Technical Note: A GBLUP to Produce ssGBLUP Solutions for Genotyped Animals

Nowadays, single-step genomic best linear unbiased prediction (ssGBLUP) is the most popular methodology for livestock genetic evaluation. It simultaneously evaluates genotyped and nongenotyped animals using an augmented genotype and pedigree relationship matrix. In most livestock species, a small proportion of the population is genotyped. A small model limited to genotyped animals (genomic best linear unbiased prediction, GBLUP) incorporating information from nongenotyped animals would be of interest, because it reduces the dimension of equations and computational costs and produces solutions equivalent to ssGBLUP. Two methods are considered in this study. First, genotyped animals, whether phenotyped or nonphenotyped, receive phenotype contributions from their nongenotyped relatives. Calculating those pseudo-phenotypes and weights for the introduced residual heterogeneity is challenging. The second method involves absorbing equations of nongenotyped animals into the equations of genotyped animals. The solutions from the resulting GBLUP model were equivalent to those of the full ssGBLUP model. However, two main reasons prevent the introduced GBLUP model from being adopted and widely implemented. First, such implementation enforces (pedigree-based) additive genetic covariances among nongenotyped animals into their residual covariance matrix. That introduces nonzero off-diagonal elements to the diagonal residual covariance matrix for nongenotyped animals, making its inversion cumbersome. Also, the inverted matrix is dense, increasing the multiplication costs to other matrices and the vector of phenotypes. Second, the breeding values of nongenotyped animals (a1) are not conditional to those of genotyped animals (a2), making back-solving of a1 from a2 challenging.