|Thallman, Richard - Mark|
|Van Vleck, Lloyd|
Submitted to: Statistical Applications in Genetics and Molecular Biology
Publication Type: Peer Reviewed Journal
Publication Acceptance Date: 3/25/2004
Publication Date: 5/1/2005
Citation: Thallman, R.M., Hanford, K.J., Kachman, S.D., Van Vleck, L.D. 2004. Sparse inverse of covariance matrix of QTL effects with incomplete marker data. Statistical Applications in Genetics and Molecular Biology 3(1)Art. 30. Interpretive Summary: Statistical models in which breeding values at quantitative trait loci (QTL) in outbred populations are fit as random effects have become popular because they require few assumptions about the number and distribution of QTL alleles segregating in the populations. Previous research has provided computationally efficient methods to use such models for the inclusion of QTL in genetic evaluations of livestock, provided that marker data is complete (available for each individual in the pedigree). However, complete marker data is almost never the case in practical applications. This paper introduces a concept that allows a computationally efficient method for the random QTL model with incomplete marker data, provided the pedigree has a simple structure, which also occurs rarely in practical applications. Nonetheless, the concept presented herein lays the foundation for future research that will allow the use of the random QTL model in complex pedigrees with incomplete marker data.
Technical Abstract: 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. It is shown that an equivalent model, in which the joint distribution of QTL breeding values and marker genotypes is considered, generates a covariance matrix with a sparse inverse that can be constructed rapidly by a simple algorithm. This makes it more feasible to include QTL as random effects in analyses that depend on sparseness of the mixed model equations. In the proposed model, each individual has two random effects for each unordered marker genotype that is possible for that individual. Therefore, individuals with marker data have two random effects, just as in 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. Extensions to the algorithm to relax these assumptions seem feasible.