About the authors xiii
Preface xv
1 Introduction 1
1.1 Observed and latent variables 1
1.2 Structural equation model 3
1.3 Objectives of the book 3
1.4 The Bayesian approach 4
1.5 Real data sets and notation 5
Appendix 1.1: Information on real data sets 7
References 14
2 Basic concepts and applications of structural equation models 16
2.1 Introduction 16
2.2 Linear SEMs 17
2.2.1 Measurement equation 18
2.2.2 Structural equation and one extension 19
2.2.3 Assumptions of linear SEMs 20
2.2.4 Model identification 21
2.2.5 Path diagram 22
2.3 SEMs with fixed covariates 23
2.3.1 The model 23
2.3.2 An artificial example 24
2.4 Nonlinear SEMs 25
2.4.1 Basic nonlinear SEMs 25
2.4.2 Nonlinear SEMs with fixed covariates 27
2.4.3 Remarks 29
2.5 Discussion and conclusions 29
References 33
3 Bayesian methods for estimating structural equation models 34
3.1 Introduction 34
3.2 Basic concepts of the Bayesian estimation and prior distributions 35
3.2.1 Prior distributions 36
3.2.2 Conjugate prior distributions in Bayesian analyses of SEMs 37
3.3 Posterior analysis using Markov chain Monte Carlo methods 40
3.4 Application of Markov chain Monte Carlo methods 43
3.5 Bayesian estimation via WinBUGS 45
Appendix 3.1: The gamma, inverted gamma, Wishart, and inverted Wishart distributions and their characteristics 53
Appendix 3.2: The Metropolis-Hastings algorithm 54
Appendix 3.3: Conditional distributions [|Y, ?] and [?|Y,] 55
Appendix 3.4: Conditional distributions [|Y, ?] and [?|Y,] in nonlinear SEMs with covariates 58
Appendix 3.5: WinBUGS code 60
Appendix 3.6: R2WinBUGS code 61
References 62
4 Bayesian model comparison and model checking 64
4.1 Introduction 64
4.2 Bayes factor 65
4.2.1 Path sampling 67
4.2.2 A simulation study 70
4.3 Other model comparison statistics 73
4.3.1 Bayesian information criterion and Akaike information criterion 73
4.3.2 Deviance information criterion 74
4.3.3 L?-measure 75
4.4 Illustration 76
4.5 Goodness of fit and model checking methods 78
4.5.1 Posterior predictive p-value 78
4.5.2 Residual analysis 78
Appendix 4.1: WinBUGS code 80
Appendix 4.2: R code in Bayes factor example 81
Appendix 4.3: Posterior predictive p-value for model assessment 83
References 83
5 Practical structural equation models 86
5.1 Introduction 86
5.2 SEMs with continuous and ordered categorical variables 86
5.2.1 Introduction 86
5.2.2 The basic model 88
5.2.3 Bayesian analysis 90
5.2.4 Application: Bayesian analysis of quality of life data 90
5.2.5 SEMs with dichotomous variables 94
5.3 SEMs with variables from exponential family distributions 95
5.3.1 Introduction 95
5.3.2 The SEM framework with exponential family distributions 96
5.3.3 Bayesian inference 97
5.3.4 Simulation study 98
5.4 SEMs with missing data 102
5.4.1 Introduction 102
5.4.2 SEMs with missing data that are MAR 103
5.4.3 An illustrative example 105
5.4.4 Nonlinear SEMs with nonignorable missing data 108
5.4.5 An illustrative real example 111
Appendix 5.1: Conditional distributions and implementation of the MH algorithm for SEMs with continuous and ordered categorical variables 115
Appendix 5.2: Conditional distributions and implementation of MH algorithm for SEMs with EFDs 119
Appendix 5.3: WinBUGS code related to section 5.3.4 122
Appendix 5.4: R2WinBUGS code related to section 5.3.4 123
Appendix 5.5: Conditional distributions for SEMs with nonignorable missing data 126
References 127
6 Structural equation models with hierarchical and multisample data 130
6.1 Introduction 130
6.2 Two-level structural equation models 131
6.2.1 Two-level nonlinear SEM with mixed type variables 131
6.2.2 Bayesian inference 133
6.2.3 Application: Filipina CSWs study 136
6.3 Structural equation models with multisample data 141
6.3.1 Bayesian analysis of a nonlinear SEM in different groups 143
6.3.2 Analysis of multisample quality of life data via WinBUGS 147
Appendix 6.1: Conditional distributions: Two-level nonlinear SEM 150
Appendix 6.2: The MH algorithm: Two-level nonlinear SEM 153
Appendix 6.3: PP p-value for two-level nonlinear SEM with mixed continuous and ordered categorical variables 155
Appendix 6.4: WinBUGS code 156
Appendix 6.5: Conditional distributions: Multisample SEMs 158
References 160
7 Mixture structural equation models 162
7.1 Introduction 162
7.2 Finite mixture SEMs 163
7.2.1 The model 163
7.2.2 Bayesian estimation 164
7.2.3 Analysis of an artificial example 168
7.2.4 Example from the world values survey 170
7.2.5 Bayesian model comparison of mixture SEMs 173
7.2.6 An illustrative example 176
7.3 A Modified mixture SEM 178
7.3.1 Model description 178
7.3.2 Bayesian estimation 180
7.3.3 Bayesian model selection using a modified DIC 182
7.3.4 An illustrative example 183
