Basic and Advanced Bayesian Structural Equation Modeling E-Book

Sik-Yum Lee

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Serie: Wiley Series in Probability and Statistics
• Sprache: Englisch

Seitenzahl: 629

Veröffentlichungsjahr: 2012

Beschreibung

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.

Leseprobe

Contents

Cover

Series

Title Page

Copyright

Dedication

About the authors

Preface

Chapter 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

Chapter 2: Basic concepts and applications of structural equation models

2.1 Introduction

2.2 Linear SEMs

2.3 SEMs with fixed covariates

2.4 Nonlinear SEMs

2.5 Discussion and conclusions

Chapter 3: Bayesian methods for estimating structural equation models

3.1 Introduction

3.2 Basic concepts of the Bayesian estimation and prior distributions

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

Chapter 4: Bayesian model comparison and model checking

4.1 Introduction

4.2 Bayes factor

4.3 Other model comparison statistics

4.4 Illustration

4.5 Goodness of fit and model checking methods

Appendix A.1 WinBUGS code

Appendix A.2 R code in Bayes factor example

Appendix A.3 Posterior predictive p-value for model assessment

Chapter 5: Practical structural equation models

5.1 Introduction

5.2 SEMs with continuous and ordered categorical variables

5.3 SEMs with variables from exponential family distributions

5.4 SEMs with missing data

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

Chapter 6: Structural equation models with hierarchical and multisample data

6.1 Introduction

6.2 Two-level structural equation models

6.3 Structural equation models with multisample data

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

Chapter 7: Mixture structural equation models

7.1 Introduction

7.2 Finite mixture SEMs

7.3 A modified mixture SEM

Appendix 7.1 The permutation sampler

Appendix 7.2 Searching for identifiability constraints

Appendix 7.3 Conditional distributions: Modified mixture SEMs

Chapter 8: Structural equation modeling for latent curve models

8.1 Introduction

8.2 Background to the real studies

8.3 Latent curve models

8.4 Bayesian analysis

8.5 Applications to two longitudinal studies

8.6 Other latent curve models

Appendix 8.1 Conditional distributions

Appendix 8.2 WinBUGS code for the analysis of cocaine use data

Chapter 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

9.4 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 -measure in equation (9.9) via MCMC samples

Chapter 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

10.6 Discussion

Appendix 10.1 Conditional distributions for parametric components

Appendix 10.2 Conditional distributions for nonparametric components

Chapter 11: Structural equation models with mixed continuous and unordered categorical variables

11.1 Introduction

11.2 Parametric SEMs with continuous and unordered categorical variables

11.3 Bayesian semiparametric SEM with continuous and unordered categorical variables

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

Chapter 12: Structural equation models with nonparametric structural equations

12.1 Introduction

12.2 Nonparametric SEMs with Bayesian P-splines

12.3 Generalized nonparametric structural equation models

12.4 Discussion

Appendix 12.1 Conditional distributions and the MH algorithm: Nonparametric SEMs

Appendix 12.2 Conditional distributions in generalized nonparametric SEMs

Chapter 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

13.6 Simulation study

13.7 A study on the intervention treatment of polydrug use

13.8 Discussion

Chapter 14: Conclusion

Index

Series

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Library of Congress Cataloging-in-Publication Data

Lee, Sik-Yum. Basic and Advanced Bayesian Structural Equation Modeling: With Applications in the Medical and Behavioral Sciences / Sik-Yum Lee and Xin-Yuan Song. p. cm. Includes bibliographical references and index. ISBN 978-0-470-66952-5 (hardback) 1. Structural equation modeling. 2. Bayesian statistical decision theory. I. Song, Xin-Yuan. II. Title. QA278.3.L439 2012 519.5′3–dc23 2012012199

A catalogue record for this book is available from the British Library.

ISBN: 978-0-470-66952-5

For our family members:

Yulin and Haotian Wu;

Mable, Anna, and Timothy Lee

About the authors

Xin-Yuan Song is an associate professor at the Department of Statistics of the Chinese University of Hong Kong. She earned her PhD in Statistics at the Chinese University of Hong Kong. She serves as an associate editor for Psychometrika, and as a member of the editorial board of Frontiers in Quantitative Psychology and Measurement and the Open Journal of Statistics. Her research interests are in structural equation models, latent variable models, Bayesian methods, survival analysis, and statistical computing. She has published over 80 papers in prestigious international journals.

