom = 10
Probability level = 0.486
3
5
187
Felson and Bohrnstedt
s Girls and Boys
Comparing Model B against Model A gives a nonsignificant chisquare of
with degrees of freedom
. Assuming that Model B
is indeed correct, the Model B estimates
are preferable over the Model A estimates.
9.4933.183
6.310
1046
Regression Weights: (girls  Default model)
Estimate
S.E.
C.R.
Label
academic

GPA
.022
.002
9.475
***
p1
attract

height
.008
.007
1.177
.239
p3
attract

weight
.003
.001
.014
p4
attract

rating
.145
.020
7.186
***
p5
attract

academic
1.448
.232
6.234
***
p6
academic

attract
.018
.039
.469
.639
p2
Covariances: (girls  Default model)
Estimate
S.E.
C.R.
Label
GPA
.5;&#x000;
rating
.526
.246
2.139
.032
height
.5;&#x000;
rating
.468
.205
.023
GPA
.5;&#x000;
weight
6.710
4.676
.151
GPA
.5;&#x000;
height
1.819
.712
2.555
.011
height
.5;&#x000;
weight
19.024
4.098
4.642
***
weight
.5;&#x000;
rating
5.243
1.395
***
error1
.5;&#x000;
error2
.004
.008
.643
Variances: (girls  Default model)
Estimate
S.E.
C.R.
Label
GPA
12.122
1.189
10.198
height
8.428
.826
10.198
weight
371.476
36.427
10.198
rating
1.015
.100
10.198
error1
.018
.003
7.111
error2
.144
.014
10.191
Example 11
Regression Weights: (boys  Default model)
Estimate
S.E.
C.R.
Label
academic

GPA
.022
.002
9.475
attract

height
.008
.007
1.177
.239
attract

weight
.003
.001
2.453
.014
attract

rating
.145
.020
7.186
attract

academic
1.448
.232
6.234
academic

attract
.018
.039
.469
.639
Covariances: (boys  Default model)
Estimate
S.E.
C.R.
Label
GPA
.2;&#x00;
rating
.507
.274
1.850
.064
height
.2;&#x00;
rating
.198
.230
.860
.390
GPA
.2;&#x00;
weight
15.645
6.899
2.268
.023
GPA
.2;&#x00;
height
1.508
.961
1.569
.117
height
.2;&#x00;
weight
42.091
6.455
6.521
weight
.2;&#x00;
rating
4.226
1.662
2.543
.011
error1
.2;&#x00;
error2
.004
.008
.466
.641
Variances: (boys  Default model)
Estimate
S.E.
C.R.
Label
GPA
16.243
1.600
10.149
height
11.572
1.140
10.149
weight
588.605
57.996
10.149
rating
.936
.092
10.149
error1
.016
.002
7.220
error2
.167
.016
10.146
189
Felson and Bohrnstedt
s Girls and Boys
The output path diagram for the girls is:
And the output for the boys is:
0
1
1
5
5
3
.
4
7
6
.
7
1
.
8
2
9
.
0
2
5
.
2
4
A nonrecursive, twogroup model
Felson and Bohrnstedt (1979) girls' data
Unstandardized estimates
0
1
1
5
5
1
2
0
1
5
.
6
4
1
.
5
1
2
.
0
9
4
.
2
3
A nonrecursive, twogroup model
Felson and Bohrnstedt (1979) boys' data
Unstandardized estimates
Example 11
It is possible to fit both Model A and Model B in the same analysis. The file
Ex11ab.amw
in the Amos
Examples
directory shows how to do this.
You might consider adding additional constrai
nts to Model B, such as requiring every
191
Felson and Bohrnstedt
s Girls and Boys
To improve the appearance of the results, from the menus, choose
and use
the mouse to arrange the six rectangl
es in a single column like this:
Drag properties
option can be used
to put the rectangles in perfect vertical
alignment.
From the menus, choose
Drag properties
In the Drag Properties dialog box, select
height
width
, and
Xcoordinate
. A check mark
will appear next to each one.
Use the mouse to drag
these properties from
academic
to
attract
This gives
attract
the same
coordinate as
academic
. In other words, it aligns them
vertically. It also makes
attract
the same size as
academic
if they are not already the
Then drag from
attract
to
height
, and so on. Keep this up until all six
variables are lined up vertically.
Example 11
193
Felson and Bohrnstedt
s Girls and Boys
Label all variances and covariances with suit
able names; for example, label them with
Test of variance/covariance homogeneity
Felson and Bohrnstedt (1979) girls' data
Model Specification
Example 11
195
Felson and Bohrnstedt
s Girls and Boys
Example 11
The following program fits both Models A and B. The program is saved in the file
Ex11ab.vb
Main()
Dim
Sem
AmosEngine
Try
Sem.TextOutput()
Sem.BeginGroup(Sem.AmosDir & "Examples\Fels_fem.sav")
Sem.GroupName("girls")
Sem.AStructure("academic
= (g1) GPA + (g2) attract + (1) error1")
Sem.AStructure("attract = " & _
"(g3) height + (g4) weight + (g5) rating + (g6) academic + (1) error2")
Sem.AStructu
&#x00;re("error2  error1")
Sem.BeginGroup(Sem.AmosDir & "Examples\Fels_mal.sav")
Sem.GroupName("boys")
Sem.AStructure("academic
= (b1) GPA + (b2) attract + (1) error1")
Sem.AStructure("attract = " & _
"(b3) height + (b4) weight + (b5) rating + (b6) academic + (1) error2")
Sem.AStructu
&#x00;re("error2  error1")
Sem.Model("Model_A")
Sem.Model("Model_B", _
"g1=b1", "g2=
b2", "g3=b3", "g4=b4"
, "g5=b5", "g6=b6")
Sem.FitAllModels()
Finally
Sem.Dispose()
Try
197
Example
Simultaneous Factor Analysis for
Example 12
Consider the hypothesis that the common fact
or analysis model of
Example 8 holds for
boys as well as for girls. The path diag
ram from Example 8 can be used as a starting
point for this twogroup model. By default,
Amos Graphics assumes that both groups
have the same path diagram, so the path di
agram does not have to be drawn a second
time for the second group.
In Example 8, where there was only one grou
p, the name of the group didnt matter.
Accepting the default name
Group number 1
was good enough. Now that there are two
groups to keep track of, the groups
should be given meaningful names.
From the menus, choose
Manage Groups
In the Manage Groups dialog box, type
Girls
for Group Name.
199
Simultaneous Factor Analysis for Several Groups
While the Manage Groups dialog box is open, create another group by clicking
in the Group Name text box.
Click
Close
to close the Manage Groups dialog box.
From the menus, choose
File
Data Files
In the Data Files dialog box, doubleclick
and specify the data file
grnt_fem.sav
Then doubleclick
and specify the data file
grnt_mal.sav
Click
to close the Data Files dialog box.
Example 12
Factor analysis: Girls' sample
Holzinger and Swineford (1939)
Model Specification
201
Simultaneous Factor Analysis for Several Groups
Model A is acceptable at any conventional
significance level. If Model A had been
rejected, we would have had to make changes
in the path diagram for at least one of the
two groups.
e 8 where the girls alone were studied.
Chisquare = 16.480
Degrees of freedom = 16
Probability level = 0.420
23.30
spatial
visperc
cubes
lozenges
wordmean
paragrap
sentence
23.87
11.60
28.28
2.83
7.97
19.93
err_w
9.68
verbal
1.00
.61
1.20
1.00
1.33
2.23
Example 12: Model A
Factor analysis: Girls' sample
Holzinger and Swineford (1939)
Unstandardized estimates
Chisquare = 16.480 (16 df)
p = .420
7.32
Example 12
The corresponding output path diagram for the 72 boys is:
Notice that the estimated regression weig
hts vary little across groups. It seems
plausible that the two populations have the
same regression weig
htsa hypothesis that
we will test in Model B.
We now accept the hypothesis that boys and gi
rls have the same path diagram. The next
16.06
spatial
visperc
cubes
lozenges
wordmean
paragrap
sentence
31.57
15.69
36.53
2.36
6.04
19.70
6.90
verbal
1.00
1.51
1.00
1.28
2.29
6.84
Example 12: Model A
Factor analysis: Boys' sample
Holzinger and Swineford (1939)
Unstandardized estimates
Chisquare = 16.480 (16 df)
203
Simultaneous Factor Analysis for Several Groups
Rightclick the arrow that points from
spatial
to
cubes
and choose
Object Properties
from the popup menu.
In the Object Properties
dialog box, click the
Example 12
The path diagram for either of the two samp
les should now look
18.29216.4801.812
205
Simultaneous Factor Analysis for Several Groups
22.00
spatial
visperc
cubes
lozenges
wordmean
paragrap
sentence
verbal
Example 12: Model B
Factor analysis: Girls' sample
Holzinger and Swineford (1939)
Unstandardized estimates
Chisquare = 18.292 (20 df)
Example 12
16.18
spatial
visperc
cubes
lozenges
wordmean
paragrap
sentence
err_v
15.25
err_c
40.97
err_l
2.36
err_p
err_s
19.94
err_w
1.00
Example 12: Model B
Factor analysis: Boys' sample
Holzinger and Swineford (1939)
Unstandardized estimates
Chisquare = 18.292 (20 df)
p = .568
207
Simultaneous Factor Analysis for Several Groups
All but two of the estimated standard errors
are smaller in Model B,
including those for
Example 12
209
Simultaneous Factor Analysis for Several Groups
Here is a program for fitting Model B, in
211
Example
Estimating and Testing Hypotheses
about Means
This example demonstrates how to estimate means and how to test hypotheses about
means. In large samples, th
Example 13
For this example, we will be using Attig
s (1983) memory data, which was described
in Example 1. We will use data from both young and old subjects. The raw data for the
two groups are contained in the Microsoft Excel workbook
UserGuide.xls
, in the
Attg_yng
and
worksheets
In this example, we will
be using only the measures
and
cued1
Model A for Young and Old Subjects
In the analysis of Model B of Example 10, we concluded that
recall1
and
cued1
have
the same variances and covariance for both old and young people. At least, the
evidence against that hypothesis was found to
be insignificant. Model A in the present
example replicates the analysis
in Example 10 of Model B
with an added twist. This
time, the
of the two variables
and
will also be estimated.
In Amos Graphics, estimating and testing hypotheses involving means is not too
different from analyzing variance and covarian
ce structures. Take Model B of Example
10 as a starting point. Young and ol
d subjects had the same path diagram:
213
Estimating and Testing Hypotheses about Means
Select
Estimate means and intercepts
Now the path diagram looks like this (t
he same path diagram for each group):
The path diagram now shows a
mean, variance
Example 13
The behavior of Amos Graphics changes in several ways when you select (put a check
mark next to)
Estimate means and intercepts
Mean and intercept fi
elds appear on the
Parameters
tab in the Object Properties
dialog box.
Constraints can be applied to means and inte
rcepts as well as regression weights,
variances, and covariances.
From the menus, choosing
Calculate Estimates
estimates means and
interceptssubject to constraints, if any.
You have to provide sample means if you provide sample covariances as input.
When you do
not
put a check mark next to
Estimate means and intercepts
Only fields for variances, covariances, an
d regression weights are displayed on the
215
Estimating and Testing Hypotheses about Means
Computation of degrees of freedom (Default model)
Number of distinct sample moments: 10
Means: (young subjects  Default model)
Estimate S.E. C.R. P Label
recall1
10.250
26.862
cued1
11.700
31.292
Covariances: (young subjects  Default model)
Estimate S.E. C.R. P Label
recall1
&#x7.2;&#x000;
cued1
4.056
.780
5.202
*** cov_rc
Variances: (young subjects  Default model)
Estimate S.E. C.R. P Label
recall1
5.678
6.245
var_rec
cued1
5.452
6.245
var_cue
Means: (old subjects  Default model)
Estimate S.E. C.R. P Label
recall1
8.675
.382
22.735
***
cued1
9.575
.374
25.609
***
Covariances: (old subjects  Default model)
Estimate S.E. C.R. P Label
recall1
&#x7.2;&#x000;
cued1
4.056
.780
5.202
*** cov_rc
Variances: (old subjects  Default model)
Estimate S.E. C.R. P Label
recall1
5.678
.909
6.245
***
var_rec
cued1
5.452
.873
6.245
***
var_cue
Example 13
Except for the means, these estimates are th
e same as those obtained in Example 10,
Model B. The estimated standard errors an
d critical ratios are also the same. This
demonstrates that merely estimating means,
without placing any constraints on them,
has no effect on the estimates of the rema
Homogenous covariance structures
Attig (1983) young subjects
Unstandardized estimates
Homogenous covariance structures
Attig (1983) old subjects
Unstandardized estimates
217
Estimating and Testing Hypotheses about Means
You can enter either a numeri
c value or a name in the
Mean
text box. For now, type the
name
mn_rec
Select
All groups
. (A check mark appears next to it.
The effect of the check mark is to
to the mean of
in every group, requiring the mean of
to be the same for all groups.)
After giving the name
mn_rec
to the mean of
, follow the same steps to give the
name
to the mean of
cued1
The path diagrams for the two groups should now look like this:
These path diagrams are saved in the file
Ex13b.amw
Invariant means and (co)variances
Attig (1983) young subjects
Model Specification
Invariant means and (co)variances
Attig (1983) old subjects
Model Specification
Example 13
With the new constraints on the means, Model B has five degrees of freedom.
Model B has to be rejected at any conventional significance level.
If Model A is correct and Model B is wrong (which is plausible, since Model A was
accepted and Model B was rejected), then
the assumption of equal means must be
wrong. A better test of the hypothesis of equal means under the assumption of equal
variances and covariances can be obtained
in the following way: In comparing Model
B with Model A, the chisquare statistics di
ffer by 14.679, with
a difference of 2 in
degrees of freedom. Since Model B is obtain
ed by placing additional constraints on
Model A, we can say that, if Model B is co
rrect, then 14.679 is
an observation on a
chisquare variable with tw
o degrees of freedom. The probability of obtaining this
large a chisquare value is 0.001. Therefore, we reject Model B in favor of Model A,
concluding that the two groups have different means.
The comparison of Model B against Model
A is as close as Amos can come to
conventional multivariate
analysis of variance. In fact, the test in Amos is equivalent
to a conventional MANOVA, except that the ch
isquare test provided by Amos is only
asymptotically correct. By contrast, MANOV
A, for this example, provides an exact
test.
It is possible to fit both Model A and Model B in a single analysis. The file
shows how to do this. One benefit of
fitting both models in a single
analysis is that Amos will recognize that the two models are nested and will
Chisquare = 19.267
Degrees of freedom = 5
Probability level = 0.002
Computation of degrees of freedom (Default model)
Number of distinct sample moments: 10
219
Estimating and Testing Hypotheses about Means
automatically compute the difference in
chisquare values as well as the
value for
testing Model B against Model A.
Example 13
exogenous variable has a mean of 0 unless yo
u specify otherwise. You need to use the
Model
221
Estimating and Testing Hypotheses about Means
Both models A and B can be fitted by
the following program. It is saved as
Main()
New
AmosEngine
Try
Sem.TextOutput()
Sem.ModelMeansAndIntercepts()
Sem.BeginGroup(Sem.AmosDir & "Examples\UserGuide.xls", "Attg_yng")
Sem.GroupName("young subjects")
Sem.AStructure("recall1 (var_rec)")
Sem.AStructure("cued1 (var_cue)")
Sem.AStructure("recall1 .70; cued1 (cov_rc)")
Sem.Mean("recall1", "yng_rec")
Sem.Mean("cued1", "yng_cue")
Sem.BeginGroup(Sem.AmosDir & "Examples\UserGuide.xls", "Attg_old")
Sem.GroupName("old subjects")
Sem.AStructure("recall1 (var_rec)")
Sem.AStructure("cued1 (var_cue)")
Sem.AStructure("recall1 .70; cued1 (cov_rc)")
Sem.Mean("recall1", "old_rec")
Sem.Mean("cued1", "old_cue")
Sem.Model("Model_A", "")
Sem.Model("Model_B", "yng_r
ec = old_rec", "yng_cue = old_cue")
Sem.FitAllModels()
Finally
Sem.Dispose()
Try
End Sub
223
Example
Regression with an Explicit Intercept
This example shows how to estimate the inte
rcept in an ordinary regression analysis.
Ordinarily, when you specify that some variable depends linearly on some others,
Amos assumes that the linear equation expressing the dependency contains an
additive constant, or intercept, but does not estimate it. For instance, in Example 4, we
specified the variable
performance
to depend linearly on three other variables:
knowledge
value
, and
. Amos assumed that the regression equation was
of the following form:
and
are regression weights, and
is the intercept. In Example 4, the
regression weights through
were estimated. Amos did not estimate
in Example
4, and it did not appear in the path diagram. Nevertheless, ,
estimated under the assumption that
was present in the regression equation.
Similarly,
knowledge
value
were assumed to have means, but their
means were not estimated and did not appear in the path diagram. You will usually be
performanceab
knowledgeb
valueb
satisfactionerror
Example 14
We will once again use the data of Warren,
White, and Fuller (1974), first used in
225
Regression with an Explicit Intercept
Notice the string
displayed above the
variable. The
to the left of the comma
indicates that the mean of the
variable is fixed at 0, a standard assumption in
linear regression models. The absence of anything to the right of the comma in
means that the variance of
is not fixed at a constant and does not have a name.
With a check mark next to
Estimate means and intercepts
, Amos will estimate a mean
for each of the predictors, an
d an intercept for the regression equation that predicts
performance
Text Output
The present analysis gives the same results
as in Example 4 but with the explicit
estimation of three means and an intercept.
The number of degrees of freedom is again
0, but the calculation of degrees of freedom
goes a little differently. Sample means are
required for this analysis; therefore, the nu
mber of distinct samp
le moments includes
the sample means as well as the sample
variances and covarian
ces. There are four
sample means, four sample va
riances, and six sample covariances, for a total of 14
sample moments. As for the parameters to
be estimated, there are three regression
weights and an intercept. Al
so, the three predictors have
among them three means,
three variances, and three covariances. Finally,
there is one error variance, for a total of
Job Performance of Farm Managers
Regression with an explicit intercept
(Model Specification)
Example 14
With 0 degrees of freedom, there is no hypothesis to be tested.
The estimates for regression weights, vari
ances, and covariances are the same as in
Example 4, and so are the associated standa
rd error estimates, critical ratios, and
values.
Chisquare = 0.000
Degrees of freedom = 0
Probability level cannot be computed
Computation of degrees of freedom (Default model)
Number of distinct sample moments: 14
Number of distinct parameters to be estimated: 14
Degrees of freedom (14  14): 0
Regression Weights: (Group number 1  Default model)
Estimate
S.E.
C.R.
Label
performance

