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Garson, G. D. (2015). Structural Equation Modeling. Asheboro, NC: Statistical Associates Publishers.

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Table of Contents
ISBN number: 978-1-62638-032-5
@c 2015 by G. David Garson and Statistical Associates Publishers. worldwide rights reserved in all languages and on all media. Permission is not granted to copy, distribute, or post e-books or passwords.



An illustrated tutorial and introduction to structural equation modeling using SPSS AMOS, SAS PROC CALIS, and Stata sem and gsem commands for examples. Suitable for introductory graduate-level study.

The 2015 edition is a major update to the 2012 edition. Among the new features are these:

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Below is the unformatted table of contents.

Table of Contents
Overview	14
Data examples in this volume	16
Key Concepts and Terms	18
The structural equation modeling process	18
Indicator variables	19
Latent variables	20
Exogenous variables	20
Endogenous variables	20
Regression models, path models, and SEM models	21
Model specification	22
Model parsimony	22
Model development	23
The measurement model versus the structural model	23
Model trimming versus model building	24
Modification indexes and parameter change	25
Path significance and critical ratios	28
Model fit	28
Software packages	29
User interfaces for SEM	30
The SPSS Amos interface	30
The SAS PROC CALIS interface	36
The Stata SEM interface	39
The Wheaton Model: A Quick Start Example	46
Overview	46
SPSS Amos Wheaton model output	48
Path weights	50
Direct and indirect path weights	53
Squared multiple correlations (SMC)	55
Factor weights	56
Goodness of fit measures	57
Modification indexes and parameter change	60
SAS Wheaton model output	61
Overview	61
Path weights	61
Direct and indirect path weights	63
Squared multiple correlations (SMC)	64
Factor weights	65
Goodness of fit measures	66
Modification indexes and parameter change	68
Stata Wheaton model output	69
Overview	69
Path weights	69
Direct and indirect path weights	71
Squared multiple correlations	74
Factor weights	77
Goodness of fit measures	77
Modification indexes and parameter change	78
Assessing Model Fit	81
Overview	81
Default, saturated, and independence models	82
Cautions regarding goodness of fit measures	83
Fit indexes are spurious when the covariance matrix is not positive definite	84
Good fit does not mean strong effect size	85
Good fit does not mean all model components fit	85
Good fit is not proof of causation	85
Good fit does not mean another model might not fit better	85
Fit measures overestimate good fit for small samples	86
Fit measures are influenced by number of indicator variables	86
Fit measures are influenced by number of model constraints	86
Fit is relative to null models, which differ	87
Good fit is relative to progress in the field	87
Should fit measures be used at all?	87
Reporting goodness of fit	88
Confirmatory Factor Analysis: Testing the Measurement Model	89
Overview	89
Data example	92
CFA in SPSS Amos	92
CFA input in Amos	92
CFA output in Amos	93
CFA input in SAS	100
CFA output in SAS	101
CFA in Stata	106
CFA input in Stata	106
CFA output in Stata	107
Further aspects of measurement models	112
Handling correlated error	112
Assigning metrics	112
Reflective vs. formative models	113
Measurement error terms	114
Simple variables and single indicator latent variables	115
Error variance when reliability is known	115
Measurement error terms vs. structural error terms	115
Testing measurement models for convergent and divergent validity	116
Measurement validity using reliability coefficients	117
Structural Equation Modeling: Testing the Structural Model	118
Overview	118
Data Example	119
Models leading to the final structural model	120
Structural modeling in Amos	125
SEM input in Amos	125
SEM output in Amos	125
Structural modeling in SAS PROC CALIS	134
SEM input in SAS	134
SEM output in SAS	136
Structural modeling in Stata	144
SEM input in Stata	144
SEM output in Stata	145
Specification Search (all possible subsets SEM)	154
Overview	154
Data example	155
