Unconstrained procedures for the estimation of positive definite covariance matrices using restricted maximum likelihood in multibreed populations

Two unconstrained procedures to ensure that intrabreed and interbreed genetic and environmental covariance estimates from multibreed populations are computed within the permissible ranges were developed. These procedures were called Partial Scoring and Cholesky Maximization. The Partial Scoring procedure uses partial steps to keep estimates of covariance matrices positive definite at each expectation-maximization (EM) iteration, and the Cholesky Maximization procedure achieves the same goal by computing the elements of the Cholesky Decomposition of each intrabreed and interbreed genetic and environmental covariance matrix. Groups of small simulated data sets containing either direct genetic effects for two traits (90 bulls, 13,500 calves) or direct and maternal genetic effects for a single trait (135 bulls, 32,400 calves) were used to test the computational feasibility of these two procedures. The overall means (and ranges) of the numbers of expectation-maximization iterations, times to convergence, and accuracy of estimation were 10 (2 to 184), 26.2 min (4.1 to 773.2 min), and 40.1% (12.7 to 81.9%) for the Partial Scoring procedure and 7 (3 to 37), 16.7 min (9.5 to 64.6 min), and 37.8% (3.1 to 67.8%) for the Cholesky Maximization procedure. Although the overall accuracy of both procedures was similar, the Cholesky Maximization procedure should be preferred because it converged faster and its covariance estimates were less affected by the values of the covariance priors than those computed using the Partial Scoring strategy. Application to large unbalanced multibreed data sets will require an iterative version of these procedures.