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.     

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.     

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.     

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.     

Mechanics of posture and gait of some large dinosaurs

Dimensions of dinosaur bones and of models of dinosaurs have been used as the basis for calculations designed to throw light on the posture and gaits of dinosaurs.
Estimates of the masses of some dinosaurs, obtained from the volumes of models, are compared with previous estimates. The positions of dinosaurs' centres of mass, derived from models, show that some large quadrupedal dinosaurs supported most of their weight on their hind legs and were probably capable of rearing up on their hind legs.
Distributions of bending moments along the backs of large dinosaurs are derived from measurements on models. The tensions required in epaxial muscles to enable Diplodocus to stand are calculated. It is likely that the long neck of this dinosaur was supported by some structure running through the notches in the neural spines of its cervical and dorsal vertebrae. The nature of this hypothetical structure is discussed.
An attempt is made to reconstruct the walking gait of sauropod dinosaurs, from the pattern of footprints in fossil tracks.
The dimensions of dinosaur leg bones are compared to predictions for mammals of equal body mass, obtained by extrapolation of allometric equations. Their dimensions are also used to calculate a quantity which is used as an indicator of strength in bending. Comparisons with values for modern animals lead to speculations about the athletic performance of dinosaurs.
Estimates of pressures exerted on the ground by the feet of dinosaurs are used in a discussion of the ability of dinosaurs to walk over soft ground.     

Energetic benefits and adaptations in mammalian limbs: Scale effects and selective pressures

Differences in limb size and shape are fundamental to mammalian morphological diversity; however, their relevance to locomotor costs has long been subject to debate. In particular, it remains unknown if scale effects in whole limb morphology could partially underlie decreasing mass‐specific locomotor costs with increasing limb length. Whole fore‐ and hindlimb inertial properties reflecting limb size and shape—moment of inertia (MOI), mass, mass distribution, and natural frequency—were regressed against limb length for 44 species of quadrupedal mammals. Limb mass, MOI, and center of mass position are negatively allometric, having a strong potential for lowering mass‐specific locomotor costs in large terrestrial mammals. Negative allometry of limb MOI results in a 40% reduction in MOI relative to isometry's prediction for our largest sampled taxa. However, fitting regression residuals to adaptive diversification models reveals that codiversification of limb mass, limb length, and body mass likely results from selection for differing locomotor modes of running, climbing, digging, and swimming. The observed allometric scaling does not result from selection for energetically beneficial whole limb morphology with increasing size. Instead, our data suggest that it is a consequence of differing morphological adaptations and body size distributions among quadrupedal mammals, highlighting the role of differing limb functions in mammalian evolution.     

Body mass estimation in xenarthra: A predictive equation suitable for all quadrupedal terrestrial placentals?

The Magnorder Xenarthra includes strange extinct groups, like glyptodonts, similar to large armadillos, and ground sloths, terrestrial relatives of the extant tree sloths. They have created considerable paleobiological interest in the last decades; however, the ecology of most of these species is still controversial or unknown. The body mass estimation of extinct species has great importance for paleobiological reconstructions. The commonest way to estimate body mass from fossils is through linear regression. However, if the studied species does not have similar extant relatives, the allometric pattern described by the regression could differ from those shown by the extinct group. That is the case for glyptodonts and ground sloths. Thus, stepwise multiple regression were developed including extant xenarthrans (their taxonomic relatives) and ungulates (their size and ecological relatives). Cases were weighted to maximize the taxonomic evenness. Twenty‐eight equations were obtained. The distribution of the percent of prediction error (%PE) was analyzed between taxonomic groups (Perissodactyla, Artiodactyla, and Xenarthra) and size groups (0–20 kg, 20–300 kg, and more than 300 kg). To assess the predictive power of the functions, equations were applied to species not included in the regression development [test set cross validation, (TSCV)]. Only five equations had a homogeneous %PE between the aforementioned groups. These were applied to five extinct species. A mean body mass of 80 kg was estimated for Propalaehoplophorus australis (Cingulata: Glyptodontidae), 594 kg for Scelidotherium leptocephalum (Phyllophaga: Mylodontidae), and 3,550.7 kg for Lestodon armatus (Phyllophaga: Mylodontidae). The high scatter of the body mass estimations obtained for Catonyx tarijensis (Phyllophaga: Mylodontidae) and Thalassocnus natans (Phyllophaga: Megatheriidae), probably due to different specializations, prevented us from predicting its body mass. Surprisingly, although obtained from ungulates and xenarthrans, these five selected equations were also able to predict the body mass of species from groups as different as rodents, carnivores, hyracoideans, or tubulidentates. This result suggests the presence of a complex common allometric pattern for all quadrupedal placentals. J. Morphol., 2008. © 2008 Wiley‐Liss, Inc.     

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, ?? , ??–??.     

