| A common statistical problem in biomedical research is to characterize the relationship between a response and predictors. The heterogeneity among subjects causes the response distribution to change across the predictor space in distributional characteristics such as skewness, quantiles and residual variation. In such settings, it would be appealing to model the conditional response distributions as flexibly changing across the predictors while conducting variable selection to identify important predictors both locally (within some local regions) and globally (across the entire range of the predictor space) for the response distribution change.;Nonparametric Bayes methods have been very useful for flexible modeling where nonparametric distributions are assumed unknown and assigned priors such as the Dirichlet process (DP). In recent years, there has been a growing interest in extending the DP to a prior model for predictor-dependent unknown distributions, so that the extended priors are applied to flexible conditional distribution modeling. However, for most of the proposed extensions, construction is not simple and computation can be quite difficult. In addition, literature has been focused on estimation and few attempts have been made to address related hypothesis testing problems such as variable selection.;Paper 1 proposes a local Dirichlet process (lDP) as a generalization of the Dirichlet process to provide a prior distribution for a collection of random probability measures indexed by predictors. The lDP involves a simple construction, results in a marginal Dirichlet process prior for the random measure at any specific predictor value, and leads to a straightforward posterior computation. In paper 2, we propose a more general approach not only estimating the conditional response distribution but also identifying important predictors for the response distribution change both with local regions and globally. This is achieved through the probit stick-breaking process mixture (PSBPM) of normal linear regressions where the PSBP is a newly proposed prior for dependent probability measures and particularly convenient to incorporate variable selection structure. In paper 3, we extend the paper 2 method for longitudinal outcomes which are correlated within subject. The PSBPM of linear mixed effect (LME) model is considered allowing for individual variability within a mixture component. |
