Statistics of sexual size dimorphism

In comparative studies of sexual size dimorphism (SSD), the methods used to quantify dimorphism are controversial. SSD is commonly expressed as a ratio between species mean values of males and females, such as M/F or (M-F)/([M+F]/2), but a number of investigators have suggested that ratios should not be used, mainly because their distributions usually violate the assumptions of parametric statistical tests, or because they lead to spurious relationships that invalidate the interpretation and statistical significance of regressions and correlations. As an alternative to ratios, the comparative study of SSD can be conducted by a combination of regression with sex-specific data and residuals from this regression. Twenty-five data sets were selected from the literature and used to duplicate a variety of statistical procedures commonly employed in studies of SSD. All analyses were repeated with five different ratios and with methods that avoid the calculation of any ratios. These data and a review of the statistical properties of ratios and residuals indicate that: (1) most of the ratios used in the SSD literature are unnecessary, and several commonly used ratios are statistically inferior to others. Only two ratios are needed, one on a logarithmic scale and one on a linear scale; (2) there is no problem with spurious correlation or non-normality when ratios are used in several types of statistical procedures commonly employed in studies of SSD; (3) residuals cannot replace ratios for the evaluation of many questions regarding the pattern of SSD among species; and (4) residuals usually are used incorrectly, leading to misspecified regression equations. Most of the questions for which residuals are used should be addressed by multiple regression. These results apply to studies using comparative methods with or without adjustments for phylogenetic effects.