  Cognizance of gene-environment interactions may help prevent or detain the
onset of complex diseases like cardiovascular disease, cancer, type2 diabetes,
autism or asthma by adjustments to lifestyle. In this regard, we extend the
Bayesian semiparametric gene-gene interaction model of Bhattacharya &
Bhattacharya (2015) to include the possibility of influencing gene-gene
interactions by environmental variables and possible mutations caused by the
environment. Our model accounts for the unknown number of genetic
sub-populations via finite mixtures composed of Dirichlet processes, which are
related to each other through a hierarchical matrix normal structure
responsible for inducing gene-gene interactions and possible mutations in
association with environmental variables. We also extend the Bayesian
hypotheses testing procedures of Bhattacharya & Bhattacharya (2015) to detect
the roles of genes and their interactions, environment and the influence of
environment on gene-gene interactions, in case-control studies. We develop an
effective parallel computing methodology, which harnesses the power of parallel
processing technology to the efficiencies of our conditionally independent
Gibbs sampling and Transformation based MCMC (TMCMC) methods.Applications of
our model and methods to simulation studies with biologically realistic
case-control genotype datasets obtained under five distinct set-ups yield
encouraging results in each case. We followed these up by application of our
ideas to a real, case-control based genotype dataset on early onset of
myocardial infarction. Beside being in broad agreement with the reported
literature on this dataset, the results obtained give some interesting insights
to the differential effect of gender on MI.
