Basic and Advanced Bayesian Structural Equation Modeling -- Contents -- About the authors -- Preface -- 1 Introduction -- 1.1 Observed and latent variables -- 1.2 Structural equation model -- 1.3 Objectives of the book -- 1.4 The Bayesian approach -- 1.5 Real data sets and notation -- Appendix 1.1: Information on real data sets -- References -- 2 Basic concepts and applications of structural equation models -- 2.1 Introduction -- 2.2 Linear SEMs -- 2.2.1 Measurement equation -- 2.2.2 Structural equation and one extension -- 2.2.3 Assumptions of linear SEMs -- 2.2.4 Model identification -- 2.2.5 Path diagram -- 2.3 SEMs with fixed covariates -- 2.3.1 The model -- 2.3.2 An artificial example -- 2.4 Nonlinear SEMs -- 2.4.1 Basic nonlinear SEMs -- 2.4.2 Nonlinear SEMs with fixed covariates -- 2.4.3 Remarks -- 2.5 Discussion and conclusions -- References -- 3 Bayesian methods for estimating structural equation models -- 3.1 Introduction -- 3.2 Basic concepts of the Bayesian estimation and prior distributions -- 3.2.1 Prior distributions -- 3.2.2 Conjugate prior distributions in Bayesian analyses of SEMs -- 3.3 Posterior analysis using Markov chain Monte Carlo methods -- 3.4 Application of Markov chain Monte Carlo methods -- 3.5 Bayesian estimation via WinBUGS -- Appendix 3.1: The gamma, inverted gamma, Wishart, and inverted Wishart distributions and their characteristics -- Appendix 3.2: The Metropolis-Hastings algorithm -- Appendix 3.3: Conditional distributions [Ω

4.3.1 Bayesian information criterion and Akaike information criterion -- 4.3.2 Deviance information criterion -- 4.3.3 Lν-measure -- 4.4 Illustration -- 4.5 Goodness of fit and model checking methods -- 4.5.1 Posterior predictive p-value -- 4.5.2 Residual analysis -- Appendix 4.1: WinBUGS code -- Appendix 4.2: R code in Bayes factor example -- Appendix 4.3: Posterior predictive p-value for model assessment -- References -- 5 Practical structural equation models -- 5.1 Introduction -- 5.2 SEMs with continuous and ordered categorical variables -- 5.2.1 Introduction -- 5.2.2 The basic model -- 5.2.3 Bayesian analysis -- 5.2.4 Application: Bayesian analysis of quality of life data -- 5.2.5 SEMs with dichotomous variables -- 5.3 SEMs with variables from exponential family distributions -- 5.3.1 Introduction -- 5.3.2 The SEM framework with exponential family distributions -- 5.3.3 Bayesian inference -- 5.3.4 Simulation study -- 5.4 SEMs with missing data -- 5.4.1 Introduction -- 5.4.2 SEMs with missing data that are MAR -- 5.4.3 An illustrative example -- 5.4.4 Nonlinear SEMs with nonignorable missing data -- 5.4.5 An illustrative real example -- Appendix 5.1: Conditional distributions and implementation of the MH algorithm for SEMs with continuous and ordered categorical variables -- Appendix 5.2: Conditional distributions and implementation of MH algorithm for SEMs with EFDs -- Appendix 5.3: WinBUGS code related to section 5.3.4 -- Appendix 5.4: R2WinBUGS code related to section 5.3.4 -- Appendix 5.5: Conditional distributions for SEMs with nonignorable missing data -- References -- 6 Structural equation models with hierarchical and multisample data -- 6.1 Introduction -- 6.2 Two-level structural equation models -- 6.2.1 Two-level nonlinear SEM with mixed type variables -- 6.2.2 Bayesian inference -- 6.2.3 Application: Filipina CSWs study.

6.3 Structural equation models with multisample data -- 6.3.1 Bayesian analysis of a nonlinear SEM in different groups -- 6.3.2 Analysis of multisample quality of life data via WinBUGS -- Appendix 6.1: Conditional distributions: Two-level nonlinear SEM -- Appendix 6.2: The MH algorithm: Two-level nonlinear SEM -- Appendix 6.3: PP p-value for two-level nonlinear SEM with mixed continuous and ordered categorical variables -- Appendix 6.4: WinBUGS code -- Appendix 6.5: Conditional distributions: Multisample SEMs -- References -- 7 Mixture structural equation models -- 7.1 Introduction -- 7.2 Finite mixture SEMs -- 7.2.1 The model -- 7.2.2 Bayesian estimation -- 7.2.3 Analysis of an artificial example -- 7.2.4 Example from the world values survey -- 7.2.5 Bayesian model comparison of mixture SEMs -- 7.2.6 An illustrative example -- 7.3 A Modified mixture SEM -- 7.3.1 Model description -- 7.3.2 Bayesian estimation -- 7.3.3 Bayesian model selection using a modified DIC -- 7.3.4 An illustrative example -- Appendix 7.1: The permutation sampler -- Appendix 7.2: Searching for identifiability constraints -- Appendix 7.3: Conditional distributions: Modified mixture SEMs -- References -- 8 Structural equation modeling for latent curve models -- 8.1 Introduction -- 8.2 Background to the real studies -- 8.2.1 A longitudinal study of quality of life of stroke survivors -- 8.2.2 A longitudinal study of cocaine use -- 8.3 Latent curve models -- 8.3.1 Basic latent curve models -- 8.3.2 Latent curve models with explanatory latent variables -- 8.3.3 Latent curve models with longitudinal latent variables -- 8.4 Bayesian analysis -- 8.5 Applications to two longitudinal studies -- 8.5.1 Longitudinal study of cocaine use -- 8.5.2 Health-related quality of life for stroke survivors -- 8.6 Other latent curve models -- 8.6.1 Nonlinear latent curve models.

