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

Abstract

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.

Dominance Genetic Variance for Traits Under Directional Selection inDrosophila serrata

“…Statistical support for the genetic dimensions underlying the additive genetic covariance matrix (A) was evaluated using nested log-likelihood ratio tests for the reduced-rank models (Hine and Blows 2006). Typically, reduced-rank estimation is carried out using factor analytic modeling (e.g., Hine and Blows 2006;McGuigan and Blows 2010;Sztepanacz and Rundle 2012): these analyses are equivalent to factor analytic models in which all specific variances are assumed to be zero (Meyer and Kirkpatrick 2008).…”

“…This is likely a consequence of restricting significant variance to such few dimensions (see Results). Assuming that all specific variances are zero when considering too few factors can result in biased estimates of residual components, with principal component models typically behaving more poorly for highly restricted ranks than factor analytic models (Meyer and Kirkpatrick 2008).…”

Comparing the intersex genetic correlation for fitness across novel environments in the fruit fly, Drosophila serrata

“…The inbred line  reference line interaction was significant (see Results) and a reduced-rank covariance matrix at this level was therefore included in all models (likelihood convergence problems in some models prevented the fitting of an unconstrained covariance matrix at this level). This reduced-rank matrix was fixed at five dimensions in all cases, corresponding to the number of dimensions with nonzero eigenvalues at this level, to avoid problems that can arise from fitting fewer dimensions than exist in the data (Meyer and Kirkpatrick, 2008).…”

Parameter expansion for estimation of reduced rank covariance matrices(Open Access publication)

“…As results show, this can increase the number of REML iterates required. Moreover, estimates of both the directions and eigenvalues of the subset of PCs fitted tend to be biassed in this case [30].…”