Bending over Backwards: Better Estimates of Genetic Covariance Matrices by Penalized REML

Knowledge of genetic parameters and variances is an essential pre-requisite for tasks such as the design of selection programmes or prediction of breeding values. Reliable estimation of these quantities is thus paramount. There is a growing trend to consider more and more complex phenotypes, necessitating multivariate analyses comprising numerous traits. Problems inherent in such analyses, arising from sampling variation and the resulting over-dispersion of sample eigenvalues, are well known. There has been longstanding interest in the 'regularization' of estimated covariance matrices. Generally, this involves a compromise between additional bias and reduced sampling variation of 'improved' estimators. Numerous simulation studies have demonstrated that this can improve the agreement between estimated and population covariance matrices; see Meyer and Kirkpatrick (2010) for a review. For instance, estimators of covariance matrices have been suggested which counter-act upwards bias of the largest and downwards bias of the smallest eigenvalues by shrinking them towards their mean. In quantitative genetic analyses, we attempt to partition covariances into their genetic and environmental components.