Allometric variation in the antlers of cervids: a comment on Lemaître et al

What is complex allometry?

Complex allometry describes a smooth, curvilinear relationship between logarithmic transformations of a biological variable and a corresponding measure for body size when the observations are displayed on a bivariate graph with linear scaling. The curvature in such a display is commonly captured by fitting a quadratic equation to the distribution; and the quadratic term is typically interpreted, in turn, to mean that the mathematically equivalent equation for describing the arithmetic distribution is a two-parameter power equation with an exponent that changes with body size. A power equation with an exponent that is itself a function of body size is virtually uninterpretable, yet numerous attempts have been made in recent years to incorporate such an exponent into theoretical models for the evolution of form and function in both plants and animals. However, the curvature that is described by a quadratic equation fitted to logarithms usually means that an explicit, non-zero intercept is required in the power equation describing the untransformed distribution - not that the exponent in the power equation varies with body size. Misperceptions that commonly accompany reports of complex allometry can be avoided by using nonlinear regression to examine untransformed data.

Measuring Complex Morphological Traits with 3D Photogrammetry: A Case Study with Deer Antlers

The increasing availability of 3D-imaging technology provides new opportunities for measuring morphology. Photogrammetry enables easy 3D-data acquisition compared to conventional methods and here we assess its accuracy for measuring the size of deer antlers, a complex morphological structure. Using a proprietary photogrammetry software, we generated 3D images of antlers for 92 individuals from 29 species of cervids that vary widely in antler size and shape and used these to measure antler volume. By repeating the process, we found that the relative error averaged 8.5% of object size. Errors in converting arbitrary voxel units into real volumetric units accounted for 70% of the measurement variance and can therefore be reduced by replicating the conversion. We applied the method to clay models of known volume and found no indication of bias. The estimation was robust against variation in imaging device, distance and operator, but approximately 40 images per specimen were necessary to achieve good precision. We used the method to show that conventional measures of main-beam length are relatively poor estimators of antler volume. Using loose antlers of known weight, we also showed that the volume may be a relatively poor predictor of antler weight due to variation in bone density across species. We conclude that photogrammetry can be an efficient and accurate tool for measuring antlers, and likely many other complex morphological traits.