A comparison of methods for fitting allometric equations to field metabolic rates of animals

We re-examined data for field metabolic rates of varanid lizards and marsupial mammals to illustrate how different procedures for fitting the allometric equation can lead to very different estimates for the allometric coefficient and exponent. A two-parameter power function was obtained in each case by the traditional method of back-transformation from a straight line fitted to logarithms of the data. Another two-parameter power function was then generated for each data-set by non-linear regression on values in the original arithmetic scale. Allometric equations obtained by non-linear regression described the metabolic rates of all animals in the samples. Equations estimated by back-transformation from logarithms, on the other hand, described the metabolic rates of small species but not large ones. Thus, allometric equations estimated in the traditional way for field metabolic rates of varanids and marsupials do not have general importance because they do not characterize rates for species spanning the full range in body size. Logarithmic transformation of predictor and response variables creates new distributions that may enable investigators to perform statistical analyses in compliance with assumptions underlying the tests. However, statistical models fitted to transformations should not be used to estimate parameters of equations in the arithmetic domain because such equations may be seriously biased and misleading. Allometric analyses should be performed on values expressed in the original scale, if possible, because this is the scale of interest.     

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.     

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, ?? , ??–??.     

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.     

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.     

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.     

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.     

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.     

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.     

Minimizing Bias in Biomass Allometry: Model Selection and Log‐Transformation of Data

Nonlinear regression is increasingly used to develop allometric equations for forest biomass estimation (i.e., as opposed to the traditional approach of log‐transformation followed by linear regression). Most statistical software packages, however, assume additive errors by default, violating a key assumption of allometric theory and possibly producing spurious models. Here, we show that such models may bias stand‐level biomass estimates by up to 100 percent in young forests, and we present an alternative nonlinear fitting approach that conforms with allometric theory.     

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.     

Grass allometry and estimation of above‐ground biomass in tropical alpine tussock grasslands

The puna/páramo grasslands span across the highest altitudes of the tropical Andes, and their ecosystem dynamics are still poorly understood. In this study we examined the above‐ground biomass and developed species specific and multispecies power‐law allometric equations for four tussock grass species in Peruvian high altitude grasslands, considering maximum height (h_max), elliptical crown area and elliptical basal area. Although these predictors are commonly used among allometric literature, they have not previously been used for estimating puna grassland biomass. Total above‐ground biomass was estimated to be of 6.7 ± 0.2 Mg ha^?1 (3.35 ± 0.1 Mg C ha^?1). All allometric relationships fitted to similar power‐law models, with basal area and crown area as the most influential predictors, although the fit improved when tussock maximum height was included in the model. Multispecies allometries gave better fits than the other species‐specific equations, but the best equation should be used depending on the species composition of the target grassland. These allometric equations provide an useful approach for measuring above‐ground biomass and productivity in high‐altitude Andean grasslands, where destructive sampling can be challenging and difficult because of the remoteness of the area. These equations can be also applicable for establishing above‐ground reference levels before the adoption of carbon compensation mechanisms or grassland management policies, as well as for measuring the impact of land use changes in Andean ecosystems.     

Pitfalls and challenges of estimating population growth rate from empirical data: consequences for allometric scaling relations

The intrinsic rate of increase is a fundamental concept in population ecology, and a variety of problems require that estimates of population growth rate be obtained from empirical data. However, depending on the extent and type of data available (e.g. time series, life tables, life history traits), several alternative empirical estimators of population growth rate are possible. Because these estimators make different assumptions about the nature of age‐dependent mortality and density‐dependence of population dynamics, among other factors, these quantities capture fundamentally different aspects of population growth and are not interchangeable. Nevertheless, they have been routinely commingled in recent ecoinformatic analyses relating to allometry and conservation biology. Here we clarify some of the confusion regarding the empirical estimation of population growth rate and present separate analyses of the frequency distributions and allometric scaling of three alternative, non‐interchangeable measures of population growth. Studies of allometric scaling of population growth rate with body size are additionally sensitive to the statistical line fitting approach used, and we find that different approaches yield different allometric scaling slopes. Across the mix of population growth estimators and line fitting techniques, we find scattered and limited support for the key allometric prediction from the metabolic theory of ecology, namely that log₁₀(population growth rate) should scale as ?0.25 power of log₁₀(body mass). More importantly, we conclude that the question of allometric scaling of population growth rate with body size is highly sensitive to previously unexamined assumptions regarding both the appropriate population growth parameter to be compared and the line fitting approach used to examine the data. Finally, we suggest that the ultimate test of allometric scaling of maximum population growth rates with body size has not been done and, moreover, may require data that are not currently available.     

