Bias in interspecific allometry: examples from morphological scaling in varanid lizards

The traditional approach to allometric analysis entails the fitting of a straight line to logarithmic transformations of the data, after which parameters in a two-parameter allometric equation are estimated by back-transformation to the original scale. We re-examined published data for dimensions of the limbs in 22 species of varanid lizards to illustrate the biases that can be introduced into allometric analyses by applying the aforementioned protocol. Statistical models fit to the original data by linear and nonlinear regression conformed better with underlying assumptions than did models obtained by back-transformation from logarithms, and the former generally were better than the latter for describing limb dimensions over the full range in body size. Allometric exponents estimated by the traditional method therefore were based on inappropriate and inaccurate statistical models and, consequently, were biased and misleading. Investigators can avoid problems such as these by performing preliminary graphical and statistical analyses on data in their original scale and by validating the fitted model. Logarithmic transformations should be used sparingly and only for cause.  © 2009 The Linnean Society of London, Biological Journal of the Linnean Society , 2009, 96 , 296–305.     

Allometric equations for predicting body mass of dinosaurs

We use data from the literature to compare two statistical procedures for estimating mass (or size) of quadrupedal dinosaurs and other extraordinarily large animals in extinct lineages. Both methods entail extrapolation from allometric equations fitted to data for a reference group of contemporary animals having a body form similar to that of the dinosaurs. The first method is the familiar one of fitting a straight line to logarithmic transformations, followed by back-transformation of the resulting equation to a two-parameter power function in the arithmetic scale. The second procedure entails fitting a two-parameter power function directly to arithmetic data for the extant forms by nonlinear regression. In the example presented here, the summed circumferences for humerus plus femur for 33 species of quadrupedal mammals was the predictor variable in the reference sample and body mass was the response variable. The allometric equation obtained by back-transformation from logarithms was not a good fit to the largest species in the reference sample and presumably led to grossly inaccurate estimates for body mass of several large dinosaurs. In contrast, the allometric equation obtained by nonlinear regression described data in the reference sample quite well, and it presumably resulted in better estimates for body mass of the dinosaurs. The problem with the traditional analysis can be traced to change in the relationship between predictor and response variables attending transformation, thereby causing measurements for large animals not to be weighted appropriately in fitting models by least squares regression. Extrapolations from statistical models obtained by back-transformation from lines fitted to logarithms are unlikely to yield reliable predictions for body size in extinct animals. Numerous reports on the biology of dinosaurs, including recent studies of growth, may need to be reconsidered in light of our findings.     

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.     

Fitting statistical models in bivariate allometry

Several attempts have been made in recent years to formulate a general explanation for what appear to be recurring patterns of allometric variation in morphology, physiology, and ecology of both plants and animals (e.g. the Metabolic Theory of Ecology, the Allometric Cascade, the Metabolic‐Level Boundaries hypothesis). However, published estimates for parameters in allometric equations often are inaccurate, owing to undetected bias introduced by the traditional method for fitting lines to empirical data. The traditional method entails fitting a straight line to logarithmic transformations of the original data and then back‐transforming the resulting equation to the arithmetic scale. Because of fundamental changes in distributions attending transformation of predictor and response variables, the traditional practice may cause influential outliers to go undetected, and it may result in an underparameterized model being fitted to the data. Also, substantial bias may be introduced by the insidious rotational distortion that accompanies regression analyses performed on logarithms. Consequently, the aforementioned patterns of allometric variation may be illusions, and the theoretical explanations may be wide of the mark. Problems attending the traditional procedure can be largely avoided in future research simply by performing preliminary analyses on arithmetic values and by validating fitted equations in the arithmetic domain. The goal of most allometric research is to characterize relationships between biological variables and body size, and this is done most effectively with data expressed in the units of measurement. Back‐transforming from a straight line fitted to logarithms is not a generally reliable way to estimate an allometric equation in the original scale.     

Is non‐loglinear allometry a statistical artifact?

