When perception isn’t reality: allometric variation in the exaggerated mandibles of male stag beetles (Coleoptera: Lucanidae)

Abstract

Abstract A variety of protocols have been used to study allometric variation in size of the exaggerated mandibles on male stag beetles. Many of these protocols entail logarithmic transformation of the original measurements followed by numerical analysis of the transformations by linear regression or some conceptual extension thereof. I reanalysed data from four such studies to show how these protocols can lead investigators to conclusions that are not well supported by the original observations. One of the data sets was originally reported to conform to simple loglinear allometry, with untransformed observations that presumably follow the path of a two-parameter power function; one was said to represent biphasic, loglinear allometry, with two distinctive morphs having different scaling relationships on the arithmetic scale; and two were originally described as cases of discontinuous, loglinear allometry caused by dimorphisms. My analyses, which were based on graphical analysis and nonlinear regression of untransformed observations, revealed that all the data sets form S-shaped distributions and that each of the distributions is well described by a four-parameter sigmoid function. None of the bivariate distributions reveals a discontinuity or dimorphism. Thus, the original authors unknowingly offered descriptions and interpretations for patterns of variation that do not exist in their data.