Taxonomic versus allometric constraints on non‐linear interaction strengths

Recently, the importance of body mass and allometric scaling for the structure and dynamics of ecological networks has been highlighted in several ground‐breaking studies. However, advances in the understanding of generalities across ecosystem types are impeded to a considerable extent by a methodological dichotomy contrasting a considerable portion of marine ecology on the one hand opposite to traditional community ecology on the other hand. Many marine ecologists are bound to the taxonomy‐neglecting size spectrum approach when describing and analysing community patterns. In contrast, the mindset of the other school is focused on taxonomies according to the Linnean system at the cost of obscuring information due to applying species or population averages of body masses and other traits. Following other pioneering studies, we addressed this lingering gap, and studied non‐linear interaction strengths (i.e. functional responses) between two taxonomically‐distinct terrestrial arthropod predators (centipedes and spiders) of varying individual body masses and their prey. We fitted three non‐linear functional response models to the data: (1) a taxonomic model not accounting for variance in body masses amongst predator individuals, (2) an allometric model ignoring taxonomic differences between predator individuals, and (3) a combined model including body mass and taxonomic effects. Ranked according to their AICs, the combined model performs better than the allometric model, which provides a superior fit to the data than the taxonomic model. These results strongly indicate that the body masses of predator and prey individuals were responsible for most of the variation in non‐linear interaction strengths. Taxonomy explained some specific patterns in allometric exponents between groups and revealed mechanistic insights in predation efficiencies. Reconciling quantitative allometric models as employed by the marine size‐spectrum approach with taxonomic information may thus yield quantitative results that are generalized across ecosystem types and taxonomic groups. Using these quantitative models as novel null models should also strengthen subsequent taxonomic analyses.     

Analyzing population growth curves

Assessing animal population growth curves is an essential feature of field studies in ecology and wildlife management. We used five models to assess population growth rates with a number of sets of population growth rate data. A 'generalized' logistic curve provides a better model than do four other popular models. Use of difference equations for fitting was checked by a comparison of that method and direct fitting of the analytical (integrated) solution for three of the models. Fits to field data indicate that estimates of the asymptote, K, from the 'generalized logistic' and the ordinary logistic agree well enough to support use of estimates of K from the ordinary logistic on data that cannot be satisfactorily fitted with the generalized logistic. Akaike's information criterion is widely used, often with a small sample version AIC_c. Our study of five models indicated a bias in the AIC_c criterion, so we recommend checking results with estimates of variance about regression for fitted models. Fitting growth curves provides a valuable supplement to, and check on computer models of populations.     

Evaluation of a two‐part regression calibration to adjust for dietary exposure measurement error in the Cox proportional hazards model: A simulation study