The allometric equation, y = ax^b, is commonly fitted to data indirectly by transforming predictor (x) and response (y) variables to logarithms, fitting a straight line to the transformations, and then back‐transforming (exponentiating) the resulting equation to the original arithmetic scale. Sometimes, however, transformation fails to linearize the observations, thereby giving rise to what has come to be known as non‐loglinear allometry. A smooth curve for observations displayed on a log–log plot is usually interpreted to mean that the scaling exponent in the allometric equation is a continuously changing function of body size, whereas a breakpoint between two (or more) linear segments on a log–log plot is typically taken to mean that the exponent changes abruptly, coincident with some important milestone in development. I applied simple graphical and statistical procedures in re‐analyses of three well‐known examples of non‐loglinear allometry, and showed in every instance that the relationship between predictor and response can be described in the original scale by simple functions with constant values for the exponent b. In no instance does the allometric exponent change during the course of development. Transformation of data to logarithms created new distributions that actually obscured the relationships between predictor and response variables in these investigations, and led to erroneous perceptions of growth. Such confounding effects of transformation are not limited to non‐loglinear allometry but are common to all applications of the allometric method. © 2012 The Linnean Society of London, Biological Journal of the Linnean Society, 2012, ?? , ??–??.     

Unanticipated consequences of logarithmic transformation in bivariate allometry

Parameters in the two-parameter allometric equation are commonly estimated by fitting a straight line to logarithmic transformations of the original data and by back-transforming the resulting model to the arithmetic scale. However, log transformation distorts the relationship between the predictor and response variables, and this distortion may be sufficient to lead unsuspecting investigators to analyze data that actually are unsuited for allometric research. Two data sets from the current literature are re-examined here to illustrate instances in which log transformation caused ugly data to look deceptively good. One of the investigations focused on the scaling of metabolism to body mass in evolutionary transitions from prokaryotic to protistan to metazoan levels of organization whereas the other addressed the scaling of intestines to body size in rodents. In both instances investigators were led to conclusions that are not supported by the original data. Problems of the sort described here can readily be avoided simply by performing preliminary graphical analysis of observations expressed in the original units and by validating the final model in the arithmetic domain.     

On the use of log‐transformation versus nonlinear regression for analyzing biological power laws

Xiao and colleagues re‐examined 471 datasets from the literature in a major study comparing two common procedures for fitting the allometric equation y = ax^b to bivariate data (Xiao et al., 2011). One of the procedures was the traditional allometric method, whereby the model for a straight line fitted to logarithmic transformations of the original data is back‐transformed to form a two‐parameter power function with multiplicative, lognormal, heteroscedastic error on the arithmetic scale. The other procedure was standard nonlinear regression, whereby a two‐parameter power function with additive, normal, homoscedastic error is fitted directly to untransformed data by nonlinear least squares. Xiao and colleagues articulated a simple (but explicit) protocol for fitting and comparing the alternative models, and then used the protocol to examine each of the datasets in their compilation. The traditional method was said to provide a better fit in 69% of the cases and an equivalent fit in another 15%, so the investigation appeared to validate findings from a large majority of prior studies on allometric variation. However, focus for the investigation by Xiao and colleagues was overly narrow, and statistical models apparently were not validated graphically in the scale of measurement. The present study re‐examined a subset of the cases using a larger pool of candidate models and graphical validation, and discovered complexities that were overlooked in their investigation. Some datasets that appeared to be described better by the traditional method actually were unsuited for use in an allometric analysis, whereas other datasets were not described adequately by a two‐parameter power function, regardless of how the model was fitted. Thus, conclusions reached by Xiao and colleagues are not well supported and their paradigm for fitting allometric equations is unreliable. Future investigations of allometric variation should adopt a more holistic approach and incorporate graphical validation on the original arithmetic scale. © 2014 The Linnean Society of London, Biological Journal of the Linnean Society, 2014, 113 , 1167–1178.     

