The bias in the G×E term can then be quantified as the difference between βG×E (the unbiased estimate from Model 1) and βG×E* (the biased estimate from Model 2). In this case: (3)βG×E*=βG×E+βC1×EσC1,GσG2 and βG×E* is biased as a function of βC1×EσC1,GσG2. Note that βC1 does not affect the bias; controlling for the main effect of the covariate does nothing to control for the covariate’s effect on the interaction. It is therefore possible that some or all of the estimated G×E effect in a model that “controls” for only the main effect of a covariate is due to the interaction between the covariate and the environmental term rather than the G×E effect itself. A similar situation occurs if the covariate is correlated with the environmental variable and interacts with the genetic polymorphism. For example, the effect of the genetic polymorphism may depend on ethnic or socioeconomic background rather than on the hypothesized environmental moderator. In this case, βG×E* is biased as a function of βG×C1σC1,EσE2. Thus, to properly control for all the potential ways k covariates might confound the G×E effect of interest, investigators should fit the following model: (4)Yi=β0+βGGi+βEEi+∑kβkCki+∑kβG×CkGiCki+∑kβCk×ECkiEi+βG×EGiEi+εi
