Bayesian influence diagnostic methods for parametric regression models

The goals of assessing the influence of individual observations in statistical analysis are not only to identify influential observations such as outliers and high leverage points, but also to determine the importance of each observation in the analysis for a better model fit. Thus, assessing the influence of individual observations on a model, choosing an appropriate dimensionality of a model and selecting the best model for a given dataset are very important and highly relevant problems in any formal statistical analysis.;Recently, Bayesian methodologies have been getting enormous attention in biomedical research due to the potential advantages of fitting a vast array of complex models posed by modern data. As the demand for Bayesian data analysis and modeling increases, we need good diagnostic methods for model assessment and selection. In this dissertation, we develop Bayesian diagnostic measures based on case-deletion to assess the influence of each observation to model fit and model complexity. First, we propose Bayesian case influence diagnostics for complex survival models. In detail, we develop case deletion influence diagnostics for both the joint and marginal posterior distributions based on the Kullback-Leibler divergence. Second, we introduce three types of Bayesian case influence measures based on case deletion, namely the &phis;- divergence, Cook's posterior mode distance and Cook's posterior mean distance to evaluate the effects of deleting a set of observations in general Bayesian parametric models. We also examine the statistical properties of these three Bayesian case influence measures and their applications to identification of influential sets and model complexity.;In any deletion diagnostic, "size matters" issue persists and it is a fundamental issue of influence analysis, because the size of the deletion diagnostic is associated with the size of the perturbation. For Cook's distance, that is Cook's distance is a monotonic function of the size of perturbation. Thus, we develop a scaled version of Cook's distance to address the size issue for deletion diagnostics in general parametric models.