An efficient exact method to obtain GBLUP and single-step GBLUP when the genomic relationship matrix is singular

摘要

BackgroundThe mixed linear model employed for genomic best linear unbiased prediction (GBLUP) includes the breeding value for each animal as a random effect that has a mean of zero and a covariance matrix proportional to the genomic relationship matrix (>Ggg), where the inverse of >Ggg is required to set up the usual mixed model equations (MME). When only some animals have genomic information, genomic predictions can be obtained by an extension known as single-step GBLUP, where the covariance matrix of breeding values is constructed by combining the pedigree-based additive relationship matrix with >Ggg. The inverse of the combined relationship matrix can be obtained efficiently, provided >Ggg can be inverted. In some livestock species, however, the number Ng of animals with genomic information exceeds the number of marker covariates used to compute >Ggg, and this results in a singular >Ggg. For such a case, an efficient and exact method to obtain GBLUP and single-step GBLUP is presented here.