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Background
Several livestock production systems are based on crossbreeding schemes (e.g., [1-3]), and take advantage of the increased performance of crossbred animals compared to purebred animals, along with breed complementarity. For such production systems based on crossbreeding, the breeding goal for the purebred populations is to optimize the performance of crossbred descendants. However, the selection of purebred animals for crossbred performance has not been extensively implemented in livestock, partly due to the difficulty of routine collection of pedigree information on crossbred animals [4].
With the advent of genomic selection, various genomic prediction models have been proposed, which use phenotypic records of crossbred animals to increase the selection response for crossbred performance in purebred animals (e.g., [2, 4-6]). These approaches predict breeding values for crossbred performance of selection candidates using the estimated allele substitution effects of many single nucleotide polymorphisms (SNPs). The SNP allele substitution effects are estimated from phenotypes of genotyped reference animals. In the context of crossbreeding, several breeds and their crosses are involved in genomic prediction, and purebred and crossbred performances are often considered to be different but correlated traits (e.g., [1, 3, 5, 7, 8]). Therefore, estimates of SNP allele substitution effects for purebred and crossbred performance traits may not be the same for purebred and crossbred populations, e.g., due to genotype by environment interactions. Assuming only additive gene action, one approach to accommodate this is to model differences between allele substitution SNP effects using a multivariate genomic model that assumes a correlation structure between the effects of SNPs across the purebred and crossbred populations, or equivalently, by assuming a genetic correlation structure across the trait measured in purebred and crossbred populations [9, 10]. These multivariate genomic models are referred to hereafter as across-breed SNP genotype models (ASGM), since the estimates of SNP allele substitution effects for the crossbred performance trait are also used to predict breeding values for crossbred performance of purebred selection candidates, regardless of their breed of origin [4, 6]. Thus, estimates of SNP effects for the crossbred performance trait using ASGM are not breed-specific. However, a number of factors may have an impact on the effect that can be measured for a SNP for the crossbred performance trait. First, the two parental alleles at a SNP in a...