A penalized likelihood approach to pooling estimates of covariance components from analyses by parts

Estimates of covariance matrices for numerous traits are commonly obtained by pooling results from a series of analyses of subsets of traits. A penalized maximum-likelihood approach is proposed to combine estimates from part analyses while constraining the resulting overall matrices to be positive definite. In addition, this provides the scope for improving' estimates of individual matrices by applying a penalty to the likelihood aimed at borrowing strength from their phenotypic counterpart. A simulation study is presented showing that the new method performs well, yielding unpenalized estimates closer to results from multivariate analyses considering all traits, than various other techniques used. In particular, combining results for all sources of variation simultaneously minimizes deviations in phenotypic estimates if sampling covariances can be approximated. A mild penalty shrinking estimates of individual covariance matrices towards their sum or estimates of canonical eigenvalues towards their mean proved advantageous in most cases. The method proposed is flexible, computationally undemanding and provides combined estimates with good sampling properties and is thus recommended as alternative to current methods for pooling.