A Comparative Analysis of Integration Indices

The degree of integration of multidimensional phenotypes has an important place in evolutionary biology, pertaining to the structure of variation that is available for natural selection to work on and therefore to the evolutionary potential of the phenotype. Various indices have been suggested in the literature for measuring integration level, yet their statistical properties have remained mostly unstudied to date. In this study, I used simulations and resampling procedures in order to compare the distributions and sampling properties of different indices. I simulated heterogeneous correlation matrices that ranged widely in their integration level. I applied non-parametric bootstrapping to explore the effect of sampling on recovering the true integration value of these matrices. In addition, I generated the statistical power space for one of the integration indices-the relative standard deviation of the eigenvalues. The results show that the relative variance of eigenvalues maps exactly onto the mean coefficient of determination, and that the index suggested by Hansen and Houle (J Evol Biol 21:1201-1219, 2008) is the same as Van Valen's (J Theor Biol 45:235-247, 1974) redundancy index, both of which have some undesirable sampling properties that render them less useful in most practical situations. Based on the power analysis, a sample of 30-40 specimens can be considered a sufficient minimum for most studies. The R codes provided here can be utilized by other researchers to yield case-specific insights.