knowledge
.258
.054
4.822
performance

value
.145
.035
4.136
performance

satisfaction
.049
.038
1.274
.203
Means: (Group number 1  Default model)
Estimate
S.E.
C.R.
Label
value
2.877
.035
81.818
knowledge
1.380
.023
59.891
satisfaction
2.461
.030
81.174
Intercepts: (Group number 1  Default model)
Estimate
S.E.
C.R.
Label
performance
.834
.140
5.951
Covariances: (Group number 1  Default model)
Estimate
S.E.
C.R.
Label
knowledge
&#x7.;
satisfaction
.007
.632
.528
value
&#x7.;
satisfaction
.006
.011
.593
.553
knowledge
&#x7.;
value
.008
3.276
.001
Variances: (Group number 1  Default model)
Estimate
S.E.
C.R.
Label
knowledge
.051
.007
6.964
value
.120
.017
6.964
satisfaction
.089
.013
6.964
error
.012
.002
6.964
227
Regression with an Explicit Intercept
Below is the path diagram that shows the un
standardized estimate
s for this example.
The intercept of 0.83 appears just above the endogenous variable
performance
Job Performance of Farm Managers
Regression with an explicit intercept
(Unstandardized estimates)
Example 14
The following program for the model of Example 14 gives all the same results, plus
mean and intercept estimates. This program is saved as
Note the
Sem.ModelMeansAndIntercepts
statement that causes Amos to treat means and
229
Regression with an Explicit Intercept
Main()
New
AmosEngine
Try
Sem.TextOutput()
Sem.Standardized()
Sem.Smc()
Sem.ImpliedMoments()
Sem.SampleMoments()
Sem.ModelMeansAndIntercepts()
Sem.BeginGroup( _
Sem.AmosDir & "Examples\UserGuide.xls", "Warren5v")
Sem.AStructure("performance  knowledge")
Sem.AStructure("performance  value")
Sem.AStructure("
performance  satisfaction")
Sem.AStructure
("performance  error (1)")
Sem.Intercept("performance")
Sem.Mean("knowledge")
Sem.Mean("value")
Sem.Mean("satisfaction")
Sem.FitModel()
Finally
Sem.Dispose()
Try
End Sub
231
Example
Factor Analysis with Structured
This example demonstrates how to estimate
factor means in a common factor analysis
of data from several populations.
Conventionally, the common factor analysis
model does not make any assumptions
about the means of any variables. In partic
ular, the model makes no assumptions about
the means of the common factors. In fact,
it is not even possible to estimate factor
means or to test hypotheses in a conv
entional, singlesample factor analysis.
However, Srbom (1974) showed that it
possible to make inferences about factor
means under reasonable assumptions, as long as you are analyzing data from more
than one population. Using Srboms approach, you cannot estimate the mean of
every factor for every population, but you can estimate
differences
in factor means
For instance, think about Example 12, where a common factor
analysis model was fitted simultaneously to
a sample of girls and a sample of boys.
Example 15
The identification status of the factor an
alysis model is a difficult subject when
estimating factor means. In
fact, Srboms accomplish
ment was to show how to
233
Factor Analysis with Structured Means
In the Object Properties
dialog box, click the
spatial
int_vis
visperc
int_cub
cubes
int_loz
lozenges
wordmean
int_par
paragrap
int_sen
sentence
err_v
err_c
err_l
err_p
err_s
err_w
verbal
cube_s
lozn_s
sent_v
word_v
Example 15
The boys path diagram sh
ould look like this:
The crossgroup constraints on intercepts
and regression weights may or may not be
satisfied in the populations. One result of fitting the model will be a test of whether
these constraints hold in the populations of girls and boys. The reason for starting out
with these constraints is that (as Srbom
points out) it is necessary to impose
constraints on the intercepts and regression weights in order to make the model
identified when estimat
ing factor means. These are not
the only constraints that would
make the model identified, but they are plausible ones.
The only difference between the boys and gi
rls path diagrams is in the constraints
on the two factor means. For
boys, the means are fixed at 0. For girls, both factor means
are estimated. The girls factor means are named
and
, but the factor means
are unconstrained because eac
h mean has a unique name.
The boys factor means were fixed at 0
in order to make the model identified.
Srbom showed that, ev
en with all the other constraint
s imposed here, it is still not
possible to estimate factor means for both boys and girls simultaneously. Take verbal
ability, for example. If
you fix the boys mean verbal ability at some constant (like 0),
you can then estimate the girls
mean verbal ability. Alternatively, you can fix the girls
mean verbal ability at some constant, and th
en estimate the boys
mean verbal ability.
The bad news is that you can
not estimate both means at on
ce. The good news is that
spatial
int_vis
visperc
int_cub
cubes
int_loz
lozenges
int_wrd
wordmean
int_par
paragrap
int_sen
sentence
err_v
err_c
err_l
err_p
err_s
err_w
verbal
cube_s
sent_v
word_v
235
Factor Analysis with Structured Means
1.07, 21.19
spatial
visperc
25.12
cubes
lozenges
wordmean
9.45
paragrap
sentence
0, 25.62
err_v
0, 12.55
err_c
0, 24.65
err_l
0, 2.84
err_p
err_s
err_w
.96, 9.95
verbal
1.00
1.00
Example 15
Here are the boys estimates:
Girls have an estimated mean spatial ability
of 1.07. We fixed the mean of boys
spatial ability at 0. Thus, girls mean spat
ial ability is estimated to be 1.07 units
below
boys mean spatial ability. This difference is
not affected by the initial decision to fix
the boys mean at 0. If we had fixed the boys mean at 10.000, the girls mean would
have been estimated to be 8.934. If we had fixed the girls mean at 0, the boys mean
would have been estimated to be 1.07.
What unit is spatial ability expressed in?
A difference of 1.07 verbal ability units
may be important or not, depending on the si
ze of the unit. Since the regression weight
visperc
on spatial ability is equal to 1,
we can say that spatial ability is
expressed in the same units as scores on the
visperc
test. Of course, this is useful
information only if you happen
to be familiar with the
visperc
test. There is another
approach to evaluati
ng the mean difference of 1.07, which does not involve
visperc
portion of the text output not reproduced here shows that
has an estimated
variance of 15.752 for boys, or a standard deviation of about 4.0. For girls, the variance
of
is estimated to be 21.188, so that its standard deviation is about 4.6. With
standard deviations this large, a difference of 1.07 would not be considered very large
The statistical significance of the 1.07 unit
difference between girls and boys is easy
to evaluate. Since the boys mean was fixed at
0, 15.75
spatial
30.14
visperc
25.12
cubes
lozenges
16.22
wordmean
9.45
paragrap
18.26
sentence
err_v
0, 15.31
err_c
0, 40.71
err_l
0, 2.35
err_p
0, 6.02
err_s
err_w
verbal
1.00
1.00
237
Factor Analysis with Structured Means
Here are the girls factor mean estimates from the text output:
The girls mean spatial ability has a critical
ratio of 1.209 and is not significantly
different from 0 (). In other words, it
is not significantly
different from the
boys mean.
Turning to verbal ability, the girl
s mean is estimated 0.96 units
above
the boys
mean. Verbal ability has a standard deviatio
n of about 2.7 among boys and about 3.15
among girls. Thus, 0.96 verbal ability units is about onethird of a standard deviation
in either group. The difference between boys
and girls approaches
significance at the
0.05 level ().
In the discussion of Model A, we used critical ratios to carry out two tests of
significance: a test for sex differences in spatial ability and a test for sex differences in
verbal ability. We will now carry out a single
test of the null hypothesis that there are
no sex differences, either in spatial ability or
in verbal ability. To do this, we will repeat
the previous analysis with th
e additional constraint that
boys and girls have the same
mean on spatial ability and on verbal abil
ity. Since the boys means are already fixed
Means: (Girls  Default model)
Estimate
S.E.
C.R.
Label
spatial
1.066
.881
1.209
.226
mn_s
verbal
.956
.521
1.836
.066
mn_v
0.226
0.066
Example 15
Click
Model B
in the Model Name text box.
Type the constraints
mn_s = 0
mn_v = 0
in the Parameter Constraints text box.
Click
Now when you choose
Analyze
Calculate Estimates
, Amos will fit both Model A and
Model B. The file
Ex15all.amw
239
Factor Analysis with Structured Means
If we did not have Model A as a basis fo
r comparison, we would now accept Model B,
using any conventional significance level.
An alternative test of Model B can be obta
ined by assuming that
Model A is correct
Assuming model Model A to be correct:
Model
DF
CMIN
NFI
Delta1
IFI
Delta2
RFI
rho1
TLI
rho2
Model B
8.030
.018
.024
.026
.021
.023
Example 15
241
Factor Analysis with Structured Means
The following program fits Model B. In this
model, the factor means are fixed at 0 for
both boys and girls. The program is saved as
Main()
New
AmosEngine
Try
Dim
dataFile
String
= Sem.AmosDir & "Examples\userguide.xls"
Sem.TextOutput()
Sem.Standardized()
Sem.Smc()
Sem.ModelMeansAndIntercepts()
Sem.BeginGroup(dataFile, "grnt_fem")
Sem.GroupName("Girls")
Sem.AStructure("visperc
= (int_vis) + (1
) spatial + (1) err_v")
Sem.AStructure("cubes = (int_cub) + (cube_s) spatial + (1) err_c")
Sem.AStructure("lozenges = (int_loz) + (lozn_s) spatial + (1) err_l")
Sem.AStructure("paragraph = (int_par) + (1) verbal + (1) err_p")
Sem.AStructure("sentence = (int_sen) + (sent_v) verbal + (1) err_s")
Sem.AStructure("wordmean
= (int_wrd) + (word_v)
verbal + (1) err_w")
Sem.Mean("spatial", "0")
Sem.Mean("verbal", "0")
Sem.BeginGroup(dataFile, "grnt_mal")
Sem.GroupName("Boys")
Sem.AStructure("visperc
= (int_vis) + (1
) spatial + (1) err_v")
Sem.AStructure("cubes = (int_cub) + (cube_s) spatial + (1) err_c")
Sem.AStructure("lozenges = (int_loz) + (lozn_s) spatial + (1) err_l")
Sem.AStructure("paragraph = (int_par) + (1) verbal + (1) err_p")
Sem.AStructure("sentence = (int_sen) + (sent_v) verbal + (1) err_s")
Sem.AStructure("wordmean
= (int_wrd) + (word_v)
verbal + (1) err_w")
Sem.Mean("spatial", "0")
Sem.Mean("verbal", "0")
Sem.FitModel()
Finally
Sem.Dispose()
Try
End Sub
Example 15
The following program (
) fits both models A and B.
Main()
Dim
Sem
AmosEngine
Try
Sem.TextOutput()
Sem.Standardized()
Sem.Smc()
Sem.ModelMeansAndIntercepts()
Sem.BeginGroup(Sem.A
mosDir & "Examples\Grnt_fem.sav")
Sem.GroupName("Girls")
Sem.AStructure("visperc
= (int_vis) + (1) spatial + (1) err_v")
Sem.AStructure("cubes = (int_cub) + (cube_s) spatial + (1) err_c")
Sem.AStructure("lozenges
= (int_loz) + (lozn_s) spatial + (1) err_l")
Sem.AStructure("paragrap =
(int_par) + (1) verbal + (1) err_p")
Sem.AStructure("sentence = (int_sen) + (sent_v) verbal + (1) err_s")
Sem.AStructure("wordmean
= (int_wrd) + (word_v) verbal + (1) err_w")
Sem.Mean("spatial", "mn_s")
Sem.Mean("verbal", "mn_v")
Sem.BeginGroup(Sem.A
mosDir & "Examples\Grnt_mal.sav")
Sem.GroupName("Boys")
Sem.AStructure("visperc
= (int_vis) + (1) spatial + (1) err_v")
Sem.AStructure("cubes = (int_cub) + (cube_s) spatial + (1) err_c")
Sem.AStructure("lozenges
= (int_loz) + (lozn_s) spatial + (1) err_l")
Sem.AStructure("paragrap =
(int_par) + (1) verbal + (1) err_p")
Sem.AStructure("sentence = (int_sen) + (sent_v) verbal + (1) err_s")
Sem.AStructure("wordmean
= (int_wrd) + (word_v) verbal + (1) err_w")
Sem.Mean("spatial", "0")
Sem.Mean("verbal", "0")
Sem.Model("Model A")
' Sex difference in factor means.
Sem.Model("Model B", "mn_s=0", "mn_v=0")
' Equal factor means.
Sem.FitAllModels()
Finally
Sem.Dispose()
Try
243
Example
Srboms Alternative to
Analysis of Covariance
This example demonstrates latent structural equation modeling with longitudinal
observations in two or more groups, models that generalize traditional analysis of
covariance techniques by in
corporating latent variables
and autocorrelated residuals
(compare to Srbom, 1978), and how assumptions employed in traditional analysis of
covariance can be tested.
Example 9 demonstrated an alternative to co
nventional analysis of covariance that
works even with unreliable covariates. Unfo
rtunately, analysis of covariance also
depends on other assumptions besides the a
ssumption of perfectly reliable covariates,
Example 16
We will again use the Olsson (1973) data in
troduced in Example 9. The sample means,
variances, and covariances from the 108 experimental subjects are in the Microsoft
the sample covariance matrix. Amos perform
s the conversion from unbiased estimates
to maximum likelihood
estimates automatically.
1
245
Srboms Alternative to Analysis of Covariance
From the menus, choose
Analysis Properties.
In the Analysis Properties
dialog box, click the
Example 16
The following path diagram is Mo
del A for the experimental group:
Means and intercepts ar
e an important part of this model, so be sure that you do the
following:
From the menus, choose
Analysis Properties
Click the
tab.
Select
Estimate means and intercepts
(a check mark appe
ars next to it).
In each group, Model A specifies that
pre_syn
and
pre_opp
are indicators of a single
latent variable called
pre_verbal
, and that
post_syn
post_opp
are indicators of
another latent variable called
. The latent variable
pre_verbal
is interpreted
An alternative to ANCOVA
Olsson (1973): control condition.
Model Specification
An alternative to ANCOVA
Olsson (1973): experimental condition.
Model Specification
247
Srboms Alternative to Analysis of Covariance
as verbal ability at the be
ginning of the study, and
post_verbal
is interpreted as verbal
ability at the conclusion of
the study. This is Srboms
measurement
model. The
structural
model specifies that
depends linearly on
opp_v1
opp_v2
require the regression weights in the measurement
model to be the same for both groups. Similarly, the labels
a_syn1
a_opp1
a_syn2
a_opp2
require the intercepts in the measur
ement model to be
the same for both
groups. These equality constr
aints are assumptions that could be wrong. In fact, one
result of the upcoming analyses will be a te
st of these assumption
s. As Srbom points
out, some assumptions have to be made
about the parameters
in the measurement
model in order to make it possible to estima
te and test hypotheses about parameters in
the structural model.
For the control subjects, the mean of
pre_verbal
and the intercept of
post_verbal
are
fixed at 0. This establishes the control group as the reference group for the group
comparison. You have to pick such a referen
ce group to make the latent variable means
and intercepts identified.
For the experimental subjects, the mean
Example 16
From the menus, choose
Analysis Properties
In the Analysis Properties dialog box, click the
Output
tab.
Select
Modification indices
and enter a suitable threshold in
the text box to its right. For
this example, the threshold will be left at its default value of 4.
Here is the modification index output from the experimental group:
The following variances are negative. (control  Default
model)
zeta
2.868
Modification Indices (experimental  Default model)
Covariances: (experimental  Default model)
M.I.
Par Change
eps2
&#x7.;က
eps4
10.508
4.700
eps2
&#x7.;က
eps3
8.980
4.021
eps1
&#x7.;က
eps4
8.339
3.908
eps1
&#x7.;က
eps3
7.058
3.310
Variances: (experimental  Default model)
M.I.
Par Change
Regression Weights: (experimental  Default model)
M.I.
Par Change
Means: (experimental  Default model)
M.I.
Par Change
Intercepts: (experimental  Default model)
M.I.
Par Change
249
Srboms Alternative to Analysis of Covariance
The largest modification index obtained w
ith Model A suggests adding a covariance
Draw a doubleheaded
and
This allows
eps2
and
to be correlated in both groups. We do not want them to be
correlated in the control group, so the cova
riance must be fixed at 0 in the control
group. To accomplish this:
Click
in the Groups panel (at the left of
the path diagram) to display the path
diagram for the control group.
Rightclick the doubleheaded arrow and choose
Object Properties
from the popup
In the Object Properties
dialog box, click the
Example 16
For Model B, the path diagram for the control group is:
An alternative to ANCOVA
Olsson (1973): control condition.
Model Specification
251
Srboms Alternative to Analysis of Covariance
For the experimental group, the path diagram is:
In moving from Model A to Model B, the chisquare statistic dropped by 17.712 (more
than the promised 10.508) while the number of degrees of freedom dropped by just 1.
Model B is an improvement over Model A but not enough of an improvement. Model
B still does not fit the data well
. Furthermore, the variance of
An alternative to ANCOVA
Olsson (1973): experimental condition.
Model Specification
Example 16
The largest modification index
(4.727) suggests allowing
eps2
eps4
correlated in the control group. (
and
are already correlated in the
experimental group.) Making this modification leads to Model C.
Model C is just like Model B except that the terms
and
are correlated in both
the control group and the experimental group.
To specify Model C, just take Model B and
remove the constraint on the covariance
Modification Indices (control  Default model)
Covariances: (control  Default model)
M.I.
Par Change
eps2
&#x15.; 6.;瀀
eps4
4.727
2.141
eps1
&#x15.; 6.;瀀
eps4
4.086
2.384
Variances: (control  Default model)
M.I.
Par Change
Regression Weights: (control  Default model)
M.I.
Par Change
Means: (control  Default model)
M.I.
Par Change
Intercepts: (control  Default model)
M.I.
Par Change
An alternative to ANCOVA
Olsson (1973): control condition.
Model Specification
253
Srboms Alternative to Analysis of Covariance
Finally, we have a model that fits.
From the point of view of statistical goodness
of fit, there is no reason to reject Model
C. It is also worth noting that all the vari
ance estimates are pos
itive. The following are
An alternative to ANCOVA
Olsson (1973): control condition.
Unstandardized estimates
Example 16
Most of these parameter estimates are not ve
ry interesting, although you may want to
check and make sure that the estimates are
reasonable. We have already noted that the
variance estimates are positive. The path
coefficients in the measurement model are
positive, which is reassuring. A mixture of po
sitive and negative regression weights in
the measurement model would have been diff
icult to interpret and would have cast
doubt on the model. The covariance between
eps2
and
is positive in both groups,
as expected.
We are primarily interest
ed in the regression of
post_verbal
on
intercept, which is fixed at 0 in the cont
rol group, is estimated to be 3.71 in the
experimental group. The regression weight is
estimated at 0.95 in the control group and
0.85 in the experimental group. The regression weights for the two groups are close
enough that they might even
be identical in the two popu
lations. Identical regression
weights would allow a greatly simplified ev
aluation of the trea
tment by limiting the
comparison of the two groups to a comparison of their intercepts. It is therefore
worthwhile to try a model in which the re
gression weights are
the same for both
groups. This will be Model D.
Model D is just like Model C except that it requires the regression weight for predicting
post_verbal
from
to be the same for both groups. This constraint can be imposed
by giving the regression weight
the same name, for example
pre2post
, in both groups. The
following is the path
diagram for Model D for
the experimental group:
An alternative to ANCOVA
Olsson (1973): experimental condition.
Unstandardized estimates
255
Srboms Alternative to Analysis of Covariance
Next is the path diagram for
Model D for the control group:
Model D would be accepted at conventional significance levels.
Chisquare = 3.976
Degrees of freedom = 5
Probability level = 0.553
An alternative to ANCOVA
Olsson (1973): experimental condition.
Model Specification
An alternative to ANCOVA
Olsson (1973): control condition.
Model Specification
Example 16
Testing Model D against Model C gives a chisquare value of 1.179 (= 3.976 2.797)
with 1(that is, 5 4) degree of freedom.
Again, you would accept the hypothesis of
equal regression weights (Model D).
With equal regression weights, the compar
ison of treated and untreated subjects
An alternative to ANCOVA
Olsson (1973): control condition.
Unstandardized estimates
An alternative to ANCOVA
Olsson (1973): experimental condition.
Unstandardized estimates
257
Srboms Alternative to Analysis of Covariance
intercept for the experi
mental group is significantly di
fferent from the in
tercept for the
control group (which is fixed at 0).
Another way of testing the difference in
post_verbal
intercepts for significance is to
repeat the Model D analysis with the addition
al constraint that the intercept be equal
across groups. Since the intercept for the control group is already fixed at 0, we need
add only the requirement that the intercept be 0 in the experimental group as well. This
restriction is used in Model E.
The path diagrams for Model E are just
like that for Model D, except that the
intercept in the regression of
on
pre_verbal
is fixed at 0 in
both
groups.
The path diagrams are not reproduced here. They can be found in
Model E has to be rejected.
Comparing Model E against Model D yields a chisquare value of 51.018 (= 55.094
3.976) with 1 (= 6 5) degree of freedom. Model E has to be rejected in favor of Model
D. Because the fit of Model E is signifi
cantly worse than that of Model D, the
hypothesis of equal intercepts again has to be
rejected. In other words, the control and
experimental groups differ at the time of th
e posttest in a way th
at cannot be accounted
for by differences that existed at the time of the pretest.
This concludes Srboms (1978) analysis of the Olsson data.
The example file
Ex16a2e.amw
fits all five models (A through E) in a single analysis.
The procedure for fitting multip
le models in a single analysis was shown in detail in
Example 6.
Chisquare = 55.094
Degrees of freedom = 6
Probability level = 0.000
Example 16
259
Srboms Alternative to Analysis of Covariance
The following is the path diagram
for Model X for the control group:
The path diagram for the experimental group is identical. Using the same parameter
names for both groups has the effect of requiring the two groups to have the same
parameter values.
Model X would be rejected at any conventional level of significance.
The analyses that follow (Models Y and Z)
are actually inappropriate now that we are
satisfied that Model X is inappropriate. We will carry out the analyses as an exercise in
order to demonstrate that they yield the
same results as obtained in Example 9.
Consider a model that is just like Mode
l D but with these additional constraints:
Verbal ability at the pretest (
pre_verbal
) has the same variance in the control and
experimental groups.
The variances of
eps3
eps4
Groupinvariant covariance structure
Olsson (1973): control condition
Model Specification
_
s
1
o
2
_
s
2
o
1
Example 16
Apart from the correlation between
and
, Model D required that
An alternative to ANCOVA
Olsson (1973): experimental condition.
261
Srboms Alternative to Analysis of Covariance
Here is the path diagram for the control group:
We must reject Model Y.
This is a good reason for being dissatisfied with the analysis of Example 9, since it
depended upon Model Y (which, in Example 9, was called Model B) being correct. If
you look back at Example 9, you will see that we accepted Model B there (
= 2.684,
df
= 2,
= 0.261). So how can we say that the
same model has to be rejected here (
= 31.816,
= 1,
= 0.001)? The answer is that, whil
e the null hypothesis is the same
in both cases (Model B in Example 9 an
d Model Y in the present example), the
alternative hypotheses are different. In Ex
ample 9, the alternative against which Model
B is tested includes the assumption that th
e variances and covariances of the observed
variables are the same fo
r both values of the
treatment
variable (also stated in the
assumptions on p. 35). In other words, the test of Model B carried out in Example 9
implicitly assumed homogeneity of varian
ces and covariances for the control and
experimental populations. This is the very
assumption that is made explicit in ModelX
of the present example.
Model Y is a restricted version of Model X. It can be shown that the assumptions of
Model Y (equal regression weights for the two populations, and equal variances and
Chisquare = 31.816
Degrees of freedom = 12
Probability level = 0.001
An alternative to ANCOVA
Olsson (1973): control condition.
Example 16
covariances of the exogenous variables) imply the assumptions of Model X (equal
covariances for the observed variab
les). Models X and Y are therefore
models,
and it is possible to carry out a
conditional
test of Model Y unde
r the assumption that
Model X is true. Of course, it will make sense to
do that test only if Model X really is
true, and we have already concluded it is
Model B of Example 9.
Finally, construct a new model (Model Z) by starting with Model Y and adding the
requirement that the intercept in the equation for predicting
post_verbal
pre_verbal
be the same in both popu
lations. This model is eq
uivalent to Model C of
Example 9. The path diagrams for Model Z are as follows:
An alternative to ANCOVA
Olsson (1973): experimental condition.
Unstandardized estimates
263
Srboms Alternative to Analysis of Covariance
Here is the path diagram for Model Z for the experimental group:
Here is the path diagram for the control group:
This model has to be rejected.
Chisquare = 84.280
Degrees of freedom = 13
Probability level = 0.000
An alternative to ANCOVA
Olsson (1973): experimental condition.
Model Specification
An alternative to ANCOVA
Olsson (1973): control condition.
Model Specification
Example 16
Model Z also has to be rejected when compared to Model Y (
= 84.280 31.816 =
= 13 12 = 1). Within rounding error, this is the same difference in
chisquare values and degrees of freedom as in Example 9, when Model C was
compared to Model B.
265
Srboms Alternative to Analysis of Covariance
To fit Model B, start with the pr
ogram for Model A and add the line
Sem.AStructure("eps2 &#x4.1;&#x4.9;&#x00; eps4")
to the model specification for the experimental group. Here is the resulting program for
Model B. It is saved as
Main()
New
AmosEngine
Try
Dim
dataFile
String
= Sem.AmosDir & "Examples\UserGuide.xls"
Sem.TextOutput()
Sem.Mods(4)
Sem.Standardized()
Sem.Smc()
Sem.ModelMeansAndIntercepts()
Sem.BeginGroup(dataFile, "Olss_cnt")
Sem.GroupName("control")
Sem.AStructure("pre_syn =
(a_syn1) + (1) pre_verbal + (1) eps1")
Sem.AStructure( _
"pre_opp = (a_opp1) + (opp_v1) pre_verbal + (1) eps2")
Sem.AStructure("post_syn = (a_syn2) + (1) post_verbal + (1) eps3")
Sem.AStructure( _
"post_opp = (a_opp2) + (opp_v2) post_verbal + (1) eps4")
Example 16
The following program fits Model C. The program is saved as
Main()
Dim
Sem
AmosEngine
Try
dataFile
String
= Sem.AmosDir & "Examples\UserGuide.xls"
Sem.TextOutput()
Sem.Mods(4)
Sem.Standardized()
Sem.Smc()
Sem.ModelMeansAndIntercepts()
Sem.BeginGroup(dataFile, "Olss_cnt")
Sem.GroupName("control")
Sem.AStructure("pre_syn = (a_syn1) + (1) pre_verbal + (1) eps1")
Sem.AStructure( _
"pre_opp = (a_opp1) + (opp_v1) pre_verbal + (1) eps2")
Sem.AStructure("post_syn = (a_syn2) + (1) post_verbal + (1) eps3")
Sem.AStructure( _
"post_opp = (a_opp2) + (opp_v2) post_verbal + (1) eps4")
Sem.AStructure("post_verbal = (0) + () pre_verbal + (1) zeta")
Sem.AS
tructure("eps2
&#x0;  eps4")
Sem.BeginGroup(dataFile, "Olss_exp")
Sem.GroupName("experimental")
Sem.AStructure("pre_syn = (a_syn1) + (1) pre_verbal + (1) eps1")
Sem.AStructure( _
"pre_opp = (a_opp1) + (opp_v1) pre_verbal + (1) eps2")
Sem.AStructure("post_syn = (a_syn2) + (1) post_verbal + (1) eps3")
Sem.AStructure( _
"post_opp = (a_opp2) + (opp_v2) post_verbal + (1) eps4")
Sem.AStructure("post_verbal = (effect) + () pre_verbal + (1) zeta")
Sem.AS
tructure("eps2
&#x0;  eps4")
Sem.Mean("pre_verbal", "pre_diff")
Sem.FitModel()
Finally
Sem.Dispose()
Try
267
Srboms Alternative to Analysis of Covariance
The following program fits Model D. The program is saved as
Main()
Dim
Sem
AmosEngine
Try
dataFile
String
= Sem.AmosDir & "Examples\UserGuide.xls"
Sem.TextOutput()
Sem.Mods(4)
Sem.Standardized()
Sem.Smc()
Sem.ModelMeansAndIntercepts()
Sem.BeginGroup(dataFile, "Olss_cnt")
Sem.GroupName("control")
Sem.AStructure("pre_syn = (a_syn1) + (1) pre_verbal + (1) eps1")
Sem.AStructure( _
"pre_opp = (a_opp1) + (opp_v1) pre_verbal + (1) eps2")
Sem.AStructure("post_syn = (a_syn2) + (1) post_verbal + (1) eps3")
Sem.AStructure( _
"post_opp = (a_opp2) + (opp_v2) post_verbal + (1) eps4")
Example 16
The following program fits Model E. The program is saved as
Main()
Dim
Sem
AmosEngine
Try
dataFile
String
= Sem.AmosDir & "Examples\UserGuide.xls"
Sem.TextOutput()
Sem.Mods(4)
Sem.Standardized()
Sem.Smc()
Sem.ModelMeansAndIntercepts()
Sem.BeginGroup(dataFile, "Olss_cnt")
Sem.GroupName("control")
Sem.AStructure("pre_syn = (a_syn1) + (1) pre_verbal + (1) eps1")
Sem.AStructure( _
"pre_opp = (a_opp1) + (opp_v1) pre_verbal + (1) eps2")
Sem.AStructure("post_syn = (a_syn2) + (1) post_verbal + (1) eps3")
Sem.AStructure( _
"post_opp = (a_opp2) + (opp_v2) post_verbal + (1) eps4")
269
Srboms Alternative to Analysis of Covariance
The following program fits all five models, A through E. The program is saved as
Ex16a2e.vb
Main()
New
AmosEngine
Try
Dim
dataFile
String
= Sem.AmosDir & "Examples\UserGuide.xls"
Sem.TextOutput()
Sem.Mods(4)
Sem.Standardized()
Sem.Smc()
Sem.ModelMeansAndIntercepts()
Sem.BeginGroup(dataFile, "Olss_cnt")
Sem.GroupName("control")
Sem.AStructure("pre_syn =
(a_syn1) + (1) pre_verbal + (1) eps1")
Sem.AStructure( _
"pre_opp = (a_opp1) + (opp_v1) pre_verbal + (1) eps2")
Sem.AStructure("post_syn = (a_syn2) + (1) post_verbal + (1) eps3")
Sem.AStructure( _
"post_opp = (a_opp2) + (opp_v2) post_verbal + (1) eps4")
Example 16
Models X, Y, and Z
271
Example
This example demonstrates the analysis of a dataset in which some values are missing.
Example 17
exclude only persons whose incomes you do not know. Similarly, in computing the
sample covariance between age and income, yo
u would exclude an observation only if
age is missing or if income is missing. Th
is approach to missing
data is sometimes
called
pairwise deletion
A third approach is
, replacing the missing values with some kind
of guess, and then proceeding with a conv
entional analysis appropriate for complete
data. For example, you might compute the m
ean income of the persons who reported
their income, and then attribute that income
to all persons who did not report their
income. Beale and Little (1975) discuss
methods for data imputation, which are
implemented in many statistical packages.
missing values, each of the 438 data values in
Grnt_fem.sav
was deleted with
probability 0.30.
273
Missing Data
Example 17
After specifying the data file to be
and drawing the above path diagram:
From the menus, choose
Analysis Properties
In the Analysis Properties dialog box, click the
Estimation
tab.
Select
Estimate means and intercepts
(a check mark appe
ars next to it).
This will give you an estimate of the interc
ept in each of the six regression equations
for predicting the measured variables. Ma
ximum likelihood estimation with missing
values works only when you estimate means
and intercepts, so you have to estimate
them even if you are not in
terested in the estimates.
Computing some fit measures requires fittin
g the saturated and independence models in
addition to your model. This is never a
Factor analysis with missing data
Holzinger and Swineford (1939): Girls' sample
Model Specification
+
2
275
Missing Data
CMIN
Model NPAR CMIN DF P CMIN/DF
Default model 19 11.547 8 .173 1.443
Saturated model 27 .000 0
Independence model 6 117.707 21 .000 5.605
Example 17
Standardized estimates and squared mu
ltiple correlations are as follows:
Regression Weights: (Group number 1  Default model)
Estimate S.E. C.R. P Label
visperc 
spatial
cubes 
spatial
.511
3.347
lozenges 
spatial
1.047
3.317
paragrap 
verbal
sentence 
verbal
1.259
6.505
wordmean