Appendix 7.1: The permutation sampler 189
Appendix 7.2: Searching for identifiability constraints 190
Appendix 7.3: Conditional distributions: Modified mixture SEMs 191
References 194
8 Structural equation modeling for latent curve models 196
8.1 Introduction 196
8.2 Background to the real studies 197
8.2.1 A longitudinal study of quality of life of stroke survivors 197
8.2.2 A longitudinal study of cocaine use 198
8.3 Latent curve models 199
8.3.1 Basic latent curve models 199
8.3.2 Latent curve models with explanatory latent variables 200
8.3.3 Latent curve models with longitudinal latent variables 201
8.4 Bayesian analysis 205
8.5 Applications to two longitudinal studies 206
8.5.1 Longitudinal study of cocaine use 206
8.5.2 Health-related quality of life for stroke survivors 210
8.6 Other latent curve models 213
8.6.1 Nonlinear latent curve models 214
8.6.2 Multilevel latent curve models 215
8.6.3 Mixture latent curve models 215
Appendix 8.1: Conditional distributions 218
Appendix 8.2: WinBUGS code for the analysis of cocaine use data 220
References 222
9 Longitudinal structural equation models 224
9.1 Introduction 224
9.2 A two-level SEM for analyzing multivariate longitudinal data 226
9.3 Bayesian analysis of the two-level longitudinal SEM 228
9.3.1 Bayesian estimation 228
9.3.2 Model comparison via the L?-measure 230
9.4 Simulation study 231
9.5 Application: Longitudinal study of cocaine use 232
9.6 Discussion 236
Appendix 9.1: Full conditional distributions for implementing the Gibbs sampler 241
Appendix 9.2: Approximation of the L?-measure in equation (9.9) via MCMC samples 244
References 245
10 Semiparametric structural equation models with continuous variables 247
10.1 Introduction 247
10.2 Bayesian semiparametric hierarchical modeling of SEMs with covariates 249
10.3 Bayesian estimation and model comparison 251
10.4 Application: Kidney disease study 252
10.5 Simulation studies 259
10.5.1 Simulation study of estimation 259
10.5.2 Simulation study of model comparison 262
10.5.3 Obtaining the L?-measure via WinBUGS and R2WinBUGS 264
10.6 Discussion 265
Appendix 10.1: Conditional distributions for parametric components 267
Appendix 10.2: Conditional distributions for nonparametric components 268
References 269
11 Structural equation models with mixed continuous and unordered categorical variables 271
11.1 Introduction 271
11.2 Parametric SEMs with continuous and unordered categorical variables 272
11.2.1 The model 272
11.2.2 Application to diabetic kidney disease 274
11.2.3 Bayesian estimation and model comparison 276
11.2.4 Application to the diabetic kidney disease data 277
11.3 Bayesian semiparametric SEM with continuous and unordered categorical variables 280
11.3.1 Formulation of the semiparametric SEM 282
11.3.2 Semiparametric hierarchical modeling via the Dirichlet process 283
11.3.3 Estimation and model comparison 285
11.3.4 Simulation study 286
11.3.5 Real example: Diabetic nephropathy study 289
Appendix 11.1: Full conditional distributions 295
Appendix 11.2: Path sampling 298
Appendix 11.3: A modified truncated DP related to equation (11.19) 299
Appendix 11.4: Conditional distributions and the MH algorithm for the Bayesian semiparametric model 300
References 304
12 Structural equation models with nonparametric structural equations 306
12.1 Introduction 306
12.2 Nonparametric SEMs with Bayesian P-splines 307
12.2.1 Model description 307
12.2.2 General formulation of the Bayesian P-splines 308
12.2.3 Modeling nonparametric functions of latent variables 309
12.2.4 Prior distributions 310
12.2.5 Posterior inference via Markov chain Monte Carlo sampling 312
12.2.6 Simulation study 313
12.2.7 A study on osteoporosis prevention and control 316
12.3 Generalized nonparametric structural equation models 320
12.3.1 Model description 320
12.3.2 Bayesian P-splines 322
12.3.3 Prior distributions 324
12.3.4 Bayesian estimation and model comparison 325
12.3.5 National longitudinal surveys of youth study 327
12.4 Discussion 331
Appendix 12.1: Conditional distributions and the MH algorithm: Nonparametric SEMs 333
Appendix 12.2: Conditional distributions in generalized nonparametric SEMs 336
References 338
13 Transformation structural equation models 341
13.1 Introduction 341
13.2 Model description 342
13.3 Modeling nonparametric transformations 343
13.4 Identifiability constraints and prior distributions 344
13.5 Posterior inference with MCMC algorithms 345
13.5.1 Conditional distributions 345
13.5.2 The random-ray algorithm 346
13.5.3 Modifications of the random-ray algorithm 347
13.6 Simulation study 348
13.7 A study on the intervention treatment of polydrug use 350
13.8 Discussion 354
References 355
14 Conclusion 358
References 360
Index 361
