Essays on Bayesian structural equations modeling

Motivated from large multilevel survey data conducted by the US Veterans Health Administration (VHA), Structural Equations Models (SEMs) are developed that involve a set of latent variables to capture dependence between different responses, a set of facility level random effects to capture facility heterogeneity and dependence between individuals within the same facility, and a set of covariates to account for individual heterogeneity. Identifiability associated with structural equations modeling is addressed and properties of proposed models are carefully examined. Specifically, three distinct SEM are developed, namely: for single survey under the normal distribution assumption; under constraint of individuals not being identifiable over the surveys, a pseudo-longitudinal model is developed under the normal distribution assumption; and considering polytomous response, an ordinal response threshold model is developed by assuming a latent continuous underlying distribution. For all the models, effective and practically useful modeling strategy are developed to deal with missing responses and to model missing covariates in the structural equations framework. Markov chain Monte Carlo sampling is used to carry out Bayesian posterior computation. Several variations of proposed models are considered and compared via the deviance information criterion (DIC). Detailed analysis of the VHA survey data are presented to illustrate the proposed methodologies. Natural extension of the thesis are briefly discussed for future research.