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PSYCHOMETRIKAVOL. 75, NO. 4, 675693
DECEMBER 2010
DOI: 10.1007/S11336-010-9174-4
BAYESIAN SEMIPARAMETRIC STRUCTURAL EQUATION MODELS WITH LATENT VARIABLES
MINGAN YANG
SAINT LOUIS UNIVERSITY
DAVID B. DUNSON
DUKE UNIVERSITY
Structural equation models (SEMs) with latent variables are widely useful for sparse covariance structure modeling and for inferring relationships among latent variables. Bayesian SEMs are appealing in allowing for the incorporation of prior information and in providing exact posterior distributions of unknowns, including the latent variables. In this article, we propose a broad class of semiparametric Bayesian SEMs, which allow mixed categorical and continuous manifest variables while also allowing the latent variables to have unknown distributions. In order to include typical identiability restrictions on the latent variable distributions, we rely on centered Dirichlet process (CDP) and CDP mixture (CDPM) models. The CDP will induce a latent class model with an unknown number of classes, while the CDPM will induce a latent trait model with unknown densities for the latent traits. A simple and efcient Markov chain Monte Carlo algorithm is developed for posterior computation, and the methods are illustrated using simulated examples, and several applications.
Key words: Dirichlet process, factor analysis, latent class, latent trait, mixture model, nonparametric Bayes, parameter expansion.
1. Introduction
In the social sciences and increasingly in other application areas, it is routine to collect multivariate data, with the individual measurements having a variety of scales (continuous, count, categorical). Often, these measurements are collected specically with the goal of studying relationships among latent variables, such as life event-induced anxiety, that can only be measured indirectly through multiple manifest variables. In such settings, structural equation models (SEMs) provide a valuable tool for obtaining insight into the relationships between different latent variables and between latent and observed variables (Bollen, 1989). In addition, SEMs provide a exible class of multivariate models for describing covariance structures in multivariate data.
In recent years, there has been increased interest in Bayesian SEMs due in part to advances in posterior computation that now allow Bayesian approaches to be implemented routinely in complex settings involving multilevel structures, missing data, censoring, and other challenges. Using Markov chain Monte Carlo (MCMC) algorithms, one can obtain samples from the exact posterior distribution of all the unknowns, including the latent variables. These samples can be used to...