Weighted Dirichlet Process Mixture Models to Accommodate Complex Sample Designs for Linear and Quantile Regression

Standard randomization-based inference conditions on the data in the population and makes inference with respect to the repeating sampling properties of the sampling indicators. In some settings these estimators can be quite unstable; Bayesian model-based approaches focus on the posterior predictive distribution of population quantities, potentially providing a better balance between bias correction and efficiency. Previous work in this area has focused on estimation of means and linear and generalized linear regression parameters; these methods do not allow for a general estimation of distributional functions such as quantile or quantile regression parameters. Here we adapt an extended Dirichlet Process Mixture model that allows the DP prior to be a mixture of DP random basis measures that are a function of covariates. These models allow many mixture components when necessary to accommodate the sample design, but can shrink to few components for more efficient estimation when the data allow. We provide an application to the estimation of relationships between serum dioxin levels and age in the US population, either at the mean level (via linear regression) or across the dioxin distribution (via quantile regression) using the National Health and Nutrition Examination Survey.