We used a GWGEI model that (1) protects against population stratification; (2) reduces gene-environment correlation; and (3) provides environmental, genetic, and gene-by-environment parameter estimates (Moreno-Macias et al. 2010). Consider Eq. (1), which describes the association between a SNP (X) and BMI (Y) for the ith person: (1)E(Yi∣X)=b0m+b1Xi. The key aspect of this simple model is that for each SNP in the genome-wide array, it conditions the estimate of this association on the mating type (m) of the parents (where m = 1, 2, . . . 6). Mating type measures the similarity between the parents for the given SNP. Parents may both have two copies of the minor allele (AA, AA), one may have two copies of the minor allele and the other two copies of its alternate form (AA, BB), they may each have one copy of each form (AB, AB), and so on. In total, there are six potential mating types: AA-AA, AA-AB, AA-BB, AB-AB, AB-BB, and BB-BB. Controlling for mating type conditions the estimate for b1 (the effect of a particular SNP on BMI) on the likelihood
