Lemaître and colleagues used conventional allometric methods to examine the relationship between length of antlers and mass of the body in males of 31 species in the mammalian family Cervidae . Logarithmic transformation failed to linearize the bivariate distribution, so the authors fitted a quadratic polynomial (and a piecewise model) that closely followed the concave curvature (figure 1a). The non-loglinear pattern characterized by this polynomial was interpreted in the context of a two-parameter allometric equation, y = a × xb, in the original arithmetic scale, with the exponent b, supposedly declining as body mass increases. Such an interpretation for functional dependence of the allometric exponent on body size is commonplace in studies reporting non-loglinear allometry , but the explanation is both unnecessary and incorrect in the case at hand.
The analysis by Lemaître et al. was seriously constrained by its reliance on logarithmic transformations . To illustrate this point, I used a combination of graphical and computational methods to study untransformed data from their online supplement. Preliminary examination of a bivariate plot revealed a curvilinear pattern in the observations (figure 1b), so I fitted two- and three-parameter power functions to the data, using the nlrwr package in R v. 2.13.2 . Two of the functions assumed additive, normal, homoscedastic error, whereas the other two assumed additive, normal, heteroscedastic error (see the electronic supplementary material, table S1). The model that described the observations best was the three-parameter power function with additive, normal, heteroscedastic error (electronic supplementary material, table S1). The mean function describes a smooth curve that closely follows the path of the observations (figure 1b); no pattern is apparent in a plot of normalized residuals and none of the residuals is so extreme as to mark it as an outlier (figure 1c); and a normal probability plot is essentially linear with an intercept of 0 and a slope of 1 (figure 1d), thereby confirming that residuals follow a normal distribution . Thus, the data are well described by a power function with an exponent that is functionally independent of body mass (figure 1b): it is not necessary to invoke a different allometric exponent for each of the 31 species in the sample.
The power function fitted to untransformed observations has important implications. For example, Lemaître et al. reported that length of the antlers increases with body size until animals attain a mass of 100–110 kg, after which antlers do not change in length. This contention is incorrect, as shown by the graph of the aforementioned power function against the backdrop of the original data (figure 1b). As animals become larger, their antlers increase in length, albeit not in direct proportion to mass.
Some readers may find the negative intercept for the three-parameter model to be problematic (figure 1b). However, instead of focusing on the y-intercept, they should consider the x-intercept [6–8], which is approximately 10 kg. Such a value indicates that antlers are unlikely to be present in species weighing less than 10 kg. Thus, in the evolution of cervids, males of a species apparently must attain a body mass of approximately 10 kg before an investment in antlers becomes beneficial; or if selection actually favours miniaturization, 10 kg is the minimum size for any species supporting antlers. It is noteworthy that male water deer (Hydropotes inermis), which lack antlers altogether, weigh about 12 kg.
Non-loglinear allometry typically causes major problems for interpretation . First, a quadratic equation follows the path of a parabola, so the model is unlikely to provide a realistic representation of biological variation under any circumstance [9–11]. Moreover, the problem is compounded when the quadratic is fitted to logarithmic transformations because the resulting equation cannot be re-expressed in log-free form on the original arithmetic scale. Consequently, investigators invoking non-loglinear allometry typically have to resort to complex interpretations that fail to yield important insights concerning the relationship between the variables of interest. More general use of nonlinear regression would help to alleviate this problem.
I declare I have no competing interests.
I received no funding for this study.
I thank J.-F. Lemaître and two anonymous referees for constructive criticisms that led to improvements in the manuscript.
The accompanying reply can be viewed at http://dx.doi.org/doi:10.1098/rsbl.2015.0144.
- Received November 1, 2014.
- Accepted January 8, 2015.
- © 2015 The Author(s) Published by the Royal Society. All rights reserved.