Contents

About the Author xi

Preface xiii

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 3.3: 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
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

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

11.3 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