Model selection and logarithmic transformation in allometric analysis

The standard approach to most allometric research is to gather data on a biological function and a measure of body size, convert the data to logarithms, display the new values in a bivariate plot, and then fit a straight line to the transformations by the method of least squares. The slope of the fitted line provides an estimate for the allometric (or scaling) exponent, which often is interpreted in the context of underlying principles of structural and functional design. However, interpretations of this sort are based on the implicit assumption that the original data conform with a power function having an intercept of 0 on a plot with arithmetic coordinates. Whenever this assumption is not satisfied, the resulting estimate for the allometric exponent may be seriously biased and misleading. The problem of identifying an appropriate function is compounded by the logarithmic transformations, which alter the relationship between the original variables and frequently conceal the presence of outliers having an undue influence on properties of the fitted equation, including the estimate for the allometric exponent. Much of the current controversy in allometric research probably can be traced to substantive biases introduced by investigators who followed standard practice. We illustrate such biases with examples taken from the literature and outline a general methodology by which the biases can be minimized in future research.     

Allometric scaling and Bayesian priors for annual survival of birds and mammals

Allometric theory predicts that instantaneous mortality rates scale with body mass with a negative quarter power. Such a relationship would mean that the survival rate of one species is partly predictable from the survival rate of other species. We develop allometric regression models for annual adult survival of birds and mammals, using data collected from the literature. These models conform to the predictions of the allometric theory; the value of negative one-quarter for the scaling parameter is within the 95% credible interval, which is [-0.31, -0.10] for birds and [-0.35, -0.15] for mammals. The predictions are very well supported when evaluated using an independent set of data. The regression models can be used to provide objective and informative Bayesian priors for annual adult survival rates of birds and mammals or to act as a point of comparison in new studies.     

Logarithmic transformation bias in allometry

It has been known for some time (DJ Finney, J. Roy. Stat. Soc. Suppl. 7:155–161, 1941) that transformation of an arithmetic data set to logarithms results in biased estimates when predicted values from a leastsquares regression are detransformed back to arithmetic units. Predicted values are estimates of the geometric mean of the dependent variable at that value of the independent variable, rather than the arithmetic mean. Since the geometric mean is always less than the arithmetic mean, detransformed predictions will underestimate the value in question. This bias affects the interpretations of allometric equations used for estimation, such as predicting fossil body mass from skeletal dimensions, and applications of allometry as a “criterion of subtraction,” in which residual variation is evaluated. A number of parametric and nonparametric corrections for transformation bias have been developed. Although this problem is relatively unexplored in mammalian morphometrics, it has received considerable attention in other disciplines that use power functions structurally identical to the allometric equation. Insights into transformation bias and the use of correction terms from economics, limnology, forestry, and hydrology are reviewed and interpreted for application to mammalian allometry. © 1993 Wiley-Liss, Inc.     

Using the Past to Predict the Present: Confidence Intervals for Regression Equations in Phylogenetic Comparative Methods

Two phylogenetic comparative methods, independent contrasts and generalized least squares models, can be used to determine the statistical relationship between two or more traits. We show that the two approaches are functionally identical and that either can be used to make statistical inferences about values at internal nodes of a phylogenetic tree (hypothetical ancestors), to estimate relationships between characters, and to predict values for unmeasured species. Regression equations derived from independent contrasts can be placed back onto the original data space, including computation of both confidence intervals and prediction intervals for new observations. Predictions for unmeasured species (including extinct forms) can be made increasingly accurate and precise as the specificity of their placement on a phylogenetic tree increases, which can greatly increase statistical power to detect, for example, deviation of a single species from an allometric prediction. We reexamine published data for basal metabolic rates (BMR) of birds and show that conventional and phylogenetic allometric equations differ significantly. In new results, we show that, as compared with nonpasserines, passerines exhibit a lower rate of evolution in both body mass and mass-corrected BMR; passerines also have significantly smaller body masses than their sister clade. These differences may justify separate, clade-specific allometric equations for prediction of avian basal metabolic rates.     

