Interspecific allometry for sexual shape dimorphism: Macroevolution of multivariate sexual phenotypes with application to Rensch's rule

Allometric trends in the degree of sexual dimorphism with body size have long fascinated evolutionary biologists. Many male-biased clades display more prominent sexual dimorphism in larger taxa (Rensch's rule), with most examples documenting this pattern for body size dimorphism. Although sexual dimorphism in traits other than body size is equally functionally relevant, characterizing allometric patterns of sexual dimorphism in such traits is hampered by lack of an analytical framework that can accommodate multivariate phenotypes. In this article, we derive a multivariate equivalency for investigating trends in sexual dimorphism-relative to overall body size-across taxa and provide a generalized test to determine whether such allometric patterns correspond with Rensch's rule. For univariate linear traits such as body size, our approach yields equivalent results to those from standard procedures, but our test is also capable of detecting trends in multivariate datasets such as shape. Computer simulations reveal that the method displays appropriate statistical properties, and an empirical example in Mediterranean lizards provides the first demonstration of Rensch's rule in a multivariate phenotype (head shape). Our generalized procedure substantially extends the analytical toolkit for investigating macroevolutionary patterns of sexual dimorphism and seeking a better understanding of the processes that underlie them.