The quantification of the bias that occurs in the interaction term in the presence of improperly modeled covariates has been derived under simplifying assumptions by Yzerbyt, Muller, and Judd (18), and so here I merely translate their conclusions to a G×E framework and refer the interested reader to their article. For simplicity, let Gi be the effects-coded (−1, 0, +1 for the aa, Aa, and AA alleles, arbitrarily coded) genetic variable where p(a)=p(A)=.5, Ei be a normally distributed, standardized environmental variable, and C1i be a mean centered covariate of interest (e.g., an ancestry score from a principal components analysis of the identity-by-state matrix) that is correlated (confounded) with either Gi or Ei. The substantive conclusions of what follows do not depend on these distributional assumptions, but the assumptions simplify the math. For the derivations below, let us first assume that C1i is confounded with Gi. In a properly specified model, the dependant variable, Yi, is therefore a function of these variables and error: (1)Yi=β0+βGGi+βEEi+βC1C1i+βG×EGiEi+βC1×EC1iEi+εi where GiEi is the product of the genetic and environmental term and C1iEi is the product of the covariate and environmental term.
