  Genome-wide association studies, in which as many as a million single
nucleotide polymorphisms (SNP) are measured on several thousand samples, are
quickly becoming a common type of study for identifying genetic factors
associated with many phenotypes. There is a strong assumption that interactions
between SNPs or genes and interactions between genes and environmental factors
substantially contribute to the genetic risk of a disease. Identification of
such interactions could potentially lead to increased understanding about
disease mechanisms; drug $\times$ gene interactions could have profound
applications for personalized medicine; strong interaction effects could be
beneficial for risk prediction models. In this paper we provide an overview of
different approaches to model interactions, emphasizing approaches that make
specific use of the structure of genetic data, and those that make specific
modeling assumptions that may (or may not) be reasonable to make. We conclude
that to identify interactions it is often necessary to do some selection of
SNPs, for example, based on prior hypothesis or marginal significance, but that
to identify SNPs that are marginally associated with a disease it may also be
useful to consider larger numbers of interactions.
