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