Residual Diagnostic Methods for Bayesian Structural Equation Models

Abstract

Thesis (Ph.D.)--University of Rochester. School of Medicine and Dentistry. Dept. of Biostatistics and Computational Biology, 2009.Often environmental epidemiological studies focus on estimating eects of highly correlated exposures on a health condition measured with multiple outcomes. Adjusting for all exposures as separate covariates in a multiple linear regression model can cause multicollinearity. Furthermore, tting separate models for each exposure and outcome combination leads to problems of multiple comparisons and may introduce confounding with the exposures left out of the model. Structural equation modeling (SEM) alleviates these issues by assuming a latent variable structure underlying the observed exposures and outcomes. Bayesian methods for estimating the parameters in SEM treat the latent variables as missing data and impute them as part of a Markov Chain Monte Carlo (MCMC) sampler, resulting in the full posterior distribution for both the parameters and the latent variables. Bayesian SEM is reviewed and illustrated with a model analyzing the eects of phthalate exposure on human semen quality. Although methods exist for checking overall goodness-of-t in SEM, little attention has been given to testing specic model assumptions. Individual-level residuals are easy to estimate in the Bayesian SEM and are used to dene posterior predictive checks for model assumptions. The empirical cumulative distribution function of the residuals is used to test the assumption that the residual error is normally distributed. Cumulative sums of the residuals are used to check the assumption that the predictors have a linear relationship to the dependent variables in the model equations. The validity of the posterior predictive checks is examined through simulation studies, and the tests are applied to the example data set