Covariances among sire by breed group of dam interaction effects in multibreed sire evaluation procedures

In multibreed populations, bulls need to be evaluated for additive and nonadditive genetic effects. When the nonadditive genetic effects associated with a bull are defined as sire x breed-group-of-dam interactions, they can be expressed as linear combinations of interactions between alleles of one or more breeds at one or more loci. If these specific allelic interactions are assumed to be independent, then variances and covariances between sire x breed-group-of-dam interaction subclasses can be shown to be linear combinations of variances and covariances of specific intra- and interlocus intra- and interbreed allelic interactions. Furthermore, covariances between sire x breed-group-of-dam interactions due to specific interactions at one, two, or more loci are zero. If dams are assumed to be unrelated to bulls and among themselves, except through their sires and maternal grandsires, efficient procedures to compute the inverse of the covariance matrices of nonadditive genetic effects can be devised, both in subclass and in regression models. Recursive procedures developed make possible the evaluation of large numbers of bulls for nonadditive genetic effects using mixed-model methodology. For completeness, recursive procedures to compute nonadditive covariance matrices in subclass and in regression models also were developed. The prediction of nonadditive genetic values for bulls, in addition to their additive genetic values, will help plan matings, make selection decisions more accurate and, possibly, make economic projections better.