Sik-Yum Lee is an emeritus professor of statistics at the Chinese University of Hong Kong. He earned his PhD in biostatistics at the University of California, Los Angeles. He received a distinguished service award from the International Chinese Statistical Association, is a former president of the Hong Kong Statistical Society, and is an elected member of the International Statistical Institute and a Fellow of the American Statistical Association. He had served as an associate editor for Psychometrika and Computational Statistics & Data Analysis, and as a member of the editorial board of the British Journal of Mathematical and Statistical Psychology, Structural Equation Modeling, and the Chinese Journal of Medicine. His research interests are in structural equation models, latent variable models, Bayesian methods, and statistical diagnostics. He is editor of the Handbook on Structural Equation Models and author of Structural Equation Modeling: A Bayesian Approach, and over 160 papers.

Preface

Latent variables that cannot be directly measured by a single observed variable are frequently encountered in substantive research. In establishing a model to reflect reality, it is often necessary to assess various interrelationships among observed and latent variables. Structural equation models (SEMs) are well recognized as the most useful statistical model to serve this purpose. In past years, even the standard SEMs were widely applied to behavioral, educational, medical, and social sciences through commercial software, such as AMOS, EQS, LISREL, and Mplus. These programs basically use the classical covariance structure analysis approach. In this approach, the hypothesized covariance structure of the observed variables is fitted to the sample covariance matrix. Although this works well for many simple situations, its performance is not satisfactory in dealing with complex situations that involve complicated data and/or model structures.

Nowadays, the Bayesian approach is becoming more popular in the field of SEMs. Indeed, we find that when coupled with data augmentation and Markov chain Monte Carlo (MCMC) methods, this approach is very effective in dealing with complex SEMs and/or data structures. The Bayesian approach treats the unknown parameter vector θ in the model as random and analyzes the posterior distribution of θ, which is essentially the conditional distribution of θ given the observed data set. The basic strategy is to augment the crucial unknown quantities such as the latent variables to achieve a complete data set in the posterior analysis. MCMC methods are then implemented to obtain various statistical results. The book Structural Equation Modeling: A Bayesian Approach, written by one of us (Sik-Yum Lee) and published by Wiley in 2007, demonstrated several advantages of the Bayesian approach over the classical covariance structure analysis approach. In particular, the Bayesian approach can be applied to deal efficiently with nonlinear SEMs, SEMs with mixed discrete and continuous data, multilevel SEMs, finite mixture SEMs, SEMs with ignorable and/or nonignorable missing data, and SEMs with variables coming from an exponential family.

The recent growth of SEMs has been very rapid. Many important new results beyond the scope of Structural Equation Modeling have been achieved. As SEMs have wide applications in various fields, many new developments are published not only in journals in social and psychological methods, but also in biostatistics and statistical computing, among others. In order to introduce these useful developments to researchers in different fields, it is desirable to have a textbook or reference book that includes those new contributions. This is the main motivation for writing this book.