Dietary questionnaires are prone to measurement error, which bias the perceived association between dietary intake and risk of disease. Short‐term measurements are required to adjust for the bias in the association. For foods that are not consumed daily, the short‐term measurements are often characterized by excess zeroes. Via a simulation study, the performance of a two‐part calibration model that was developed for a single‐replicate study design was assessed by mimicking leafy vegetable intake reports from the multicenter European Prospective Investigation into Cancer and Nutrition (EPIC) study. In part I of the fitted two‐part calibration model, a logistic distribution was assumed; in part II, a gamma distribution was assumed. The model was assessed with respect to the magnitude of the correlation between the consumption probability and the consumed amount (hereafter, cross‐part correlation), the number and form of covariates in the calibration model, the percentage of zero response values, and the magnitude of the measurement error in the dietary intake. From the simulation study results, transforming the dietary variable in the regression calibration to an appropriate scale was found to be the most important factor for the model performance. Reducing the number of covariates in the model could be beneficial, but was not critical in large‐sample studies. The performance was remarkably robust when fitting a one‐part rather than a two‐part model. The model performance was minimally affected by the cross‐part correlation.     

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.     

Underestimating the frequency,strength and cost of antipredator responses with data from GPS collars: an example with wolves and elk

Field studies that rely on fixes from GPS‐collared predators to identify encounters with prey will often underestimate the frequency and strength of antipredator responses. These underestimation biases have several mechanistic causes. (1) Step bias: The distance between successive GPS fixes can be large, and encounters that occur during these intervals go undetected. This bias will generally be strongest for cursorial hunters that can rapidly cover large distances (e.g., wolves and African wild dogs) and when the interval between GPS fixes is long relative to the duration of a hunt. Step bias is amplified as the path travelled between successive GPS fixes deviates from a straight line. (2) Scatter bias: Only a small fraction of the predators in a population typically carry GPS collars, and prey encounters with uncollared predators go undetected unless a collared group‐mate is present. This bias will generally be stronger for fission–fusion hunters (e.g., spotted hyenas, wolves, and lions) than for highly cohesive hunters (e.g., African wild dogs), particularly when their group sizes are large. Step bias and scatter bias both cause underestimation of the frequency of antipredator responses. (3) Strength bias: Observations of prey in the absence of GPS fix from a collared predator will generally include a mixture of cases in which predators were truly absent and cases in which predators were present but not detected, which causes underestimation of the strength of antipredator responses. We quantified these biases with data from wolves and African wild dogs and found that fixes from GPS collars at 3‐h intervals underestimated the frequency and strength of antipredator responses by a factor >10. We reexamined the results of a recent study of the nonconsumptive effects of wolves on elk in light of these results and confirmed that predation risk has strong effects on elk dynamics by reducing the pregnancy rate.     

Testing metabolic scaling theory using intraspecific allometries in Antarctic microarthropods

Quantitative scaling relationships among body mass, temperature and metabolic rate of organisms are still controversial, while resolution may be further complicated through the use of different and possibly inappropriate approaches to statistical analysis. We propose the application of a modelling strategy based on the theoretical approach of Akaike's information criteria and non‐linear model fitting (nlm). Accordingly, we collated and modelled available data at intraspecific level on the individual standard metabolic rate of Antarctic microarthropods as a function of body mass (M), temperature (T), species identity (S) and high rank taxa to which species belong (G) and tested predictions from metabolic scaling theory (mass‐metabolism allometric exponent b = 0.75, activation energy range 0.2–1.2 eV). We also performed allometric analysis based on logarithmic transformations (lm). Conclusions from lm and nlm approaches were different. Best‐supported models from lm incorporated T, M and S. The estimates of the allometric scaling exponent linking body mass and metabolic rate resulted in a value of 0.696 ± 0.105 (mean ± 95% CI). In contrast, the four best‐supported nlm models suggested that both the scaling exponent and activation energy significantly vary across the high rank taxa (Collembola, Cryptostigmata, Mesostigmata and Prostigmata) to which species belong, with mean values of b ranging from about 0.6 to 0.8. We therefore reached two conclusions: 1, published analyses of arthropod metabolism based on logarithmic data may be biased by data transformation; 2, non‐linear models applied to Antarctic microarthropod metabolic rate suggest that intraspecific scaling of standard metabolic rate in Antarctic microarthropods is highly variable and can be characterised by scaling exponents that greatly vary within taxa, which may have biased previous interspecific comparisons that neglected intraspecific variability.