Allometric scaling of seed retention time in seed dispersers and its application to estimation of seed dispersal potentials of theropod dinosaurs

Seed retention time (SRT), the time interval between seed ingestion and defaecation, is a critical parameter that determines the spatial pattern of seed dispersal created by an animal, and is therefore, an essential component of trait‐based modelling of seed dispersal functions. However, no simple predictive model of SRT for any given animal exists. We explored the linkage between animal traits and SRT. We collected previously published data on mean SRT for 112 species of birds, mammals, reptiles and fishes and investigated the general allometric scaling of mean SRT with body mass for each taxon. Moreover, we analysed the effects of food habit and digestive strategy on mean SRT for birds and mammals. In general, mean SRT increased with body mass in all four taxa, whereas the pattern of allometric scaling varied greatly among the taxa. Birds had a smaller intercept and larger slope than those of mammals, whereas reptiles had a much larger intercept and smaller slope than those of either birds or mammals. For birds, food habit was also detected as an important factor affecting SRT. We applied the allometric scaling that was obtained for birds to estimate mean SRT of extinct Mesozoic dinosaurs (Theropoda) – few of which are assumed to have acted as seed dispersers. SRT for large carnivorous theropods was estimated to be 4–5 days, when considering only body mass. The present study provides allometric scaling parameters of mean SRT for a variety of seed‐dispersing animals, and highlights large variations in scaling among taxa. The allometric scaling obtained could be a critical component of further trait‐based modelling of seed dispersal functions. Further, the potential and limitations of the scaling of animal SRT with body mass and a future pathway to the development of trait‐based modelling are discussed.     

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.     

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.     

Sauropod dinosaurs evolved moderately sized genomes unrelated to body size

Sauropodomorph dinosaurs include the largest land animals to have ever lived, some reaching up to 10 times the mass of an African elephant. Despite their status defining the upper range for body size in land animals, it remains unknown whether sauropodomorphs evolved larger-sized genomes than non-avian theropods, their sister taxon, or whether a relationship exists between genome size and body size in dinosaurs, two questions critical for understanding broad patterns of genome evolution in dinosaurs. Here we report inferences of genome size for 10 sauropodomorph taxa. The estimates are derived from a Bayesian phylogenetic generalized least squares approach that generates posterior distributions of regression models relating genome size to osteocyte lacunae volume in extant tetrapods. We estimate that the average genome size of sauropodomorphs was 2.02 pg (range of species means: 1.77–2.21 pg), a value in the upper range of extant birds (mean = 1.42 pg, range: 0.97–2.16 pg) and near the average for extant non-avian reptiles (mean = 2.24 pg, range: 1.05–5.44 pg). The results suggest that the variation in size and architecture of genomes in extinct dinosaurs was lower than the variation found in mammals. A substantial difference in genome size separates the two major clades within dinosaurs, Ornithischia (large genomes) and Saurischia (moderate to small genomes). We find no relationship between body size and estimated genome size in extinct dinosaurs, which suggests that neutral forces did not dominate the evolution of genome size in this group.     

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.     

Scaling of skeletal mass to body mass in fishes

The few available observations are consistent with the supposition that the relative weightlessness of fishes leads to isometric scaling of skeletal mass to body mass. To explore further this pattern we studied scaling in ontogeny with freshwater tilapia, Oreochromis nilotica, and in phylogeny with adult coral reef fishes. Body mass and skeletal mass were measured for freshly caught fishes. Data were transformed to logarithms and fitted to a power function with least-square linear regression. Whereas slope for all O. nilotica combined was consistent with isometry (b = 1.00; 95% CI = 0.02), slopes calculated separately for juveniles (b = 1.16; CI = 0.07) and adults (b = 1.10; CI = 0.07) indicated positive allometric scaling of the skeleton during ontogeny. The scaling pattern was isometric for a multispecies sample of perciform fishes from coral reefs (b = 0.82; CI = 0.21). However, the single perciform species with the largest number of individuals in the sample, Epinephelus guttatus, was positively allometric (b = 1.13; CI = 0.12), whereas the tetraodontiform, Balistes vetula, was isometric (b = 1.05; CI = 0.12). Instead of leading to isometry, weightlessness may increase the range of possibilities for the scaling of skeleton mass to body mass in fishes compared to terrestrial vertebrates. The scaling of the skeleton in fishes may be related to foraging style and manner of locomotion in water rather than be driven by the need to resist gravity. © 1996 Wiley-Liss, Inc.     

Turning Semicircular Canal Function on Its Head: Dinosaurs and a Novel Vestibular Analysis

Previous investigations have correlated vestibular function to locomotion in vertebrates by scaling semicircular duct radius of curvature to body mass. However, this method fails to discriminate bipedal from quadrupedal non-avian dinosaurs. Because they exhibit a broad range of relative head sizes, we use dinosaurs to test the hypothesis that semicircular ducts scale more closely with head size. Comparing the area enclosed by each semicircular canal to estimated body mass and to two different measures of head size, skull length and estimated head mass, reveals significant patterns that corroborate a connection between physical parameters of the head and semicircular canal morphology. Head mass more strongly correlates with anterior semicircular canal size than does body mass and statistically separates bipedal from quadrupedal taxa, with bipeds exhibiting relatively larger canals. This morphologic dichotomy likely reflects adaptations of the vestibular system to stability demands associated with terrestrial locomotion on two, versus four, feet. This new method has implications for reinterpreting previous studies and informing future studies on the connection between locomotion type and vestibular function.     

