Sigmoid allometries generate male dimorphism in secondary sexual traits: a comment on Packard (2023)

The detection of male dimorphism has seen numerous statistical advances. Packard has recently criticized a widely used method, reanalyzing data from beetles and harvestmen using an alternative method. We disagree with Packard conclusions, probably due to different implicit definitions of male dimorphism. We consider that male dimorphism manifests in a distribution when it is significantly better described by a model with two values of central tendency (bimodality), rather than a model with only one (unimodality). Thus, while Packard suggests sigmoid allometries as alternatives to male dimorphism, we argue that such allometries are manifestations of mechanisms that generate bimodal distributions. Instead of focusing on this dichotomy, we propose an approach to test whether bimodality in a trait simply arises from its allometry by: (1) characterizing the trait static allometry, (2) simulating body size values based on original data parameters, and (3) generating new trait sizes using the static allometries. The percentage of simulations generating equal or greater bimodality than the data represents the likelihood that the bimodality can be explained by the allometry alone. Our method offers a null model linking sigmoid allometries and bimodal distributions, providing a test for mechanisms that accentuate trait bimodality beyond what the trait allometry generates.