Nonparametric Bayesian Regression for Categorical Responses: Novel Methodology for Modeling, Inference and Applications

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The investigator develops flexible Bayesian mixture models and corresponding methods for inference for a number of regression problems with ordinal categorical responses. More specifically, methodology is developed for: nonparametric mixture regression for binary responses; ordinal regression, including multivariate ordinal responses and mixed ordinal-continuous responses; dynamic modeling for ordinal regression relationships; and modeling and risk assessment for bioassay dose-response studies with an ordinal classification. All the methods build from nonparametric priors, including both general mixture prior models as well as more structured forms as motivated by the application area and inferential objectives of the model. The key research activity involves flexible regression modeling based on either dependent nonparametric priors for the process of response distributions indexed by covariate values or nonparametric mixture models for the joint distribution of the response(s) and covariates. These classes of models enable rich inference for both the regression relationship and for the conditional response distribution. Hence, they improve model fit and predictive performance compared to standard parametric models, but also relative to existing Bayesian semiparametric work. For all the modeling approaches under development, the investigator studies relevant theoretical properties, model specification, prior elicitation, Markov chain MonteCarlo posterior simulation techniques, and model checking and comparison.Regression problems with ordinal categorical responses -- involving data on response variables recorded on an ordinal scale and on associated explanatory variables -- are of key importance in various fields of the biomedical, environmental and social sciences. As researchers from these fields collect more and more data involving ordinal responses, especially over time or time and space, the need for analyses that enhance theirunderstanding of underlying processes grows. This inspires the need for sufficiently rich statistical models that can accommodate general ordinal regression relationships. The primary motivation for this research is to expand the catalog of ordinal regression modeling tools available to such scientists, in the process expanding the methodology in the field of Bayesian nonparametrics, a burgeoning area of Bayesian statistics. Due to their generality, the statistical methods developed under this research project have the potential for substantive applications in several scientific fields. A particularly promising area of application involves evolutionary biology problems on estimation of the form of natural selection as it relates phenotypic traits to ordinal fitness measures, such as survival, maturity or reproductive success. For such settings, improved estimation of the fitness surface as well as understanding of its temporal and/or spatial evolution can have an impact on effective decision making for the population under study.