title: Genetic evaluation for three-way crossbreeding.
abstract: BACKGROUND
  Commercial pig producers generally use a terminal crossbreeding system with three breeds. Many pig breeding organisations have started to use genomic selection for which genetic evaluation is often done by applying single-step methods for which the pedigree-based additive genetic relationship matrix is replaced by a combined relationship matrix based on both marker genotypes and pedigree. Genomic selection is implemented for purebreds, but it also offers opportunities for incorporating information from crossbreds and selecting for crossbred performance. However, models for genetic evaluation for the three-way crossbreeding system have not been developed.
  
  
  RESULTS
  Four-variate models for three-way terminal crossbreeding are presented in which the first three variables contain the records for the three pure breeds and the fourth variable contains the records for the three-way crossbreds. For purebred animals, the models provide breeding values for both purebred and crossbred performances. Heterogeneity of genetic architecture between breeds and genotype by environment interactions are modelled through genetic correlations between these breeding values. Specification of the additive genetic relationships is essential for these models and can be defined either within populations or across populations. Based on these two types of additive genetic relationships, both pedigree-based, marker-based and combined relationships based on both pedigree and marker information are presented. All these models for three-way crossbreeding can be formulated using Kronecker matrix products and therefore fitted using Henderson's mixed model equations and standard animal breeding software.
  
  
  CONCLUSIONS
  Models for genetic evaluation in the three-way crossbreeding system are presented. They provide estimated breeding values for both purebred and crossbred performances, and can use pedigree-based or marker-based relationships, or combined relationships based on both pedigree and marker information. This provides a framework that allows information from three-way crossbred animals to be incorporated into a genetic evaluation system.
authors.affiliation: Department of Molecular Biology and Genetics, Center for Quantitative Genetics and Genomics, Aarhus University, Blichers Allé 20, P.O. BOX 50, 8830, Tjele, Denmark. olef.christensen@mbg.au.dk.
authors.forename: Ole F
authors.initials: OF
authors.lastname: Christensen
authors.affiliation: INRA, UMR 1388 GenPhySE, BP52627, 31326, Castanet Tolosan, France. andres.legarra@toulouse.inra.fr.
authors.forename: Andres
authors.initials: A
authors.lastname: Legarra
authors.affiliation: Department of Molecular Biology and Genetics, Center for Quantitative Genetics and Genomics, Aarhus University, Blichers Allé 20, P.O. BOX 50, 8830, Tjele, Denmark. mogens.lund@mbg.au.dk.
authors.forename: Mogens S
authors.initials: MS
authors.lastname: Lund
authors.affiliation: Department of Molecular Biology and Genetics, Center for Quantitative Genetics and Genomics, Aarhus University, Blichers Allé 20, P.O. BOX 50, 8830, Tjele, Denmark. guosheng.su@mbg.au.dk.
authors.forename: Guosheng
authors.initials: G
authors.lastname: Su
mesh_terms: D000465:Algorithms; D000818:Animals; D001947:Breeding; D003433:Crosses, Genetic; D019143:Evolution, Molecular; D005260:Female; D023281:Genomics / Q000379:methods; D008297:Male; D008957:Models, Genetic; D010375:Pedigree; D015203:Reproducibility of Results; D013552:Swine