verbal
2.140
6.572
Intercepts: (Group number 1  Default model)
Estimate S.E. C.R. P Label
visperc
28.885
cubes
24.998
lozenges
15.153
wordmean
18.097
paragrap
10.987
sentence
18.864
Covariances: (Group number 1  Default model)
Estimate S.E. C.R. P Label
verbal
&#x8.4;&#x000;
spatial
7.993
3.211
2.490
.013
Variances: (Group number 1  Default model)
Estimate S.E. C.R. P Label
spatial
11.600
.011
verbal
2.743
err_v
8.518
.028
err_c
2.669
.003
err_l
11.662
.002
err_p
1.277
.027
err_s
2.403
.001
err_w
6.883
Standardized Regression Weights: (Group number 1 
Default model)
Estimate
visperc 
spatial .782
cubes 
spatial .700
lozenges 
spatial .685
paragrap 
verbal .890
sentence 
verbal .828
wordmean

verbal .828
Correlations: (Group number 1  Default model)
Estimate
verbal
&#x8.3;&#x000;
spatial .447
Squared Multiple Correlations: (Group number 1 
Default model)
Estimate
wordmean
.686
sentence
.685
paragrap
.793
lozenges
.469
cubes
.490
visperc
.612
277
Missing Data
Here is the path diagram showing the standardized estimates and the squared multiple
correlations for the en
dogenous variables:
Factor analysis with missing data
Holzinger and Swineford (1939): Girls' sample
Standardized estimates
df = 8
p = .173
Example 17
This section outlines three steps necessa
likelihood ratio chi
Fitting the factor model
Fitting the saturated model
Computing the likelihood ratio chisquare statistic and its
value
First, the three steps are performed by thre
e separate programs. After that, the three
steps will be combined into a single program.
The following program fits the confirmatory
factor model (Model A). It is saved as
Notice that the
ModelMeansAndIntercepts
279
Missing Data
intercepts do not have to ap
pear in the model unless you
want to estimate them or
constrain them.
The fit of Model A is summarized as follows:
Function of log likelihood
value is displayed instead of the chisquare fit statistic
Example 17
The following program fits the saturated model (Model B). The program is saved as
Following the
BeginGroup
line, there are six uses of the
Mean
281
Missing Data
Means: (Group number 1  Model 1)
Estimate
S.E. C.R. P Label
visperc 28.883
.910
31.756
cubes 25.154
.540
46.592
lozenges 14.962
1.101
13.591
paragrap 10.976
.466
23.572
sentence 18.802
.632
29.730
wordmean
18.263
1.061
17.211
Covariances: (Group number 1  Model 1)
Estimate
S.E. C.R. P Label
visperc&#x00; 
cubes 17.484
4.614
3.789
visperc&#x00; 
lozenges 31.173
9.232
3.377
cubes&#x00; 
lozenges 17.036
5.459
3.121
.002
visperc&#x00; 
paragrap 8.453
3.705
2.281
.023
cubes&#x00; 
paragrap 2.739
2.179
1.257
.209
lozenges
paragrap 9.287
4.596
2.021
.043
visperc&#x00; 
sentence 14.382
5.114
2.813
.005
cubes&#x00; 
sentence 1.678
2.929
.573
.567
lozenges
sentence 10.544
6.050
1.743
.081
sentence 13.470
2.945
4.574
visperc&#x00; 
wordmean
14.665
8.314
1.764
.078
cubes&#x00; 
wordmean
3.470
4.870
.713
.476
lozenges
wordmean
29.655
10.574
2.804
.005
wordmean
23.616
5.010
4.714
sentence
wordmean
29.577
6.650
4.447
Variances: (Group number 1  Model 1)
Estimate
C.R. P Label
visperc 49.584
9.398
5.276
cubes 16.484
3.228
5.106
lozenges 67.901
13.404
5.066
paragrap 13.570
2.515
5.396
sentence 25.007
4.629
5.402
wordmean
73.974
13.221
5.595
Example 17
AllImpliedMoments
method in the program displays the following table of
estimates:
These estimates, even the estimated means,
are different from the sample values
computed using either pairwise or listwise
Implied (for all variables) Covariances (Group number 1  Model 1)
wordmean sentence paragrap lozenges cubes visperc
wordmean 73.974
sentence 29.577 25.007
paragrap 23.616 13.470 13.570
lozenges 29.655 10.544 9.287 67.901
cubes 3.470 1.678 2.739 17.036 16.484
visperc 14.665 14.382 8.453 31.173 17.484 49.584
Implied (for all variables) Mean
s (Group number 1  Model 1)
wordmean sentence paragrap lozenges cubes visperc
18.263 18.802 10.976 14.962 25.154 28.883
283
Missing Data
Instead of consulting a chisquare table, you can use the
ChiSquareProbability
Example 17
value is 0.173; therefore, we accept the
hypothesis that Model A is correct at the
0.05 level.
As the present example illustrates, in order
to test a model with incomplete data, you
have to compare its fit to that of another, al
ternative model. In this example, we wanted
to test Model A, and it was ne
cessary also to fit Model B as a standard against which
Model A could be compared. The alternative model has to meet two requirements.
First, you have to be satisfied that it is co
285
Example
More about Missing Data
This example demonstrates the analysis of
data in which some va
lues are missing by
design and then explores the benefits of intentionally collecting incomplete data.
Researchers do not ordinarily like missing data
. They typically take great care to avoid
these gaps whenever possible. But sometimes it is actually better
not
to observe every
variable on every occasion. Matthai (1951) and Lord (1955) described designs where
certain data values are intentionally not observed.
The basic principle employed in such design
s is that, when it is impossible or too
costly to obtain sufficient observations
on a variable, estimates with improved
accuracy can be obtained by
taking additional observations on other correlated
variables.
Such designs can be highly useful, but be
cause of computational difficulties, they
have not previously been employed except
in very simple situations. This example
describes only one of many possible designs where some data are intentionally not
collected. The method of analysis
is the same as in Example 17.
Example 18
For this example, the Attig data (intro
duced in Example 1) was modified by
eliminating some of the data values and
treating them as missing. A portion of the
modified data file for young people,
Atty_mis.sav
, is shown below as it appears in the
SPSS Statistics Data Editor. The file contai
ns scores of Attigs 40 young subjects on
the two vocabulary tests
v_short
and
vocab
. The variable
vocab
is the WAIS vocabulary
V_short
is the score on a small subset of
items on the WAIS vocabulary test.
scores were deleted for 30 randomly picked subjects.
A second data file,
Atto_mis.sav
, contains vocabulary test scores for the 40 old
subjects, again with 30 randomly picked
vocab
287
More about Missing Data
Of course, no sensible person deletes data th
at have already been collected. In order for
this example to make sense, imagine this
pattern of missing data arising in the
following circumstances.
Suppose that
vocab
is the best vocabulary test you kn
ow of. It is highly reliable and
valid, and it is the vocabulary test that you want to use. Unfortunately, it is an
expensive test to administer. Maybe it takes a long time to give the test, maybe it has
to be administered on an individual basis,
or maybe it has to be scored by a highly
trained person.
is not as good a vocabulary test
, but it is short, inexpensive,
and easy to administer to a large number of
people at once. You administer the cheap
test,
to 40 young and 40 old subjects. Then you randomly pick 10 people from
each group and ask them to take the expensive test,
vocab
Suppose the purpose of
the research is to:
Estimate the average
vocab
test score in the population of young people.
Estimate the average
vocab
score in the population of old people.
Test the hypothesis that young people and old people have the same average
In this scenario, you are not
interested in the average
score. However, as will
be demonstrated below, the
v_short
scores are still useful because they contain
information that can be used to es
timate and test hypotheses about
vocab
The fact that missing values are missing by
Example 18
In the Analysis Properties dialog box, click the
Estimation
tab.
Select
Estimate means and intercepts
(a check mark ap
pears next to it).
While the Analysis Properties dialog box is open, click the
tab.
Select
Standardized estimates
Critical ratios for differences
Because this example focuses on group differences in the mean of
vocab
, it will be
useful to have names for the mean of the young group and the mean of the old group.
To give a name to the mean of
in the young group:
Rightclick the
vocab
rectangle in the path diagram for the young group.
Object Properties
from the popup menu.
In the Object Properties dialog box, click the
289
More about Missing Data
Here are the two path diagrams containing
means, variances, and covariances for the
young and old subjects respectively:
Text Output
In the Amos Output window, click
Notes for Model
in the upper left pane.
The text output shows that Model A is saturated, so that the model is not testable.
Number of distinct sample moments:10
Example 18
looking at estimated standard errors. For
the young subjects, the standard error for 56.891 shown above is about 1.765, whereas
the standard error of the sa
mple mean, 58.5, is about 2.21. For the old subjects, the
standard error for 65.001 is about 2.167 wh
ile the standard error
of the sample mean,
Means: (young subjects  Default model)
Estimate
S.E.
C.R.
Label
vocab
56.891
1.765
32.232
m1_yng
v_short
7.950
.627
12.673
par_4
Covariances: (young subjects  Default model)
Estimate
S.E.
C.R.
Label
vocab
&#x14.;5.;退
v_short
32.916
8.694
3.786
***
par_3
Correlations: (young subjects  Default model)
Estimate
vocab
&#x14.;5.;退
v_short
.920
Variances: (young subjects  Default model)
Estimate
S.E.
C.R.
Label
vocab
83.320
25.639
3.250
.001
par_7
v_short
15.347
3.476
4.416
par_8
Means: (old subjects  Default model)
Estimate
S.E.
C.R.
Label
vocab
65.001
2.167
29.992
m1_old
v_short
10.025
19.073
par_6
Covariances: (old subjects  Default model)
Estimate
S.E.
C.R.
Label
vocab
&#x70;
v_short
31.545
8.725
3.616
***
par_5
Correlations: (old subjects  Default model)
Estimate
vocab
&#x70;
v_short
.896
Variances: (old subjects  Default model)
Estimate
S.E.
C.R.
Label
vocab
115.063
37.463
3.071
.002
par_9
v_short
10.774
2.440
4.416
par_10
291
More about Missing Data
62, is about 4.21. Although the standard errors just mentioned are only approximations,
they still provide a rough basi
s for comparison. In the case of the young subjects, using
the information contained in the
andard error of the
estimated
vocab
mean by about 21%. In the case of
the old subjects, th
e standard error
was reduced by about 49%.
Another way to evaluate the additional information that can be attributed to the
scores is by evaluating the sample size requirements. Suppose you did not use
the information in the
v_short
scores. How many more young examinees would have
to take the
test to reduce the st
andard error of its mean by 21%? Likewise, how
many more old examinees would have to take the
test to reduce the standard
error of its mean by 49%? The answer is th
at, because the standard
error of the mean
is inversely proportional to the square root
of the sample size, it would require about
1.6 times as many young subjects and about 3.8 times as many old subjects. That is, it
would require about 16 young subjects and 38 old subjects taking the
vocab
test,
instead of 10 young and 10 old subjects taking both tests, and 30 young and 30 old
subjects taking the short test
alone. Of course, this calc
ulation treats the estimated
standard errors as though they were exact standard errors, and so it gives only a rough
idea of how much is gained by using scores on the
v_short
test.
Do the young and old populations have different mean
vocab
scores? The estimated
mean difference is 8.110 (65.001 56.891). A critical ratio for testing this difference
for significance can be found in the following table:
m1_yng
m1_old
par_3
par_4
par_5
par_6
par_7
m1_yng
.000
m1_old
2.901
.000
par_3
2.702
3.581
.000
par_4
36.269
25.286
2.864
.000
par_5
2.847
3.722
.111
2.697
.000
par_6
25.448
30.012
2.628
2.535
2.462
.000
par_7
1.028
.712
2.806
2.939
1.912
2.858
.000
par_8
10.658
12.123
2.934
2.095
1.725
1.514
2.877
par_9
1.551
1.334
2.136
2.859
2.804
2.803
.699
par_10
15.314
16.616
2.452
1.121
3.023
.300
2.817
par_8
par_9
par_10
par_8
.000
par_9
2.650
.000
par_10
1.077
2.884
.000
Example 18
The first two rows and columns, labeled
m1_yng
and
, refer to the group means
of the
test. The critical ratio for the mean di
fference is 2.901, according to which
the means differ significantly at the 0.05 level; the older population scores higher on
the long test than the younger population.
Another test of the hypothesis of equal
vocab
group means can be obtained by
refitting the model with equa
lity constraints imposed on
these two means. We will do
that next.
In Model B,
is required to have the same mean for young people as for old
people. There are two ways to impose this
293
More about Missing Data
To specify Model B, click
New
In the Model Name text box, change
Model Number 2
Model B
m1_old = m1_yng
CMIN
Model
NPAR
CMIN
DF
CMIN/DF
Model A
10
.000
Model B
7.849
.005
7.849
Saturated model
10
.000
Independence model
33.096
.000
5.516
Example 18
295
More about Missing Data
Here is a program for fitting Model B. In
this program, the
Main()
New
AmosEngine
Try
Sem.TextOutput()
Sem.Crdiff()
Sem.ModelMeansAndIntercepts()
Sem.BeginGroup(Sem.AmosDir & "Examples\atty_mis.sav")
Sem.GroupName("young_subjects")
Sem.Mean("vocab", "mn_vocab")
Sem.Mean("v_short")
Sem.BeginGroup(Sem.AmosDir & "Examples\atto_mis.sav")
Sem.GroupName("old_subjects")
Sem.Mean("vocab", "mn_vocab")
Sem.Mean("v_short")
Sem.FitModel()
Finally
Sem.Dispose()
Try
End Sub
Function of log likelihood = 437.813
297
Example
This example demonstrates how to obtain
robust standard error estimates by the
Example 19
The bootstrap has its own shortcomings, incl
uding the fact that it can require fairly
large samples. For readers who are new to bootstrapping, we recommend the
Scientific
article by Diaconis and Efron (1983).
The present example demonstrates the bootst
rap with a factor analysis model, but,
of course, you can use the bootstrap with an
Holzinger and Swineford (1939) Girls' sample
Model Specification
299
Click the
Bootstrap
tab.
Select
Perform bootstrap
500
in the
Number of bootstrap samples
text box.
You can monitor the progress of the bootstrap algorithm by watching the
Computation
summary
panel at the left of the path diagram.
The model fit is, of course, the same as in Example 8.
Example 19
Regression Weights: (Group number 1  Default
model)
Estimate S.E. C.R. P Label
visperc 
spatial
1.000
cubes 
spatial
.610
.143
4.250
***
lozenges 
spatial
1.198
.272
4.405
***
verbal
1.000
sentence 
verbal
1.334
.160
8.322
***
wordmean

verbal
2.234
.263
8.482
***
Standardized Regression Weights: (Group number 1 
Default model)
Estimate
visperc 
spatial .703
cubes 
spatial .654
lozenges 
spatial .736
verbal .880
sentence 
verbal .827
wordmean

verbal .841
Covariances: (Group number 1  Default model)
Estimate S.E. C.R. P Label
spatial
&#x7.;00;
verbal
7.315
2.571
2.846
.004
Correlations: (Group number 1  Default model)
Estimate
spatial
&#x7.;00;
verbal .487
Variances: (Group number 1  Default model)
Estimate S.E. C.R. P Label
spatial
23.302
8.123
2.868
.004
verbal
9.682
2.159
4.485
err_v
23.873
5.986
3.988
err_c
11.602
2.584
4.490
28.275
7.892
3.583
2.834
.868
3.263
.001
err_s
7.967
1.869
4.263
19.925
4.951
4.024
Squared Multiple Correl
ations: (Group number 1 
Default model)
Estimate
wordmean
.708
sentence
.684
.774
lozenges
.542
cubes
.428
visperc
.494
301
The bootstrap output begins with a table of di
agnostic information th
at is similar to the
following:
It is possible that one or more bootstrap sa
mples will have a singular covariance matrix,
or that Amos will fail to find a solution fo
r some bootstrap sample
s. If any such samples
occur, Amos reports their oc
currence and omits them from the bootstrap analysis. In
the present example, no bootstrap sample had a singular covariance matrix, and a
solution was found for each of the 500 bootstrap samples. The bootstrap estimates of
standard errors are:
0 bootstrap samples were unused because of a singular covariance matrix.
0 bootstrap samples were unused because a solution was not found.
500 usable bootstrap samples were obtained.
Example 19
Scalar Estimates (Group number 1  Default model)
Regression Weights: (Group number 1  Default
model)
Parameter SE SESE Mean Bias SEBias
visperc 
spatial
.000
.000
1.000
.000 .000
cubes 
spatial
.140
.004
.609
.001 .006
lozenges 
spatial
.373
.012
1.216
.018 .017
paragrap 
verbal
.000
.000
1.000
.000 .000
sentence 
verbal
.176
.006
1.345
.011 .008
wordmean

verbal
.254
.008
2.246
.011 .011
Standardized Regression Weights: (Group number 1 
Default model)
Parameter SE SESE Mean Bias SEBias
visperc 
spatial
.123
.004
.709
.006 .005
cubes 
spatial
.101
.003
.646
.008 .005
lozenges 
spatial
.121
.004
.719
.017 .005
paragrap 
verbal
.047
.001
.876
.004 .002
sentence 
verbal
.042
.001
.826
.000 .002
wordmean