Specification search in Amos	155
Overview	155
Specification search user interface	155
Example	156
Tools	157
The Options button	158
Perform Specification Search button	158
Other tools for specification search	160
Specification search in SAS	160
Specification search in Stata	160
Multi-group Analysis	160
Overview	160
Introduction	160
To standardize or not to standardize?	161
Model invariance	163
Examining non-invariance across groups	165
Critical ratios of differences test	166
Example and data for multi-group analysis	166
Multi-group analysis in SPSS Amos	168
Baseline multi-group testing	168
Data setup for multi-group analysis in Amos	170
Viewing group models	172
Testing for measurement invariance across groups (multi-group modeling)	173
Parameters to constrain to be equal	173
Goodness of fit	177
Path and covariance significance	179
Modification indexes	180
No negative variances	182
Critical ratios of differences tests	182
Multi-group analysis in SAS	189
Overview	189
Example	189
Data setup for multi-group analysis in SAS	190
Baseline multi-group testing	190
The unconstrained multigroup model	192
The measurement weights model	194
Other models	197
Multi-group analysis in Stata	198
Overview	198
Baseline multi-group testing	198
Multi-group models in Stata	200
The measurement weights model	201
Other models	205
Failure to converge	208
Latent Growth Curve Modeling	209
Example data	209
Overview	211
The LGC model in Amos	211
Amos input for the LGC model	211
Amos output for the LGC model	224
Amos linear growth model with a time-invariant predictor	225
AMOS summary	232
The LGC model in SAS	234
Overview	234
SAS input for the LGC model	234
SAS LGC output	236
SAS LGC model with a time-invariant predictor	239
SAS summary	241
The LGC model in Stata	242
Overview	242
Stata input for the LGC model	243
Stata LGC output	246
Stata LGC model with a time-invariant predictor	249
Stata summary	251
Multiple linear growth models	252
Ordinal data in SEM	253
Overview	253
Treating ordinal variables as interval in data level	253
Conversion to dummy variables	254
Using an appropriate correlation matrix as input	255
Bayesian estimation	256
Generalized structural equation modeling	256
Statistical packages' treatment of ordinal data	257
SPSS Amos	257
SAS	257
Stata	257
Bayesian SEM	257
Overview	257
Key concepts and terms	259
Prior distributions	259
Markov chain Monte Carlo (MCMC) methods	260
Posterior predictive p	260
DIC (deviance information criterion)	261
Effective number of parameters	261
Combining Bayesian and ML methods	261
Cross-validation	261
Residual analysis	262
Bayesian SEM in SPSS AMOS	262
Data levels	262
Nominal-level data in Amos with Bayesian estimation	262
Ordinal-level data in Amos with Bayesian estimation	262
Entering ordinal data	262
Censored data	264
Data imputation	264
Warning regarding mixture modeling	264
Warning regarding binning numerical variables	265
Warning regarding variable names	265
Bayesian SEM in SPSS AMOS	265
Example	265
To estimate means and intercepts	266
The Bayesian estimation window	266
Prior parameter distributions	269
Posterior parameter distributions	270
Diagnostic graphs	272
Fit measures	272
Additional estimates	273
Mixture modeling / latent class analysis	274
Overview	274
Example	275
Mixture modeling in SPSS Amos	275
Amos input	275
Amos output	282
Latent structure analysis	285
Mixture regression modeling	285
Mean Structure Analysis	286
Overview	286
Example	286
Mean structure analysis in SPSS Amos	287
Amos input	287
Amos output	296
Obtaining output	296
Model fit criteria	298
Upholding the baseline model	300
Analysis of mean structure	301
Estimates of latent means	302
Other output tables	304
Mean structure analysis in SAS	304
Overview	304
Example	304
SAS input	305
The MEANSTR option	305
SAS syntax for mean structure analysis	305
SAS output	309
Overview	309
Model fit criteria	309
Upholding the baseline model	310
Analysis of mean structure	311
Estimates of latent means	312
Other output tables	314
Mean structure analysis in Stata	314
Overview	314
Example	314
Stata input	315
Putting the example dataset in use	315
Running the measurement intercepts model	315
