Restricted maximum likelihood procedures for the estimation of additive and nonadditive genetic variances and covariances in multibreed populations.

Restricted maximum-likelihood procedures were developed to estimate additive and nonadditive genetic and environmental covariances for multiple traits in multibreed populations. The computational procedure follows the expectation-maximization (EM) algorithm, where the set of equations in the maximization step is solved by successive approximations. This computational procedure does not guarantee convergence to a symmetric positive-definite covariance matrix. Thus, computer programs will need to incorporate restrictions in the maximization step to ensure positive definiteness of each covariance matrix. Additive genetic and environmental covariances were modeled in subclass form (zeros and ones in the design matrices). Nonadditive genetic covariances were modeled in regression form (any value between and including zero and one in the design matrices). Computational requirements will be larger than for intrabreed analyses. Appropriate simplifying assumptions and numerical techniques (e.g., sparse and iterative numerical techniques) will be required for the implementation of these multibreed covariance estimation procedures. Number of iterations (5 to 12) and computing times (57 to 113 min) to achieve convergence when estimating 21 genetic and environmental covariances in five small simulated multibreed data sets (two breeds, 25,200 to 50,400 calves, 120 to 135 unrelated bulls) suggest that these procedures are computationally feasible.