
STRUCTURAL EQUATION MODELING
Overview
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 graduatelevel study.
The 2015 edition is a major update to the 2012 edition. Among the new features are these:
The full content is now available from Statistical Associates Publishers. Click here.
Below is the unformatted table of contents.
STRUCTURAL EQUATION MODELING 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 in SAS PROC CALIS 100 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 Multigroup Analysis 160 Overview 160 Introduction 160 To standardize or not to standardize? 161 Model invariance 163 Examining noninvariance across groups 165 Critical ratios of differences test 166 Example and data for multigroup analysis 166 Multigroup analysis in SPSS Amos 168 Baseline multigroup testing 168 Data setup for multigroup analysis in Amos 170 Viewing group models 172 Testing for measurement invariance across groups (multigroup 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 Multigroup analysis in SAS 189 Overview 189 Example 189 Data setup for multigroup analysis in SAS 190 Baseline multigroup testing 190 The unconstrained multigroup model 192 The measurement weights model 194 Other models 197 Multigroup analysis in Stata 198 Overview 198 Baseline multigroup testing 198 Multigroup 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 timeinvariant 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 timeinvariant 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 timeinvariant 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 Crossvalidation 261 Residual analysis 262 Bayesian SEM in SPSS AMOS 262 Data levels 262 Nominallevel data in Amos with Bayesian estimation 262 Ordinallevel 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 Twostage least squares estimation (2SLS) 346 Asymptotically distributionfree 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 Goodnessoffit measures and tests based on predicted vs. observed covariances 348 Overview 348 Chisquare 348 Hoelter's critical N 352 Relative chisquare 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 Goodnessoffit index (GFI) 363 Adjusted goodnessoffit 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 TuckerLewis index (TLI) or nonnormed fit index (NNFI) 371 Normed fit index (NFI) 372 Relative fit index (RFI) 373 Incremental fit index (IFI) 374 BentlerBonett index (BBI) 375 Bollen86 Fit index (B86 or BFI) 375 Goodnessoffit 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 Noncentralitybased 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 overdispersed data (QAIC) 384 Bayesian Information Criterion (BIC) 385 Samplesize adjusted BIC (SABIC) 387 Hannan & Quinn information criterion (HQIC) 387 BrowneCudeck criterion (BCC) 387 Expected crossvalidation index (ECVI) 388 Modified expected crossvalidation index (MECVI) 388 Crossvalidation 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 Oneindictor 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 underidentified 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 Nonzero 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 fourstep 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 modelimplied 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 writeups 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 nonexperimental 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 Pagecount: 487