The phenotype, or dependent variable, is hereafter denoted by P, with G and E used to represent the genotypic and environmental variables, respectively. The traditional regression equation with an interaction term has the form: (1)P=β0+β1G+β2E+β3G×E, which determines four unknown β coefficients (to simplify formulas we omit covariates in the regression model other than the gene and environment variables, and do not use a subscript or error term specific to an individual in the sample). The interaction effect is explained by the slopes resulting from the above equation for different levels of the genotypic variable. The phenotype’s regression on the environment at each fixed levelof the genotype can bemodeled as: P = δi0 + δi1E, where i is the genotypic level, δi0 is intercept term for the level, δi1 is the slope coefficient (i takes the value of the genotype, e.g., 0, 1, 2). Figure 1 shows an example of a three level genetic variant and the regression lines for three genotypes in three-dimensional space and corresponding projections in two-dimensional space.
