Computing procedures for genetic evaluation including phenotypic, full pedigree, and genomic information

摘要

Currently, genomic evaluations use multiple-step procedures, which are prone to biases and errors. A single-step procedure may be applicable when genomic predictions can be obtained by modifying the numerator relationship matrix A to H = A + A_Δ, where A_Δ includes deviations from expected relationships. However, the traditional mixed model equations require H~(-1) which is usually difficult to obtain for large pedigrees. The computations with H are feasible when the mixed model equations are expressed in an alternate form that also applies for singular H and when those equations are solved by the conjugate gradient techniques. Then the only computations involving H are in the form of Aq or A_Δq, where q is a vector. The alternative equations have a nonsymmetric left-hand side. Computing A_Δq is inexpensive when the number of nonzeros in A_Δ is small, and the product Aq can be calculated efficiently in linear time using an indirect algorithm. Generalizations to more complicated models are proposed. The data included 10.2 million final scores on 6.2 million Holsteins and were analyzed by a repeatability model. Comparisons involved the regular and the alternative equations. The model for the second case included simulated A_Δ. Solutions were obtained by the preconditioned conjugate gradient algorithm, which works only with symmetric matrices, and by the bi-conjugate gradient stabilized algorithm, which also works with nonsymmetric matrices. The convergence rate associated with the nonsymmetric solvers was slightly better than that with the symmetric solver for the original equations, although the time per round was twice as much for the nonsymmetric solvers. The convergence rate associated with the alternative equations ranged from 2 times lower without A_Δ to 3 times lower for the largest simulated A_Δ. When the information attributable to genomics can be expressed as modifications to the numerator relationship matrix,rnthe proposed methodology may allow the upgradingrnof an existing evaluation to incorporate the genomicrninformation.