verbal
.050
.002
.841
.001 .002
Covariances: (Group number 1  Default model)
Parameter SE SESE Mean Bias SEBias
spatial
&#x9.4;&#x000;
verbal
2.393
.076
7.241
.074 .107
Correlations: (Group number 1  Default model)
Parameter SE SESE Mean Bias SEBias
spatial
&#x9.4;&#x000;
verbal
.132
.004
.495
.008 .006
Variances: (Group number 1  Default model)
Parameter
SE SESE Mean Bias SEBias
spatial
9.086
.287
23.905
.603
.406
verbal
2.077
.066
9.518
.164
.093
err_v
9.166
.290
22.393
1.480
.410
err_c
3.195
.101
11.191
.411
.143
err_l
9.940
.314
27.797
.478
.445
err_p
.878
.028
2.772
.062
.039
err_s
1.446
.046
7.597
.370
.065
err_w
5.488
.174
19.123
.803
.245
Squared Multiple Correlations: (Group number 1 
Default model)
Parameter SE SESE Mean Bias SEBias
wordmean
.083
.003
.709
.001
.004
sentence
.069
.002
.685
.001
.003
paragrap
.081
.003
.770
.004
.004
lozenges
.172
.005
.532
.010
.008
cubes
.127
.004
.428
.000
.006
visperc
.182
.006
.517
.023
.008
303
The first column, labeled
S.E.
, contains bootstrap estimates of standard errors.
These estimates may be compared to th
e approximate standard error estimates
obtained by maximum likelihood.
The second column, labeled
S.E.S.E.
, gives an approximate
standard error for the
bootstrap standard error estimate itself.
The column labeled
Mean
305
Example
Bootstrapping for Model Comparison
This example demonstrates the use of the bootstrap for model comparison.
The problem addressed by this method is not that of evaluating an individual model
in absolute terms but of choosing among two or more competing models. Bollen and
Stine (1992), Bollen (1982), and Stine (1989) suggested the possibility of using the
bootstrap for model selection in analysis of
moment structures. Linhart and Zucchini
(1986) described a general schema for bo
otstrapping and model selection that is
appropriate for a large class of models, incl
uding structural modeling. The Linhart and
Zucchini approach is employed here.
The bootstrap approach to model comparison can be summarized as follows:
p samples by sampling with replacement from the
original sample. In
other words, the
original sample
serves as the
population
for
purposes of bootstrap sampling.
Example 20
The present example uses th
e combined male an
d female data from the GrantWhite
high school sample of the Holzinger and Swineford (1939) study, previously discussed
in Examples 8, 12, 15, 17, and 19. The 145 combined observations are given in the file
Five measurement models will be fitted to
the six psychological tests. Model 1 is a
factor analysis model with one factor.
Onefactor model
Holzinger and Swineford (1939) data
Model Specification
307
Bootstrapping for Model Comparison
Model 2 is an unrestricted factor analysis w
ith two factors. Note that fixing two of the
regression weights at 0 does not constrain the model but serves only to make the model
identified (Anderson, 1984; Bollen and Jreskog, 1985; Jreskog, 1979).
Model 2R is a restricted factor analysis mode
l with two factors, in which the first three
tests depend upon only one of the factors while the remaining three tests depend upon
only the other factor.
Two unconstrained factors
Holzinger and Swineford (1939) data
Model Specification
Restricted twofactor model
Holzinger and Swineford (1939) data
Model Specification
Example 20
The remaining two models provide customary po
ints of reference for evaluating the fit
of the previous models. In the saturated mo
del, the variances and covariances of the
observed variables are unconstrained.
In the independence model, the variances of
the observed variables are unconstrained
and their covariances are required to be 0.
Example 20: Saturated model
Variances and covariances
Holzinger and Swineford (1939) data
Model Specification
Example 20: Independence model
Only variances are estimated
Holzinger and Swineford (1939) data
Model Specification
309
Bootstrapping for Model Comparison
You would not ordinarily fit the saturated and independence models separately, since
Amos automatically reports fi
t measures for those two models in the course of every
analysis. However, it is necessary to spec
ify explicitly the satu
rated and independence
models in order to get bootstrap results
for those models. Five separate bootstrap
analyses must be performed, one for each model. For
each
of the five analyses:
From the menus, choose
Analysis Properties
In the Analysis Properties
dialog box, click the
Bootstrap
tab.
Select
Perform bootstrap
(a check mark ap
pears next to it).
1000
in the
Number of bootstrap samples
text box.
Click the
Random #
tab and enter a value for
Seed for random numbers
It does not matter what seed you choose, bu
Example 20
Click the
Numerical
tab and limit the number of iteratio
ns to a realistic figure (such as
40) in the
Iteration limit
Amos Graphics input files for the five
models have been saved with the names
Ex202.amw
Ex20sat.amw
Ex20ind.amw
Text Output
In viewing the text output for Model 1, click
Summary of Bootstrap Iterations
in the tree
diagram in the upper left pane of the Amos Output window.
The following message shows
that it was not necessary to discard any bootstrap
samples. All 1,000 bootstrap samples were used.
Click
Bootstrap Distributions
in the tree diagram to see a histogram of
where
contains sample moments from the
original sample of 145 GrantWhite
students (that is, the moments in the bootstrap population), and contains the
0 bootstrap samples were unused because of a singular covariance matrix.
0 bootstrap samples were unused because a solution was not found.
1000 usable bootstrap samples were obtained.
,
,
1
,
h
KL
C
b
KL
C
b
ML
C
a
a,
a
,
a
,
311
Bootstrapping for Model Comparison
implied moments obtained fro
m fitting Model 1 to the
bth
bootstrap sample. Thus,
is a measure of how much the po
pulation moments differ from the
moments estimated from the
bootstrap sample using Model 1.
The average of over 1,000 bootstrap samples was 64.162 with a standard
error of 0.292. Similar histograms, along wi
th means and standard errors, are displayed
for the other four models but are not repr
oduced here. The average discrepancies for
ML discrepancy (implied vs pop) (Default model)

48.268 **
52.091 *********
55.913 *************
59.735 *******************
63.557 *****************
67.379 ************
71.202 ********
N = 1000 75.024 ******
Mean = 64.162 78.846 ***
S. e. = .292 82.668 *
86.490 **
90.313 **
94.135 *
97.957 *
101.779 *

Example 20
fail for models that fit poorly. If some way could be found to successfully fit Model 2
to these 19 samplesfor ex
ample, with handpicked start values or a superior
algorithmit seems likely that the discrepancie
s would be large. According to this line
of reasoning, discarding bootstrap samples fo
r which estimation failed would lead to a
downward bias in the mean discrepancy. Th
us, you should be concerned by estimation
failures during bootstrapping, primarily when they occur for the model with the lowest
mean discrepancy.
In this example, the lowest mean di
screpancy (26.57) oc
curs for Model 2R,
confirming the model choice based on th
e BCC, AIC, and CAIC criteria. The
differences among the mean discrepancies are large compared to their standard errors.
Since all models were fitted to the same
bootstrap samples (ex
cept for samples where
Model 2 was not successfully fitted), you woul
d expect to find positive correlations
313
Example
Bootstrapping to Compare
Example 21
The HolzingerSwineford (1939) data from Example 20 (in the file
) are used
in the present example.
315
Bootstrapping to Compare Estimation Methods
Finally, click the
tab.
Select
Perform bootstrap
and type
1000
for
Number of bootstrap samples
Select
Bootstrap ADF
Bootstrap ML
Bootstrap GLS
Bootstrap ULS
Example 21
Selecting
Bootstrap ADF
Bootstrap ML
Bootstrap GLS
ootstrap SLS
, and
Bootstrap
specifies that each of C
ADF
GLS
, and C
ULS
is to be used to measure the
317
Bootstrapping to Compare Estimation Methods
Select the
Maximum likelihood
discrepancy to repeat the analysis.
Select the
Generalized least squares
discrepancy to repeat the analysis again.
Select the
Unweighted least squares
discrepancy to repeat the analysis one last time.
The four Amos Graphics inpu
t files for this example are
and
Ex21uls.amw
Text Output
In the first of the four
analyses (as found in
), estimation using ADF
produces the following histogram output. To view this histogram:
Click
Bootstrap Distributions
ADF Discrepancy (implied vs pop)
in the tree diagram in
the upper left pane of the Amos Output window.
Example 21
This portion of the output shows the dist
ribution of the population discrepancy
across 1,000 bootstrap samples, where contains the implied moments
obtained by minimizing , that is, th
e sample discrepancy. The average of
across 1,000 bootstrap samples is 20.6
01, with a standard error of 0.218.
The following histogram shows the distribution of . To view this histogram:
Click
Bootstrap Distributions
ML Discrepancy (implied vs pop)
in the tree diagram in the
upper left pane of the Amos Output window.
ADF discrepancy (implied vs pop) (Default model)

7.359 *
10.817 ********
14.274 ****************
17.732 ********************
21.189 *******************
24.647 *************
28.104 ********
N = 1000 31.562 ****
Mean = 20.601 35.019 **
S. e. = .218 38.477 **
41.934 *
45.392 *
48.850 *
52.307 *
55.765 *

ML discrepancy (implied vs pop) (Default model)

11.272 ****
22.691 ********************
34.110 ********************
45.530 ***********
56.949 *****
68.368 ***
79.787 **
N = 1000 91.207 *
Mean = 36.860 102.626 *
S. e. = .571 114.045 *
125.464 *
136.884 
148.303 
159.722 
171.142 *

319
Bootstrapping to Compare Estimation Methods
The following histogram shows the distribution of . To view this histogram:
Click
Bootstrap Distributions
GLS Discrepancy (implied vs pop)
in the tree diagram in
the upper left pane of the Amos Output window.
The following histogram shows the distribution of . To view this histogram:
Click
Bootstrap Distributions
ULS Discrepancy (implied vs pop)
in the tree diagram in
the upper left pane of the Amos Output window.
GLS discrepancy (implied vs pop) (Default model)

7.248 **
11.076 *********
14.904 ***************
18.733 ********************
22.561 **************
26.389 ***********
30.217 *******
N = 1000 34.046 ****
Mean = 21.827 37.874 **
S. e. = .263 41.702 ***
45.530 *
49.359 *
53.187 *
57.015 *
60.844 *

ULS discrepancy (implied vs pop) (Default model)

5079.897 ******
30811.807 ********************
56543.716 ********
82275.625 ****
108007.534 **
133739.443 *
159471.352 *
N = 1000 185203.261 *
Mean = 43686.444 210935.170 
S. e. = 1011.591 236667.079 *
262398.988 
288130.897 
313862.806 
339594.715 
365326.624 *

