Use of the numerator relationship matrix in genetic analysis of autopolyploid species

Mixed models incorporating the inverse of a numerator relationship matrix (NRM) are widely used to estimate genetic parameters and to predict breeding values in animal breeding. A simple and quick method to directly calculate the inverse of the NRM has been historically developed for diploid animal species. Mixed models are less used in plant breeding partly because the existing method for diploids is not applicable to autopolyploid species. This is because of the phenomenon of double reduction and the possibility that gametes carry alleles which are identical by descent. This paper generalises the NRM and its inverse for autopolyploid species, so it can be easily incorporated into their genetic analysis. The technique proposed is to first calculate the kinship coefficient matrix and its inverse as a precursor to calculating the NRM and its inverse. This allows the NRM to be calculated for populations containing individuals of mixed ploidy levels. This generalization can also accommodate uncertain parentage by generating the "average" relationship matrix. The possibility that non-inbred parents can produce inbred progeny (double reduction) is also discussed. Rules are outlined that are applicable for any level of ploidy. Examples of use of the matrix are provided using simulated pedigrees.