Bayesian variable selection for hierarchical gene–environment and gene–gene interactions

We propose a Bayesian hierarchical mixture model framework that allows us to investigate the genetic and environmental effects, gene by gene interactions and gene by environment interactions in the same model. Our approach incorporates the natural hierarchical structure between the main effects and interaction effects into a mixture model, such that our methods tend to remove the irrelevant interaction effects more effectively, resulting in more robust and parsimonious models. We consider both strong and weak hierarchical models. For a strong hierarchical model, both the main effects between interacting factors must be present for the interactions to be considered in the model development, while for a weak hierarchical model, only one of the two main effects is required to be present for the interaction to be evaluated. Our simulation results show that the proposed strong and weak hierarchical mixture models work well in controlling false-positive rates and provide a powerful approach for identifying the predisposing effects and interactions in gene–environment interaction studies, in comparison with the naive model that does not impose this hierarchical constraint in most of the scenarios simulated. We illustrate our approach using data for lung cancer and cutaneous melanoma.