Structural Equation Modeling

Preface ix

1 Introduction to structural equation modeling 1

1.1 Introduction 1

1.2 Model formulation 3

1.3 Model identification 11

1.4 Model estimation 14

1.5 Model fit evaluation 19

1.6 Model modification 27

1.7 Computer programs for SEM 28

Appendix 1.A Expressing variances and covariances among observed variables as functions of model parameters 30

Appendix 1.B Maximum likelihood function for SEM 32

2 Confirmatory factor analysis 33

2.1 Introduction 33

2.2 Basics of CFA models 34

2.3 CFA models with continuous indicators 45

2.4 CFA models with non-normal and censored continuous indicators 61

2.5 CFA models with categorical indicators 70

2.6 The item response theory (IRT) model and the graded response model (GRM) 77

2.7 Higher-order CFA models 91

2.8 Bifactor models 96

2.9 Bayesian CFA models 102

2.10 Plausible values of latent variables 110

Appendix 2.A BSI-18 instrument 113

Appendix 2.B Item reliability 114

Appendix 2.C Cronbach's alpha coefficient 116

Appendix 2.D Calculating probabilities using probit regression coefficients 117

3 Structural equation models 119

3.1 Introduction 119

3.2 Multiple indicators, multiple causes (MIMIC) model 120

3.3 General structural equation models 137

3.4 Correcting for measurement error in single indicator variables 144

3.5 Testing interactions involving latent variables 150

3.6 Moderated mediating effect models 153

3.7 Using plausible values of latent variables in secondary analysis 164

3.8 Bayesian structural equation modeling (BSEM) 167

Appendix 3.A Influence of measurement errors 173

Appendix 3.B Fraction of missing information (FMI) 175

4 Latent growth modeling (LGM) for longitudinal data analysis 177

4.1 Introduction 177

4.2 Linear LGM 178

4.3 Nonlinear LGM 192

4.4 Multiprocess LGM 216

4.5 Two-part LGM 221

4.6 LGM with categorical outcomes 229

4.7 LGM with individually varying times of observation 238

4.8 Dynamic structural equation modeling (DSEM) 241

5 Multigroup modeling 253

5.1 Introduction 253

5.2 Multigroup CFA models 254

5.3 Multigroup SEM 316

5.4 Multigroup latent growth modeling (LGM) 327

6 Mixture modeling 339

6.1 Introduction 339

6.2 Latent class analysis (LCA) modeling 340

6.3 Extending LCA to longitudinal data analysis 373

6.4 Growth mixture modeling (GMM) 392

6.5 Factor mixture modeling (FMM) 411

Appendix 6.A Including covariates in LTA model 418

Appendix 6.B Manually implementing three-step mixture modeling 434

7 Sample size for structural equation modeling 443

7.1 Introduction 443

7.2 The rules of thumb for sample size in SEM 444

7.3 The Satorra-Saris method for estimating sample size 445

7.4 Monte Carlo simulation for estimating sample sizes 458

7.5 Estimate sample size for SEM based on model fit indexes 473

7.6 Estimate sample sizes for latent class analysis (LCA) model 479

References 483

Index 507