Statistics of eigenvalue dispersion indices: quantifying the magnitude of phenotypic integration

Abstract

Quantification of the magnitude of covariation plays a major role in the studies of phenotypic integration, for which statistics based on dispersion of eigenvalues of a covariance or correlation matrix—eigenvalue dispersion indices—are commonly used. However, their use has been hindered by a lack of clear understandings on their statistical meaning and sampling properties such as the magnitude of sampling bias and error. This study remedies these issues by investigating properties of these statistics with both analytic and simulation-based approaches. The relative eigenvalue variance of a covariance matrix is known in the statistical literature as a test statistic for sphericity, thus is an appropriate measure of eccentricity of variation. The same of a correlation matrix is exactly equal to the average squared correlation, thus is a clear measure of overall integration. Exact and approximate expressions for the mean and variance of these statistics are analytically derived for the null and arbitrary conditions under multivariate normality, clarifying the effects of sample size N, number of variables p, and parameters on the sampling bias and error. Accuracy of the approximate expressions are evaluated with simulations, confirming that most of them work reasonably well with a moderate sample size (N ≥ 16–64). Importantly, sampling properties of these indices are not adversely affected by high p:N ratio, promising their utility in high-dimensional phenotypic analyses. These statistics can potentially be applied to shape variables and phylogenetically structured data, for which necessary assumptions and modifications are presented.

Detecting (non)parallel evolution in multidimensional spaces: angles, correlations, and eigenanalysis

“…Equation A.8 holds for any correlation matrix in general (e.g., Gleason & Staelin, 1975;Durand & Le Roux, 2017;Watanabe, 2021). Also, by equation A.3, n i=1 l m i = p i=1 k m i holds for any integer m. Therefore,…”

“…where V is a p × p matrix of eigenvectors, and Concentration of variation in a trait space is often measured by dispersion of eigenvalues (Cheverud et al, 1983;Wagner, 1984;Pavlicev et al, 2009a;Haber, 2011;Watanabe, 2021)…”

“…10) and hence has the mean 1/p and variance 2(p − 1)/p 2 (p + 2). Furthermore, it is possible to show that r 2 ij 's are uncorrelated with one another under the null hypothesis (Schott, 2005;Watanabe, 2021). Therefore, the expectation and variance of the sum of squared correlations are:…”

Evolvability and Macroevolution: Overview and Synthesis

“…Such instability seems to limit the macroevolutionary role of this phenomenon, but the several potential mechanisms behind it have not been explored. Further, the orientation and eccentricity of larger-scale multivariate variation is stable at higher levels within larger clades, and is also associated with evolutionary trajectories (Haber, 2016;Watanabe, 2018Watanabe, , 2022. Thus, the potential role of speciation rates in evolvability may depend on the eccentricity of the variational envelope around the taxa within a clade, the resistance of that envelope to external pressures, and the relative constancy of phenotypic changes at speciation within and among clades.…”

Solving the grand challenge of phenotypic integration: allometry across scales

“…Phenotypic integration (i.e. how complex organisms are built and how phenotypic traits are connected and coordinated) has been examined from a wide variety of approaches and research fields, including physiology, evolutionary biology, ecology and allometry (see for instance Schlichting and Pigliucci 1995;Pigliucci et al 1996;Klingenberg 1996;Murren 2002;Pigliucci 2003;Klingenberg and Marugan-Lobon 2013;Klingenberg 2014;Klingenberg 2016;Watanabe 2022). However, joint empirical efforts at the interface of those disciplines warrant further study.…”