Example 21
Below is a table showing the mean of
across 1,000 bootstrap samples with
the standard errors in parentheses. Th
e four distributions
just displayed are
summarized in the first row of the table. Th
e remaining three rows show the results of
estimation by minimizing
, and
, respectively.
The first column, labeled
ADF
, shows the relative performance of the four estimation
321
Example
This example takes you through two sp
ecification searches: one is largely
confirmatory (with few optional arrows), and the other is largely exploratory (with
many optional arrows).
This example uses the Felson and Bohrnstedt (
1979) girls data, also used in Example 7.
The initial model for the specification
search comes from Felson and Bohrnstedt
(1979), as seen in Figure 221:
Example 22
Figure 221:
Felson and Bohrnstedts model for girls
Felson and Bohrnstedt were primarily inte
rested in the two singleheaded arrows,
academic
attract
and
academic
323
Click on the Specification Search toolbar, and then click the doubleheaded arrow
that connects
error1
and
. The arrow changes color to
indicate that the arrow is
optional.
Tip:
If you want the optional arrow to be dash
ed as well as colored, as seen below,
choose
Interface Properties
from the menus, click the
tab, and
select the
Alternative to color
check box.
To make the arrow required again, click
on the Specification
Search toolbar, and
then click the arrow. When you move the po
inter away, the arrow will again display as
a required arrow.
Click again, and then click the arrows in
the path diagram until it looks like this:
GPA
height
rating
weight
academic
attract
error1
error2
GPA
height
rating
weight
academic
attract
error1
error2
Example 22
When you perform the exploratory analysis
later on, the program
will treat the three
colored arrows as optional and
325
dels reported can speed up a specification search
significantly. However, only eight models
in total will be en
countered during the
specification search for this exampl
e, and specifying a nonzero value for
Example 22
The following table summarizes fit measures
for the eight models
and the saturated
column contains an arbi
trary index number from 1
through 8 for each of the
models fitted during
the specification search.
Sat
identifies the saturated model.
327
Figure 222:
Path diagram for Model 7
GPA
height
rating
weight
academic
attract
error1
error2
12.12
GPA
8.43
height
1.02
rating
371.48
weight
academic
attract
.02
error1
.14
error2
.53
.47
6.71
1.82
19.02
5.24
1.44
Chisquare = 3.071 (4 df)
p = .546
Example 22
In the Specification Search window, click the column heading
The table sorts according to
so that the best model according to
(that is, the
model with the smallest
) is at the top of the list.
Based on a suggestion by Burnham and Anderson (1998), a constant has been added
to all the
values so that the smallest
value is 0. The
subscript on
serves as a reminder of this rescaling.
(not shown in the above figure) and
have been similarly rescaled. As a rough
guideline, Burnham and Anderson (1998,
329
Viewing the Akaike Weights
Click the
Options
button on the Specification Search toolbar.
In the Options dialog box, click the
tab.
In the BCC, AIC, BIC group, select
Akaike weights / Bayes factors (sum=1)
In the table of fit measures, the column that was labeled
is now labeled
BCC
and
contains Akaike weights. (See Appendix G.)
Example 22
probabilities. With equal prior probabili
ties, the Akaike weights are themselves
posterior probabilities, so that one can say that Model 7 is the
KL
best model with
probability 0.494, Model 6 is the
best model with probability 0.205, and so on. The
four most probable models are Models 7, 6, 8, and 1. After adding their probabilities
(0.494 + 0.205 + 0.192 + 0.073 = 0.96), one
can say that there is
a 96% chance that the
best model is among those four. (Burnham and Anderson, 1998, pp. 127129). The
subscript on
serves as a reminder that
can be interpreted as a probability
Current results
tab of the Options dialog box, select
Zerobased (min=0)
in the
BCC, AIC, BIC group.
331
In the Specification Search wind
ow, click the co
The table is now sorted according to
so that the best model according to
(that
is, the model with the smallest
) is at the top of the list.
Model 7, with the smallest
, is the model with the hi
ghest approximate posterior
probability (using equal prior
probabilities for the models and using a particular prior
Example 22
In the table of fit measures, the column that was labeled
is now labeled
and
contains Bayes factors scaled so that they sum to 1.
With equal prior probabilities for the models
and using a particular prior distribution
333
Madigan and Raftery (1994) suggest that only models in
Occams window
be used
for purposes of model averaging (a
topic not discussed here). The
0.86020
0.043
120
0.05
Example 22
Examining the Short List of Models
Click on the Specification Search toolbar. This displays a short list of models.
In the figure below, the short list shows th
e best model for each
335
Viewing a Scatterplot of Fit and Complexity
Click on the Specification Search toolba
r. This opens the Plot window, which
displays the following graph:
The graph shows a scatterplot of fit (measured by C) versus complexity (measured by
Example 22
Choose one of the models from the popup menu to see that model highlighted in the
table of model fit statistics and, at the same
time, to see the path diagram of that model
in the drawing area.
In the following figure, the cursor points
to two overlapping points that represent
Model 6 (with a discrepancy of 2.76) and
Model 8 (with a discrepancy of 2.90).
The graph contains a horizontal
line representing points for which
is constant.
Initially, the line is centered at 0 on the vertical axis. The
Fit values
panel at the lower
left shows that, for points
on the horizontal line,
= 0 and also
= 0. (
is referred to
in Amos output.)
and
are two versions of NFI that use two different
baseline models (see Appendix F).
Initially, both
are equal to 1 for points on the horizontal line. The
location of the horizont
al line is adjustable.
You can move the line by dragging it with
the mouse. As you move the line, you can se
e the changes in the location of the line
reflected in the fit measures in the lower left panel.
337
Adjusting the Line Representing Constant Fit
Move your mouse over the adjustable line. When the pointer changes to a hand, drag
the line so that
is equal to 0.900. (Keep an eye on
in the lower left panel while
you reposition the
adjustable line.)
is the familiar form of the NFI statisti
c for which the baseline model requires the
observed variables to be uncorrelated with
out constraining their
means and variances.
Points that are below the line have
� 0.900 and those above the line have
0.900. That is, the adjustable line se
parates the acceptable models from the
unacceptable ones according to a widely used
convention based on a remark by Bentler
Example 22
Viewing the Line Repres
In the Plot window, select
C df
in the Fit measure group. This displays the following:
The scatterplot remains unchan
ged except for the position of
the adjustable line. The
adjustable line now cont
ains points for which
is constant. Whereas the line was
previously horizontal, it is now
tilted downward, indicating that
gives some
weight to complexity in assessing model adeq
uacy. Initially, the adjustable line passes
through the point for which
is smallest.
Click that point, and then choose
from the popup menu.
This highlights Model 7 in th
e table of fit measures and al
so displays the path diagram
for Model 7 in the drawing area.
The panel in the lower left corner shows th
e value of some fit measures that depend
only on
and that are therefore, like
itself, constant along the adjustable
line.
and
are two versions of
that use two different baseline models (see
339
Appendix G). Initially, both
and
are equal to 1 for poin
ts on the adjustable
line. When you move the adjust
able line, the fit measures in
the lower left panel change
to reflect the changing position of the line.
Adjusting the Line Representing Constant C
Drag the adjustable line so that
is equal to 0.950.
is the usual CFI statistic for which th
e baseline model requires the observed
variables to be uncorrelated
without constraining their mean
s and variances. Points that
are below the line have
� 0.950 and those above the line have
0.950. That
is, the adjustable line separates the accep
table models from the unacceptable ones
according to the recommendation of Hu and Bentler (1999).
Example 22
Viewing Other Lines Representing Constant Fit
Click
, and
in turn.
Notice that the slope of the adjustable line
becomes increasingly negative. This reflects
the fact that the five measures (
C df
) give increasing weight
to model complexity. For each of these five
measures, the adjustable line has constant
slope, which you can confirm by dragging the line with the mouse. By contrast, the
slope of the adjustable line for
C / df
is not constant (the slop
e of the line changes when
you drag it with the mous
e) and so the slope for
C / df
cannot be compar
ed to the slopes
C df
Viewing the BestFit Graph for C
In the Plot window, select
in the Plot type group.
In the Fit measure group, select
Figure 225:
Smallest value of C for each number of parameters
341
Each point in this graph re
presents a model for which
is less than or equal to that of
any other model that has the same number of
parameters. The graph shows that the best
67.342
3.071
67.3423.071
64.271
Example 22
is the measure among
that imposes the greatest
penalty for complexity. The high penalty fo
r complexity is reflected in the steep
positive slope of the graph as the number of
343
In the Fit measure group, select
The Plot window displays the following graph:
Figure 226:
Scree plot for C
In this
scree plot, the point with coordinate 17
on the horizontal axis has coordinate
64.271 on the vertical axis.
3.071
67.342
67.3423.07164.271
0
=
2.761
2.7610
Example 22
The figure on either p. 340 or p. 343 can be used to support a heuristic
components to the model, one component at
a time. The scree plot
presented here for
shows the improvement in model fit that
is obtained by incrementing the number
1
345
For
C df
, and
BIC
, the units and the origin of the vertical axis are different
than for
, but the graphs are otherwise identica
l. This means that the final model
selected by the scree test
is independent of which measure of fit is used (unless
is used). This is the advantage of the scree plot over the
bestfit
plot demonstrated
earlier in this example (see Viewing th
e BestFit Graph for C on p.340, and
Viewing the BestFit Graph for Other Fit M
easures on p.341). The bestfit plot and
the scree plot contain nearly
the same information, but the shape of the bestfit plot
depends on the choice of fit measure while
the shape of the scree plot does not (with
the exception of
C / df
Both the bestfit plot and the scree plot
are independent of sample size in the sense
that altering the sample size without alteri
ng the sample moments
has no effect other
than to rescale the vertical axis.
Example 22
The previous specification search was largel
y confirmatory in th
at there were only
three optional arrows. You can take a much
more exploratory approach to constructing
a model for the Felson and Bohrnstedt data. Suppose that your only hypothesis about
the six measured variables is that
academic
depends on the other five variables, and
depends on the other five variables.
The path diagram shown in Figure 227 w
ith 11 optional arrows implements this
hypothesis. It specifies which variables are endogenous, and nothing more. Every
observedvariable model that is consistent
specification search. The covariances among
the observed, exogenous variables could
have been made optional, but doing so would have increased the number of optional
arrows from 11 to 17, increasing the number
of candidate models from 2,048 (that is, 2
to 131,072 (that is, 2
). Allowing the covariances
among the observed, exogenous
variables to be optional would have been costly
, and there would seem to be little interest
in searching for models in which some pairs of those variables are uncorrelated.
Figure 227:
Highly exploratory model for Felson and Bohrnstedts girls data
GPA
height
rating
weight
academic
attract
347
Open
Ex22b.amw
. If you performed a typical installation, the path will be
C:\Program Files\IBM\SPSS\Amos\23\Examples\languag&#xl4.9; n4.;gu4;&#x.1ag;.1e;e\Ex22b.amw
Tip:
If the last file you opened was in the
Examples
folder, you can open the file by
doubleclicking it in the Files list
to the left of the drawing area.
From the menus, choose
Analyze
Specification Search
Click on the Specification Search toolbar,
and then click the arrows in the path
diagram until it looks like the diagram on p. 346.
Tip:
You can change multiple arrows at once by clicking and dragging the mouse
pointer through them.
1
Example 22
This restores the default setting we altered earlier in this example. With the default
349
In the Specification Search window, click the
column heading. This sorts the table
according to
Figure 228:
The 10 best models according to BIC
The sorted table shows that Model 22
is the best model according to
(Model
numbers depend in part on the order in wh
ich the objects in the path diagram were
drawn; therefore, if you draw your own pa
th diagram, your mode
l numbers may differ
from the model numbers here.) The secondbest model according to
BIC
, namely
Model 32, is the best according to
BCC
These models are shown below:
Model 22Model 32
1
1
Example 22
Viewing the Scree Plot
Click on the Specification Search toolbar.
In the Plot window, select
in the Plot type group.
The scree plot strongly suggests that mode
351
Example
Exploratory Factor Analysis by
This example demonstrates exploratory fact
or analysis by means of a specification
search. In this approach to exploratory f
actor analysis, any measured variable can
(optionally) depend on any factor. A specif
ication search is performed to find the
subset of singleheaded arrows that provid
es the optimum combin
ation of simplicity
and fit. It also demonstrates a heuristic spec
ification search that is practical for models
that are too big for an exhaustive specification search.
This example uses the Holzinger and Swineford girls (1939) data from Example 8.
The initial model is shown in
Figure 231 on p. 352. Duri
ng the specification search,
all singleheaded arrows that point from fa
ctors to measured variables will be made
optional. The purpose of the specification se
arch is to obtain guidance as to which
singleheaded arrows are essential to the
model; in other words, which variables
depend on which factors.
The two factor variances are both fixed at
1, as are all the regression weights
associated with residual variables. W
ithout these constraints, all the models
encountered during the specification
search would be unidentified.
Example 23
Figure 231:
Exploratory factor analysis
model with two factors
Open the file
. If you performed a typical in
stallation, the path will be
C:\Program Files\IBM\SPSS\Amos\
23\Examples\languag&#xlan;.3g;&#xu4.;:g;.3e;e\Ex23.amw
Initially, the path diagram appears
as in Figure 231. There is no
point in trying to fit this
model as it stands because it is not identified,
even with the factor variances fixed at 1.
To open the Specification
Search window, choose
Specification Search
Initially, only the toolbar is visible, as seen here:
1
353
Exploratory Factor
Analysis by Specification Search
Making All Regression Weights Optional
Click on the Specification Search toolba
r, and then click all the singleheaded
arrows in the path diagram.
Figure 232:
Twofactor model with all regression weights optional
During the specification sear
ch, the program will attempt
to fit the model using every
possible subset of the optional arrows.
visperc
cubes
lozenges
wordmean
paragraph
sentence
err_v
err_c
err_l
err_p
err_s
err_w
Example 23
Next search
tab. Notice that the default value for
Retain only the best ___
models
is
355
Exploratory Factor
Analysis by Specification Search
With this setting, the program will display only the 10 best models according to
whichever criterion you use for sorting the columns of the model list. For example, if
you click the column heading
C / df
, the table will show the 10 models with the smallest
values of
C / df
, sorted according to
C / df
. Scatterplots will disp
lay only the 10 best
4096
Example 23
In the Specification Search window, click the column heading
The table sorts according to
so that the best model according to
(that is, the
model with the smallest
) is at the top of the list.
Figure 233:
The 10 best models according to BCC
The two best models according to
BCC
(Models 52 and 53) have identical fit measures
(out to three decimal places anyway). The
explanation for this ca
n be seen from the
path diagrams for the two models.
In the Specification Search window, doublec
lick the row for Mode
l 52. This displays
its path diagram in
the drawing area.
357
Exploratory Factor
Analysis by Specification Search
To see the path diagram for Model 53, doubleclick its row.
Figure 234:
Reversing F1 and F2 yields another candidate model
This is just one pair of models where reversing the roles of
and
changes one
member of the pair into the other. There are other such pairs. Models 52 and 53 are
equivalent, although they are counted separa
tely in the list of 4,
096 candidate models.
The 10 models in Figure 233 on p. 356 come in five pairs, but candidate models do
not always come in equivalent pairs, as Figure
235 illustrates. The model in that figure
1
1
1
Example 23
The occurrence of equivalent candidate
models makes it unclear how to apply
Bayesian calculations to select
a model in this example. Similarly, it is unclear how to
use Akaike weights. Furthermore, Burnham and Andersons guidelines (see p. 328) for
1
359
Exploratory Factor
Analysis by Specification Search
Viewing the Scree Plot
Click on the Specification Search toolbar.
In the Plot window, select
Scree in
the Plot type group.
The scree plot strongly suggests the use of
13 parameters because of the way the graph
drops abruptly and then levels off immediately after the 13
parameter. Click the point
with coordinate 13 on the horizontal axis. A popup shows that the point represents
Models 52 and 53, as shown in Figure 234 on p. 357.
Viewing the Short List of Models
Click on the Specification Search toolbar. Take note of the short list of models for
future reference.
Example 23
The number of models that must be fitted
in an exhaustive specification search grows
rapidly with the number of optional arrows. There are 12 optional arrows in Figure
232 on p. 353 so that an exhaustive specification search requires fitting
models. (The number of models will be somewhat smaller if you specify a small
positive number for
Retain only the best___models
on the
Next search
tab of the Options
dialog box.) A number of heuristic search procedures have been proposed for reducing
the number of models that have to be fitted
(Salhi, 1998). None of these is guaranteed
to find the best model, but they have the
advantage of being computationally feasible
in problems with more than, say, 20 optional arrows where an exhaustive specification
search is impossible.
Amos provides three heuristi
c search strategies in addi
tion to the option of an
exhaustive search. The heuristic strategies do not attempt to find the overall best model
because this would require choosing a definition of
best
in terms of the minimum or
maximum of a specific fit measure. Instead, th
e heuristic strategies attempt to find the
4096
361
Exploratory Factor
Analysis by Specification Search
and Backward searches are alternated un
til one Forward or Backward search is
completed with no improvement.
Click the
Options
button on the Specification Search toolbar.
In the Options dialog box, click the
tab.
Select
Stepwise
On the Specification Search toolbar, click .
237 suggest examining the 13p
Example 23
Viewing the Scree Plot
Click on the Specification Search toolbar.
In the Plot window, select
in the Plot type group.
The scree plot confirms that adding a 13
parameter provides a substantial reduction
in discrepancy and that adding ad
363
Exploratory Factor
Analysis by Specification Search
A heuristic specification search can fail to
find any of the best models for a given
365
Example
MultipleGroup Factor Analysis
This example demonstrates a twogroup fact
or analysis with au
tomatic specification
of crossgroup constraints.
This example uses the Holzinger and Swineford girls and boys (1939) data from
Examples 12 and 15.
The presence of means and intercepts as
Example 24
Figure 241:
Twofactor model for girls and boys
This is the same twogroup factor analysis
problem that was considered in Example 12.
The results obtained in Example 12 w
ill be obtained here automatically.
From the menus, choose
Open
In the Open dialog box, doubleclick the file
Ex24a.amw
. In a typical installation, the
path will be:
C:\Program Files\IBM\SPSS\Amos\
23Examples\langua&#xla4;&#x.5ng;&#x4.5;&#xua4;&#x.5ge;ge\Ex24a.amw
The path diagram is the same for boys as fo
r girls and is shown in Figure 241. Some
regression weights are fixed at 1. These re
gression weights will
remain fixed at 1
throughout the analysis to follow. The
up analysis adds
constraints to the model you specify but does not remove any constraints.
From the menus, choose
MultipleGroup Analysis
Click
in the message box that appears. Th
is opens the Multip
leGroup Analysis
dialog box.
367
MultipleGroup Factor Analysis
Figure 242:
The MultipleGroup Analysis dialog box
Most of the time, you will simply click
. This time, however, let's take a look at some
parts of the MultipleGroup Analysis dialog box.
There are eight columns of check boxes.
Check marks appear on
ly in the columns
labeled
. This means that the program will generate three models, each with
Example 24
spatial
visperc
cubes
lozenges
wordmean
paragrap
sentence
err_v
err_c
err_p
err_s
err_w
369
MultipleGroup Factor Analysis
Viewing the Generated Models
In the MultipleGroup Analysis dialog box, click
Example 24
Fitting All the Models and Viewing the Output
From the menus, choose
Calculate Estimates
to fit all models.
From the menus, choose
Text Output
In the navigation tree of th
e output viewer, click the
Model Fit
node to expand it, and
then click
The CMIN table shows the likelihood ratio ch
isquare statistic for each fitted model.
The data do not depart significantly from an
y of the models. Furthermore, at each step
up the hierarchy from the Unconstrained model to the Measurement residuals model,
the increase in chisquare is never much larg
er than the increase in degrees of freedom.
There appears to be no significant evidence
371
MultipleGroup Factor Analysis
Here is the CMIN table:
In the navigation tree, click
AIC
under the
Model Fit
node.
values indicate that the best trad
eoff of model fit and parsimony is
Example 24
Introducing explicit means and intercepts
into a model raises
additional questions
From the menus, choose
Open
In the Open dialog box, doubleclick the file
Ex24b.amw
. In a typical installation, the
path will be:
C:\Program Files\IBM\SPSS\Amos\
23\Examples\languag&#xlan;gu;Jg;一e\Ex24b.amw
The path diagram is the same for boys as for girls and is shown below. Some regression
weights are fixed at 1. The means of all the
unobserved variables are fixed at 0. In the
following section, you will remove the constraints on the girls factor means. The other
constraints (the ones that you do not remove) will remain in effect throughout the
analysis.
Figure 244:
Twofactor model with explicit means and intercepts
373
MultipleGroup Factor Analysis
Initially, the factor means are fixed at 0 for both boys and girls. It is not possible to
estimate factor means for both groups. However, Srbom (1974) showed that, by fixing
the factor means of a single group to constant values and placing suitable constraints
on the regression weights and intercepts in a factor model, it is possible to obtain
meaningful estimates of the factor means fo
r all of the other groups. In the present
example, this means picking one group, say boys, and fixing their factor means to a
constant, say 0, and then removing the constr
aints on the factor means of the remaining
group, the girls. The constraints on regression weights and intercepts required by
Srboms approach will be ge
nerated automatically by Amos.
The boys factor means are already fixed at 0. To remove the constraints on the girls'
factor means, do the following:
In the drawing area of the Amos
Graphics window, rightclick
and choose
Object Properties
from the popup menu.
In the Object Properties
dialog box, click the
Example 24
Now that the constraints on the girls factor means have been removed, the girls and
boys path diagrams look like this:
Tip:
375
MultipleGroup Factor Analysis
The default settings, as shown above, will generate the following nested hierarchy of
five models:
Click
From the menus, choose
Analyze
Calculate Estimates
The panel at the left of the path diagram sh
ows that two models could not be fitted to
the data. The two models that could not be fitted, the
Unconstrained
model with no
ModelConstraints
Model 1 (column 1)
Measurement weights (factor
Model 2 (column 2)
All of the above, and measurement intercepts (intercepts in
the equations for predicting measured variables) are equal
Model 3 (column 3)
All of the above, and structural means (factor means) are
equal across groups.
Model 4 (column 4)
All of the above, and structural covariances (factor variances
and covariances) are equal across groups.
Example 24
crossgroup constraints, and the
Measurement weights
model with factor loadings held
equal across groups, are unidentified.
Viewing the Output
From the menus, choose
Text Output
In the navigation tree of the output viewer, expand the
Model Fit
node.
Some fit measures for the four automati
cally generated and identified models are
shown here, along with fit measures for the saturated and independence models.
Click
under the Model Fit node.
The CMIN table shows that none of the gene
rated models can be rejected when tested
against the saturated model.
On the other hand, the change in chisquare () when introducing
the equalfactormeans constraint looks larg
e compared to the ch
ange in degrees of
freedom ().
In the navigation tree, click the
Model Comparison
node.
ModelNPARCMINDFPCMIN/DF
Measurement intercepts3022.593240.5440.941
Structural means2830.624260.2431.178
Structural covariances2534.381290.2261.186
Measurement residuals1938.459350.3161.099
Saturated model540.000
Independence model24337.553300.0011.252
30.6222.598.03
26242
377
MultipleGroup Factor Analysis
Assuming model
Measurement intercepts
to be correct, the following table shows that
this chisquare difference is significant:
In the preceding two tables, two chisquare
statistics and their associated degrees of
freedom are especially important. The first
with , allowed
accepting the hypothesis of equal intercepts and equal regression weights in the
measurement model. It was important to establish the credibility of this hypothesis
because, without equal intercepts and equal regression weights, it would be unclear
that the factors have the same meaning for boys as for girls and so there would be no
interest in compar
ing
their means. The other important chisquare statistic,
with , leads to rejection of the hypothes
is that boys and girls have the same
factor means.
Group differences between the boys and gi
22.59
=
379
Example
This example shows you how
to automatically implement
Srboms alternative to
analysis of covariance.
Example 16 demonstrates the benefits of
Srboms approach to analysis of
covariance with latent variables. Unfort
unately, as Example 16 also showed, the
Srbom approach is difficult to apply, involving many steps. This example
automatically obtains the same results as Example 16.
The Olsson (1973) data from Example 16 will be used here. The sample moments can
be found in the workbook
m the experimental
group are in the worksheet
. Sample moments from th
e control group are in
the worksheet
The model was described in Example 16.
Example 25
Figure 251:
Srbom model for Olsson data
. If you performed a typical installation, the path will be
C:\Program Files\IBM\SPSS\Amos\
23\Examples\languag&#xlan;.3g;&#xu4.;:g;.3e;e\Ex25.amw
The path diagram is the same for the control and experimental groups and is shown in
Figure 251. Some regression weights are fi
xed at 1. The means of all the residual
(error) variable means are fixed at 0. These co
nstraints will remain in effect throughout
the analysis.
Constraining the Latent Variable Means and Intercepts
The model in Figure 251, Srboms mode
l for Olsson data, is unidentified and will
remain unidentified for every set of cross
group constraints that Amos automatically
381
MultipleGroup Analysis
In the drawing area, rightclick
pre_verbal
and choose
Object Properties
from the pop
up menu.
In the Object Properties
dialog box, click the
Example 25
Click
to generate the following nested hierarchy of eight models:
ModelConstraints
Model 1 (column 1)
Measurement weights (factor lo
adings) are constant across
groups.
Model 2 (column 2)
All of the above, and measurem
ent intercepts (intercepts in
the equations for predicting measured variables) are constant
across groups.
Model 3 (column 3)
All of the above, and the structural weight (the regression
weight for predicting
post_verbal
) is constant across groups.
Model 4 (column 4)
All of the above, and the structural intercept (the intercept in
the equation for predicting
post_verbal
groups.
Model 5 (column 5)
All of the above, and the structural mean (the mean of
pre_verbal
) is constant across groups.
Model 6 (column 6)
All of the above, and the structural covariance (the variance
of
pre_verbal
) is constant across groups.
Model 7 (column 7)
All of the above, and the structural residual (the variance of
383
MultipleGroup Analysis
From the menus, choose
Analyze
Calculate Estimates
The panel to the left of the path diagram sh
ows that two models could not be fitted to
the data. The two models that could not be fitted, the
Unconstrained
model and the
Measurement weights
model, are unidentified.
Viewing the Text Output
From the menus, choose
Text Output
In the navigation tree of the output vi
ewer, expand the Model Fit node, and click
This displays some fit measures for the se
ven automatically generated and identified
models, along with fit measures for the saturated and independence models, as shown
in the following CMIN table:
ModelNPARCMINDFPCMIN/DF
Measurement intercepts2234.77560.0005.796
Structural weights2136.34070.0005.191
Structural intercepts2084.06080.00010.507
Structural means1994.97090.00010.552
Structural covariances1899.976100.0009.998
Structural residuals17112.143110.00010.195
Measurement residuals13122.366150.0008.158
Saturated model280.0000
Independence model16682.638120.00056.887
Example 25
There are many chisquare stat
istics in this table, but on
