Comparing Covariance Matrices by Relative Eigenanalysis, with Applications to Organismal Biology

Abstract

Most biologists are familiar with principal component analysis as an ordination tool for questions about within-group or between-group variation in systems of quantitative traits, and with multivariate analysis of variance as a tool for one useful description of the latter in the context of the former. Less familiar is the mathematical approach of relative eigenanalysis of which both of these are special cases: computing linear combinations for which two variance–covariance patterns have maximal ratios of variance. After reviewing this common algebraic–geometric core, we demonstrate the effectiveness of this exploratory approach in studies of developmental canalization and the identification of divergent and stabilizing selection. We further outline a strategy for statistical classification when group differences in variance dominate over differences in group averages.

Appendix

Corresponding to the maneuvers here is a null model that is often helpful in deciding whether a relative eigenanalysis reveals anything worth thinking about. We’re not testing the ratio, only asking that it be at least as large as the value expected on the absence of any signal. The approach parallels the analogous decision regarding eigenvectors arising from successive eigenvalues of one single matrix (Coquerelle et al. 2011; Bookstein 2014). That approach, in turn, is a modification of the stepdown test for sphericity that is already in the advanced textbooks (cf. Morrison 1976, pp. 336–337). As this argument has not apparently been published before, we sketch it here for any interested reader. Our notation is borrowed from Anderson (1963), where several of the corresponding asymptotic likelihood ratio tests were published.

Suppose we are observing a covariance matrix on a sample of size N for a list of p variables that are all actually independent Gaussians of mean 0 and variance 1. Let the matrix $\mathbf{U}$ be $\sqrt{N}$ times the deviation of the empirically observed covariance matrix from the correct answer, which is the identity matrix of rank p. As samples grow large, $\mathbf{U}$ becomes Gaussian with mean zero for every element and variance 2 along its diagonal, 1 elsewhere.

That theorem corresponds to a $\mathbf{U}$ that was generated to describe some multivariate Gaussian distribution’s ordinary eigenvalues. The relative eigenvalues, which we have been depicting here as the eigenvalues of $\mathbf{T}^{-1/2}\mathbf{S}\mathbf{T}^{-1/2}$, are also the eigenvalues of $\mathbf{S}\mathbf{T}^{-1}$. For large samples, the distribution of the deviation of $\mathbf{T}^{-1}$ from the identity is the same as the distribution of the deviation of $\mathbf{T}$. The effect of this additional factor of $\mathbf{T}^{-1}$ on what was already the deviation of the sample described by $\mathbf{S}$ is to alter $\mathbf{U}$ by another term, additive in this metric, of exactly the same distribution. Their sum, scaled by $\sqrt{N}$, thus is in the limit a set of Gaussians with terms of mean zero and variances 4 down the diagonal, 2 elsewhere.

At equation 3.9, pages 132–133 of this same paper, Anderson shows, by expressing the eigenvalues themselves in terms of the elements of $\mathbf{U}$, that for the ordinary eigenproblem the log likelihood ratio for the null hypothesis of sphericity for any q consecutive eigenvalues (not necessarily the full set of all p corresponding to the p original variables) is the quantity log a/g, where a is the arithmetic mean of the eigenvalues and g is their geometric mean. Using the theorem about $\mathbf{U}$ for ordinary eigenvalues, he shows that in the limit of large samples this log likelihood ratio is distributed approximately as 1/Nq times a χ² on (q − 1)(q + 2)/2 degrees of freedom, where q is the number of eigenvalues being compared (in our applications, usually, q = 2). It follows, then, that in the relative eigenanalysis application the same quantity log a/g is distributed as 2/Nq times the same χ² (Fig. 11).

Fig. 11

From the expected value of the corresponding χ² distributions comes a permissive criterion for narrative validity of relative eigenvectors: restrict the text only to those for which the relative eigenvalue λ _i has a ratio to its successor λ _i+1 as least as large as the value given in this diagram. Solid line curve for ordinary eigenvalues. Dotted line curve for relative eigenvalues