TOOLS AND TECHNIQUES A Comparative Analysis of Integration Indices Annat Haber Received: 15 February 2011 / Accepted: 18 August 2011 Ó Springer Science+Business Media, LLC 2011 Abstract 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 sug- gested 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 pro- cedures 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 stan- dard 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. Keywords Correlation Á Morphological integration Á Modularity Á Quantitative genetics Á Statistical power Introduction The study of multivariate covariation between characters has received increasing attention in evolutionary biology in the last three decades, relating to issues such as recon- structing selection, genotype–phenotype mapping, con- straints, evolvability, modularity, and morphological integration (e.g., Lande and Arnold 1983; Cheverud 1982, 1984; Wagner 1984a, 1988; Zelditch 1988; Zeng 1988; Schluter 1996; Wagner and Altenberg 1996; Magwene 2001; Steppan et al. 2002; Schlosser 2004; Hallgrı ´msson et al. 2005; Draghi and Wagner 2008; Zelditch et al. 2008, 2009; Agrawal and Stinchcombe 2009), as well as phylo- genetic reconstruction (e.g., Emerson and Hastings 1998; Fu and Murphy 1999; O’Keefe and Wagner 2001). The overall level of correlation of a multivariate distribution, also known as the integration level, has been of particular interest, because it reflects the tendency of the system to generate variation that is biased in particular directions (Van Valen 1974; Wagner 1984b, 1988; Hansen and Houle 2008). A more spherical, less eccentric, distribution, reflecting a lower integration level, describes a system in which variation occurs more equally in all directions. In the context of evolutionary biology, such a system has the potential to respond to a wider range of selection vectors, but at the same time entails less coordination between characters; this perhaps leads to a reduced ability to Electronic supplementary material The online version of this article (doi:10.1007/s11692-011-9137-4) contains supplementary material, which is available to authorized users. A. Haber Committee on Evolutionary Biology, University of Chicago, 1025 E. 55th St Culver Hall, Chicago, IL 60637, USA Present Address: A. Haber (&) Department of Zoology, Tel Aviv University, Tel-Aviv 69978, Israel e-mail: annat22@gmail.com 123 Evol Biol DOI 10.1007/s11692-011-9137-4