RT Journal Article
SR Electronic
T1 Statistics of eigenvalue dispersion indices: quantifying the magnitude of phenotypic integration
JF bioRxiv
FD Cold Spring Harbor Laboratory
SP 2021.06.19.449119
DO 10.1101/2021.06.19.449119
A1 Junya Watanabe
YR 2021
UL http://biorxiv.org/content/early/2021/06/19/2021.06.19.449119.abstract
AB 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.Competing Interest StatementThe authors have declared no competing interest.