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J Dairy Sci. 2009 Sep;92(9):4648-55. doi: 10.3168/jds.2009-2064.

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

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Department of Animal and Dairy Science, University of Georgia, Athens, Georgia 30602, USA.


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(Delta), where A(Delta) 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(Delta)q, where q is a vector. The alternative equations have a nonsymmetric left-hand side. Computing A(Delta)q is inexpensive when the number of nonzeros in A(Delta) 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(Delta). 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(Delta) to 3 times lower for the largest simulated A(Delta). When the information attributable to genomics can be expressed as modifications to the numerator relationship matrix, the proposed methodology may allow the upgrading of an existing evaluation to incorporate the genomic information.

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