Discontinuous, biphasic, ontogenetic shifts in the metabolic allometry of aquatic animals?

Several investigations in recent years have reported patterns of discontinuous, biphasic, loglinear variation in the metabolic allometry of aquatic animals. These putative shifts in pattern of allometry have been attributed to changes in the primary site for gas exchange from cutaneous to branchial as animals undergo ontogenetic changes in size, shape, and surface area. Because of the important implications of the earlier research with regard to both physiology and evolution, I re-examined data that purportedly support claims of discontinuous, biphasic allometry in oxygen consumption versus body size of American eels (Anguilla rostrata) and spiny lobsters (Sagmariasus verreauxi). I used ANCOVA to fit three different statistical models to each set of logarithmic transformations and then assessed the fits by Akaike's Information Criterion. The observations for both species were described better by a single straight line fitted to the full distribution than by a biphasic model. Eels, lobsters, and other aquatic animals undergo changes in shape and surface area as they grow, but such changes are not necessarily accompanied by changes in the pattern of metabolic allometry.

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.

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.

A new perspective on the static metabolic allometry of carabid beetles.

The conventional allometric method entails fitting a straight line to logarithmic transformations of the original bivariate data and then back-transforming the resulting equation (at least implicitly) to form a two-parameter power function, Y = a × Xb , on the arithmetic scale. Although the protocol is widely used in contemporary research, it commonly performs poorly and thereby leads investigators to form inaccurate impressions of the dominant pattern in their data. Here I re-examine the metabolic allometry for six species of carabid beetle to illustrate the problem that arises when pattern in the original data can be described by two (or more) statistically equivalent equations with different functional form. Whereas conventional analyses of data for the beetles yielded only a single descriptive model for each dataset (i.e., the two-parameter power equation), the more versatile protocol used here fitted two to four statistically equivalent equations (including the two-parameter power function) to the same sets of observations. Conclusions based on just the power equation estimated by conventional allometry would be misleading because the equation does not afford a unique description for pattern in the data.

A new research paradigm for bivariate allometry: combining ANOVA and non-linear regression.

A novel statistical routine is presented here for exploring and comparing patterns of allometric variation in two or more groups of subjects. The routine combines elements of the analysis of variance (ANOVA) with non-linear regression to achieve the equivalent of an analysis of covariance (ANCOVA) on curvilinear data. The starting point is a three-parameter power equation to which a categorical variable has been added to identify membership by each subject in a specific group or treatment. The protocol differs from earlier ones in that different assumptions can be made about the form for random error in the full statistical model (i.e. normal and homoscedastic, normal and heteroscedastic, lognormal and heteroscedastic). The general equation and several modifications thereof were used to study allometric variation in field metabolic rates of marsupial and placental mammals. The allometric equations for both marsupials and placentals have an explicit, non-zero intercept, but the allometric exponent is higher in the equation for placentals than in that for marsupials. The approach followed here is extraordinarily versatile, and it has wider application in allometry than standard ANCOVA performed on logarithmic transformations.

Misconceptions about logarithmic transformation and the traditional allometric method.

Logarithmic transformation is often assumed to be necessary in allometry to accommodate the kind of variation that accompanies multiplicative growth by plants and animals; and the traditional approach to allometric analysis is commonly believed to have important application even when the bivariate distribution of interest is curvilinear on the logarithmic scale. Here I examine four arguments that have been tendered in support of these perceptions. All the arguments are based on misunderstandings about the traditional method for allometric analysis and/or on a lack of familiarity with newer methods of nonlinear regression. Traditional allometry actually has limited utility because it can be used only to fit a two-parameter power equation that assumes lognormal, heteroscedastic error on the original scale. In contrast, nonlinear regression can fit two- and three-parameter power equations with differing assumptions about structure for error directly to untransformed data. Nonlinear regression should be preferred to the traditional method in future allometric analyses.

Why allometric variation in mammalian metabolism is curvilinear on the logarithmic scale.