ly two of them matter. The
Srbom procedure comes down to two basi
c questions. First, does the Structural
weights model fit? This model specifies th
at the regression weight for predicting
post_verbal
be constant across groups.
If the Structural weights model is accepted,
one follows up by asking whether the
next model up the hierarchy, the Structural
intercepts model, fits significantly worse.
On the other hand, if the Structural weights
model has to be rejected, one never gets to
the question about the Structural
intercepts model. Unfortunately, that is the case here.
The Structural weights
and , is rejected at any
conventional sign
ificance level.
To see if it is possible to improve th
e fit of the Structural weights model:
Close the output viewer.
From the Amos Graphics menus, choose
Analysis Properties
Click the
tab and select the
Modification Indices
check box.
Close the Analysis Properties dialog box.
From the menus, choose
Calculate Estimates
to fit all models.
Only the modification indices for the Struct
ural weights model need to be examined
because this is the only model whos
e fit is essential to the analysis.
From the menus, choose
Text Output
, select
Modification Indices
in the navigation
tree of the output viewer, then select
Structural weights
in the lower left panel.
Expand the
Modification Indices
node and select
As you can see in the following covariance
table for the control group, only one
modification index exceeds the default threshold of 4:
M.I.Par Change
eps2 &#x5.4;&#x4.1;&#x000; eps44.5532.073
36.34
=
385
MultipleGroup Analysis
Now click
experimental
in the panel on the left. As
you can see in the following
covariance table for the experimental group,
there are four modifica
tion indices greater
than 4:
Of these, only two modifications have an
=
Example 25
387
Example
This example demonstrates Ba
yesian estimation using Amos.
In maximum likelihood estimati
on and hypothesis testing,
the true values of the
Example 26
becomes as simple as plotting histograms and computing sample means and
percentiles.
389
Bayesian Estimation
A prior distribution quantifies the research
ers belief concerni
ng where the unknown
parameter may lie. Knowledge of how a variable is distributed in the population can
3.410
Example 26
Performing Bayesian Estimation Using Amos Graphics
To illustrate Bayesian estimation using Am
os Graphics, we revisit Example 3, which
shows how to test the null hyp
391
Bayesian Estimation
This is the resulting path diagram (you can also find it in
Before performing a Bayesian analysis of this model, we perform a maximum
likelihood analysis for
comparison purposes.
From the menus, choose
Analyze
Calculate Estimates
parameter estimates and standard errors:
Covariances: (Group number 1  Default model)
EstimateS.E.C.R.PLabel
age &#x5.4;&#x4.1;&#x000; vocabulary5.0148.5600.5860.558
Variances: (Group number 1  Default model)
EstimateS.E.C.R.PLabel
age21.5744.8864.416***
vocabulary131.29429.7324.416***
Example 26
Bayesian analysis requires estimation
of explicit means and
intercepts. Before
performing any Bayesian analysis in Amos,
you must first tell Amos to estimate means
and intercepts.
From the menus, choose
Analysis Properties
Select
Estimate means and intercepts
. (A check mark will appear next to it.)
To perform a Bayesian analys
is, from the menus, choose
Analyze
Bayesian Estimation
or press the keyboard combination
Ctrl+B
393
Bayesian Estimation
The Bayesian SEM window appears, and
the MCMC algorithm immediately begins
generating samples.
The Bayesian SEM window has a toolbar near the top of the window and has a results
summary table below. Each row of the summa
ry table describes th
e marginal posterior
distribution of a single model pa
Example 26
The multiple imputation and Bayesian estim
ation algorithms implemented in Amos
make extensive use of a stream of random numbers that depends on an initial
random
number seed
. The default behavior of Amos is to change the random number seed
every time you perform Bayesian estimation,
Bayesian data imputation, or stochastic
regression data imputation. Consequently, when you try to replicate one of those
analyses, you can expect to get slightly di
fferent results because
of using a different
random number seed.
If, for any reason, you need an exact replication of an earlier analysis, you can do
so by starting with the same random number
seed that was used in the earlier analysis.
To find out what the current random num
ber seed is or to change its value:
From the menus, choose
Tools
Seed Manager
By default, Amos incremen
ts the current random number seed by one for each
395
Bayesian Estimation
maintains a log of previous seeds used, so it
is possible to match the file creation dates
of previously generated analysis results or
Example 26
Record the value of this seed in a safe place
so that you can replicat
e the results of your
analysis at a later date.
Tip:
We use the same seed value of 14942405 for all examples in this guide so that you
can reproduce our results.
We mentioned earlier that the MCMC algori
thm used by Amos draws random values
of parameters from highdimensional join
t posterior distributi
ons via Monte Carlo
simulation of the posterior di
averaging across the random samples prod
uced by the MCMC procedure. It is
important to have at least a rough idea of
how much uncertainty in the posterior mean
is attributable to Monte Carlo sampling.
The second column, labeled
, reports an estimated stan
dard error that suggests
how far the MonteCarlo esti
mated posterior mean may lie from the true posterior
mean. As the MCMC procedure continues to
generate more samples, the estimate of
the posterior mean
becomes more precise, and the
gradually drops. Note that
S.E. is not an estimate of how far the po
sterior mean may lie from the unknown true
397
Bayesian Estimation
50058316331
Example 26
You can change the refresh interval to something other than the default of 1,000
observations. Alternatively, you can refresh th
e display at a regular time interval that
you specify.
If you select
Refresh the display manually
, the display will never be updated
automatically. Regardless of
what you select on the
Refresh
tab, you can refresh the
display manually at any time by clicking the
Refresh
button on the Bayesian SEM
toolbar.
Are there enough samples to yield stable
399
Bayesian Estimation
samples, one may ask whether there are enough of these samples to accurately estimate
the summary statistics, such
as the posterior mean.
That question pertains to the second ty
pe of convergence, which we may call
convergence of po
sterior summaries
Convergence of posterior summaries is
complicated by the fact that the analysis
samples are not independent but are actually
an autocorrelated time series. The 1001
sample is correlated with the 1000
, which,
in turn, is correl
ated with the 999
, and so on. These correl
ations are an inherent
feature of MCMC, and because of these co
rrelations, the summary statistics from
5,500 (or whatever number of) analysis samp
les have more variability than they would
if the 5,500 samples had been independent. Nevertheless, as we continue to accumulate
more and more analysis sa
mples, the posterior summa
ries gradually stabilize.
Amos provides several diagnostics that
help you check
convergence. Notice the
value 1.0083 on the toolbar of the Bayesian
SEM window on p. 393. This is an overall
convergence statistic based on a measure
suggested by Gelman, Carlin, Stern, and
Rubin (2004). Each time the screen refreshes, Amos updates the
C.S.
for each
parameter in the su
mmary table; the
value on the toolbar is the largest of the
individual
values. By default, Amos judges th
e procedure to have converged if the
C.S.
values is less than 1.002. By this standard, the maximum
C.S.
of
1.0083 is not small enough. Amos displa
ys an unhappy face when the overall
is not small enough. The
C.S.
compares the variability wi
thin parts of the analysis
sample to the variability across these parts. A value of 1.000 represents perfect
convergence, and larger values indicate that
the posterior summaries
can be made more
precise by creating mo
re analysis samples.
Pause Sampling
button a second time instructs Amos to resume the
sampling process. You can also pause and resume sampling by choosing
Sampling
from the Analyze menu, or by using the keyboard combination
Ctrl+E
. The
next figure shows the results after resuming the sampling for a while and pausing again.
Example 26
At this point, we have 22,501 analysis samples, although the display was most recently
updated at the 22,500
sample. The largest
is 1.0012, which is below the 1.002
criterion that indicates acceptable conv
ergence. Reflecting the satisfactory
convergence, Amos now displays a happy f
ace . Gelman et al. (2004) suggest that,
for many analyses, values of 1.10 or smaller are sufficient. The default criterion of
1.002 is conservative. Judging that the MC
MC chain has converged by this criterion
does not mean that the summary table will stop changing. The summary table will
continue to change as long
as the MCMC algorithm keeps running. As the overall
value on the toolbar approaches 1.000, however, there is not much more precision to
be gained by taking additional samples, so we might as well stop.
In addition to the
value, Amos offers several
plots that can help you check
convergence of the Bayesian MCMC method. To view these plots:
From the menus, choose
Amos displays the Posterior dialog box.
401
Bayesian Estimation
Select the
age  &#x4.7;&#x  ;.70;vocabulary
parameter from the Bayesian SEM window.
Example 26
The Posterior dialog box now displays a frequency polygon of the distribution of the
agevocabulary
covariance across th
e 22,500 samples.
One visual aid you can use to judge whether it
is likely that Amos has converged to the
posterior distribution is a si
multaneous display of two estimates of the distribution, one
obtained from the first third
of the accumulated samples and another obtained from the
last third. To display the two estimates of the marginal posterior on the same graph:
Select
First and last
. (A check mark will appear next to the option.)
403
Bayesian Estimation
In this example, the distributions of the first and last thirds of the analysis samples are
almost identical, which suggests that Amos
tified the important
features of the posterior distribution of the
covariance. Note that this
posterior distribution appears to be centered
at some value near 6, which agrees with
Mean
Example 26
To view the trace plot, select
Trace
The plot shown here is quite ideal. It ex
hibits rapid upanddown variation with no
longterm trends or drifts. If we were to
mentally break up this plot into a few
horizontal sections, the trace within any
section would not look much different from
the trace in any other section. This indicate
s that the convergence in distribution takes
place rapidly. Longterm trends
or drifts in the plot indicate slower convergence. (Note
that
is relative to the horizontal scale of this plot, which depends on the
number of samples. As we take
405
Bayesian Estimation
To display this plot, select
Autocorrelation
Lag
, along the horizontal axis, refers to
the spacing at which
the correlation is
estimated. In ordinary situations, we expe
ct the autocorrelation coefficients to die
down and become close to 0, and remain
near 0, beyond a certain lag. In the
autocorrelation plot shown above, the lag10
correlationthe co
rrelation between any
sampled value and the value drawn 10 iter
ations lateris approximately 0.50. The
lag35 correlation lies below 0.20, and at
lag 90 and beyond, the correlation is
effectively 0. This indicates that by 90 it
erations, the MCMC procedure has essentially
forgotten its starting position, at least as fa
Example 26
when the missing values fall in a peculiar patt
407
Bayesian Estimation
Select
to display a similar plot using vertical blocks.
Select
Contour
to display a twodimensional plot
of the bivariate posterior density.
Example 26
Ranging from dark to light,
the three shades of gray re
present 50%, 90%, and 95%
credible regions
, respectively. A credible region is
conceptually similar to a bivariate
confidence region that is fa
miliar to most data analys
ts acquainted with classical
409
Bayesian Estimation
Recall that the summary table in the Bayesian SEM window displays the lower and
upper endpoints of a Bayesian credible inte
rval for each estimand. By default, Amos
presents a 50% interval, which is similar
to a conventional 50% confidence interval.
Researchers often report 95% confidence in
tervals, so you may want to change the
boundaries to correspond to a posterior probability content of 95%.
Click the
Display
tab in the Options dialog box.
as the Confidence level value.
Click the
Close
button. Amos now displays
95% credible intervals.
Example 26
Gill (2004) provides a readab
le overview of Bayesian estima
tion and its advantages in
a special issue of
Political Analysis
. Jackman (2000) offers a
more technical treatment
of the topic, with examples
, in a journal article format
. The book by Gelman, Carlin,
Stern, and Rubin (2004) addresses a multitude of practical issues with numerous
examples.
411
Example
This example demonstrates using a nondiffuse prior distribution.
Example 26 showed how to perform Bayesian estimation for a simple model with the
uniform prior distribution that Amos uses by default. In the present example, we
consider a more complex model and make use of a nondiffuse prior distribution. In
particular, the example shows how to specif
y a prior distribution so that we avoid
negative variance estimates and other improper estimates.
In the discussion of the previous example,
we noted that Bayesian estimation depends
on information supplied by
the analyst in conjunction
likelihood estimation maximizes the likelihood of an unknown parameter
given the observed data
through the relationship
estimation approximates the
of
y,
y), where
the
of
y) is the posterior density of
given
Conceptually, this means that
the posterior density of
given
is the product of the
prior distribution of
and the likelihood of the observed data (Jackman, 2000, p. 377).
Example 27
lihood function becomes more and more
tightly concentrated about the
ML estimate. In that case, a diffuse prior tends to be
nearly flat or constant over
the region where the likelihood is high; the shape of the
413
Bayesian Estimation Using a NonDiffuse Prior Distribution
experimental condition, measured their leve
ls of depression, treated the experimental
group, and then remeasured participants depression. The researchers did not rely on
a single measure of depression. Instead, they used two wellknown depression scales,
the Beck Depression Inventory (Beck, 19
67) and the Hamilton Rating Scale for
Depression (Hamilton, 1960). We will call them
BDI
and
for short. The data
are in the file
feelinggood.sav
The following figure shows th
e results of using maximum likelihood estimation to fit
a model for the effect of treatment (
) on depression at Time 2. Depression at
Time 1 is used as a covariate. At Time 1 and then again at Time 2,
and
are
modeled as indicators of a single
underlying variable, depression (
The path diagram for
this model is in
. The chisquare statistic of 0.059 with
one degree of freedom indicates a good fit, but the negative residual variance for post
therapy
makes the solution improper.
Example 27
Does a Bayesian analysis with a diffuse prio
r distribution yield results similar to those
of the maximum likelihood solution? To find ou
t, we will do a Bayesian analysis of the
same model. First, we will show how to in
crease the number of
burnin observations.
This is just to show you how to do it. Noth
ing suggests that the default of 500 burnin
observations needs to be changed.
To change the number of burnin observations to 1,000:
From the menus, choose
Options
In the Options dialog box, select the
MCMC
tab.
Change
Number of burnin observations
to 1000.
Click
and allow MCMC sampling to proc
eed until the unhappy face turns
happy .
415
Bayesian Estimation Using a NonDiffuse Prior Distribution
The summary table should look something like this:
Example 27
In this analysis, we allowed Amos to
reach its default limit of 100,000 MCMC
samples. When Amos reaches this lim
it, it begins a process known as
1000
88000
53000
8424000
417
Bayesian Estimation Using a NonDiffuse Prior Distribution
Fortunately, there is a remedy for this pr
oblem: Assign a prior density of 0 to any
parameter vector for which the variance of
is negative. To change the prior
distribution of the variance of
From the menus, choose
Alternatively, click the
button on the Bayesian SEM toolbar, or enter the
keyboard combination
Ctrl+R
. Amos displays the Prior dialog box.
Example 27
Select the variance of
in the Bayesian SEM window
to display the default prior
distribution for
Replace the default lower bound of with 0.
419
Bayesian Estimation Using a NonDiffuse Prior Distribution
Click
to save this change.
Amos immediately discards the accumulate
d MCMC samples and begins sampling all
over again. After a while, the Bayesian SE
Example 27
The posterior mean
of the variance of
is now positive. Examining its posterior
distribution confirms that no sampled values fall below 0.
421
Bayesian Estimation Using a NonDiffuse Prior Distribution
Is this solution proper? The post
erior mean of each variance is positive, but a glance at
Min
column shows that some of the sa
mpled values for
the variance of
and the
variance of
are negative. To avoid negative variances for
and
, we can modify
their prior distribution
It is not too difficult to impose such co
Example 27
From the menus, choose
Options
In the Options dialog box, click the
tab.
Select
. (A check mark will
appear next to it.)
Selecting
Admissibility test
423
Bayesian Estimation Using a NonDiffuse Prior Distribution
Notice that the analysis took only 73,000
425
Example
Bayesian Estimation of Values Other
Example 28
Suppose we are interested in the indirect effect of
on
alienation71
through the
mediation of
alienation67
. In other words, we suspect th
at socioeconomic status exerts
an impact on alienation in 1967, which
in turn influences
alienation in 1971.
427
Bayesian Estimation of Values
Example 28
From the menus, choose
Calculate Estimates
to obtain the maximum
likelihood chisquare test of model fit and the parameter estimates.
The results are identical to
those shown in Example 6, Model C. The standardized
direct effect of
on
alienation71
is 0.19. The standardized indirect effect of
on
alienation71
is defined as the product of two
standardized direct effects: the
standardized direct effect of
on
alienation67
(0.56) and the standardized direct
effect of
alienation67
on
alienation71
(0.58). The product of
these two standardized
direct effects is .
0.580.32
429
Bayesian Estimation of Values
Example 28
The MCMC algorithm converges quite rapidly within 22,000 MCMC samples.
The summary table displays results for mo
del parameters only. To estimate the
posterior of quantities derived from the mode
l parameters, such as indirect effects:
From the menus, choose
Additional Estimands
431
Bayesian Estimation of Values
Example 28
To print the results, select the items you want
to print. (A check mark will appear next
to them).
From the menus, choose
Be careful because it is possible to generate a lot of printed output. If you put a check
mark in every check box in this example, the program will print
matrices.
To view the posterior m
eans of the standardized direct effects, select
Standardized
Direct Effects
and
Mean
in the panel at the left.
The posterior means of the st
andardized direct and indirect effects of socioeconomic
status on alienation in 1971 are almost id
entical to the maximum likelihood estimates.
1811
433
Bayesian Estimation of Values
Example 28
The lower boundary of the 95% credible interval for the indirect effect of
socioeconomic status on alienation in 1971 is 0.382. The corresponding upper
boundary value is 0.270, as shown below:
We are now 95% certain that the true value of
this standardized indirect effect lies
435
Bayesian Estimation of Values
Example 28
Amos then displays the posterior distribution
of the indirect effe
ct of socioeconomic
status on alienation in 1971. The distribution of the indirect effect is approximately, but
not exactly, normal.
437
Bayesian Estimation of Values
439
Example
Estimating a UserDefined Quantity
in Bayesian SEM
This example shows how to estimate a us
erdefined quantity: in this case, the
difference between a direct effect and an indirect effect.
In the previous example, we showed how
to use the Additional
Estimands feature of
Amos Bayesian analysis to estimate an i
ndirect effect. Suppose you wanted to carry
the analysis a step further and address a
commonly asked research question: How does
an indirect effect compare to the corresponding direct effect?
You can use the Custom Estimands feature of
Amos to estimate and draw inferences
about an arbitrary function of the model
parameters. To illustrate the Custom
Estimands feature, let us revisit the previo
us example. The path diagram for the model
is shown on p. 440 and can be found in the file
. The model allows
socioeconomic status to exert a direct effect on alienation experienced in 1971. It also
allows an indirect effect that is medi
ated by alienation experienced in 1967.
The remainder of this example focuses on the direct effect, the indirect effect, and
Example 29
) and the two components of the indirect effect (
and
). Although not
required, parameter labels make it easier to specify custom estimands.
To begin a Bayesian analysis of this model:
From the menus, choose
Bayesian Estimation
After a while, the Bayesian SEM window
441
Estimating a UserDefined
Quantity in Bayesian SEM
From the menus, choose
Additional Estimands
In the Additional Estimands window, select
Standardized Direct Effects
and
The posterior mean for the direct effect of
alienation71
is 0.195.
Example 29
Select
Standardized Indirect Effects
The indirect effect of socioeconomic st
atus on alienation in 1971 is 0.320.
443
Estimating a UserDefined
Quantity in Bayesian SEM
The posterior distribution of the indirect effect
lies entirely to the
left of 0, so we are
practically certain that the indirect effect is less than 0.
Example 29
You can also display the poster
ior distribution of the direct effect. The program does
not, however, have any builtin way to
examine the posterior distribution of the
445
Estimating a UserDefined
Quantity in Bayesian SEM
In this section, we show how to write
a Visual Basic program for estimating the
Example 29
447
Estimating a UserDefined
Quantity in Bayesian SEM
computing the direct effect an
d the indirect individually, but this is only to show how
to do it. The direct effect and the indirect
effect individually can be estimated without
defining them as custom estimands. To define each estimand, we use the keyword
newestimand
, as shown below:
The words
indirect
are estimand labels. You can use different
labels.
CalculateEstimands
computes the values of the estimands defined in
DeclareEstimands
subroutine. Only the first line of the function is shown.
To display all of th
e lines, doubleclick
Function CalculateEstimands
or click the
sign
in the little box at the beginning of the line.
Example 29
The placeholder
TODO: Your code goes here
needs to be replaced with lines for
evaluating the estimands called
direct
and
We start by writing Visual Basic code
for computing the direct effect. In the
following figure, we have already typed part of a Visual Basic statement:
estimand(direct) .value =
449
Estimating a UserDefined
Quantity in Bayesian SEM
We need to finish the statement by adding ad
ditional code to the right of the equals (
sign, describing how to compute the direct eff
ect. The direct effect is to be calculated
for a set of parameter values that are accessi
ble through the AmosEngine object that is
supplied as an argument to the
CalculateEstimands
function. Unless you are an expert
Amos programmer, you would not know how to use the AmosEngine object; however,
there is an easy way to get the needed Visual Basic syntax by dragging and dropping.
Find the direct effect in the Bayesian SEM
window and click to select its row. (Its row
is highlighted in th
e following figure.)
Move the mouse pointer to an
edge of the selected row.
Either the top edge or the
bottom edge will do.
Example 29
Tip:
When you get the mouse pointer on the right spot, a plus (+) symbol will appear
next to the mouse pointer.
Hold down the left mouse button, drag the mouse pointer into the Visual Basic window
to the spot where you want the expression fo
r the direct effect to go, and release the
mouse button.
451
Estimating a UserDefined
Quantity in Bayesian SEM
Example 29
Using the same draganddrop process as previously described, start dragging things
from the Bayesian SEM window to the Unnamed.vb window.
First, drag the direct effect of socioeconomic status on alienation in 1967 to the right
side of the equals sign in the unfinished statement.
453
Estimating a UserDefined
Quantity in Bayesian SEM
Next, drag and drop the direct effect of 1967 alienation on 1971 alienation
This second direct effect appears in the Unnamed.vb window as
Example 29
Finally, use the keyboard to insert an asterisk (
455
Estimating a UserDefined
Quantity in Bayesian SEM
For complicated custom estimands,
you can also drag and drop from the
Additional Estimands window to
the Custom Estimands window.
Example 29
To find the posterior distribution
of all three custom estimands, click
The results will take a few seconds. A stat
us window keeps you informed of progress.
457
Estimating a UserDefined
Quantity in Bayesian SEM
The marginal posterior distributions of the three custom estimands are summarized in
the following table:
can also be found in the Bayesian SEM summary table, and the
indirect
can be found in the Additional Estimands table. We are really
interested in
difference
. Its posterior mean is 0.132. Its minimum is 0.412, and its
maximum is 0.111.
To see its marginal posterio
r, from the menus, choose
View
Select the
difference
row in the Custom Estimands table.
Example 29
Most of the area lies to the left
of 0, meaning that the difference is almost sure to be
negative. In other words, it is almost certai
n that the indirect effect is more negative
than the direct effect. Eyeballi
ng the posterior, perhaps 95% or
so of the area lies to the
left of 0, so there is about a 95% chance that
the indirect effect is larger than the direct
effect. It is not necessary to rely on
eyeballing
the posterior, however. There is a way
to find any area under a marginal posterior or, more generally, to estimate the
probability that any proposition
459
Estimating a UserDefined
Quantity in Bayesian SEM
Visual inspection of the frequency po
lygon reveals that the majority of
difference
values are negative, but it does not tell
us exactly what proportion of values are
negative. That proportion is our estimate of
the probability that
the indirect effect
exceeds the direct effect. To estimate
probabilities like these, we can use
dichotomous
Visual Basic (or C#) pr
ograms, dichotomous estimands are just like
numeric estimands except that dichotomous estimands take on only two values: true
and false. In order to estimate the probabilit
y that the indirect effect is more negative
than the direct effect, we need to define a
Example 29
Add lines to the
CalculateEstimates
function specifying
how to compute them.
In this example, the first dichotomous custom
estimand is true wh
en the value of the
indirect effect is less than 0. The second
dichotomous custom estimand is true when
the indirect effect is smaller than the direct effect.
Click the
Run
button.
Amos evaluates the truth of each logical expression for each MCMC sample drawn.
When the analysis finishes, Amos reports the proportion of MCMC samples in which
each expression was found to be true. These proportions appear in the
Dichotomous
section of the Custom Estimands summary table.
461
Estimating a UserDefined
Quantity in Bayesian SEM
column shows the proportio
n of times that each evaluated expression was true
in the whole series of MCMC samples. In
this example, the number of MCMC samples
was 29,501, so
is based on approximately 30,000 samples. The
, and
columns show the proportion of
times each logical expression was true in the first third,
the middle third, and the final third of the MC
MC samples. In this
illustration, each of
these proportions is based upon approximately 10,000 MCMC samples.
Based on the proportions in the
Dichotomous Estimands
area of the Custom
Estimands window, we can say with near certain
ty that the indirect effect is negative.
This is consistent with the frequency po
lygon on p. 444 that showed no MCMC
samples with an indirect effect va
lue greater than or equal to 0.
Similarly, the probability is about 0.975 th
at the indirect effect is larger (more
negative) than the direct effect. The 0.975 is on
ly an estimate of the probability. It is a
proportion based on 29,501 correlated observ
ations. However it appears to be a good
estimate because the proportions from the fi
rst third (0.974), midd
le third (0.979) and
final third (0.971) are so close together.
463
Example
Data Imputation
This example demonstrates multiple im
putation in a factor analysis model.
Example 17 showed how to fit a model using maximum likelihood when the data
contain missing values. Amos can also
values for those that are missing. In
data imputation, each missing value is re
placed by some numeric guess. Once each
missing value has been replaced by an impu
Example 30
Bayesian imputation
is like stochastic regression imputation except that it takes
imputation. Eventually, all three of the
following steps need to be carried out.
465
Data Imputation
Step 1: Use the Data Imputatio
n feature of Amos to create
complete data files.
Step 2: Perform an an
alysis of each of the
Example 30
From the menus, choose
Data Imputation
Amos displays the Amos Data Imputation window.
Make sure that
Bayesian imputation
is selected.
467
Data Imputation
SPSS Statistics to analyze the
Example 30
Click
Click
Options
in the Data Imputation window to display the available imputation
options.
The online help explains these options. To get an explanation of an option, place your
mouse pointer over the option in question and press the
key. The figure below shows
how the number of observations can be changed from 10,000 (the default) to 30,000.
Close the Options dialog box and click the
Impute
button in the Data Imputation
window. After a short time, the following message appears:
Click
Amos lists the names of the completed data files.
469
Data Imputation
Each completed data file contains 73 comple
te cases. Here is a view of the first few
records of the first completed data file,
Grant_Imp1.sav
Example 30
471
Example
Example 31
Step 2: Ten Separate Analyses
sentence
Dependent Variable: wordmean
473
.
1
1
m
t
t
m
Q
Example 31
To obtain a standard error for the comb
0.0233

