Basic and Advanced Bayesian Structural Equation Modeling

Xin-Yuan Song and Sik-Yum Lee, Department of Statistics, The Chinese University of Hong Kong

“These programming files and data files are very useful for understanding and applying the presented methodology.”  (Psychometrika, 1 March 2015)

• About the authors xiiiPreface xv1 Introduction 11.1 Observed and latent variables 11.2 Structural equation model 31.3 Objectives of the book 31.4 The Bayesian approach 41.5 Real data sets and notation 5Appendix 1.1: Information on real data sets 7References 142 Basic concepts and applications of structural equation models 162.1 Introduction 162.2 Linear SEMs 172.2.1 Measurement equation 182.2.2 Structural equation and one extension 192.2.3 Assumptions of linear SEMs 202.2.4 Model identification 212.2.5 Path diagram 222.3 SEMs with fixed covariates 232.3.1 The model 232.3.2 An artificial example 242.4 Nonlinear SEMs 252.4.1 Basic nonlinear SEMs 252.4.2 Nonlinear SEMs with fixed covariates 272.4.3 Remarks 292.5 Discussion and conclusions 29References 333 Bayesian methods for estimating structural equation models 343.1 Introduction 343.2 Basic concepts of the Bayesian estimation and prior distributions 353.2.1 Prior distributions 363.2.2 Conjugate prior distributions in Bayesian analyses of SEMs 373.3 Posterior analysis using Markov chain Monte Carlo methods 403.4 Application of Markov chain Monte Carlo methods 433.5 Bayesian estimation via WinBUGS 45Appendix 3.1: The gamma, inverted gamma, Wishart, and inverted Wishart distributions and their characteristics 53Appendix 3.2: The Metropolis–Hastings algorithm 54Appendix 3.3: Conditional distributions [|Y, θ] and [θ|Y,] 55Appendix 3.4: Conditional distributions [|Y, θ] and [θ|Y,] in nonlinear SEMs with covariates 58Appendix 3.5: WinBUGS code 60Appendix 3.6: R2WinBUGS code 61References 624 Bayesian model comparison and model checking 644.1 Introduction 644.2 Bayes factor 654.2.1 Path sampling 674.2.2 A simulation study 704.3 Other model comparison statistics 734.3.1 Bayesian information criterion and Akaike information criterion 734.3.2 Deviance information criterion 744.3.3 Lν-measure 754.4 Illustration 764.5 Goodness of fit and model checking methods 784.5.1 Posterior predictive p-value 784.5.2 Residual analysis 78Appendix 4.1: WinBUGS code 80Appendix 4.2: R code in Bayes factor example 81Appendix 4.3: Posterior predictive p-value for model assessment 83References 835 Practical structural equation models 865.1 Introduction 865.2 SEMs with continuous and ordered categorical variables 865.2.1 Introduction 865.2.2 The basic model 885.2.3 Bayesian analysis 905.2.4 Application: Bayesian analysis of quality of life data 905.2.5 SEMs with dichotomous variables 945.3 SEMs with variables from exponential family distributions 955.3.1 Introduction 955.3.2 The SEM framework with exponential family distributions 965.3.3 Bayesian inference 975.3.4 Simulation study 985.4 SEMs with missing data 1025.4.1 Introduction 1025.4.2 SEMs with missing data that are MAR 1035.4.3 An illustrative example 1055.4.4 Nonlinear SEMs with nonignorable missing data 1085.4.5 An illustrative real example 111Appendix 5.1: Conditional distributions and implementation of the MH algorithm for SEMs with continuous and ordered categorical variables 115Appendix 5.2: Conditional distributions and implementation of MH algorithm for SEMs with EFDs 119Appendix 5.3: WinBUGS code related to section 5.3.4 122Appendix 5.4: R2WinBUGS code related to section 5.3.4 123Appendix 5.5: Conditional distributions for SEMs with nonignorable missing data 126References 1276 Structural equation models with hierarchical and multisample data 1306.1 Introduction 1306.2 Two-level structural equation models 1316.2.1 Two-level nonlinear SEM with mixed type variables 1316.2.2 Bayesian inference 1336.2.3 Application: Filipina CSWs study 1366.3 Structural equation models with multisample data 1416.3.1 Bayesian analysis of a nonlinear SEM in different groups 1436.3.2 Analysis of multisample quality of life data via WinBUGS 147Appendix 6.1: Conditional distributions: Two-level nonlinear SEM 150Appendix 6.2: The MH algorithm: Two-level nonlinear SEM 153Appendix 6.3: PP p-value for two-level nonlinear SEM with mixed continuous and ordered categorical variables 155Appendix 6.4: WinBUGS code 156Appendix 6.5: Conditional distributions: Multisample SEMs 158References 1607 Mixture structural equation models 1627.1 Introduction 1627.2 Finite mixture SEMs 1637.2.1 The model 1637.2.2 Bayesian estimation 1647.2.3 Analysis of an artificial example 1687.2.4 Example from the world values survey 1707.2.5 Bayesian model comparison of mixture SEMs 1737.2.6 An illustrative example 1767.3 A Modified mixture SEM 1787.3.1 Model