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.