0.0085
0.0233

0.0085
0.0326
0.03260.1807

1.172
0.1807



101
0.0233
0.0085

109
475
477
Example
This example demonstrates parameter estima
tion, estimation of posterior predictive
distributions, and data imputation with censored data.
For this example, we use the censored data
from 103 patients who were accepted into
the Stanford Heart Transplantation Program during the years 1967 through 1974. The
data were collected by Crowley and Hu (1977) and have been reanalyzed by
Example 32
Reading across the first visible row in the
figure above, Patient 17 was accepted into
the program in 1968. The patient at that time was 20.33 years old. The patient died 35
days later. The next number, 5.916, is th
e square root of 35
. Amos assumes that
censored variables are normally distributed.
The square root of survival time will be
used in this example in the belief that it is
probably more nearly normally distributed
than is survival time itself.
Uncensored
simply means that we know how long the
patient lived. In other words, the patient ha
s already died, and that is how we are able
to tell that he lived for 35 days
after being admitted into the program.
Some patients were still alive when last
seen. For example, Patient 25 entered the
program in 1969 at the age of 33.22 years.
The patient was last seen 1,799 days later.
The number 42.415 is the square root of 1,799. The word
censored
in the Status column
means that the patient was still alive 1,799 da
ys after being accepted into the program,
and that is the last time the patient was seen
. So, we cant say that the patient survived
for 1,799 days. In fact, he survived for longer than that; we just dont know how much
longer. There are more cases like that. Patien
t number 26 was last seen 1,400 days after
acceptance into the prog
ram and, at that time, was still alive, so we know that that
patient lived for at least 1,400 days.
It is not clear what is to be done with a
censored value like Patient 25s survival time
of 1,799 days. You cant just discard the 1,799 and all the other censored values
because that amounts to discarding the patients who lived a long time. On the other
hand, you cant keep the 1,799 and treat it as an ordinary score because you know the
patient really lived for more than 1,799 days.
In Amos, you can use the information that Patient 25 lived for more than 1,799 days,
479
Censored Data
known more precisely than it is. Of course,
wherever the data provide an exact numeric
value, as in the case of Patient 24 who is known to have survived for 218 days, that
exact numeric value is used.
The data file needs to be recoded before Am
os reads it. The next figure shows a portion
Example 32
Then in the Data Files dialog box, click the
File Name
button.
Select the data file
Select
Allow nonnumeric data
(a check mark appears next to it).
Recoding the data as sh
own above and selecting
Allow nonnumeric data
are the only
extra steps that are required for analyzing censored data. In all other respects, fitting a
model with censored data and in
timesqr
age
acceptyr
481
Censored Data
To fit the model:
Click on the toolbar.
From the menus, choose
Analyze
The button is disabled because, with nonnumeric data, you can perform only
Bayesian estimation.
After the Bayesian SEM window opens, wa
it until the unhappy face changes into
a happy face . The table of estimates in
the Bayesian SEM wi
ndow should look
something like this:
(Only a portion of the table is shown in the figure.) The
column contains point
Example 32
The Posterior dialog box opens, displaying th
e posterior distribution of the regression
The posterior distribution of the regression weight is indeed centered around 0.29.
483
Censored Data
Example 32
The posterior distribution for Patient 25s
timesqr
lies entirely to
the right of 42.415.
Of course, we knew from the data alone that
timesqr
exceeds 42.415, but now we also
know that there is practically
no chance that Patient 25s
timesqr
exceeds 70. For that
matter, there is only
a slim chance that
timesqr
exceeds even 55.
To see a posterior predictive distribution th
at looks very different from Patient 25s:
Click the
symbol in the 100
row of the Posterior Pred
ictive Distributions table.
485
Censored Data
Patient 100 was still alive wh
en last observed on the 38
day after acceptance into the
program, so that his
timesqr
is known to exceed 6.164. The posterior distribution of that
patients
timesqr
shows that it is practically guarante
ed to be between 6.164 and 70, and
Example 32
You can use this model to impute
values for the censored values.
Close the Bayesian SEM window if it is open.
From the Amos Graphics menu, choose
Data Imputation
Notice that
Regression imputation
Stochastic regression imputation
are disabled.
When you have nonnumeric data such as censored data,
Bayesian imputation
is the only
choice.
We will accept the options shown in the pr
eceding figure, crea
ting 10 completed
487
Censored Data
Wait until th
Data Imputation dialog box displays a happy face to indicate that
Example 32
Doubleclick the file name to display the contents of the single completed data file,
which contains 10 completed datasets.
The file contains 1,030 cases because each of
489
Censored Data
The first row of the completed data file contains a
value of 7. Because that was
not a censored value, 7 is not an imputed value.
It is just an ordinary numeric value that
was present in the original data fi
le. On the other hand, Patient 25s
timesqr
censored, so that patient has an imputed
(in this case, 49.66.) The value of
49.66 is a value drawn randomly from the poster
ior predictive distribution in the figure
on p. 484.
Example 32
Normally, the next step would be to
491
Example
This example shows how to fit a factor anal
ysis model to orderedcategorical data. It
also shows how to find the posterior predictive distribution for the numeric variable
that underlies a categorical response and how to impute a numeric value for a
categorical response.
This example uses data on attitudes towa
rd environment issues obtained from a
questionnaire administered to 1,017 respon
dents in the Netherlands. The data come
from the European Values Study Group (see
the bibliography for a citation). The data
file
string.sav
contains responses to six questionnaire items with
categorical responses
strongly disagree
disagree
agree
agree
Example 33
One way to analyze these data is to assign
numbers to the four categorical responses;
for example, using the assignment 1 =
, 4 =
. If you assign numbers
to categories in that wa
493
OrderedCategorical Data
It may be slightly easier to use
environmentnlnumeric.sav
because Amos will
assume by default that the numbered categories go in the order 1, 2, 3, 4, with 1 being
the lowest category. That happens
to be the correct order. With
environmentnl
string.sav
, by contrast, Amos will assume by de
fault that the categories are arranged
Example 33
The ordinal properties of the data cannot be inferred from the data file alone. To give
Amos the additional information it needs so
that it can interpret the data values
From the Amos Graphics menus, choose
Select
in the list of variables in the upperle
ft corner of the Data Recode window.
This displays a frequency distribution of the responses to
item1
at the bottom of the
window.
495
OrderedCategorical Data
In the box labeled Recoding rule, the notation
No recoding
means that Amos will read
the responses to
item1
as is. In other words, it will read either
, or an empty
string. We cant leave things that way because Amos doesnt know what to do with
, and so on.
Click
No recoding
and select
Orderedcategorical
from the dropdown list.
Example 33
The frequency table at the bottom of the window now has a
New Value
column that
shows how the
item1
values in the data file will be recoded before Amos reads the data.
The first row of the frequency table shows that
empty strings in the original data file
will be treated as missing values
. The second row shows that the
497
OrderedCategorical Data
column, based on the assumpti
on that scores on the underlying numeric variable are
normally distributed with a mean of 0 and a standard deviation of 1.
The ordering of the categories in the
Original Value
column needs to be changed. To
change the ordering:
Click the
Example 33
You can rearrange the categories and the boundaries. To do this:
Drag and drop with the mouse.
Select a category or boundary w
ith the mouse and then click the
button.
After putting the categories and boundari
es in the correct
order, the Ordered
499
OrderedCategorical Data
You cant drag and drop between the Ordered categories list box and the
Unordered categories list box. You have to use the
and
Down
buttons to move a
category from one box to the other.
We could stop here and close the Ordered
Categorical Details dialog box because we
have the right number of boundaries and ca
tegories and we have the categories going
in the right order. However, we will make a further change based on a suggestion by
Example 33
distributed with a mean of 0 and a standard
deviation of 1. Alternatively, you can
assign a value to a boundary instead of letti
ng Amos estimate it. To assign a value:
Select the boundary with the mouse.
Type a numeric value in the text box.
The following figure shows the result of assigning values 0 and 1 to the two boundaries.
Although it may not be obvious, it is perm
issible to assign 0 and 1, or any pair of
numbers, to the two boundaries, as long as
the higher boundary is
assigned a larger
value than the lower one. No matter how ma
ny boundaries there are (as long as there
are at least two), assigning values to two of the boundaries amounts to choosing a zero
point and a unit of measurement for the underlying numeric variable. The scaling of
the underlying numeric variable is discusse
d further in the Help file under the topic
Choosing boundaries when
there are three categories.
Click
501
OrderedCategorical Data
The frequency table shows how the values that
appear in the data file will be recoded
before Amos reads them. Reading the
frequency table from top to bottom:
An empty string will be treated as a missing value.
and
will be recoded as 0, meaning that the underlying numeric
score is less than 0.
will be recoded as 01, meaning that
the underlying numeric score is between
0 and 1.
will be recoded as �1, meaning that the underlying numeric score is greater
than 1.
That takes care of
item1
. What was just done for
item1
has to be repeated for each of
the five remaining observed variables. Af
Example 33
observed variables, you can view the original
dataset along with the recoded variables.
To do this:
Click the
button.
The table on the left shows the contents of
the original data file before recoding. The
table on the right shows the recoded variables after recoding. When Amos performs an
analysis, it reads the recoded va
lues, not the or
iginal values.
You can create a raw data file in wh
ich the data recoding
has already been
performed. In other words, you can create a ra
w data file that contains the inequalities
on the righthand side of the figure above.
In that case, you woul
dnt need to use the
Data Recode window in Amos. Indeed, th
at approach was used in Chapter 32.
Finally, close the Data Recode wi
ndow before specifying the model.
After you have specified the rules for data
recoding as shown above, the analysis
proceeds just like any Bayesian analysis. For this example, a factor analysis model will
be fitted to the six questionnaire items in
the environment dataset. The first three items
were designed to be measures of willingness to spend money to take care of the
environment. The other three items were designed to be measures of awareness of
503
OrderedCategorical Data
environmental issues. This design of the qu
estionnaire is reflected in the following
factor analysis model, which is saved in the file
The path diagram is drawn exactly as it would be drawn for numeric data. This is one
of the good things about having at least three categories for each orderedcategorical
variable: You can specify a model in the way th
at you are used to, just as though all the
variables were numeric, and
the model will work for any combination of numeric and
orderedcategorical variables. If variables are dichotomous, you will need to impose
0, 1
WILLING
item1
item2
item3
item6
item4
item5
0, 1
AWARE
Example 33
After the Bayesian SEM window opens, wait until the unhappy face changes into a
505
OrderedCategorical Data
The Posterior window displays the poster
ior distribution. The appearance of the
distribution confirms what was concluded
above from the mean, standard deviation,
skewness, and kurtosis of the distribution.
The shape of the distribution is nearly
normal, and it looks like roughly 95% of the area lies between 0.53 and 0.65 (that is,
within 0.06 of 0.59).
Example 33
If you know how to interpret the diagnostic output from MCMC algorithms (for
507
OrderedCategorical Data
The First and last plot provides another
diagnostic. It shows
two estimates of the
posterior distribution (two superimposed plot
s), one estimate from th
e first third of the
MCMC sample and another estimate from
the last third of the MCMC sample.
Example 33
509
OrderedCategorical Data
We are in an even
better position to guess at Person 1s score on the numeric
variable that underlies
item1
because Person 1 gave a response to
item1
. This persons
response places his or her score in the mi
ddle interval, between
the two boundaries.
Since the two boundaries were
arbitrarily fixed at 0 and 1,
we know that the score is
Example 33
The Posterior Predictive Distributions wind
ow contains a table with a row for every
person and a column for every observed variab
le in the model. An asterisk (*) indicates
a missing value, while indi
cates a response that places in
equality constraints on the
underlying numeric variable
. To display the posterior distribution for an item:
Click on the table entry in the upperl
eft corner (Person 1s response to
item1
The Posterior window opens, displaying the posterior distribution of Person 1s
underlying numeric score. At first, the posterior distribution looks jagged and random.
511
OrderedCategorical Data
That is because the program is building up an
estimate of the posterior distribution as
MCMC sampling proceeds. The longer you wait, the better the estimate of the posterior
distribution will be. After a while, the esti
mate of the posterior
distribution stabilizes
Example 33
Next, click the table entry in the first column of the 22
row to estimate Person 22s
score on the numeric variable that
underlies his or her response to
item1
513
OrderedCategorical Data
The mean of the posterior distribution (0.52)
can be taken as an estimate of Person 1s
score on the underlying variable if a point estimate is required. Looking at the plot of
the posterior distribution, we can be nearly
100% sure that the score is between 1 and
Example 33
That takes care of the path diagram. It is
also necessary to make a change to the data
because if
WILLING
is an observed variable, then there has to be a
WILLING
column
in the data file. You can directly modify the da
ta file. Since this is a data file in SPSS
Statistics format, you would use SPSS Statistics to add a
WILLING
variable to the data
file, making sure that
all the scores on
are missing.
To avoid changing the original data file:
Rightclick the
variable in the path diagram
from the popup menu to open the Data Recode window.
In the Data Recode window, click
Create Variable
. A new variable with the default
, appears in the New and recoded variables list box.
0, 1
WILLING
item1
item2
item3
item6
item4
item5
0, 1
AWARE
515
OrderedCategorical Data
Change
to
WILLING
. (If necessary, click the
Rename Variable
button.)
Example 33
You can optionally view the recoded dataset that includes the new
WILLING
variable
by clicking the
button.
517
OrderedCategorical Data
The table on the left shows the original
Example 33
Data imputation works the same way for or
deredcategorical data as it does for
numeric data. With orderedcategorical da
ta, you can impute numeric values for
missing values, for scores on latent variables, and for scores on the unobserved numeric
variables that underlie observed
orderedcategorical measurements.
You need a model in order to perform imputation. You could use the factor analysis
model that was used earlier. There are se
veral advantages and one disadvantage to
using the factor analysis model for imputati
on. One advantage is that, if the model is
correct, you can impute values for the factors.
model. The disadvantage of using the factor analysis model is that it may be wrong. To
be on the safe side, the present example wi
ll use the model that has the biggest chance
of being correct, the saturated model sh
own in the following
figure. (See the file
519
OrderedCategorical Data
After drawing the path diagram for the saturated model, you can begin the imputation.
From the Amos Grap
hics menu, choose
Analyze
Data Imputation
item1
item2
item3
item6
item4
item5
Example 33
In the Amos Data Imputa
tion window, notice that
Regression imputation
and
Stochastic
regression imputation
are disabled. When you have nonnumeric data,
Bayesian
imputation
is the only choice.
We will accept the options shown in the pr
eceding figure, crea
ting 10 completed
521
OrderedCategorical Data
Click
in the Data Imputation dialog box.
The Summary window shows a list of the comple
ted data files that were created. In this
case, only one completed data file was created.
Doubleclick the file name in the Summary
window to display the contents of the
single completed data file, which contains 10 completed data sets.
The file contains 10,170 cases because each
Example 33
523
Example
Mixture Modeling with Training Data
Mixture modeling is appropriate when you have
a model that is inco
rrect for an entire
population, but where the population can be divided into subgroups in such a way that
the model is correct in each subgroup.
Mixture modeling is discussed in the context of structural equation modeling by
Arminger, Stein, and Wittenberg (1999), Hoshino (2001), Lee (2007, Chapter 11),
Loken (2004), Vermunt and Magidson (2005), and Zhu and Lee (2001), among
The present example demonstrates mixtur
e modeling for the situation in which
some cases have already been assigned to gr
oups while other cases have not. It is up
to Amos to learn from the cases that are alr
eady classified and to classify the others.
We begin mixture modeling with an exam
ple in which some cases have already
Example 34
525
Mixture Modeling with Training Data
0.51.01.52.02.5
4.0
6.0
L
e
n
g
Example 34
Species information is available for 10 of the
527
Mixture Modeling with Training Data
Click
New
to create a second group.
Change the name in the Group Name text box from
Group number 2
to
PossiblyVersicolor
Click
New
to create a third group.
Change the name in the Group Name text box from
Group number 3
to
Click
Close
Example 34
From the menus, choose
File
Data Files
Click
529
Mixture Modeling with Training Data
In the Data Files dialog box, click
Group Value
and then doubleclick
Example 34
The Data Files dialog box should now look like this:
531
Mixture Modeling with Training Data
Repeat the precedi
ng steps for the
PossiblyVersicolor
group, but this time doubleclick
versicolor
in the Choose Value for Group dialog box.
Repeat the preceding st
eps once more for the
PossiblyVirginica
group, but this time
doubleclick
in the Choose Value for Group dialog box. The Data Files dialog
box will end up looking like this:
Example 34
Click
to close the Data Files dialog box.
We will use a saturated model for the variables
533
Mixture Modeling with Training Data
From the menus, choose
Analysis Properties
Select
Estimate means and intercepts
(a check mark will appear next to it).
Example 34
Click on the toolbar.
From the menus, choose
The button is disabled because, in mixture modeling, you can perform only
Bayesian estimation.
535
Mixture Modeling with Training Data
After the Bayesian SEM window opens, wa
it until the unhappy face changes into
a happy face . The table of estimates in
the Bayesian SEM wi
ndow should look
something like this:
Example 34
537
Mixture Modeling with Training Data
The Posterior window shows that the proportion of flowers that belong to the
Example 34
For each flower, the Posterior Predictive
Distributions window
shows the probability
that that flower is
539
Mixture Modeling with Training Data
Latent Structure Analysis
It was mentioned earlier that you are not
limited to saturated
models when doing
mixture modeling. You can use a factor analysis model, a regression model, or any
model at all. You may want to become familiar with an important variation of the
Latent structure analysis
(Lazarsfeld and Henry, 1968) is a variation
of mixture modeling in which the measured
variables are required to be independent
within each group. When the
measured variables are multivariate normal, they are
required to be uncorrelated.
To require that the measured variables be
uncorrelated, delete th
e doublehead
ed arrow
in the path diagram of the saturated mo
del. (This path diagram is saved as
b.amw
Click the
button to perform the latent st
ructure analysis. The results of the
latent structure analysis will not be presented here.
541
Example
Mixture Modeling without
Training Data
Mixture modeling is appropriate when you have
a model that is inco
rrect for an entire
population, but where the population can be divided into subgroups in such a way that
the model is correct in each subgroup.
When Amos performs mixture modeling, it allows you to assign some cases to
groups before the analysis starts. Example
34 shows how to do
that. In the present
example, all cases are unclassified at th
e start of the mixtur
e modeling analysis.
This example uses the Anderson (1935) iris data that was used in Example 34. This
time, however, we will not use the
dataset, which contains species
information for 30 of the 150 flowers. Instead, we will use the
dataset, which
Example 35
543
Mixture Modeling without Training Data
Click
New
to create a second group.
Click
New
once more to create a third group.
Click
Close
This example fits a threegroup mixture model. When you arent sure how many
groups there are, you can run the program multiple times. Run the program once to fit
a twogroup model, then again to fit a threegroup model, and so on.
Example 35
From the menus, choose
File
Data Files
Click
Group number 1
to select the first row.
Click
File Name
, select the
file that is in the Amos
directory, and
click
Open
Click
Grouping Variable
and doubleclick
Species
in the Choose a Grouping Variable
dialog box. This tells the program that the
Species
variable will be used to distinguish
one group from another.
545
Mixture Modeling without Training Data
Repeat the preceding steps for
Group number 2
, specifying the same data file (
and the same grouping variable (
Species
Repeat the preceding
steps once more for
Group number 3
, specifying the same data file
iris2.sav
) and the same grouping variable (
Species
Example 35
Select
Assign cases to groups
(a check mark will appear next to it).
So far, this has been just like any ordinary multiplegroup analysis except for the check
mark next to
Assign cases to groups
. That check mark turns this into a mixture
modeling analysis. The check mark tells Amos
to assign a flower to a group if the
grouping variable in the data file does not
already assign it to a group. Notice that it
547
Mixture Modeling without Training Data
was not necessary to click
Group Value
to specify a value for the grouping variable. The
data file contains no values for the grouping variable (
Species
), so the program
automatically constructed the following
Species
values for the three groups:
Cluster1
, and
Cluster3
Click
to close the Data Files dialog box.
We will use a saturated model for the variables
Example 35
549
Mixture Modeling without Training Data
Rightclick
Example 35
In the Object Properties dialog
551
Mixture Modeling without Training Data
After the Bayesian SEM window opens, wa
it until the unhappy face changes into
a happy face . The table of estimates in th
e Bayesian SEM window should then look
something like this:
Example 35
553
Mixture Modeling without Training Data
To obtain probabilities of group membership for each individual flower:
Click the Posterior
Predictive button .
From the menus, choose
For each flower, the Posterior Predictive Di
stributions window shows the probability
that the value of the
Species
variable is
Cluster2
Example 35
The first 50 cases, which we know to be examples of
555
Mixture Modeling without Training Data
Latent Structure Analysis
There is a variation of mixture modeling called
latent structure analysis
in which
observed variables are required to be independent within each group.
To require that
PetalLength
PetalWidth
be uncorrelated and therefore (because
they are multivariate normally
distributed) independent, remove the doubleheaded
arrow that connects th
em in the path diagram. The resulting path diagram is shown
here. (This path diagram is saved as the file,
Optionally, remove the constraints on the va
Example 35
If you attempt to replicate the analysis in th
557
Mixture Modeling without Training Data
Label switching can be revealed by a multi
model posterior distribution for one or
559
Example
Mixture Regression Modeling
Mixture regression modeling (Ding, 2006) is
appropriate when you have a regression
model that is incorrect for an entire population, but where the population can be
divided into subgroups in such a way that the regression model is correct in each
Example 36
A scatterplot of
dosage
and
performance
shows two distinct groups of people in the
sample. In one group,
performance
improves as
dosage
goes up. In the other group,
performance
2.000.002.004.006.00
0.00
10.00
20.00
561
Mixture Regression Modeling
It would be a mistake to try to fit a single
regression line to the whole sample. On the
other hand, two straight lines, one for each group, would fit the data well. This is a job
for mixture regression modeling. A mixtur
e regression analysis would attempt to
divide the sample up into groups and to fi
t a separate regression line to each group.
Example 36
0.005.0010.00
10.00
15.00
20.00
25.00
e
r
f
o
r
m
a
n
c
e
563
Mixture Regression Modeling
The program will then use the five cases that have been preclassified to assist in
classifying the remaining cases. Preassignin
g selected individual cases to groups is
mentioned here only as a possibility. In the present example, no cases will be pre
assigned to groups.
DosageAndPerformance2.sav
Example 36
Click
to create a second group.
Click
This example fits a twogroup mixture regression model. When you arent sure how
many groups there are, you
can run the program multiple
times. Run the program once
to fit a twogroup model, then again to fit a threegroup model, and so on.
565
Mixture Regression Modeling
From the menus, choose
File
Data Files
Click
Group number 1
to select that row.
Click
File Name
, select the
DosageAndPerformance2.sav
file that is in the Amos
Examples
directory, and click
Open
Click
Grouping Variable
and doubleclick
group
in the Choose a Grouping Variable
dialog box. This tells the program that the variable called
group
will be used to
distinguish one group from another.
Example 36
Repeat the preceding steps for
Group number 2
, specifying the same data file
DosageAndPerformance2.sav
) and the same grouping variable (
group
567
Mixture Regression Modeling
Select
Assign cases to groups
(a check mark will appear next to it).
So far, this has been just like any ordinary
multiplegroup analysis
except for the check
mark next to
Assign cases to groups
. That check mark turn
s this into a mixture
Example 36
modeling analysis. The check mark tells Am
os to assign a case to a group if the
grouping variable in the data file does not
already assign it to a group. Notice that it
was not necessary to click
Group Value
to specify a value for the grouping variable. The
data file contains no values for the grouping variable (
group
), so the program
automatically construc
ted values for the
group
variable:
Cluster1
Group
number 1
, and
Cluster2
Group number 2
Click
to close the Data Files dialog box.
Draw a path diagram for the re
gression model, as follows. (This path diagram is saved
From the menus, choose
Analysis Properties
Select
Estimate means and intercepts
(a check mark will
appear next to it).
569
Mixture Regression Modeling
Click on the toolbar.
From the menus, choose
Analyze
The button is disabled because, in
mixture modeling, you can perform only
Bayesian estimation.
Example 36
After the Bayesian SEM window opens, wait until the unhappy face changes into
a happy face . The table of estimates in th
e Bayesian SEM window should then look
something like this:
The Bayesian SEM window contains all of the parameter estimates that you would get
in an ordinary multiplegroup
regression analysis. There is a separate table of estimates
for each group. You can switch from group to group by clicking the tabs just above the
table of estimates.
571
Mixture Regression Modeling
The bottom row of the table contains an estimate of the proportion of the population
that lies in an individual group. The preceding figure, which displays estimates for
Group number 1
, shows that the proportion of the population in
Group number 1
estimated to be 0.247. To see the estimated
posterior distribution
of that population
proportion, rightclick the proportions row in the table and choose
Show Posterior
from the popup menu.
Example 36
The graph in the Posterior window shows that the proportion of the population in
Group number 1
is practically guaranteed to be
somewhere between 0.15 and 0.35.
Lets compare the regression weight and the intercept in
Group number 1
with the
corresponding
estimates in
Group number 2
Group number 1
, the regression weight
estimate is 2.082 and the intercept estimate is 5.399. In
Group number 2
, the regression
weight estimate (1.999) is about the same as in
Group number 1
while the intercept
estimate (9.955) is substa
ntially higher than in
Group number 1
573
Mixture Regression Modeling
Example 36
To obtain probabilities of group membership for each individual case:
Click the Posterior Predictive button .
From the menus, choose
For each case, the Posterior Predictive Dist
ributions window shows the probability that
group
variable takes on the value
Cluster1
or
. Case 1 is estimated to have
a 0.88 probability of being in
Group number 1
and a 0.12 probability of being in
Group
number 2
. Recall that the first group has an inte
rcept of about 5.399 while the second
group has an intercept of about 9.955, so
Group number 1
is the low performing group.
Therefore, there is an 88 per
cent chance that the first person
in the sample is in the low
performing group and a 12 percent chance that that person is in the high performing
group.
575
Mixture Regression Modeling
Example 36
The path diagram should now
look like the following figure. (This path diagram is
saved as
After constraining the slope and error varian
ce to be the same for the two groups, you
can repeat the mixture modeling analysis by clicking the Bayesian button . The
results of that analysis will not be presented here.
577
Mixture Regression Modeling
For the prior distribution of group proportions, Amos uses a Dirichlet distribution with
Example 36
It is possible that the result
s reported here for Group number 1 will match the results
that you get for Group number 2, and that
the results repo
rted here for
Group number
579
Example
Using Amos Graphics without
Drawing a Path Diagram
People usually specify models in Amos Graphics by drawing path diagrams; however,
Amos Graphics also provides a nongraphical method for model specification. If you
don't want to draw a path diagram, you can specify a model by entering text in the
form of a Visual Basic or C# program. In
such a program, each object in a path
diagram (for example, each rectangle, elli
pse, singleheaded arrow, doubleheaded
arrow, and figure caption) corresponds to a single program statement. Usually, a
program statement is one line of text.
Here are some reasons why you might choose to specify a model by entering text
rather than by drawing a path diagram.
Your model is so big that drawing
its path diagram would be difficult.
You prefer using a keyboard to using a mouse, or prefer working with text to
working with graphics.
You need to generate a lot of similar mode
Example 37
The Holzinger and Swineford (1939) dataset
from Example 8 is used for this example.
The factor analysis model from Example 8 is used for this example. Whereas the model
was specified in Example 8 by drawing its
path diagram, the same model will be
specified in the current example by
writing a Visual Basic program.
From the menus, choose
Plugins
In the Plugins dialog box, click
581
Using Amos Graphics without Drawing a Path Diagram
The Program Editor window opens.
In the Program Editor window, change the
Name
and
Description
functions so that they
return meaningful strings.
You may find it helpful at this point to refe
r to the first path diagram in Example 8. We
are going to add one line to the
Mainsub
function for each rectangle, ellipse and arrow
Example 37
In the Program Editor, enter the line
pd.Observed("visperc")
as the first line in the Mainsub
If you save the plugin now, you can use it later on to draw a rectangle representing a
variable called
visperc
. The rectangle will be drawn with
arbitrary height
and width at
a random location in the path diagram. You can specify its height, width and location.
For example,
pd.Observed("visperc", 400, 300, 200, 100)
draws a rectangle for a variable called
visperc
. The rectangle will be centered 400
logical pixels from the left edge of the pa
th diagram, 300 logical pixels from the top
edge. It will be 200 logical pixels wide and
100 logical pixels high. (A logical pixel is
1/96 of an inch.) The online help gives other variations of the
583
Using Amos Graphics without Drawing a Path Diagram
Enter the following ad
ditional lines in the Mainsub fu
nction so that the plugin will
draw five more rectangles for the five remaining observed variables:
pd.Observed("cubes")
pd.Observed("lozenges")
pd.Observed("paragrap")
pd.Observed("sentence")
pd.Observed("wordmean")
Enter the following lines so that the plugin
will draw eight ellipses for the eight
unobserved variables:
pd.Unobserved("err_v")
pd.Unobserved("err_c")
pd.Unobserved("err_l")
pd.Unobserved("err_p")
pd.Unobserved("err_s")
pd.Unobserved("err_w")
pd.Unobserved("spatial")
pd.Unobserved("verbal")
Example 37
Enter the following lines so
that the plugin will draw
the 12 singleheaded arrows:
pd.Path("visperc", "spatial", 1)
pd.Path("cubes", "spatial")
pd.Path("lozenges", "spatial")
pd.Path("paragrap", "verbal", 1)
pd.Path("sentence", "verbal")
pd.Path("wordmean", "verbal")
pd.Path("visperc", "err_v", 1)
pd.Path("cubes", "err_c", 1)
pd.Path("lozenges", "err_l", 1)
pd.Path("paragrap", "err_p", 1)
pd.Path("sentence", "err_s", 1)
pd.Path("wordmean", "err_w", 1)
Notice that in some of the lines above, the
585
Using Amos Graphics without Drawing a Path Diagram
Specify a height, width and location each time you use the Observed, Unobserved and
Example 37
The Mainsub function now looks lik
e this in the Program Editor:
587
Using Amos Graphics without Drawing a Path Diagram
This completes the plugin fo
r specifying the factor anal
ysis model from Example 8.
You can find a prewritten copy of the plugin in a file called
Ex37aplugin.vb
located
in a subfolder of Amoss plugins folder. If you performed a typical installation of
Amos,
Ex37aplugin.vb
is in the location:
Files\IBM\SPSS\Amos\23\Plugins\language&#xlang;&#xuage;.60;
Click
in the Program Editor window. An
y compilation errors will be
displayed on the Syntax errors tab of the Program Editor window.
After you fix any compilation errors, click
Close
in the Program Editor window. You
will be asked if you want to save the file:
Click
. The Save As dialog box will be displayed.
In the Save As dialog box, type a
filename for your plugin and click
. Your plugin
must be saved in the Save As dialogs default folder location. In a typical Amos
installation, that folder location is
C:\Program Files\IBM\SPSS\Amos\23\Plugins
Example 37
After you have saved your plugin, its name,
Example 37a
, appears on the list of plugins
in the Plugins window. (Recall that
Example 37a
is the string returned by the plugins
function.)
Close the Plugins window.
From the menus, choose
File
New
to start with an empty path diagram.
589
Using Amos Graphics without Drawing a Path Diagram
Example 37
In Example 8, the data file
Grnt_fem.sav
was specified interactively (by choosing
File
on the menus). You can do the same thing here as well. As an
alternative, you can specify the
Grnt_fem.sav
data file within the plugin by adding the
following lines to the Mainsub function:
591
Using Amos Graphics without Drawing a Path Diagram
Then you can use the program variable
wordmean
to refer to the model variable called
, and use the program variable
verbal
del variable called
. If you want to draw a si
ngleheaded arrow from the
variable to the
variable, you can write either
pd.Path(wordmean, verbal)
pd.Path(wordmean, "verbal")
The advantage of the unquoted version ove
r the quoted version is that, with the
unquoted version, typing errors are likely to be detected when you click the
593
Example
Simple UserDefined Estimands I
This example shows how to estimate user
Example 38
The latent variable called
67alienation
in Example 6 is here called
alienation67
Similarly,
71alienation
has been changed to
alienation71
. The reason for the change
of variable names is that these names are
going to appear in expressions where names
are not allowed to begin with a numeric character.
Five of the regression weights
in this model have been named
, in
order to make it easy to di
scuss the indirect effect of
ses
on
powles71
. There are two
such indirect eff
ects: the product
and the product
CDB
. You can estimate the sum
of the two indirect effects,
+
, by clicking
�View Analysis Pro�perties Output
and putting a check mark next to
Indirect, direct & total effects
. This capability is built
into Amos and does not require you to specify a userdefined estimand. Suppose,
however, that you want to estimate both
of the individual indirect effects,
and
as well as their sum. All three can be es
timated as userdefined estimands in the
following way.
595
Simple UserDefined Estimands I
Click
Not estimating any userdefined estimand
on the status bar in the lowerleft corner
of the Amos Graphics window. Then click
Define new estimands
from the popup menu.
In the new window that opens, enter thre
e lines to define three custom estimands:
Example 38
The names of the three custom estimands are
Indirect_AB
Indirect_CDB
and
Sum
You can make up other names instead. Names for estimands must be made up of letters
597
Simple UserDefined Estimands I
Optionally, add lines and comments, as shown here:
Click the
Close
button.
Click
in the following dialog.
Example 38
In the Save As dialog, type
indirect effects
in the File name box. Then click the
button.
Click
�View Analysis Pro�perties Bootstrap
, and put check marks next to
Perform
bootstrap
Biascorrected confidence intervals
. Also, since the data file contains
599
Simple UserDefined Estimands I
sample moments and not raw data, put a check mark next to
Example 38
In the Amos Output
window, doubleclick
, then doubleclick
, then
click
Userdefined estimands
The estimand called
Indirect_AB
is estimated to be 0.205. This is the product of the
regression weight
(0.212) and the regression weight
601
Simple UserDefined Estimands I
Click
Bootstrap standard errors
Indirect_AB
is approximately normally distributed with a standard error of about
Click
Bootstrap Confidence
Example 38
The population value of the
Indirect_AB
603
Simple UserDefined Estimands I
605
Example
Simple UserDefined Estimands II
This example shows how to estimate the difference between two standardized
regression weights, along with a bootstrap st
andard error, a confidence interval, and a
significance test for the difference.
Four quizzes were administered to a cl
ass of 39 students. The quizzes were
approximately equally spaced thro
ughout the semester. The file
Example 39
The following path diagram shows the standa
rdized regression we
ights estimated for
this model.
Let's compare two standardized regressi
on weights, say the weight for using
predict
, and the weight for using
to predict
0.390.35
0.04
607
Simple UserDefined Estimands II
In the window that opens, enter one line to specify the new estimand, as follows:
You can choose a name other than
StandardizedWeightDiff
if you wish.
Click the
Check Syntax
button. If you have made no typing mistakes, the message
Syntax is OK
will be displayed in the Description box.
Click the
Close
button.
Click
in the following dialog.
Example 39
In the Save As dialog, type
StandardizedDifference
in the File name box. Then click
button.
609
Simple UserDefined Estimands II
Click
�View Analysis Properties � Bootstrap
, and put check marks next to
bootstrap
and
Biascorrected confidence intervals
Click
Analyze� Calculate Estimates
Click
�View Text Output
In the Amos Output
window, doubleclick
, then doubleclick
, then
click
Userdefined estimands
The estimand called
StandardizedWeightDiff
is estimated to be 0.047.
611
Simple UserDefined Estimands II
Click
Bootstrap standard errors
The difference is approximately normally di
stributed with a standard error of about
Example 39
Click
Bootstrap Confidence
613
1
=
G
g
g
+
=
g
+
=
g
1
dpq
g
Appendix A
= the covariance matrix for group
, according to the model
= the mean vector for group
, according to the model
= the population covariance matrix for group
= the population mean vector for group
= the distinct elements of arranged in a single column
vector
= the nonnegative integer specified by the
ChiCorrect
g
vec
S
x
a
615
Discrepancy Functions
Amos minimizes discrepancy functions (Browne, 1982, 1984) of the form:
Different discrepancy functions are obtained by changing the way
is defined. If
means and intercepts are un
constrained and do not appear as explicit model
to be:
S
,
x
;
,
a
,
,
r
N
N
f
N
r
N
C
g
g
g
g
g
g
S
g
g
g
g
g
g
g
g
g
g
g
KL
x
S
S
,
x
;
,
1
log
Appendix B
For
generalized least squares
estimation (
), , and are obtained by
taking
to be:
For
asymptotically di
estimation (
), , and are obtained
by taking
to be:
where the elements of are given by Browne (1984, Equations 3.13.4):
GLS
2
1
g
g
g
g
GLS
S
S
;
g
g
g
g
g
g
g
g
ADF
(
)
(
;
s
U
s
S
r
g
ir
g
g
i
N
x
)
(
)
(
(
)
(
1
)
(
)
(
)
(
j
g
jr
N
r
g
i
g
ir
g
g
ij
x
x
x
N
w
(
)
(
)
(
)
(
)
(
)
(
1
)
(
)
(
)
(
,
l
g
lr
g
k
g
kr
g
j
g
jr
N
r
g
i
g
ir
g
g
kl
ij
x
x
x
x
x
x
x
N
w
617
Discrepancy Functions
For
scalefree least squares
estimation (
), , and are obtained by taking
to be:
(D5)
For
unweighted least squares
estimation (
), , and are obtained by
taking
(D6)
Emulisrel6
(
)
(
)
(
,
,
g
kl
g
ij
g
kl
ij
kl
ij
g
w
w
U
2
1
g
g
g
g
SLS
D
S
;
ULS
2
1
g
g
g
ULS
S
S
;
g
g
g
N
C
FCNG
N
F
N
C
1
(
1
1
Appendix B
Suppose you have two independent samples and a model for each. Furthermore,
suppose that you analyze the two samples simultaneously, but that, in doing so, you
the simultaneous analysis of both groups will
be the same as from separate analyses of
each group alone.
Furthermore, the discrepancy function
(D1a) obtained from the simultaneous
analysis will be the sum of the discrepancy functions from the two separate analyses.
Formula (D1) does not have this property when
is nonzero. Using formula (D1) to do
a simultaneous analysis of the two groups
will give the same parameter estimates as
two separate analyses, but the discrepancy function from the simultaneous analysis
will not be the sum of the in
dividual discrepancy functions.
On the other hand, suppose you have a single
sample to which you have fitted some
model using Amos. Now suppose that you arbitrarily split the sample into two groups
of unequal size and perform a simultaneous analysis of both groups, employing the
original model for both groups and constraini
619
Measures of Fit
Appendix C
Models with relatively few pa
p
d
621
Measures of Fit
is the number of
sample moments and
is the number of distinct parameters.
Rigdon (1994a) gives a detailed explanation
of the calculation and interpretation of
degrees of freedom.
Use the
\df
text macro to display the degrees of freedom in the output path
diagram.
PRATIO
The parsimony ratio (James, Mulaik
d
PRATIO
Appendix C
specified model). That is,
is a
value for testing the hypothesis that the model fits
perfectly in the population.
One approach to model selection employs st
atistical hypothesis testing to eliminate
from consideration those models that are
inconsistent with the available data.
Hypothesis testing is a widely accepted pro
cedure, and there is a
lot of experience in
its use. However, its unsuitability as a de
vice for model selection was pointed out early
in the development of analysis of moment st
ructures (Jreskog, 1969). It is generally
acknowledged that most models are useful ap
proximations that do
not fit perfectly in
the population. In other words, the null hypoth
esis of perfect fit is
not credible to begin
with and will, in the end, be accepted only if the sample is not allowed to get too big.
If you encounter resistance to the foregoing view of the role of hypothesis testing in
model fitting, the following quotations may
come in handy. The first two predate the
development of structural modeling and
The power of the test to detect an underlying disagreement between theory and
data is controlled largely by the size
of the sample. With a small sample an
alternative hypothesis whic
h departs violently from th
e null hypothesis may still
have a small probability of yielding a significant value of . In a very large
sample, small and unimportant departures
from the null hypothesis are almost
no doubt, indicate that any such nontrivia
l hypothesis is statistically untenable.
(Jreskog, 1969, p. 200)
...in very large samples virtually all mode
ls that one might consider would have
to be rejected as statistically untenable.
... In effect, a nonsignificant chisquare
value is desired, and one attempts to in
fer the validity of the hypothesis of no
difference between model and data. Such logic is wellknown in various
statistical guises as attempting to pr
ove the null hypothesis. This procedure
cannot generally be justified, since the chisquare variate
can be made small by
simply reducing sample size. (Bentler and Bonett, 1980, p. 591)
623
Measures of Fit
Our opinion...is that this nu
ll hypothesis [of perfect fit] is implausible and that it
C
Appendix C
is the minimum value,
, of the discrepancy,
(see Appendix B).
\fmin
text macro to display the minimum value of the discrepancy
in the output path diagram.
Steiger and Lind (1980) introduced the use of the population discrepancy function as
a measure of model adequacy. The population discrepancy function, , is the value
of the discrepancy function obtained by
fitting a model to the population moments
rather than to sample moments. That is,
in contrast to
Steiger, Shapiro, and Browne (1985) sh
owed that, under certain conditions,
has a noncentral chisquare distribution with
degrees of freedom and noncentrality
=
,
=
=
CnF
NCPmax
.
,
d
C
625
Measures of Fit
for , and is obtained by solving
for , where is the distribution function of the noncentral chisquared
distribution with noncentrality parameter and
degrees of freedom.
.
,

