Soft Bayesian Additive Regression Trees (SBART) for correlated survey response with non-Gaussian error

Complex surveys are important data resources across diverse domains, including social sciences, public health, and market research. When faced with unknown effects and interactions of multiple covariates, analysis using usual parametric regression functions may not be adequate. Moreover, mean regression assuming a Gaussian response is often inappropriate when the response distribution is heavy-tailed or highly skewed. While the ensemble of regression trees has received much attention in recent years, there has been limited attention to nonparametric Bayesian regression functions - specifically for quantile regression and modelling of skewed clustered complex survey data. For clustered survey data with subject-specific survey weights, this paper introduces the Soft Bayesian Additive Regression Trees (SBART) framework for doing quantile regression as well as full modelling of the heavy-tailed and skewed response distribution. We illustrate the advantages of our methods through simulation studies and an analysis of multivariate periodontal responses from the National Health and Nutrition Examination Survey.