Running the structural means model	315
A likelihood ratio test of model differences	316
Stata output	316
Overview	316
Model fit criteria	316
Upholding the baseline model	318
Analysis of mean structure	319
Estimates of latent means	319
Other output tables	320
Generalized SEM (GSEM) in Stata	320
Overview	320
Why "generalized"?	320
Data distributions and link functions	321
GSEM postestimation commands	325
GSEM limitations in Stata	325
Example	326
Stata input	326
Stata output	331
Default GSEM output	331
Postestimation GSEM output	333
Multilevel SEM (MSEM) in Stata	335
Overview	335
When multilevel modeling is needed	335
Multilevel SEM software	337
Example	338
Stata input	338
Structural GSEM	338
Multilevel GSEM (MGSEM)	338
Stata output	340
Structural GSEM	340
Multilevel GSEM (MGSEM)	340
Estimation options in SEM	342
Maximum likelihood estimation (ML)	343
Full information maximum likelihood (FIML)	344
Weighted least squares estimation (WLS)	344
Generalized least squares estimation (GLS)	345
Ordinary least squares estimation (OLS)	345
Unweighted least squares estimation (ULS)	345
Two-stage least squares estimation (2SLS)	346
Asymptotically distribution-free estimation (ADF)	346
Elliptical distribution theory estimation (EDT)	346
Bayesian estimation	346
Bootstrapped vs. Bayesian estimates	347
Goodness of fit measures	348
A helpful spreadsheet	348
Goodness-of-fit measures and tests based on predicted vs. observed covariances	348
Overview	348
Chi-square	348
Hoelter's critical N	352
Relative chi-square	353
Minimum fit function (FMIN)	353
Root mean square residual (SRMR and RMR)	354
The standardized residual matrix	356
Root mean square error of approximation (RMSEA) and PCLOSE	360
Goodness-of-fit index (GFI)	363
Adjusted goodness-of-fit index, AGFI	365
Goodness of fit tests involving comparison with the null model	365
Overview	365
Likelihood ratio test	366
Wald tests	368
Comparative fit index (CFI)	369
Tucker-Lewis index (TLI) or non-normed fit index (NNFI)	371
Normed fit index (NFI)	372
Relative fit index (RFI)	373
Incremental fit index (IFI)	374
Bentler-Bonett index (BBI)	375
Bollen86 Fit index (B86 or BFI)	375
Goodness-of-fit tests penalizing for lack of parsimony	375
Overview	375
Parsimony ratio (PRATIO)	376
Parsimony comparative fit index (PCFI)	377
Parsimony normed fit index  (PNFI)	378
Parsimony normed fit index 2 (PNFI2)	378
Parsimony goodness of fit index  (PGFI)	378
Parsimony index (PI)	378
Information complexity index (ICOMP)	379
Noncentrality-based goodness of fit	379
Overview	379
Noncentrality parameter (NCP)	379
McDonald noncentrality index (NCI)	380
Information theory goodness of fit measures	381
Overview	381
Akaike Information Criterion (AIC)	382
Corrected AIC (AICC)	383
Consistent AIC (CAIC)	384
AIC for over-dispersed data (QAIC)	384
Bayesian Information Criterion (BIC)	385
Sample-size adjusted BIC (SABIC)	387
Hannan & Quinn information criterion (HQIC)	387
Browne-Cudeck criterion (BCC)	387
Expected cross-validation index (ECVI)	388
Modified expected cross-validation index (MECVI)	388
Cross-validation index (CVI)	388
Assumptions	389
Data level	389
Overview	389
Dichotomous data	389
Ordinal data	389
Nominal data	389
Ordinal data	390
Dichotomous data	392
Multinomial data	392
Sample size	393
Linearity	395
Outliers	396
Multiple indicators	398
One-indictor regression models	398
Low measurement error	399
Complete data or appropriate data imputation	399
Multivariate normal distribution of the indicators	399
Multivariate normal distribution of the latent dependent variables	401
Correlated indicators	401
Not theoretically under-identified or just identified	401
Recursivity	404
Not empirically identified due to high multicollinearity	404
High precision	405
Small, random residuals	405
Uncorrelated error terms	405
Multicollinearity	405
Non-zero covariances	406
Frequently Asked Questions	407
SEM analysis	407
What are common guidelines for conduction SEM research and reporting it?	407
How do I write up a SEM analysis?	407
What is a "structural equation model" and how is it diagrammed?	412
How do I save latent variable (factor) scores for use in other procedures?	413
