An effect size for comparing the strength of morphological integration across studies

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, $$ . First, we evaluate several eigenvalue dispersion indices under various conditions, and show that only $$ remains stable across samples size and the number of variables. We then demonstrate that $$ accurately characterizes input patterns of covariation, so long as redundant dimensions are excluded from the calculations. However, we also show that the variance of the sampling distribution of $$ depends on input levels of trait covariation, making $$ unsuitable for direct comparisons. As a solution, we propose transforming $$ 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.