Optimization in integrated biochemical systems

As of yet, steady-state optimization in biochemical systems has been limited to a few studies in which networks of fluxes were optimized. These networks of fluxes are represented by linear (stoichiometric) equations that are used as constraints in a linear program, and a flux or a sum of weighted fluxes is used as the objective function. In contrast to networks of fluxes, systems of metabolic processes have not been optimized in a comparable manner. The primary reason is that these types of integrated biochemical systems are full of synergisms, antagonisms, and regulatory mechanisms that can only be captured appropriately with nonlinear models. These models are mathematically complex and difficult to analyze. In most cases it is not even possible to compute, let alone optimize, steady-state solutions analytically. Rare exceptions are S-system representations. These are nonlinear and able to represent virtually all types of dynamic behaviors, but their steady states are characterized by linear equations that can be evaluated both analytically and numerically. The steady-state equations are expressed in terms of the logarithms of the original variables, and because a function and its logarithms of the original variables, and because a function and its logarithm assume their maxima for the same argument, yields or fluxes can be optimized with linear programs expressed in terms of the logarithms of the original variables. (c) 1992 John Wiley & Sons, Inc.     

Morphological variability and allometric relationships of the herb Panax notoginseng in Yunnan,China

A plant's morphology changes throughout its ontogeny. Investigating the allometric relationships between different morphological traits could provide useful information for cultivation of medicinal plants. Here we collected 698 individuals of Panax notoginseng for allometric analysis from seven populations cultivated in Yunnan, Southwest China. The slopes and intercepts of the allometric relationships were estimated by Standardized Major Axis regression. Significant differences (p < 0.05) were found in each morphological variable considered among populations. Allometric analysis showed that all of the log-log relationships had different slopes or shared a common slope but differed in intercept (p < 0.001). The morphological traits showed flexible allometric relationships. However, the root biomass that considered as a target trait showed the least allometric variability (slope = 1.068–1.378) when compared to other variables. This could be because of the hundreds of years of cultivation and artificial selection.     

Incisor size and diet revisited: The view from a platyrrhine perspective

Allometric relationships between incisor size and body size were determined for 26 species of New World primates. While previous studies have suggested that the incisors of Old World primates, and anthropoids in general, scale isometrically with body size, the data presented here indicate a negative allometric relationship between incisor size and body size among New World species. This negative allometry was exhibited by platyrrhines when either upper or lower incisor row length was regressed against body weight, and when either least-squares or bivariate principal axis equations were used. When upper incisor length was plotted against skull length, negative allometry could be sustained using both statistical techniques only when the full sample of 26 species was plotted. The choice of variables to represent incisor size and body size, and the choice of a statistical technique to effect the allometric equation, had a more pronounced impact on the location of individual species with regard to lines of best fit. Platyrrhines as a group have smaller incisors relative to body size than do catarrhines, regardless of diet. Among New World primates, small incisors represent a plausible primitive condition; species with relatively large incisors manifest a phyletic change associated with a dietary shift to foods that require increased incisal preparation. The opposite trend characterizes Old World primates. In spite of the taxonomic differences in relative incisor size between platyrrhine and catarrhine primates, inferences about diet derived from an allometric equation for all anthropoids should prove reliable as long as the species with unknown diet does not lie at the upper end of the body size range for platyrrhines or catarrhines.     

Ontogenetic scaling of fish metabolism in the mouse-to-elephant mass magnitude range

Intraspecific or ontogenetic analyses of mass-metabolism relationships do not often conform to the same allometric correlations as those seen in interspecific analyses. A commonly cited reason for this discrepancy is that ontogenetic studies examine smaller mass ranges than interspecific studies, and are therefore not statistically comparable. In this study the metabolic rate of yellowtail kingfish was measured from 0.6 mg-2.2 kg, a mass range comparable to that between a mouse and an elephant. Linear regression of the log transformed data resulted in a scaling exponent of 0.90 and high correlation coefficient. Statistical and information theory comparisons of three other models showed that a segmented linear regression and curvilinear quadratic function were an improvement over a simple linear regression. This confirmed previous observations that the metabolic scaling exponent of fish changes during ontogeny. Ammonia excretion rates were also measured and scaled linearly with an exponent of 0.87. The data showed that the metabolism of yellowtail kingfish during ontogeny did not scale with the commonly cited 2/3 or 3/4 mass exponent. This demonstrates that differences between interspecific and ontogenetic allometries are not necessarily statistical artefacts.     