Similar to Structural Equation Modeling, the theme of this book is the Bayesian analysis of SEMs. Chapter 1 provides an introduction. Chapter 2 presents the basic concepts of standard SEMs and provides a detailed discussion on how to apply these models in practice. Materials in this chapter should be useful for applied researchers. Note that we regard the nonlinear SEM as a standard SEM because some statistical results for analyzing this model can be easily obtained through the Bayesian approach. Bayesian estimation and model comparison are discussed in Chapters 3 and 4, respectively. Chapter 5 discusses some practical SEMs, including models with mixed continuous and ordered categorical data, models with variables coming from an exponential family, and models with missing data. SEMs for analyzing heterogeneous data are presented in Chapters 6 and 7. Specifically, multilevel SEMs and multisample SEMs are discussed in Chapter 6, while finite mixture SEMs are discussed in Chapter 7. Although some of the topics in Chapters 3–7 have been covered by Structural Equation Modeling, we include them in this book for completeness. To the best of our knowledge, materials presented in Chapters 8–13 do not appear in other textbooks. Chapters 8 and 9 respectively discuss second-order growth curve SEMs and a dynamic two-level multilevel SEM for analyzing various kinds of longitudinal data. A Bayesian semiparametric SEM, in which the explanatory latent variables are modeled through a general truncated Dirichlet process, is introduced in Chapter 10. The purposes for introducing this model are to capture the true distribution of explanatory latent variables and to handle nonnormal data. Chapter 11 deals with SEMs with unordered categorical variables. The main aim is to provide SEM methodologies for analyzing genotype variables, which play an important role in developing useful models in medical research. Chapter 12 introduces an SEM with a general nonparametric structural equation. This model is particularly useful when researchers have no idea about the functional relationships among outcome and explanatory latent variables. In the statistical analysis of this model, the Bayesian P-splines approach is used to formulate the nonparametric structural equation. As we show in Chapter 13, the Bayesian P-splines approach is also effective in developing transformation SEMs for dealing with extremely nonnormal data. Here, the observed nonnormal random vector is transformed through the Bayesian P-splines into a random vector whose distribution is close to normal. Chapter 14 concludes the book with a discussion. In this book, concepts of the models and the Bayesian methodologies are illustrated through analyses of real data sets in various fields using the software WinBUGS, R, and/or our tailor-made C codes. Chapters 2–4 provide the basic concepts of SEMs and the Bayesian approach. The materials in the subsequent chapters are basically self-contained. To understand the material in this book, all that is required are some fundamental concepts of statistics, such as the concept of conditional distributions.

We are very grateful to organizations and individuals for their generous support in various respects. The Research Grant Council of the Hong Kong Special Administration Region has provided financial support such as GRF446609 and GRF404711 for our research and for writing this book. The World Value Study Group, World Values Survey, 1981–1984 and 1990–1993, the World Health Organization WHOQOL group, Drs. D. E. Morisky, J. A. Stein, J. C. N. Chan, Y. I. Hser, T. Kwok, H. S. Ip, M. Power and Y. T. Hao have been kind enough to let us have their data sets. The Department of Epidemiology and Public Health of the Imperial College, School of Medicine at St. Mary’s Hospital (London, UK) provided their WinBUGS software. Many of our graduate students and research collaborators, in particular J. H. Cai, J. H. Pan, Z. H. Lu, P. F. Liu, J. Chen, D. Pan, H. J. He, K. H. Lam, X. N. Feng, B. Lu, and Y. M. Xia, made very valuable comments which led to improvements to the book. We are grateful to all the wonderful people on the John Wiley editorial staff, particularly Richard Leigh, Richard Davies, Heather Kay, Prachi Sinha Sahay, and Baljinder Kaur for their continued assistance, encouragement and support of our work. Finally, we owe deepest thanks for our family members for their constant support and love over many years.

This book features an accompanying website:

www.wiley.com/go/medical_behavioral_sciences

2

Basic concepts and applications of structural equation models

2.1 Introduction

Structural equation models (SEMs) are a flexible class of models that allow complex modeling of correlated multivariate data for assessing interrelationships among observed and latent variables. It is well known in the fields of social and psychological sciences that this class of models subsumes many widely used statistical models, such as regression, factor analysis, canonical correlations, and analysis of variance and covariance. Traditional methods for analyzing SEMs were mainly developed in psychometrics, and have been extensively applied in behavioral, educational, social, and psychological research in the past twenty years. Recently, SEMs have begun to attract a great deal of attention in public health, biological, and medical sciences. Today, due to strong demand in various disciplines, there are more than a dozen SEM software packages, such as AMOS, EQS6, LISREL, and Mplus. Among the various ways to specify SEMs in these software packages, we choose the key idea of the LISREL (Jöreskog and Sörbom, 1996) formulation in defining the basic model through a measurement equation and a structural equation. The main reasons for this choice are as follows: (i) The measurement and structural equations are very similar to the familiar regression models, hence more direct interpretation can be achieved and the common techniques in regression such as outlier and residual analyses can be employed. (ii) It makes a clear distinction between observed and latent variables. (iii) It directly models raw individual observations with latent variables, hence it can be naturally generalized to handle complex situations, and results in a direct estimation of latent variables. (iv) The development of statistical methodologies for subtle SEMs is more natural and comparatively easier.