Understanding how and why phenotypic traits covary is a major interest in evolutionary biology. Biologists have long sought to characterize the extent of morphological integration in organisms, but comparing levels of integration for a set of traits across taxa has been hampered by the lack of a reliable summary measure and testing procedure. Here we propose a standardized effect size for this purpose, calculated from the relative eigenvalue variance, Vrel. First we evaluate several eigenvalue dispersion indices under various conditions, and show that only Vrel remains stable across samples size and the number of variables. We then demonstrate that Vrel accurately characterizes input patterns of covariation, so long as redundant dimensions are excluded from the calculations. However, we also show that the variance of the sampling distribution of Vrel depends on input levels of trait covariation, making Vrel unsuitable for direct comparisons. As a solution, we propose transforming Vrel to a standardized effect size (Z-score) for representing the magnitude of integration for a set of traits. We also propose a two-sample test for comparing the strength of integration between taxa, and show that this test displays appropriate statistical properties. We provide software for implementing the procedure, and an empirical example illustrates its use.

An HDI-120 surface scanner was used to capture 3D surface scans of individual os coxa from adult specimens, and from each surface scan, the locations of eight three-dimensional landmarks were digitized using the program Landmark. Landmark configurations were then subjected to a Generalized Procrustes Analysis using the R-package geomorph, and a set of seven linear distance measures were then derived from the Procrustes-aligned coordinates. Additionally, we multiplied the Procrustes coordinates by size, and then calculated a second set of linear distance measures from these size-incorporated coordinates. 

Files may be opened and processed in R, using the package geomorph.