What is four-step SEM modeling ?	416
How can I use SEM to test for the unidimensionality of a construct?	417
How does one test for modifier or covariate control variables in a structural model?	417
How do you test for interaction effects and use crossproduct interaction terms in SEM?	418
Can one use simple variables in lieu of latent variables in SEM models?	421
How is the model-implied covariance matrix computed to compare with the sample one in model fit measures in SEM?	423
What are "replacing rules" for equivalent models?	423
Differentiate ML, FIML, and EM estimation. Are these the same?	424
Goodness of fit	424
What is the relation of goodness of fit measures to the null model?	424
Instead of using SEM to test alternative models, could I just use it to identify important variables even when fit is poor?	425
Can incremental goodness of fit (GOF) measures be used with any model?	426
Are LISREL goodness of fit measures the same as other packages?	426
SEM tests and computational issues	427
How do I determine if the difference between two structural path coefficients is significant?	427
Why is it that this and other write-ups of SEM give little emphasis to the concept of significance testing?	427
If I run a SEM model for two subgroups of my sample, can I compare the path coefficients?	428
How can I tell beforehand if my model is identified and thus can have a unique solution?	428
When is a confirmatory factor analysis (CFA) model identified in SEM?	431
Data issues	432
Can I use SEM with archival and secondary data?	432
How is matrix input used instead of raw data?	433
How do I use polychoric correlation for ordinal and binary variables?	435
What is a matrix in Lisrel?	441
Should one standardize variables prior to structural equation modeling, or use standardized regression coefficients as an input matrix?	442
Can I use dichotomous, multinomial, and ordinal data in SEM?	442
Can SEM handle longitudinal data?	444
How should one handle missing data in SEM?	444
I've heard SEM is just for non-experimental data, right?	446
Types of SEM models	446
How and why is SEM used for confirmatory factor analysis, often as a preliminary step in SEM?	446
What is a second order factor model in SEM?	446
What is confirmatory tetrad analysis (CTA) in SEM?	446
Error conditions in SEM	448
What is the difference in the handling of error between regression and SEM?	448
What does it mean when I get negative error variance estimates?	448
What is a "Heywood case"?	449
Relation of SEM to other procedures	449
What is the historical origin of path analysis?	449
Can I compute OLS regression with SEM software?	451
Given the advantages of SEM over OLS regression, when would one ever want to use OLS regression?	451
Why do I get different results in SEM compared to OLS or logistic regression?	451
I have heard SEM is like factor analysis. How so?	452
SPSS Amos questions	452
How do run a SEM model in Amos?	452
What is the baseline model in Amos and why does this matter?	454
What is the Amos toolbar?	454
How are data files linked to SEM in Amos?	456
In Amos, how do you enter a label in a variable (in an oval or rectangle)?	456
How do you vertically align latent variables (or other objects) in Amos?	457
In Amos, what do you do if the diagram goes off the page?	457
In Amos, how to you move a parameter label to a better location?	457
How is an equality constraint added to a model in Amos?	457
How do you test for normality and outliers in Amos?	457
How do you interpret Amos output when bootstrapped estimates are requested?	458
Amos keeps telling me I am specifying a data file which is not my working file, yet the correct data file IS in the SPSS worksheet.	460
What is a matrix in Amos?	460
What text macros are available to display fit measures in Amos?	461
How do you get the actual case scores for latent variables in Amos?	463
SAS questions	463
How was multigroup analysis done in SAS prior to SAS Version 9.22?	463
Stata questions	466
Failure to converge in Stata	466
How may the intmethod() option help with gsem failure to converge?	468
Acknowledgments	469
Bibliography	469

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