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 (), where the inverse of 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 . The inverse of the combined relationship matrix can be obtained efficiently, provided can be inverted. In some livestock species, however, the number of animals with genomic information exceeds the number of marker covariates used to compute , and this results in a singular . 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 is singular, but these require working with large dense matrices. Another approach is to modify 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 being dense and difficult to compute as 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 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.

The impact of truncating data on the predictive ability for single‐step genomic best linear unbiased prediction

“…The current study has investigated the impact of removing data, in the form of phenotype, genotype and pedigree information, from older animals based on simulated and empirical data. Due to the rapid uptake of genotyping animals across the majority of livestock species, a large amount of research has been conducted on methods to minimize the computational load and alleviate issues that arise when the number of genotyped animals becomes large (Fernando, Cheng, & Garrick, ; Fragomeni et al., ; Misztal et al., ). The concept of truncating data in the form of pedigree information is not new within animal breeding, although the value of utilizing genotypes from older animals is currently not well understood.…”

Large-scale genomic prediction using singular value decomposition of the genotype matrix

“…Fernando et al [ 5 ] suggested a QR decomposition of the matrix, which is generally faster than SVD. A QR decomposition of matrix , of dimension , with , is: where is a matrix with orthogonal columns (i.e.…”

“…The GRM can thus be approximated as: where is a matrix, which is considerably smaller than the original ( ). This approach is equivalent to strategy IV of Fernando et al [ 5 ] (except that core animals are assumed to explain nearly all genomic variation rather than all genomic variation exactly). Here, genotypes of all animals are expressed as linear functions of genotypes of a reduced set of animals (rows in the genotype matrix).…”

“…However, in some cases, the total GRM does not have full rank, and thus no inverse. Therefore, Fernando et al [ 5 ] suggested exact methods to obtain ssGBLUP solutions. One of the options that they proposed was to model animal genetic effects as linear combinations of independent factors.…”

Snp_blup_rel: software for calculating individual animal SNP-BLUP model reliabilities

“…Calculation of genomic reliability for individual EBV by GBLUP requires inverting the coefficient matrix of the mixed model equations (MME) that include the inverse of the genomic relationship matrix. These matrix inversions become infeasible as the number of genotyped animals increases (Fernando et al 2016). In contrast, the MME matrix size of SNP-BLUP is bounded by the number of SNP markers.…”