Studies performed over the last 20 years have repeatedly documented a slight convex curvature (relative to the x-axis) in double-logarithmic plots of basal metabolic rate (BMR) versus body mass in mammals. This curvilinear pattern has usually been interpreted in the context of a simple, two-parameter power function on the arithmetic scale, y  =  a  ×  xb , with the exponent in the equation supposedly increasing systematically with body size. An equation of this form has caused concern among ecologists because a variable exponent is inconsistent with an assumption underlying the metabolic theory of ecology (MTE). However, the appearance of an exponent that varies with body size is an artifact resulting from the widespread use of logarithmic transformations in allometric analyses. Curvature in the distribution on the logarithmic scale actually is caused by a requirement for an explicit, non-zero intercept-and not a variable exponent-in the model describing the distribution on the arithmetic scale. Thus, the MTE need not be revised to accommodate an exponent that varies with body size in the scaling of mammalian BMR, but the theory may need to be tweaked to accommodate an intercept in the allometric equation. In general, any bivariate dataset that is well described by a three-parameter power equation on the arithmetic scale will follow a curvilinear path when displayed on the logarithmic scale. Consequently, reports of curvilinearity in log domain (i.e., "complex allometry") need to be revisited because conclusions from those investigations are likely to be flawed.

Is complex allometry in field metabolic rates of mammals a statistical artifact?

Recent reports indicate that field metabolic rates (FMRs) of mammals conform to a pattern of complex allometry in which the exponent in a simple, two-parameter power equation increases steadily as a dependent function of body mass. The reports were based, however, on indirect analyses performed on logarithmic transformations of the original data. I re-examined values for FMR and body mass for 114 species of mammal by the conventional approach to allometric analysis (to illustrate why the approach is unreliable) and by linear and nonlinear regression on untransformed variables (to illustrate the power and versatility of newer analytical methods). The best of the regression models fitted directly to untransformed observations is a three-parameter power equation with multiplicative, lognormal, heteroscedastic error and an allometric exponent of 0.82. The mean function is a good fit to data in graphical display. The significant intercept in the model may simply have gone undetected in prior analyses because conventional allometry assumes implicitly that the intercept is zero; or the intercept may be a spurious finding resulting from bias introduced by the haphazard sampling that underlies "exploratory" analyses like the one reported here. The aforementioned issues can be resolved only by gathering new data specifically intended to address the question of scaling of FMR with body mass in mammals. However, there is no support for the concept of complex allometry in the relationship between FMR and body size in mammals.

Relative Growth by the Elongated Jaws of Gars: A Perspective on Polyphasic Loglinear Allometry.

Nonlinear regression was used to fit power functions with different forms for random error to data for length of the upper jaw versus length of the rest of body in two species of gars. Growth by the jaws of these species was reported earlier to conform to a pattern of polyphasic loglinear allometry indicative of relatively rapid growth by the jaw in small individuals (allometric exponent greater than 1) and relatively slow growth by the jaw in larger ones (allometric exponent less than 1). The new analyses revealed, however, that the pattern of relative growth by jaws of these fishes does not change with body size: the jaws grow more slowly than the rest of the body throughout life. The flaw in earlier investigations was the unrealistic assumption that data comprising a single sample in log domain form two (or more) statistically distinct distributions when expressed in arithmetic domain. The paradigm used in the current study provides a simple way to fit and compare different power functions, with different forms for random error, directly to raw data and thereby obviates the use of logarithmic transformations in allometric research.

Allometric scaling of mammalian blood pressure: A comment on White and Seymour (2014).

White and Seymour examined the scaling of central arterial blood pressure against body mass in mammals ranging in size from a 30 g mouse to a 4080 kg elephant. Exponents in power functions fitted to each of three datasets (systolic, diastolic, and mean arterial pressure) were reported to be significantly greater than zero and indistinguishable from 0.33. The first of these outcomes would indicate that blood pressure increases with body size, whereas the second is consistent with the heart working against gravity to move blood to the head. Taken together, these results seemingly refute the notion that the cephalic circulation functions as an energy-neutral siphon. However, the main findings by White and Seymour were presented in the form of graphs that distorted the relationships between the variables of interest. I use simple graphics to show that the data were unsuited from the outset for use in allometric analyses and that conclusions of the investigation are not well supported.

Quantifying the curvilinear metabolic scaling in mammals.

A perplexing problem confronting students of metabolic allometry concerns the convex curvature that seemingly occurs in log-log plots of basal metabolic rate (BMR) vs. body mass in mammals. This putative curvilinearity has typically been interpreted in the context of a simple power function, Y=a*Xb, on the arithmetic scale, with the allometric exponent, b, supposedly increasing steadily as a dependent function of body size. The relationship can be quantified in arithmetic domain by exponentiating a quadratic equation fitted to logarithmic transformations of the original data, but the resulting model is not in the form of a power function and it is unlikely to describe accurately the pattern in the original distribution. I therefore re-examined a dataset for 636 species of mammal and discovered that the relationship between BMR and body mass is well-described by a power function with an explicit, non-zero intercept and lognormal, heteroscedastic error. The model has an invariant allometric exponent of 0.75, so the appearance in prior investigations of a steadily increasing exponent probably was an aberration resulting from undue reliance on logarithmic transformations to estimate statistical models in arithmetic domain. Theoretical constructs relating BMR to body mass in mammals may need to be modified to accommodate a positive intercept in the statistical model, but they do not need to be revised, or rejected, at present time on grounds that the allometric exponent varies with body size. New data from planned experiments will be needed to confirm any hypothesis based on data currently available.

