Genotype by environment interactions and genetic correlations involving two environmental factors.

The concept of the genetic correlation for one trait across environments was extended to two environmental factors B and C. Three additive genetic correlations for the same trait were defined: rG, across both environmental factors; rG(B), across C and within B; and rG(C), across B and within C. As genotype x environment variances increase, the genetic correlations across environments decrease. These three genetic correlations are biased downward in the presence of heterogeneity of genetic variances within environments when they are calculated from the usual analysis of variance (r*G, r*G(B), r*G(C)). Correction factors were derived to remove the bias. The two-way genotype by environment interaction variances can be biased upward or downward by the heterogeneity, but the three-way interaction variance is always biased upward. Correction factors for the interaction variances were also derived. Four additive genetic correlations between two traits (X and Y) were derived: rG xy, across both B and C; rG(B) xy, and rG(C) xy, across one environmental factor and within the other; and rG(BC) xy, within both B and C. These concepts were extended to genetic correlations for dominance and maternal effects. Paternal half-sib and factorial mating designs were used to obtain the various genetic correlations. An example of a paternal half-sib design with beef cattle was used to illustrate the methodology.