description 1787.3.2 Bayesian estimation 1807.3.3 Bayesian model selection using a modified DIC 1827.3.4 An illustrative example 183Appendix 7.1: The permutation sampler 189Appendix 7.2: Searching for identifiability constraints 190Appendix 7.3: Conditional distributions: Modified mixture SEMs 191References 1948 Structural equation modeling for latent curve models 1968.1 Introduction 1968.2 Background to the real studies 1978.2.1 A longitudinal study of quality of life of stroke survivors 1978.2.2 A longitudinal study of cocaine use 1988.3 Latent curve models 1998.3.1 Basic latent curve models 1998.3.2 Latent curve models with explanatory latent variables 2008.3.3 Latent curve models with longitudinal latent variables 2018.4 Bayesian analysis 2058.5 Applications to two longitudinal studies 2068.5.1 Longitudinal study of cocaine use 2068.5.2 Health-related quality of life for stroke survivors 2108.6 Other latent curve models 2138.6.1 Nonlinear latent curve models 2148.6.2 Multilevel latent curve models 2158.6.3 Mixture latent curve models 215Appendix 8.1: Conditional distributions 218Appendix 8.2: WinBUGS code for the analysis of cocaine use data 220References 2229 Longitudinal structural equation models 2249.1 Introduction 2249.2 A two-level SEM for analyzing multivariate longitudinal data 2269.3 Bayesian analysis of the two-level longitudinal SEM 2289.3.1 Bayesian estimation 2289.3.2 Model comparison via the Lν-measure 2309.4 Simulation study 2319.5 Application: Longitudinal study of cocaine use 2329.6 Discussion 236Appendix 9.1: Full conditional distributions for implementing the Gibbs sampler 241Appendix 9.2: Approximation of the Lν-measure in equation (9.9) via MCMC samples 244References 24510 Semiparametric structural equation models with continuous variables 24710.1 Introduction 24710.2 Bayesian semiparametric hierarchical modeling of SEMs with covariates 24910.3 Bayesian estimation and model comparison 25110.4 Application: Kidney disease study 25210.5 Simulation studies 25910.5.1 Simulation study of estimation 25910.5.2 Simulation study of model comparison 26210.5.3 Obtaining the Lν-measure via WinBUGS and R2WinBUGS 26410.6 Discussion 265Appendix 10.1: Conditional distributions for parametric components 267Appendix 10.2: Conditional distributions for nonparametric components 268References 26911 Structural equation models with mixed continuous and unordered categorical variables 27111.1 Introduction 27111.2 Parametric SEMs with continuous and unordered categorical variables 27211.2.1 The model 27211.2.2 Application to diabetic kidney disease 27411.2.3 Bayesian estimation and model comparison 27611.2.4 Application to the diabetic kidney disease data 27711.3 Bayesian semiparametric SEM with continuous and unordered categorical variables 28011.3.1 Formulation of the semiparametric SEM 28211.3.2 Semiparametric hierarchical modeling via the Dirichlet process 28311.3.3 Estimation and model comparison 28511.3.4 Simulation study 28611.3.5 Real example: Diabetic nephropathy study 289Appendix 11.1: Full conditional distributions 295Appendix 11.2: Path sampling 298Appendix 11.3: A modified truncated DP related to equation (11.19) 299Appendix 11.4: Conditional distributions and the MH algorithm for the Bayesian semiparametric model 300References 30412 Structural equation models with nonparametric structural equations 30612.1 Introduction 30612.2 Nonparametric SEMs with Bayesian P-splines 30712.2.1 Model description 30712.2.2 General formulation of the Bayesian P-splines 30812.2.3 Modeling nonparametric functions of latent variables 30912.2.4 Prior distributions 31012.2.5 Posterior inference via Markov chain Monte Carlo sampling 31212.2.6 Simulation study 31312.2.7 A study on osteoporosis prevention and control 31612.3 Generalized nonparametric structural equation models 32012.3.1 Model description 32012.3.2 Bayesian P-splines 32212.3.3 Prior distributions 32412.3.4 Bayesian estimation and model comparison 32512.3.5 National longitudinal surveys of youth study 32712.4 Discussion 331Appendix 12.1: Conditional distributions and the MH algorithm: Nonparametric SEMs 333Appendix 12.2: Conditional distributions in generalized nonparametric SEMs 336References 33813 Transformation structural equation models 34113.1 Introduction 34113.2 Model description 34213.3 Modeling nonparametric transformations 34313.4 Identifiability constraints and prior distributions 34413.5 Posterior inference with MCMC algorithms 34513.5.1 Conditional distributions 34513.5.2 The random-ray algorithm 34613.5.3 Modifications of the random-ray algorithm 34713.6 Simulation study 34813.7 A study on the intervention treatment of polydrug use 35013.8 Discussion 354References 35514 Conclusion 358References 360Index 361