General allometric equations and biomass allocation of <Emphasis Type="Italic">Pinus massoniana</Emphasis> trees on a regional scale in southern China

Applying allometric equations in combination with forest inventory data is an effective approach to use when qualifying forest biomass and carbon storage on a regional scale. The objectives of this study were to (1) develop general allometric tree component biomass equations and (2) investigate tree biomass allocation patterns for Pinus massoniana, a principal tree species native to southern China, by applying 197 samples across 20 site locations. The additive allometric equations utilized to compute stem, branch, needle, root, aboveground, and total tree biomass were developed by nonlinear seemingly unrelated regression. Results show that the relative proportion of stem biomass to tree biomass increased while the contribution of canopy biomass to tree biomass decreased as trees continued to grow through time. Total root biomass was a large biomass pool in itself, and its relative proportion to tree biomass exhibited a slight increase with tree growth. Although equations employing stem diameter at breast height (dbh) alone as a predictor could accurately predict stem, aboveground, root, and total tree biomass, they were poorly fitted to predict the canopy biomass component. The inclusion of the tree height (H) variable either slightly improved or did not in any way increase model fitness. Validation results demonstrate that these equations are suitable to estimate stem, aboveground, and total tree biomass across a broad range of P. massoniana stands on a regional scale.     

Allometric and morphological characteristics of Tonicella marmorea (Fabricius, 1780) populations (Mollusca: Polyplacophora: Ischnochitonidae)

Allometric and morphological characteristics of population samples of Tonicella marmorea from three sea lochs on the west coast of Scotland are described. The suitability of allometric relationships as taxonomic criteria are investigated by statistical comparisons of various regression constants, scatter diagrams and comparisons of curvature indices. Morphological characteristics such as ctenidia number and numbers of notches in the head and tail valves proved too variable to be regarded as diagnostic but they are useful taxonomic indicators. The need for all taxonomic characteristics to be derived from population samples is stressed and the use of valve and girdle sculpturing together with radula structure is emphasized.     

Allometry, antilog transformations, and the perils of prediction on the original scale

Biologists often use allometric equations that take the form of power functions (e.g., Y = aM(b), where M stands for mass and a and b are empirically fitted constants). Typically, these allometric equations are fitted by taking the antilog of log-log regressions. Predictions from these allometric equations are biased, and the bias my be appreciable. Methods for making predictions that correct for the bias are available, but they have rarely, if ever, been used by ecological and evolutionary physiologists. Just as physiologists would not use an instrument that was not properly calibrated, they should not use allometric equations to make predictions unless they account for the bias of those predictions. We analyzed 20 interspecific and 10 intraspecific data sets. We compared predictions from standard allometric equations with those from several alternative methods. Our analyses suggest that the bias of predictions from interspecific data sets may be substantial. For the intraspecific data sets we analyzed, the bias was likely to be small. Biologists, including ecological and evolutionary physiologists, should exercise care when using allometric equations to make predictions, particularly given that methods to adjust for bias are easily implemented.     

Environmental temperature effec on heart weight of chickens

Broiler chicks were reared to 7 weeks of age at 7.2, 20.0 and 32.2°C. Heart ventricle weight, plasma volume, and blood volume were inversely related to environmental temperature. Allometric equations were determined for heart weight for chicks at 7.2 and 32.2°C. The results obtained demonstrate the need to specify environmental conditions when reporting allometric equations. Right ventricle weight as a percentage of total ventricle weight was not always significantly affected by environmental temperature, but there was a trend to greater right ventricular weight at lower temperatures.