# Basic and Advanced Bayesian Structural Equation Modeling : With Applications in the Medical and Behavioral Sciences.

Series: Wiley Series in Probability and Statistics SerPublisher: New York : John Wiley & Sons, Incorporated, 2012Copyright date: ©2012Edition: 1st edDescription: 1 online resource (397 pages)Content type:- text

- computer

- online resource

- 9781118359433

- 519.53

- QA278.3 -- .L44 2012eb

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Intro -- 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|Y,θ] and [θ|Y,Ω] -- Appendix 3.4: Conditional distributions [Ω|Y,θ] and [θ|Y,Ω] in nonlinear SEMs with covariates -- Appendix 3.5: WinBUGS code -- Appendix 3.6: R2WinBUGS code -- References -- 4 Bayesian model comparison and model checking -- 4.1 Introduction -- 4.2 Bayes factor -- 4.2.1 Path sampling -- 4.2.2 A simulation study -- 4.3 Other model comparison statistics.

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.

This book provides clear instructions to researchers on how to apply Structural Equation Models (SEMs) for analyzing the inter relationships between observed and latent variables. Basic and Advanced Bayesian Structural Equation Modeling introduces basic and advanced SEMs for analyzing various kinds of complex data, such as ordered and unordered categorical data, multilevel data, mixture data, longitudinal data, highly non-normal data, as well as some of their combinations. In addition, Bayesian semiparametric SEMs to capture the true distribution of explanatory latent variables are introduced, whilst SEM with a nonparametric structural equation to assess unspecified functional relationships among latent variables are also explored. Statistical methodologies are developed using the Bayesian approach giving reliable results for small samples and allowing the use of prior information leading to better statistical results. Estimates of the parameters and model comparison statistics are obtained via powerful Markov Chain Monte Carlo methods in statistical computing. Introduces the Bayesian approach to SEMs, including discussion on the selection of prior distributions, and data augmentation. Demonstrates how to utilize the recent powerful tools in statistical computing including, but not limited to, the Gibbs sampler, the Metropolis-Hasting algorithm, and path sampling for producing various statistical results such as Bayesian estimates and Bayesian model comparison statistics in the analysis of basic and advanced SEMs. Discusses the Bayes factor, Deviance Information Criterion (DIC), and L_\nu-measure for Bayesian model comparison. Introduces a number of important generalizations of SEMs, including multilevel and mixture SEMs, latent curve models and longitudinal SEMs, semiparametric SEMs and those with various types of discrete data, and

nonparametric structural equations. Illustrates how to use the freely available software WinBUGS to produce the results. Provides numerous real examples for illustrating the theoretical concepts and computational procedures that are presented throughout the book. Researchers and advanced level students in statistics, biostatistics, public health, business, education, psychology and social science will benefit from this book.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2019. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.

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