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

Abstract

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 ({mathbf {G}}_{gg}), where the inverse of {mathbf {G}}_{gg} 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 {mathbf {G}}_{gg}. The inverse of the combined relationship matrix can be obtained efficiently, provided {mathbf {G}}_{gg} can be inverted. In some livestock species, however, the number N_{g} of animals with genomic information exceeds the number of marker covariates used to compute {mathbf {G}}_{gg}, and this results in a singular {mathbf {G}}_{gg}. For such a case, an efficient and exact method to obtain GBLUP and single-step GBLUP is presented here.ResultsExact methods are already available to obtain GBLUP when {mathbf {G}}_{gg} is singular, but these require working with large dense matrices. Another approach is to modify {mathbf {G}}_{gg} to make it nonsingular by adding a small value to all its diagonals or regressing it towards the pedigree-based relationship matrix. This, however, results in the inverse of {mathbf {G}}_{gg} being dense and difficult to compute as N_{g} grows. The approach presented here recognizes that the number r of linearly independent genomic breeding values cannot exceed the number of marker covariates, and the mixed linear model used here for genomic prediction only fits these r linearly independent breeding values as random effects.ConclusionsThe exact method presented here was compared to Apy-GBLUP and to Apy single-step GBLUP, both of which are approximate methods that use a modified {mathbf {G}}_{gg} that has a sparse inverse which can be computed efficiently. In a small numerical example, predictions from the exact approach and Apy were almost identical, but the MME from Apy had a condition number about 1000 times larger than that from the exact approach, indicating ill-conditioning of the MME from Apy. The practical application of exact SSGBLUP is not more difficult than implementation of Apy.Electronic supplementary materialThe online version of this article (doi:10.1186/s12711-016-0260-7) contains supplementary material, which is available to authorized users.