Sparse inverse of covariance matrix of QTL effects with incomplete marker data

Gametic models for fitting breeding values at QTL as random effects in outbred populations have become popular because they require few assumptions about the number and distribution of QTL alleles segregating. The covariance matrix of the gametic effects has an inverse that is sparse and can be constructed rapidly by a simple algorithm, provided that all individuals have marker data, but not otherwise. An equivalent model, in which the joint distribution of QTL breeding values and marker genotypes is considered, was shown to generate a covariance matrix with a sparse inverse that can be constructed rapidly with a simple algorithm. This result makes more feasible including QTL as random effects in analyses of large pedigrees for QTL detection and marker assisted selection. Such analyses often use algorithms that rely upon sparseness of the mixed model equations and require the inverse of the covariance matrix, but not the covariance matrix itself. With the proposed model, each individual has two random effects for each possible unordered marker genotype for that individual. Therefore, individuals with marker data have two random effects, just as with the gametic model. To keep the notation and the derivation simple, the method is derived under the assumptions of a single linked marker and that the pedigree does not contain loops. The algorithm could be applied, as an approximate method, to pedigrees that contain loops.