An efficient exact method to obtain GBLUP and single-step GBLUP when the genomic relationship matrix is singular. Issue 1 (December 2016) - Fernando, Rohan; Cheng, Hao; Garrick, Dorian
Abstract Background The 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 ( G_gg G g g ), where the inverse of G_gg G g g 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 G_gg G g g . The inverse of the combined relationship matrix can be obtained efficiently, provided G_gg G g g can be inverted. In some livestock species, however, the number N_g N g of animals with genomic information exceeds the number of marker covariates used to compute G_gg G g g, and this results in a singular G_gg G g g . For such a case, an efficient and exact method to obtain GBLUP and single-step GBLUP is presented here. Results Exact methods are already available to obtain GBLUP when G_gg G g g is singular, but these require working with large dense matrices. Another approach is to modify G_gg G g g 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 G_gg G g g being dense and difficult to compute as N_g N g grows. The approach presented here recognizes that the numberr 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 theser linearly independent breeding values as random effects. Conclusions The 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 G_gg G g g 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.