Basic and Advanced Bayesian Structural Equation Modeling: With Applications in

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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.

About the authors xiii Preface xv 1 Introduction 1 1.1 Observed and latent variables 1 1.2 Structural equation model 3 1.3 Objectives of the book 3 1.4 The Bayesian approach 4 1.5 Real data sets and notation 5 2 Basic concepts and applications of structural equation models 16 2.1 Introduction 16 2.2 Linear SEMs 17 2.3 SEMs with fixed covariates 23 2.4 Nonlinear SEMs 25 2.5 Discussion and conclusions 29 3 Bayesian methods for estimating structural equation models 34 3.1 Introduction 34 3.2 Basic concepts of the Bayesian estimation and prior distributions 35 3.3 Posterior analysis using Markov chain Monte Carlo methods 40 3.4 Application of Markov chain Monte Carlo methods 43 3.5 Bayesian estimation via WinBUGS 45 4 Bayesian model comparison and model checking 64 4.1 Introduction 64 4.2 Bayes factor 65 4.3 Other model comparison statistics 73 4.4 Illustration 76 4.5 Goodness of fit and model checking methods 78 5 Practical structural equation models 86 5.1 Introduction 86 5.2 SEMs with continuous and ordered categorical variables 86 5.3 SEMs with variables from exponential family distributions 95 5.4 SEMs with missing data 102 6 Structural equation models with hierarchical and multisample data 130 6.1 Introduction 130 6.2 Two-level structural equation models 131 6.3 Structural equation models with multisample data 141 7 Mixture structural equation models 162 7.1 Introduction 162 7.2 Finite mixture SEMs 163 7.3 A Modified mixture SEM 178 8 Structural equation modeling for latent curve models 196 8.1 Introduction 196 8.2 Background to the real studies 197 8.3 Latent curve models 199 8.4 Bayesian analysis 205 8.5 Applications to two longitudinal studies 206 8.6 Other latent curve models 213 9 Longitudinal structural equation models 224 9.1 Introduction 224 9.2 A two-level SEM for analyzing multivariate longitudinal data 226 9.3 Bayesian analysis of the two-level longitudinal SEM 228 9.4 Simulation study 231 9.5 Application: Longitudinal study of cocaine use 232 9.6 Discussion 236 10 Semiparametric structural equation models with continuous variables 247 10.1 Introduction 247 10.2 Bayesian semiparametric hierarchical modeling of SEMs with covariates 249 10.3 Bayesian estimation and model comparison 251 10.4 Application: Kidney disease study 252 10.5 Simulation studies 259 10.6 Discussion 265 11 Structural equation models with mixed continuous and unordered categorical variables 271 11.1 Introduction 271 11.2 Parametric SEMs with continuous and unordered categorical variables 272 11.3 Bayesian semiparametric SEM with continuous and unordered categorical variables 280 12 Structural equation models with nonparametric structural equations 306 12.1 Introduction 306 12.2 Nonparametric SEMs with Bayesian P-splines 307 12.3 Generalized nonparametric structural equation models 320 12.4 Discussion 331 13 Transformation structural equation models 341 13.1 Introduction 341 13.2 Model description 342 13.3 Modeling nonparametric transformations 343 13.4 Identifiability constraints and prior distributions 344 13.5 Posterior inference with MCMC algorithms 345 13.6 Simulation study 348 13.7 A study on the intervention treatment of polydrug use 350 13.8 Discussion 354 14 Conclusion 358 References 360 Index 361