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It is becoming increasingly clear that complex interactions among genes andenvironmental factors play crucial roles in triggering complex diseases. Thus,understanding such interactions is vital, which is possible only throughstatistical models that adequately account for such intricate, albeit unknown,dependence structures. Bhattacharya & Bhattacharya (2016b) attempt suchmodeling, relating finite mixtures composed of Dirichlet processes thatrepresent unknown number of genetic sub-populations through a hierarchicalmatrix-normal structure that incorporates gene-gene interactions, and possiblemutations, induced by environmental variables. However, the product dependencestructure implied by their matrix-normal model seems to be too simple to beappropriate for general complex, realistic situations. In this article, wepropose and develop a novel nonparametric Bayesian model for case-controlgenotype data using hierarchies of Dirichlet processes that offers a morerealistic and nonparametric dependence structure between the genes, induced bythe environmental variables. In this regard, we propose a novel and highlyparallelisable MCMC algorithm that is rendered quite efficient by thecombination of modern parallel computing technology, effective Gibbs samplingsteps, retrospective sampling and Transformation based Markov Chain Monte Carlo(TMCMC). We use appropriate Bayesian hypothesis testing procedures to detectthe roles of genes and environment in case-control studies. We apply our ideasto 5 biologically realistic case-control genotype datasets simulated underdistinct set-ups, and obtain encouraging results in each case. We finally applyour ideas to a real, myocardial infarction dataset, and obtain interestingresults on gene-gene and gene-environment interaction, while broadly agreeingwith the results reported in the literature.

A Non-Gaussian, Nonparametric Structure for Gene-Gene and  Gene-Environment Interactions in Case-Control Studies Based on Hierarchies of  Dirichlet Processes