Originally published as Genetics Published Articles Ahead of Print on August 30, 2008.

Genetics, Vol. 180, 1153-1166, October 2008, Copyright © 2008
doi:10.1534/genetics.108.090159

Perils of Parsimony: Properties of Reduced-Rank Estimates of Genetic Covariance Matrices

Karin Meyer*,1 and Mark Kirkpatrick{dagger}

* Animal Genetics and Breeding Unit,2, University of New England, Armidale NSW 2351, Australia and {dagger} Section of Integrative Biology, University of Texas, Austin, Texas 78712

1 Corresponding author: Animal Genetics and Breeding Unit, University of New England, Armidale NSW 2351, Australia.
E-mail: kmeyer{at}didgeridoo.une.edu.au

Eigenvalues and eigenvectors of covariance matrices are important statistics for multivariate problems in many applications, including quantitative genetics. Estimates of these quantities are subject to different types of bias. This article reviews and extends the existing theory on these biases, considering a balanced one-way classification and restricted maximum-likelihood estimation. Biases are due to the spread of sample roots and arise from ignoring selected principal components when imposing constraints on the parameter space, to ensure positive semidefinite estimates or to estimate covariance matrices of chosen, reduced rank. In addition, it is shown that reduced-rank estimators that consider only the leading eigenvalues and -vectors of the "between-group" covariance matrix may be biased due to selecting the wrong subset of principal components. In a genetic context, with groups representing families, this bias is inverse proportional to the degree of genetic relationship among family members, but is independent of sample size. Theoretical results are supplemented by a simulation study, demonstrating close agreement between predicted and observed bias for large samples. It is emphasized that the rank of the genetic covariance matrix should be chosen sufficiently large to accommodate all important genetic principal components, even though, paradoxically, this may require including a number of components with negligible eigenvalues. A strategy for rank selection in practical analyses is outlined.