Covariance Structures for Quantitative Genetic Analyses

INTRODUCTION Covariance matrices in quantitative genetic analyses have, by and large, been considered ‘unstructured’, i.e. for q random variables, there are q(q + 1)/2 distinct covariance components. This implies that the number of parameters to be estimated increases quadratically with the number of variables. Multivariate analyses involving more than a few traits have been hampered by computational problems. Recent improvements in computer hardware, both speed and memory available, have made analysis of larger data sets and models feasible. In addition, methodology to estimate covariance components has seen substantial progress. For restricted maximum likelihood (REML) estimation in particular, there are now fast and reliable algorithms available, capable of dealing with analyses involving higher dimensional covariance matrices among numerous traits or random regression coefficients. However, computational problems aside, an inherent problem remains : with many parameters to be estimated we rarely have sufficient data to support accurate estimation of all the elements of unstructured covariance matrices. Attempts to improve efficiency of multivariate estimation fall into two broad categories, ‘shrinkage’ and estimation assuming covariance matrices have a certain structure. This paper reviews approaches to structured estimation relevant to quantitative genetic analyses.