Long-bone circumference and weight in mammals, birds and dinosaurs

The mid-shaft circumferences of the humerus and femur are closely related to body weight in living terrestrial vertebrates. Because these elements are frequently preserved in subfossil and fossil vertebrate skeletal materials, the relationship can be used to estimate body weight in extinct vertebrates. When the allometric equations are applied to the mid-shaft circumferences of these elements in dinosaurs, the weights calculated for some giant sauropods ( Brachiosaurus ) are found to be lighter than previous estimates.     

A quantitative method for inferring locomotory shifts in amniotes during ontogeny,its application to dinosaurs and its bearing on the evolution of posture

Evolutionary transitions between quadrupedal and bipedal postures are pivotal to the diversification of amniotes on land, including in our own lineage (Hominini). Heterochrony is suggested as a macroevolutionary mechanism for postural transitions but understanding postural evolution in deep time is hindered by a lack of methods for inferring posture in extinct species. Dinosaurs are an excellent natural laboratory for understanding postural transitions because they demonstrate at least four instances of quadrupedality evolving from bipedality, and heterochronic processes have been put forward as an explanatory model for these transitions. We extend a quantitative method for reliably inferring posture in tetrapods to the study of ontogenetic postural transitions using measurements of proportional limb robusticity. We apply this to ontogenetic series of living and extinct amniotes, focusing on dinosaurs. Our method correctly predicts the general pattern of ontogenetic conservation of quadrupedal and bipedal postures in many living amniote species and infers the same pattern in some dinosaurs. Furthermore, it correctly predicts the ontogenetic postural shift from quadrupedal crawling to bipedal walking in humans. We also infer a transition from early ontogenetic quadrupedality to late-ontogenetic bipedality in the transitional sauropodomorph dinosaur Mussaurus patagonicus and possibly in the early branching ceratopsian Psittacosaurus lujiatunensis but not in the sauropodomorph Massospondylus carinatus. The phylogenetic positions of these ontogenetic shifts suggest that heterochrony may play a role in the macroevolution of posture, at least in dinosaurs. Our method has substantial potential for testing evolutionary transitions between locomotor modes, especially in elucidating the role of evolutionary mechanisms like heterochrony.     

Footprints pull origin and diversification of dinosaur stem lineage deep into Early Triassic

The ascent of dinosaurs in the Triassic is an exemplary evolutionary radiation, but the earliest phase of dinosaur history remains poorly understood. Body fossils of close dinosaur relatives are rare, but indicate that the dinosaur stem lineage (Dinosauromorpha) originated by the latest Anisian (ca 242-244 Ma). Here, we report footprints from the Early-Middle Triassic of Poland, stratigraphically well constrained and identified using a conservative synapomorphy-based approach, which shifts the origin of the dinosaur stem lineage back to the Early Olenekian (ca 249-251 Ma), approximately 5-9 Myr earlier than indicated by body fossils, earlier than demonstrated by previous footprint records, and just a few million years after the Permian/Triassic mass extinction (252.3 Ma). Dinosauromorph tracks are rare in all Polish assemblages, suggesting that these animals were minor faunal components. The oldest tracks are quadrupedal, a morphology uncommon among the earliest dinosauromorph body fossils, but bipedality and moderately large body size had arisen by the Early Anisian (ca 246 Ma). Integrating trace fossils and body fossils demonstrates that the rise of dinosaurs was a drawn-out affair, perhaps initiated during recovery from the Permo-Triassic extinction.     

Biomechanics of Running Indicates Endothermy in Bipedal Dinosaurs

Background

One of the great unresolved controversies in paleobiology is whether extinct dinosaurs were endothermic, ectothermic, or some combination thereof, and when endothermy first evolved in the lineage leading to birds. Although it is well established that high, sustained growth rates and, presumably, high activity levels are ancestral for dinosaurs and pterosaurs (clade Ornithodira), other independent lines of evidence for high metabolic rates, locomotor costs, or endothermy are needed. For example, some studies have suggested that, because large dinosaurs may have been homeothermic due to their size alone and could have had heat loss problems, ectothermy would be a more plausible metabolic strategy for such animals.

Methodology/Principal Findings

Here we describe two new biomechanical approaches for reconstructing the metabolic rate of 14 extinct bipedal dinosauriforms during walking and running. These methods, well validated for extant animals, indicate that during walking and slow running the metabolic rate of at least the larger extinct dinosaurs exceeded the maximum aerobic capabilities of modern ectotherms, falling instead within the range of modern birds and mammals. Estimated metabolic rates for smaller dinosaurs are more ambiguous, but generally approach or exceed the ectotherm boundary.

Conclusions/Significance

Our results support the hypothesis that endothermy was widespread in at least larger non-avian dinosaurs. It was plausibly ancestral for all dinosauriforms (perhaps Ornithodira), but this is perhaps more strongly indicated by high growth rates than by locomotor costs. The polarity of the evolution of endothermy indicates that rapid growth, insulation, erect postures, and perhaps aerobic power predated advanced “avian” lung structure and high locomotor costs.