Use of the simple cross-product term when three genotype levels exist in the data is not recommended either. Only under highly specific circumstances does the cross-product term accurately reflect the interaction. This is because for the three-category genetic variant (yielding three “phenotype by environment” dependencies) there are six unknowns, these being δij, where j = 0,1 and i = 0,1,2. The four βi coefficients in the interaction regression Eq. (1), cannot in general, determine all six unknown δij coefficients. Hence, regression with the traditional cross-product interaction term cannot reliably recover the exact levels for all three genotypic groups. In Appendix A we delineate the specific conditions (which are unlikely to occur in practice) under which the traditional cross-product approach yields equivalent estimates.