8.6.2 Multilevel latent curve models -- 8.6.3 Mixture latent curve models -- Appendix 8.1: Conditional distributions -- Appendix 8.2: WinBUGS code for the analysis of cocaine use data -- References -- 9 Longitudinal structural equation models -- 9.1 Introduction -- 9.2 A two-level SEM for analyzing multivariate longitudinal data -- 9.3 Bayesian analysis of the two-level longitudinal SEM -- 9.3.1 Bayesian estimation -- 9.3.2 Model comparison via the Lν-measure -- 9.4 Simulation study -- 9.5 Application: Longitudinal study of cocaine use -- 9.6 Discussion -- Appendix 9.1: Full conditional distributions for implementing the Gibbs sampler -- Appendix 9.2: Approximation of the Lν-measure in equation (9.9) via MCMC samples -- References -- 10 Semiparametric structural equation models with continuous variables -- 10.1 Introduction -- 10.2 Bayesian semiparametric hierarchical modeling of SEMs with covariates -- 10.3 Bayesian estimation and model comparison -- 10.4 Application: Kidney disease study -- 10.5 Simulation studies -- 10.5.1 Simulation study of estimation -- 10.5.2 Simulation study of model comparison -- 10.5.3 Obtaining the Lν-measure via WinBUGS and R2WinBUGS -- 10.6 Discussion -- Appendix 10.1: Conditional distributions for parametric components -- Appendix 10.2: Conditional distributions for nonparametric components -- References -- 11 Structural equation models with mixed continuous and unordered categorical variables -- 11.1 Introduction -- 11.2 Parametric SEMs with continuous and unordered categorical variables -- 11.2.1 The model -- 11.2.2 Application to diabetic kidney disease -- 11.2.3 Bayesian estimation and model comparison -- 11.2.4 Application to the diabetic kidney disease data -- 11.3 Bayesian semiparametric SEM with continuous and unordered categorical variables -- 11.3.1 Formulation of the semiparametric SEM.

11.3.2 Semiparametric hierarchical modeling via the Dirichlet process -- 11.3.3 Estimation and model comparison -- 11.3.4 Simulation study -- 11.3.5 Real example: Diabetic nephropathy study -- Appendix 11.1: Full conditional distributions -- Appendix 11.2: Path sampling -- Appendix 11.3: A modified truncated DP related to equation (11.19) -- Appendix 11.4: Conditional distributions and the MH algorithm for the Bayesian semiparametric model -- References -- 12 Structural equation models with nonparametric structural equations -- 12.1 Introduction -- 12.2 Nonparametric SEMs with Bayesian P-splines -- 12.2.1 Model description -- 12.2.2 General formulation of the Bayesian P-splines -- 12.2.3 Modeling nonparametric functions of latent variables -- 12.2.4 Prior distributions -- 12.2.5 Posterior inference via Markov chain Monte Carlo sampling -- 12.2.6 Simulation study -- 12.2.7 A study on osteoporosis prevention and control -- 12.3 Generalized nonparametric structural equation models -- 12.3.1 Model description -- 12.3.2 Bayesian P-splines -- 12.3.3 Prior distributions -- 12.3.4 Bayesian estimation and model comparison -- 12.3.5 National longitudinal surveys of youth study -- 12.4 Discussion -- Appendix 12.1: Conditional distributions and the MH algorithm: Nonparametric SEMs -- Appendix 12.2: Conditional distributions in generalized nonparametric SEMs -- References -- 13 Transformation structural equation models -- 13.1 Introduction -- 13.2 Model description -- 13.3 Modeling nonparametric transformations -- 13.4 Identifiability constraints and prior distributions -- 13.5 Posterior inference with MCMC algorithms -- 13.5.1 Conditional distributions -- 13.5.2 The random-ray algorithm -- 13.5.3 Modifications of the random-ray algorithm -- 13.6 Simulation study -- 13.7 A study on the intervention treatment of polydrug use -- 13.8 Discussion -- References.

14 Conclusion.