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Genome-wide association studies, in which as many as a million singlenucleotide polymorphisms (SNP) are measured on several thousand samples, arequickly becoming a common type of study for identifying genetic factorsassociated with many phenotypes. There is a strong assumption that interactionsbetween SNPs or genes and interactions between genes and environmental factorssubstantially contribute to the genetic risk of a disease. Identification ofsuch interactions could potentially lead to increased understanding aboutdisease mechanisms; drug $\times$ gene interactions could have profoundapplications for personalized medicine; strong interaction effects could bebeneficial for risk prediction models. In this paper we provide an overview ofdifferent approaches to model interactions, emphasizing approaches that makespecific use of the structure of genetic data, and those that make specificmodeling assumptions that may (or may not) be reasonable to make. We concludethat to identify interactions it is often necessary to do some selection ofSNPs, for example, based on prior hypothesis or marginal significance, but thatto identify SNPs that are marginally associated with a disease it may also beuseful to consider larger numbers of interactions.

Structures and Assumptions: Strategies to Harness Gene $\times$ Gene and Gene $\times$ Environment Interactions in GWAS