Multiplicative by nature: Logarithmic transformation in allometry.

The traditional allometric method, which is at the heart of research paradigms used by comparative biologists around the world, entails fitting a straight line to logarithmic transformations of the original bivariate data and then back-transforming the resulting equation to form a two-parameter power function in the arithmetic scale. The method has the dual advantages of enabling investigators to fit statistical models that describe multiplicative growth while simultaneously addressing the multiplicative nature of residual variation in response variables (heteroscedasticity). However, important assumptions of the traditional method seldom are assessed in contemporary practice. When the assumptions are not met, mean functions may fail to capture the dominant pattern in the original data and incorrect form for error may be imposed upon the fitted model. A worked example from metabolic allometry in doves and pigeons illustrates both the power of newer statistical procedures and limitations of the traditional allometric method.

Fitting statistical models in bivariate allometry: scaling metabolic rate to body mass in mustelid carnivores.

The ongoing debate about methods for fitting the two-parameter allometric equation y=ax(b) to bivariate data seemed to be resolved recently when three groups of investigators independently reported that statistical models fitted by the traditional allometric method (i.e., by back-transforming a linear model fitted to log-log transformations) typically are superior to models fitted by standard nonlinear regression. However, the narrow focus for the statistical analyses in these investigations compromised the most important of the ensuing conclusions. All the investigations focused on two-parameter power functions and excluded from consideration other simple functions that might better describe pattern in the data; and all relied on Akaike's Information Criterion instead of graphical validation to identify the better statistical model. My re-analysis of data from one of the studies (BMR vs. body mass in mustelid carnivores) revealed (1) that the best descriptor for pattern in the dataset is a straight line and not a two-parameter power function; (2) that a model with additive, normal, heteroscedastic error is superior to one with multiplicative, lognormal, heteroscedastic error; and (3) that Akaike's Information Criterion is not a generally reliable metric for discriminating between models fitted to different distributions. These findings have apparent implications for interpreting the outcomes of all three of the aforementioned studies. Future investigations of allometric variation should adopt a more holistic approach to analysis and not be wedded to the traditional allometric method.

Julian Huxley, Uca pugnax and the allometric method.

The allometric method, which often is attributed to Julian Huxley, entails fitting a straight line to logarithmic transformations of the original bivariate data and then back-transforming the resulting equation to form a power function in the arithmetic scale. Development of the technique was strongly influenced by Huxley's own research on growth by the enlarged 'crusher' claw in male fiddler crabs (Uca pugnax). Huxley reported a discontinuity in the log-log plot of chela mass vs body mass, which he interpreted as an abrupt change in relative growth of the chela at about the time crabs attain sexual maturity. My analysis of Huxley's arithmetic data indicates, however, that the discontinuity was an artifact caused by logarithmic transformation and that dynamics of growth by the crusher claw do not change at any point during development. Arithmetic data are well described by a power function fitted by nonlinear regression but not by one estimated by back-transforming a line fitted to logarithms. This finding and others like it call into question the continued reliance on the allometric method in contemporary research.

Rotational distortion in conventional allometric analyses.

Three data sets from the recent literature were submitted to new analyses to illustrate the rotational distortion that commonly accompanies traditional allometric analyses and that often causes allometric equations to be inaccurate and misleading. The first investigation focused on the scaling of evaporative water loss to body mass in passerine birds; the second was concerned with the influence of body size on field metabolic rates of rodents; and the third addressed interspecific variation in kidney mass among primates. Straight lines were fitted to logarithmic transformations by Ordinary Least Squares and Generalized Linear Models, and the resulting equations then were re-expressed as two-parameter power functions in the original arithmetic scales. The re-expressed models were displayed on bivariate graphs together with tracings for equations fitted directly to untransformed data by nonlinear regression. In all instances, models estimated by back-transformation failed to describe major features of the arithmetic distribution whereas equations fitted by nonlinear regression performed quite well. The poor performance of equations based on models fitted to logarithms can be traced to the increased weight and leverage exerted in those analyses by observations for small species and to the decreased weight and leverage exerted by large ones. The problem of rotational distortion can be avoided by performing exploratory analysis on untransformed values and by validating fitted models in the scale of measurement.