d
C
x


===
F
90
LO
90
HI
Appendix C
error of approximation
, called
RMS
by Steiger and Lind, and
RMSEA
by Browne and
Cudeck (1993).
The columns labeled
LO 90
and
HI 90
contain the lower limit and upper limit of a 90%
confidence interval on the population value of
. The limits are given by
Rule of Thumb
Practical experience has made us feel that a value of the
of about 0.05 or
less would indicate a close fit of the mode
l in relation to the degrees of freedom.
This figure is based on subjective judgment. It cannot be regarded as infallible or
correct, but it is more reasonable than
the requirement of
exact fit with the
= 0.0. We are also of the opinion that a value of about 0.08 or less for the
would indicate a reasonable error of approximation and would not want
to employ a model with a
RMSEA
greater than 0.1. (Browne and Cudeck, 1993)
Use the
text macro to display the estimated root mean square error of
approximation in the output path diagram,
\rmsealo
for its lower 90% confidence
estimate, and
\rmseahi
for its upper 90% confidence estimate.
F
population
F
RMSEA
estimated
n
90
LO
n
90
HI
627
Measures of Fit
is a
value for testing the null hypothesis that the
population
is no greater than 0.05.
By contrast, the
value in the
column (see P on p.621) is for testing the hypothesis
that the population
RMSEA
Based on their experience with
, Browne and Cudeck (1993) suggest that a
of 0.05 or less indicates a
. Employing this definition of close fit,
gives a test of close fit while
gives a test of exact fit.
\pclose
text macro to display the
value for close fit of the population
in the output path diagram.
PCLOSE1
ndd
.
RMSEA
:
RMSEA
:
H
C
2
AIC
Appendix C
See also ECVI on p.629.
Use the
text macro to display the value of
the Akaike information criterion
in the output path diagram.
The BrowneCudeck (1989) criterion is given by
where if the
Emulisrel6
command has been used, or if it
has not.
imposes a slightly greater penalty for model complexity than does
is the only measure in this section that
was developed specifically for analysis of
moment structures. Browne and Cudeck pr
ovided some empirical evidence suggesting
that
may be superior to more generally
applicable measur
es. Arbuckle (in
preparation) gives an alternative justification for
BCC
and derives the above formula
for multiple groups.
See also MECVI on p.630.
Use the
text macro to display the value of the BrowneCudeck criterion in
the output path diagram.
The Bayes information criterion (Schwarz, 1978; Raftery, 1993) is given by the
In comparison to the
BCC
and
, the
assigns a greater penalty to model
complexity and, therefore, has a greater tendency to pick parsimonious models. The
is reported only for the case of a single
group where means and intercepts are not
explicit model parameters.
g
g
g
G
g
g
g
g
g
g
p
p
N
p
p
b
q
C
1
2
3
2
BCC
=

=
629
Measures of Fit
Use the
\bic
text macro to display the value of the Bayes information criterion in
the output path diagram.
Bozdogans (1987)
CAIC
) is given by the formula
assigns a greater penalty to model complexity than either
but not as
great a penalty as does
is reported only for the case of a single group where
ln
CAIC
N
q
C
q
F
n
2
AIC
1
ECVI
q
d
90
LO
q
d
90
HI
Appendix C
Except for a scale factor,
MECVI
is identical to
where if the
Emulisrel6
command has been used, or if it
has not.
See also BCC on p.628.
Use the
text macro to display the modified
ECVI
statistic in the output
Several fit measures encourage you to reflect
on the fact that, no matter how badly your
model fits, things could always be worse.
g
g
g
G
g
g
g
g
g
g
p
p
N
p
p
a
q
F
n
1
2
3
2
1
BCC
MECVI


71.544
631
Measures of Fit
This thingscouldbemuchworse philosophy of
model evaluation is incorporated into
a number of fit measures. All of the measur
Model
NPAR
CMIN/DF
71.544
Model B: Most General
6.383
Model C: TimeInvariance
7.501
Model D: A and C Combined
73.077
Saturated model
0.000
Independence model
2131.790
142.119
b
F
C
C
1
1
NFI
=
=
71.544
=
2131.790
Appendix C
Looked at in this way, the fit of Model A is
a lot closer to the fit of the saturated model
than it is to the fit of th
e independence model. In fact,
you might say that Model A has
a discrepancy that is 96.6% of the way between the (terribly fitting) independence
model and the (perfectly
fitting) saturated model.
Rule of Thumb
Model
NPAR
Model A: No Autocorrelation
0.000
Model B: Most General
0.271
Model C: TimeInvariance
0.484
Model D: A and C Combined
0.000
Independence model
0.000
.
790
.
2131
54
.
71
1
790
.
2131
54
.
71
790
.
2131
b
b
b
F
d
F
d
C
d
C
1
1
RFI
633
Measures of Fit
where and
are the discrepancy and the degrees of freedom for the model being
evaluated, and and are the discrepa
ncy and the degrees of freedom for the
baseline model.
is obtained from the
by substituting
for
RFI
values close to 1
indicate a very good fit.
text macro to display the relative
fit index value in the output path
diagram.
IFI
Bollens (1989b) incr
emental fit index (
) is given by:
where and
are the discrepancy and the degrees of freedom for the model being
evaluated, and and are the discrepa
ncy and the degrees of freedom for the
baseline model.
values close to 1 indicate a very good fit.
Use the
\ifi
text macro to display the incrementa
l fit index value in the output path
diagram.
The TuckerLewis coefficient (
in the notation of Bollen, 1989b) was discussed by
C
C
C
b
IFI
TLI
2
b
b
b
C
d
C
d
C
Appendix C
The comparative fit index (
; Bentler, 1990) is given by
where ,
, and
NCP
are the discrepancy, the de
grees of freedom, and the
b
b
C
d
C
NCP
NCP
1
0
,
max
0
,
max
1
CFI
b
C
d
C
1
RNI
635
Measures of Fit
is the result of ap
d
NFI
PRATIO
NFI
PNFI
d
CFI
=
PRATIO
CFI
PCFI
Appendix C
is given by
where is the minimum value of the discrepancy function defined in Appendix B and
is obtained by evaluating
with ,
g = 1, 2,...,G
. An exception has to be
made for maximum likelihood estimation, since (D2) in Appendix B is not defined for
. For the purpose of computing
in the case of maximum likelihood
estimation, in Appendix B is calculated as
with , where is the maximum likelihood estimate of .
always less than or equal to 1.
= 1 indicates a perfect fit.
Use the
text macro to display the value
of the goodnessoffit index in the
output path diagram.
(adjusted goodnessoffit index) takes into account the degrees of freedom
available for testing the model. It is given by
where
is bounded above by 1, which indicates a perfect fit. It is not, however,
bounded below by 0, as the
Use the
\agfi
text macro to display the value of the adjusted
in the output path
F
1
GFI
=
=
S
2
1
g
g
g
g
S
K
S
;
d
1
1
AGFI
g
g
b
d
*
637
Measures of Fit
d
GFI
PGFI
g
g
b
d
*
Appendix C
Here are the critical
s displayed by Amos for each of the models in Example 6:
Model A, for instance, would have been acc
epted at the 0.05 level if the sample
moments had been exactly as they were found to be in the Wheaton study but with a
sample size of 164. With a sample size of 165, Model A would have been rejected.
Hoelter argues that a critical
under the assumption that your model is correct.
Model
HOELTER
HOELTER
0.01
Model A: No Autocorrelation
Model B: Most General
Model C: TimeInvariance
Model D: A and C Combined
Independence model
0.05
0.01
¦¦¦
111
639
Measures of Fit
The smaller the
Model
GFI
AGFI
PGFI
0.913
Model B: Most General
0.990
Model C: TimeInvariance
0.993
Model D: A and C Combined
0.941
Saturated model
Independence model
0.292
641
643
Using Fit Measures to Rank Models
In general, it is hard to pick a fit measure because there are so many from which to
choose. The choice gets easier when the pu
rpose of the fit measure is to compare
models to each other rather th
an to judge the merit of mode
ls by an absolute standard.
For example, it turns out that
d
d
RMSEA


RFI

TLI




d
d
Appendix E
The following fit measures depend monotonically on and not at all on
specification search procedure reports on
ly as representative of them all.
Each of the following fit measur
es is a weighted sum of and
and can produce a
distinct rank order of models. The specificati
on search procedure reports each of them
except for
NCPmax

CFI1

RNI1

CMIN

NFI1

BCC
CAIC
645
Using Fit Measures to Rank Models
Each of the following fit measures is capable of providing a unique rank order of
models. The rank order depends on the choice of baseline model as well. The
specification search procedure do
es not report these measures.
The following fit measures are the only ones reported by Amos that are not functions
of and
in the case of maximum likelihood
estimation. The specification search
procedure does not re
port these measures.
GFI
AGFI
647
Baseline Models for Descriptive Fit
Seven measures of fit (
TLI
) require a
null
or
baseline
bad model against which other models can be compared. The specification
search procedure offers a choice of four null, or baseline, models:
The observed variables are required
to be uncorrelated. Their means and
variances are unconstrained. This is the baseline
Independence
model in an ordinary
Amos analysis when you do not perform a specification search.
The correlations among the observed va
riables are required to be equal. The
means and variances of the observed variables are unconstrained.
Null 3:
The observed variables are required to
be uncorrelated and to have means of 0.
Their variances are unconstrai
ned. This is the baseline
Independence
model used by
Amos 4.0.1 and earlier for
models where means and intercepts are explicit model
Appendix F
From the menus, choose
Specification Search
Click the
button on the Specifi
cation Search toolbar.
In the Options dialog box, click the
tab.
The four null models and the saturated mo
del are listed in the Benchmark models
group.
649
Rescaling of AIC, BCC, and BIC
The fit measures,
, are defined in Appendix C. Each measure is of
the form , where
takes on the same value for all models. Small values are
good, reflecting a combination of good fit to the data (small ) and parsimony
). The measures are used for comparin
g models to each other and not for
judging the merit of a single model.
The specification search proced
ure in Amos provides three ways of rescaling these
measures, which were illustrated in Examples 22 and 23. This appendix provides
formulas for the rescaled fit measures.
AIC
BIC
=
BCC
BCC
BIC
min
Appendix G
The rescaled values are either 0 or positive
. For example, the best model according to
AIC has , while inferior models have positive values that reflect how
much worse they are than the best model.
To display , , and after a specification search, click on the
Specification Search toolbar.
tab of the Options dialog box, click
Zerobased (min = 0)
Akaike Weights and Bayes Factors (Sum = 1)
To obtain the following rescaling, select
Akaike weights and Bayes factors (sum = 1)
on
tab of the Options dialog box.
Each of these rescaled measures sums to
1 across models. The rescaling is performed
only after an exhaustive specification search.
If a heuristic search is carried out or if a
positive value is specified for
Retain only the best ___ models
, then the summation in
the denominator cannot be calculated, an
d rescaling is not performed. The are
called
Akaike weights
by Burnham and Anderson (1998). has the same
AIC
AIC
AIC
BIC
AIC

BCC
BCC


i
BCC
i
i
651
Rescaling of AIC, BCC, and BIC
Akaike Weights and Bayes Factors (Max = 1)
To obtain the following rescaling, select
Akaike weights and Ba
yes factors (max = 1)
on
tab of the Options dialog box.
For example, the best model according to
has , while inferior models
have
between 0 and 1. See Burnham and Anderson (1998) for further discussion
of , and Raftery (1993, 1995) and Ma
digan and Raftery (1
discussion of .

BCC

BIC

AIC
AIC
AIC
BIC
653
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669
223
ADF, asymptotically distributionfree,
admissibility test in Bayesian estimation,
Akaike information criterion,
311
Burnham and Andersons guidelines for,
Akaike weights,
category boundaries,
CFI, comparative fit index,
change
orientation of drawing area,
671
Index
draw covariances,
drawing area
add covariance paths,
add unobserved variable,
change orientation of,
viewing measurement weights,
duplicate measurement model,
ECVI, expected crossvalidation index,
endogenous variables,
EQS (SEM program),
independence model,
274
indirect effects,
finding a confidence interval for,
viewing standardized,
429
inequality constraints on data,
673
Index
move objects,
multiple models in a single analysis,
multiplegroup analysis,
multiplegroup factor analysis,
multiply imputed data file, combining results,
473
PNFI, parsimonious normed fit index,
675
Index
benefits of equal parameters,
equal paramaters,
group name in figure caption,
specifying group differences
conventions,
stability test in Bayesian estimation,
obtain,
view,
statistical hypothesis testing,
stochastic regression imputation,
structural covariances,
structural equation modeling,
methods for estimating,
structural model,
structure specification,