Structural Equation Modelling A Bayesian Approach
1 Introduction 1
1.1 Standard Structural Equation Models 1
1.2 Covariance Structure Analysis 2
1.3 Why a New Book? 3
1.4 Objectives of the Book 4
1.5 Data Sets and Notations 6
Appendix 1.1 7
References 10
2 Some Basic Structural Equation Models 13
2.1 Introduction 13
2.2 Exploratory Factor Analysis 15
2.3 Confirmatory and Higher-order Factor Analysis Models 18
2.4 The LISREL Model 22
2.5 The Bentler–Weeks Model 26
2.6 Discussion 27
References 28
3 Covariance Structure Analysis 31
3.1 Introduction 31
3.2 Definitions, Notations and Preliminary Results 33
3.3 GLS Analysis of Covariance Structure 36
3.4 ML Analysis of Covariance Structure 41
3.5 Asymptotically Distribution-free Methods 44
3.6 Some Iterative Procedures 47
Appendix 3.1: Matrix Calculus 53
Appendix 3.2: Some Basic Results in Probability Theory 57
Appendix 3.3: Proofs of Some Results 59
References 65
4 Bayesian Estimation of Structural Equation Models 67
4.1 Introduction 67
4.2 Basic Principles and Concepts of Bayesian Analysis of SEMs 70
4.3 Bayesian Estimation of the CFA Model 81
4.4 Bayesian Estimation of Standard SEMs 95
4.5 Bayesian Estimation via WinBUGS 98
Appendix 4.1: The Metropolis–Hastings Algorithm 104
Appendix 4.2: EPSR Value 105
Appendix 4.3: Derivations of Conditional Distributions 106
References 108
5 Model Comparison and Model Checking 111
5.1 Introduction 111
5.2 Bayes Factor 113
5.3 Path Sampling 115
5.4 An Application: Bayesian Analysis of SEMs with Fixed Covariates 120
5.5 Other Methods 127
5.6 Discussion 130
Appendix 5.1: Another Proof of Equation (5.10) 131
Appendix 5.2: Conditional Distributions for Simulating Y t 133
Appendix 5.3: PP p-values for Model Assessment 136
References 136
6 Structural Equation Models with Continuous and Ordered Categorical Variables 139
6.1 Introduction 139
6.2 The Basic Model 142
6.3 Bayesian Estimation and Goodness-of-fit 144
6.4 Bayesian Model Comparison 155
6.5 Application 1: Bayesian Selection of the Number of Factors in EFA 159
6.6 Application 2: Bayesian Analysis of Quality of Life Data 164
References 172
7 Structural Equation Models with Dichotomous Variables 175
7.1 Introduction 175
7.2 Bayesian Analysis 177
7.3 Analysis of a Multivariate Probit Confirmatory Factor Analysis Model 186
7.4 Discussion 190
Appendix 7.1: Questions Associated with the Manifest Variables 191
References 192
8 Nonlinear Structural Equation Models 195
8.1 Introduction 195
8.2 Bayesian Analysis of a Nonlinear SEM 197
8.3 Bayesian Estimation of Nonlinear SEMs with Mixed
Continuous and Ordered Categorical Variables 215
8.4 Bayesian Estimation of SEMs with Nonlinear Covariates
and Latent Variables 220
8.5 Bayesian Model Comparison 230
References 239
9 Two-level Nonlinear Structural Equation Models 243
9.1 Introduction 243
9.2 A Two-level Nonlinear SEM with Mixed Type Variables 244
9.3 Bayesian Estimation 247
9.4 Goodness-of-fit and Model Comparison 255
9.5 An Application: Filipina CSWs Study 259
9.6 Two-level Nonlinear SEMs with Cross-level Effects 267
9.7 Analysis of Two-level Nonlinear SEMs using WinBUGS 275
Appendix 9.1: Conditional Distributions: Two-level Nonlinear
SEM 279
Appendix 9.2: MH Algorithm: Two-level Nonlinear SEM 283
Appendix 9.3: PP p-value for Two-level NSEM with Mixed
Continuous and Ordered-categorical Variables 285
Appendix 9.4: Questions Associated with the Manifest Variables 286
Appendix 9.5: Conditional Distributions: SEMs with Cross-level
Effects 286
Appendix 9.6: The MH algorithm: SEMs with Cross-level Effects 289
References 290
10 Multisample Analysis of Structural Equation Models 293
10.1 Introduction 293
10.2 The Multisample Nonlinear Structural Equation Model 294
10.3 Bayesian Analysis of Multisample Nonlinear SEMs 297
10.4 Numerical Illustrations 302
Appendix 10.1: Conditional Distributions: Multisample SEMs 313
References 316
11 Finite Mixtures in Structural Equation Models 319
11.1 Introduction 319
11.2 Finite Mixtures in SEMs 321
11.3 Bayesian Estimation and Classification 323
11.4 Examples and Simulation Study 330
11.5 Bayesian Model Comparison of Mixture SEMs 344
Appendix 11.1: The Permutation Sampler 351
Appendix 11.2: Searching for Identifiability Constraints 352
References 352
12 Structural Equation Models with Missing Data 355
12.1 Introduction 355
12.2 A General Framework for SEMs with Missing Data that are
MAR 357
12.3 Nonlinear SEM with Missing Continuous and Ordered
Categorical Data 359
12.4 Mixture of SEMs with Missing Data 370
12.5 Nonlinear SEMs with Nonignorable Missing Data 375
12.6 Analysis of SEMs with Missing Data via WinBUGS 386
Appendix 12.1: Implementation of the MH Algorithm 389
References 390
13 Structural Equation Models with Exponential Family of
Distributions 393
13.1 Introduction 393
13.2 The SEM Framework with Exponential Family of
Distributions 394
13.3 A Bayesian Approach 398
13.4 A Simulation Study 402
13.5 A Real Example: A Compliance Study of Patients 404
13.6 Bayesian Analysis of an Artificial Example using WinBUGS 411
13.7 Discussion 416
Appendix 13.1: Implementation of the MH Algorithms 417
Appendix 13.2 419
References 419
14 Conclusion 421
References 425
Index 427 |
