Structural Equation Modeling

Jichuan Wang, PhD, is Professor in the Department of Pediatrics, Epidemiology, and Biostatistics at the George Washington University (GWU) School of Medicine. He also serves as Senior Biostatistician in the National Children's Medical Center (CNMC) in Washington, DC. Xiaoqian Wang, PhD, is a Principle Consultant at Mobley Group Pacific Ltd., P.R. China.

Preface ix

1 Introduction to structural equation modeling 1

1.1 Introduction 1

1.2 Model formulation 3

1.2.1 Measurement models 4

1.2.2 Structural models 6

1.2.3 Model formulation in equations 7

1.3 Model identification 11

1.4 Model estimation 14

1.4.1 Bayes estimator 17

1.5 Model fit evaluation 19

1.5.1 The model 𝜒2 statistic 20

1.5.2 Comparative fit index (CFI) 20

1.5.3 Tucker Lewis index (TLI) or non-normed fit index (NNFI) 21

1.5.4 Root mean square error of approximation (RMSEA) 22

1.5.5 Root mean-square residual (RMR), standardized RMR (SRMR), and weighted RMR (WRMR) 22

1.5.6 Information criteria indices 24

1.5.7 Model fit evaluation with Bayes estimator 25

1.5.8 Model comparison 26

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

2.2.1 Latent variables/factors 39

2.2.2 Indicator variables 39

2.2.3 Item parceling 40

2.2.4 Factor loadings 42

2.2.5 Measurement errors 42

2.2.6 Item reliability 44

2.2.7 Scale reliability 44

2.3 CFA models with continuous indicators 45

2.3.1 Alternative methods for factor scaling 52

2.3.2 Model estimated item reliability 57

2.3.3 Model modification based on modification indices 57

2.3.4 Model estimated scale reliability 58

2.3.5 Item parceling 60

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

2.4.1 Testing non-normality 61

2.4.2 CFA models with non-normal indicators 62

2.4.3 CFA models with censored data 67

2.5 CFA models with categorical indicators 70

2.5.1 CFA models with binary indicators 72

2.5.2 CFA models with ordinal categorical indicators 76

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

2.6.1 The item response theory (IRT) model 77

2.6.2 The graded response model (GRM) 86

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

3.2.1 Interaction effects between covariates 126

3.2.2 Differential item functioning (DIF) 127

3.3 General structural equation models 137

3.3.1 Testing indirect effects 141

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

3.6.1 Bootstrap confidence intervals 159

3.6.2 Estimating counterfactual-based causal effects in Mplus 160

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

4.2.1 Unconditional linear LGM 178

4.2.2 LGM with time-invariant covariates 184

4.2.3 LGM with time-invariant and time-varying covariates 189

4.3 Nonlinear LGM 192

4.3.1 LGM with polynomial time functions 192

4.3.2 Piecewise LGM 203

4.3.3 Free time scores 210

4.3.4 LGM with distal outcomes 211

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

4.8.1 DSEM using observed centering for covariates 241

4.8.2 Residual DSEM (RDSEM) using observed centering for covariates 245

4.8.3 Residual DSEM (RDSEM) using latent variable centering for covariates 248

5 Multigroup modeling 253

5.1 Introduction 253

5.2 Multigroup CFA models 254

5.2.1 Multigroup first-order CFA 258

5.2.2 Multigroup second-order CFA 289

5.2.3 Multigroup CFA with categorical indicators 306

5.3 Multigroup SEM 316

5.3.1 Testing invariance of structural path coefficients across groups 322

5.3.2 Testing invariance of indirect effects across groups 326

5.4 Multigroup latent growth modeling (LGM) 327

5.4.1 Testing invariance of the growth function 332

5.4.2 Testing invariance of latent growth factor means 335

6 Mixture modeling 339

6.1 Introduction 339

6.2 Latent class analysis (LCA) modeling 340

6.2.1 Description of LCA models 341

6.2.2 Defining the latent classes 347

6.2.3 Predicting class membership 347

6.2.4 Unconditional LCA 348

6.2.5 Directly including covariates into LCA models 360

6.2.6 Approaches for auxiliary variables in LCA models 363

6.2.7 Implementing the PC, three-step, Lanza’s, and BCH methods 365

6.2.8 LCA with residual covariances 370

6.3 Extending LCA to longitudinal data analysis 373

6.3.1 Longitudinal latent class analysis (LLCA) 373

6.3.2 Latent transition analysis (LTA) models 375

6.4 Growth mixture modeling (GMM) 392

6.4.1 Unconditional growth mixture modeling (GMM) 394

6.4.2 GMM with covariates and a distal outcome 402

6.5 Factor mixture modeling (FMM) 411

6.5.1 LCFA models 417

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

7.3.1 Application of The Satorra-Saris method to CFA models 446

7.3.2 Application of the Satorra-Saris’s method to latent growth models 454

7.4 Monte Carlo simulation for estimating sample sizes 458

7.4.1 Application of a Monte Carlo simulation to CFA models 459

7.4.2 Application of a Monte Carlo simulation to latent growth models 463

7.4.3 Application of a Monte Carlo simulation to latent growth models with covariates 467

7.4.4 Application of a Monte Carlo simulation to latent growth models with missing values 469

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

7.5.1 Application of the MacCallum–Browne–Sugawara’s method 474

7.5.2 Application of Kim’s method 477

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

References 483

Index 507