Nonparametric Bayesian inference in biostatistics
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Traditional parametric linear models are subject to several limiting constraints. In biomedical data analysis, parametric assumptions are often inappropriate because of multimodality and skewness arising from patient heterogeneity, presence of outliers, lack of important covariates, etc. For these reasons it is desirable to relax the parametric assumption, leading to a nonparametric approach to statistical modelling that accommodates these non-standard relationships in data. This dissertation is a ¯rst step to understand the suitability of Polya tree priors and other nonparametric models for modeling biomedical data. In particular, the Polya tree prior is applied to repeated fractional data, cell line data, and microarray data. For repeated fractional data with a range of possible values from the unit interval and positive probability masses on 0 and 1, a latent variable is introduced to address probability point masses at 0 and 1. Posterior simulations for Polya tree priors on residual distributions, random e®ects distributions, and gene expression distributions are discussed. We propose new models and introduce Polya tree priors in these applications and develop novel algorithms to facilitate posterior inference. Three case studies highlight aspects of inference with Polya trees. In one of the case studies we develop a nonparametric approach to inference about di®erential gene expression in microarray group comparison experiments. Future directions for research are discussed.