title: An efficient exact method to obtain GBLUP and single-step GBLUP when the genomic relationship matrix is singular.
abstract: 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 ([Formula: see text]), where the inverse of [Formula: see text] 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 [Formula: see text]. The inverse of the combined relationship matrix can be obtained efficiently, provided [Formula: see text] can be inverted. In some livestock species, however, the number [Formula: see text] of animals with genomic information exceeds the number of marker covariates used to compute [Formula: see text], and this results in a singular [Formula: see text]. For such a case, an efficient and exact method to obtain GBLUP and single-step GBLUP is presented here.
  
  
  
  Exact methods are already available to obtain GBLUP when [Formula: see text] is singular, but these require working with large dense matrices. Another approach is to modify [Formula: see text] 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 [Formula: see text] being dense and difficult to compute as [Formula: see text] 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.
  
  
  
  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 [Formula: see text] 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.
authors: Department of Animal Science, Iowa State University, Ames, IA, 50011, USA. rohan@iastate.edu.
authors: Rohan L
authors: 0000-0001-5821-099X
authors: RL
authors: Fernando
authors: Department of Animal Science, Iowa State University, Ames, IA, 50011, USA.
authors: Hao
authors: H
authors: Cheng
authors: Department of Animal Science, Iowa State University, Ames, IA, 50011, USA.|Institute of Veterinary, Animal and Biomedical Sciences, Massey University, Palmerston North, New Zealand.
authors: Dorian J
authors: DJ
authors: Garrick
mesh_terms: D000818:Animals; D003198:Computer Simulation; D016678:Genome; D023281:Genomics / Q000379:methods*; D016014:Linear Models; D058751:Livestock / Q000235:genetics; D008957:Models, Genetic; D010375:Pedigree; D000068618:Selective Breeding / Q000235:genetics