Julian Huxley and the quantification of relative growth

Data transformation and model selection in bivariate allometry

Students of biological allometry have used the logarithmic transformation for over a century to linearize bivariate distributions that are curvilinear on the arithmetic scale. When the distribution is linear, the equation for a straight line fitted to the distribution can be back-transformed to form a two-parameter power function for describing the original observations. However, many of the data in contemporary studies of allometry fail to meet the requirement for log-linearity, thereby precluding the use of the aforementioned protocol. Even when data are linear in logarithmic form, the two-parameter power equation estimated by back-transformation may yield a misleading or erroneous perception of pattern in the original distribution. A better approach to bivariate allometry would be to forego transformation altogether and to fit multiple models to untransformed observations by nonlinear regression, thereby creating a pool of candidate models with different functional form and different assumptions regarding random error. The best model in the pool of candidate models could then be identified by a selection procedure based on maximum likelihood. Two examples are presented to illustrate the power and versatility of newer methods for studying allometric variation. It always is better to examine the original data when it is possible to do so.

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.

Commentary: Allometric analyses of data with outlying observations: The ontogenetic shift in metabolic allometry of American eels (Anguilla rostrata)

Authors of a recent report concluded that different patterns of metabolic allometry characterize juvenile and subadult stages in the life cycle of American eels (Anguilla rostrata). This conclusion was based on a comparison of straight lines fitted to logarithmic transformations of the original observations for metabolic rate and body mass, with the line fitted to transformations for 30 juveniles having a substantially lower slope than the line describing observations for 30 subadults. However, the authors failed to account for an influential outlier in the sample of juvenile eels, and this one outlier was determinative for the outcome of the analysis. When the outlier is removed from the combined data set for juveniles and subadults, the resulting sample of 59 observations is well described by a single straight line, which implies, in turn, that untransformed observations can be described by a two-parameter power equation with lognormal error. This supposition is confirmed by a graph of the two-parameter equation against the backdrop of the untransformed data. Thus, no change in the pattern of metabolic allometry occurs during the ontogeny of American eels: the same pattern of allometric variation characterizes both juvenile and subadult animals.

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

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.

Back to the basics: allometric growth by the horns of bovid mammals

I used the equivalent of nonlinear analysis of covariance (ANCOVA) to re-examine relative growth by the horns on males and females of alpine ibex (Capra ibex) and mouflon sheep (Ovis gmelini). A prior study of allometric growth by the horns on these animals described a pattern of biphasic allometry for both sexes, with two different mathematical equations being required to capture the pattern of variation over the full range in body size. However, the investigation in question used conventional analytical methods based on logarithmic transformations, which alter bivariate distributions and commonly introduce problems with analysis and interpretation. My new analyses of data for both species revealed that untransformed observations for both males and females are monophasic and that they are described quite well by three-parameter power equations with negative intercepts. Equations for males follow a steep upward trajectory whereas those for females follow much shallower paths. The negative intercepts indicate that males and females of both species must attain a minimum body size before horns begin to develop. Conclusions from the earlier investigation were based on inaccurate perceptions of pattern in the data. Future studies should be based on graphical and analytical analysis of observations expressed on the original arithmetic scale.

Allometric growth in mass by the brain of mammals

I re-examined published data for ontogenetic change in relative mass of the brain in six species of mammal (i.e., sheep, pig, cow, horse, rat, cat) to illustrate an insidious problem with conventional analyses of brain-body allometry. Graphical displays of logarithmic transformations of the original data for each species give the appearance of two discrete mathematical distributions, but untransformed observations nonetheless conform to a single distribution that is well described by a single, nonlinear equation. The concept of biphasic, allometric growth by the brain consequently is an artifact of transformation. The notion of Rapid and Slow phases in relative growth by the brain also is an artifact, because the notion is based explicitly on the concept of biphasic growth allometry. Relative growth by the brain in sheep, pigs, cows, and horses follows the path of a power curve with an exponent less than 1, so relative growth declines progressively as animals grow to their maximum size, at which point growth effectively ends for both brain and body. Relative growth by the brain in rats and cats follows the path of an exponential curve and consequently is more like relative growth by the brain of odontocoete cetaceans and primates, with the brain growing rapidly relative to the body early in ontogeny and attaining maximum (cats) or near-maximum (rats) mass well before the body reaches its maximum. An exponential pattern of relative growth by the brain appears to have evolved independently in rodents, carnivores, odontocoetes, and primates.

Is relative growth by the mammalian heart biphasic or monophasic?

I re-examined data for relative growth by the heart in four species of mammal to reconcile divergent reports that appear in the literature. Raw data for heart and body mass for Horro sheep, humans, gray kangaroos, and tammar wallabies were studied by linear and nonlinear regression, thereby enabling me to avoid the confounding effects of logarithmic transformation and to evaluate multiple statistical models for describing pattern in each set of observations. My analyses indicate that relative growth by the heart is monophasic in all four species and either isometric or near isometric on the arithmetic scale. The heart in these mammals consequently grows in mass in approximate proportion to growth in mass by the body. The appearance of biphasic allometric growth in prior studies was an artifact resulting from logarithmic transformation. Although parturition in sheep and humans is accompanied by a change in the distribution of blood out of the heart and into pulmonary and systemic circuits, the challenge is met without marked increases in absolute or relative size of the heart.