In summary, modeling gene-environment interactions in the common way using the cross-product term, with three categories of the interacting variable (i.e., three levels of G) yields three critical problems. Because the regression lines for a three category genotypic variable have more unknowns than are in the traditional regression Eq. (1) with a cross-product term, only under very specific circumstances (when the coefficients satisfy Eqs. (A6) and (A7) in Appendix A) will the regression lines adequately capture the nature of the interaction in the data. Accordingly, modeling gene-environment interaction with an interaction regression line with four parameters produces lines with several constraints: first, the lines will always be ordered (e.g., 0, 1, 2 but never 0, 2, 1); secondly, differences between the lines must be the same (e.g., the difference between G = 0 and G = 1 will always be the same as the slope difference between G = 1 and G = 2); thirdly, the lines will always cross at a single point (or always will be parallel in the event of no interaction). None of these are necessarily true for any given G × E, making the cross-product parameterization of G × E problematic.
