Genetic Model and Notation
Let Y be a quantitative trait and G be the minor allele count for the SNP under investigation (G = 0, 1, or 2); the additive assumption is not critical to the method development. Let E be an exposure variable interacting with the genetic factor. This exposure E could reflect continuous or categorical measures of environmental or genetic background. The underlying true genetic model can include main effects of both G (βG) and E (βE) on Y, as well as the interaction effect (βGE):(Equation 1) Y∼β0+βGG+βEE+βGEGE+ε.We assume that the trait Y is normally distributed with unit variance conditional upon G and E, in other words, Var(Y |G = g, E = e) = 1 and ε∼N(0,1).
When considering only G, the working model would reduce to(Equation 2) Y∼β0+βGG+εG.Pare et al.13 showed that the conditional variance of Y conditional on G alone could be expressed as σG2 = Var(Y|G = g) = (βE+βGEg)2+1. Thus, if an interaction effect was present (i.e., βGE≠0), the trait variance would differ between genotypes.