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

Abstract

Quantification of the magnitude of trait covariation plays a pivotal role in the study of phenotypic evolution, for which statistics based on dispersion of eigenvalues of a covariance or correlation matrix--eigenvalue dispersion indices--are commonly used. This study remedies major issues over the use of these statistics, namely, a lack of clear understandings on their statistical justifications and sampling properties. The relative eigenvalue variance of a covariance matrix is known in the statistical literature a test statistic for sphericity, thus is an appropriate measure of eccentricity of variation. The same of a correlation matrix is equal to the average squared correlation, which has a straightforward interpretation as a measure of integration. Expressions for the mean and variance of these statistics are analytically derived under multivariate normality, clarifying the effects of sample size N, number of variables p, and parameters on sampling bias and error. Simulations confirmed that approximations involved are reasonably accurate with a moderate sample size (N [≥] 16-64). Importantly, sampling properties of these indices are not adversely affected by a high p:N ratio, promising their utility in high-dimensional phenotypic analyses. They can furthermore be applied to shape variables and phylogenetically structured data with appropriate modifications.