Recent Developments in Variance and Covariance Estimations

Estimation of variances and covariances by equating quadratics and bilinear forms from analysis of variance and covariance to their expectations have been widely used in animal breeding. These methods lead to biased estimators if the data arise from populations that have been undergoing selection. Further, newer methods exist that are capable of yielding smaller sampling variance and mean squared errors. These are maximum likelihood, restricted maximum likelihood and minimum variance quadratic unbiased estimation. Widely used statistical packages incorporate these methods, but only for the traditional mixed model for variance components. They do not include methods for correlated variables (related animals, for example) nor do they handle multitrait models. This paper describes very general and powerful methods for these more general models. They appear to control very well on bias due to selection, and two of them guarantee that estimates of variances will be positive and estimated covariance matrices will be positive definite. Several examples of these methods are presented.