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.

The POTATOPLAN project aims to adapt comprehensive genetic evaluation systems used in forest tree and livestock breeding to potatoes. These systems use Best Linear Unbiased Prediction (BLUP) to incorporate all measurements from all relatives and correlated traits to best predict the additive and total genetic values of all genotypes for all traits. This enables optimal selection of genotypes for breeding and deployment as any genotype at any stage of testing can be compared with any other, including commercial varieties. The method can remove biases due to natural and artificial environmental variation while accounting for GxE, genetic trends over time due to selection, differences in data amount and quality, and varying sampling procedures. The models have been adapted to potatoes by estimating additive genetic effects using an additive relationship matrix that recognises the polyploid nature of most potato varieties. Family merit score counts (number of selected progeny) are converted into a binary selection trait for the genotypes in the family, with a value of 1 for selected named progeny and 0 for unselected progeny. This trait is used to predict the value of the selected progeny for all traits that contribute to the merit score through the genetic correlations determined on an experimental basis. Measurements of genotype performance using different plot or sample sizes is accounted for by using weighted analysis with the individual plant or standard sample size error variance as the reference point. These approaches should allow the more widespread adoption of BLUP methods in potato and other crop breeding. POTATOPLAN also includes a comprehensive database of pedigree, measurements, trial designs, genetic and other model parameters, and resultant genetic values to make them useable for breeders and growers.