In extant taxa, body size is recognized as one of the most important biological properties because it strongly correlates with numerous physiological and ecological factors, such as metabolic rate [1–3], growth rate [4, 5], fecundity , diversity , and population density [8, 9], as well as home range and land area [6, 10, 11], which are related to the productivity of the host environment . Due to these relationships, estimates of body mass (the standard measure of body size) are essential for inferring the paleobiology of extinct taxa, and investigating large-scale evolutionary and ecological patterns in the history of life.
Due to the biological implications of body size, it is not surprising that numerous paleontological studies have used body mass estimates to reconstruct and interpret: patterns of body size evolution [13–22], brain-size allometry and evolution [23–26], the evolution of reproduction [27–29], growth rates [30, 31], postural allometry and locomotion [14, 32, 33], metabolism [34–36], paleotemperature , visceral organ size , and community and trophic structures [10, 39, 40]. In order to infer these biological properties, studies require the use of an estimate or proxy of body size, which can have a large effect on the final interpretation. As a result, it is important to understand the set of assumptions/errors incurred by body size estimates and proxies.
Currently, there are two types of methods used to estimate body mass in extinct animals: volumetric reconstructions and skeletal scaling relationships. The latter method is commonly used to predict body mass in extinct members of relatively recent crown clades (that is, of Mesozoic origin) such as Mammalia and Aves [21, 41–45]. However, in stem groups (for example, non-avian dinosaurs and non-mammalian synapsids), estimations are often based on volumetric reconstructions, which involve physical three-dimensional scale models [46, 47], graphic double integration of two-dimensional reconstructions [48–50], or computer-generated life reconstructions [51–55]. Such estimates are widely used in the literature (for example, [35, 38]) despite the fact that they are prone to a considerable amount of error. In a typical example, body mass estimates for a single mounted skeleton of Brachiosaurus brancai recently published by the same research group have resulted in estimates of 38 tonnes and 74.4 tonnes [54, 56]. Such differences in estimates are the result of differing interpretations of a multitude of factors associated with the mass and proportion of an organism's tissues and organs , or, perhaps most importantly, the effects of air sacs and lungs, which will likely have a large effect on specific gravity (the total body density of the animal in relation to water), needed to estimate mass from a volume. Within non-avian reptiles specific gravity has been noted to range from 0.8 to 1.2 [46, 48]; however, given the varying levels of bone pneumaticity observed in saurischian dinosaurs [58, 59], and the fact that birds typically exhibit lower densities than mammals and other reptiles , it is almost certain that the specific gravity of extinct animals also varied . As a result, assumptions based on a set density parameter will considerably affect a mass estimate [54, 56]. Perhaps more importantly, the numerous assumptions about soft tissue properties and body shape (for example, muscle sizes) in many of the models make it difficult to control for sources of error and to determine the confidence associated with a given mass estimate, although recent computational modelling advances attempt to outline maximum and minimum body mass bounds (for example, [54, 61, 62]). Despite the complications associated with life reconstructions of extinct taxa, models are important for testing numerous biomechanical hypotheses [61, 63–68]. Therefore, it is important that models be constrained by data derived from extant taxa, such as those obtained from scaling relationships.
An alternative method to reconstructions, and one that can be used to test and constrain scale and computational models (), is the use of scaling relationships between body mass and skeletal dimensions derived from extant taxa. A skeletal measure, if strongly related to body mass, will provide an estimate that controls for the sources of error associated with making a reconstruction, such as determination of tissue volume and specific gravity, which are virtually impossible to constrain in life-reconstructions. Furthermore, skeletal measurements are generally easier to obtain than full body scale reconstructions, especially for taxa that are only partially preserved, and are therefore more practical estimators in large-scale evolutionary and ecological studies (for example, [15–17, 20]). Finally, the variation in the extant dataset can be used to quantify the degree of confidence in the estimated parameter, and can thus provide a range in which a particular body mass is likely to fall, thereby providing a constraint for estimates produced by reconstructed models. Scaling methods are almost universally accepted as a means to estimate body mass accurately for extinct taxa of crown groups, such as mammals and birds (for example, [17, 42]), but have been extensively criticized when applied to more distantly related stem taxa that fall outside the body size range observable in extant representatives, such as Indricotherium , xenarthrans , and non-avian dinosaurs [70–72]. For the first two groups, studies have since shown that scaling relationships still provide the most reliable mass estimates [43, 69].
Dinosaurian body masses are still generally estimated using reconstructions, with the exception of two studies [45, 73]. The pioneering work completed by Anderson et al. , herein referred to as the Anderson method, suggested that the body mass of dinosaurs could be estimated using the measured scaling relationship between live mass and total circumference of the stylopodia (humerus + femur) derived from a sample of 33 species of extant terrestrial mammals. Although the Anderson method provides a more objective way to estimate body mass in extinct taxa, it has been criticized by numerous authors (for example, [49, 56, 61, 70, 71, 74–76]). Here we use an extensive dataset of extant mammals and non-avian reptiles compiled from individual skeletons of live-weighed animals, in order to directly test the three main criticisms made towards the use of a universal limb scaling relationship to estimate body mass in extinct terrestrial amniotes:
1. The widely cited Anderson method, especially among non-avian dinosaur researchers, is criticized based on its use of a taxonomically biased sample towards ungulates (for example, ). Studies examining limb-scaling patterns in mammals have noted that the limb proportions of ungulates differ from those of other mammals [70, 77, 78]. However, whether ungulates differ from other groups of mammals in their scaling patterns of limb circumference to body mass has not been directly tested.
2. Differences in gait and limb posture impart different stress regimes on the limbs [79, 80]. These differences may affect limb morphology, thereby negating the applicability of a single equation to estimate body mass in a variety of extinct vertebrates. Given different stress regimes, we test for differential limb scaling between animals of various gaits and limb posture by comparing differently sized sub-samples of mammals, and parasagittal mammals to sprawling reptiles.
3. Residual outliers (large residual values) and extreme outliers (values at the upper and lower extremes of the dataset) can have a large effect on regression coefficients . The problem of residual outliers in the large-bodied mammalian sample of Anderson et al.  was discussed by Packard et al. . We have expanded the sample size of the large-bodied dataset and will address the effect that potential residual outliers have on the circumference to body mass relationship. The effect of extreme outliers on limb scaling is, in part, mediated by logarithmic transformation of the data, but will also be assessed through size class comparisons. Although the issue of body mass extrapolation to giant extinct taxa (for example, Sauropoda; [50, 72]) will always exist, the vast majority of extinct animals, including most non-avian dinosaurs, fall within the body mass range of extant taxa.
All three of these criticisms are tested for the first time, within the context of 200 mammal and 47 non-avian reptile species [See Additional file 1, Dataset]. Based on our results we develop a universal scaling equation between the total circumference of the stylopodia and body mass that is applicable to all terrestrial quadrupeds, and permits estimation of body mass in extinct taxa along with an error factor that can constrain estimates for use in future paleobiological studies.