An approach to scoring cursorial limb proportions in carnivorous dinosaurs and an attempt to account for allometry

From an initial dataset of 53 theropod species, the general relationship between theropod lower-leg length and body mass is identified. After factoring out this allometric relationship, theropod hindlimb proportions are assessed irrespective of body mass. Cursorial-limb-proportion (CLP) scores derived for each of the considered theropod taxa offer a measure of the extent to which a particular species deviates in favour of higher or lower running speeds. Within the same theropod species, these CLP scores are found to be consistent across multiple adult specimens and across disparate ontogenetic stages. Early theropods are found to have low CLP scores, while the coelurosaurian tyrannosauroids and compsognathids are found to have high CLP scores. Among deinonychosaurs, troodontids have consistently high CLP scores, while many dromaeosaur taxa, including Velociraptor and Deinonychus, have low CLP scores. This indicates that dromaeosaurs were not, overall, a particularly cursorily adapted group. Comparisons between the CLP scores of Tyrannosaurus and specimens referred to the controversial genus Nanotyrannus indicate a strong discrepancy in cursorial adaptations, which supports the legitimacy of Nanotyrannus and the previous suggestions of ecological partitioning between Nanotyrannus and the contemporaneous Tyrannosaurus.

muscle capabilities and on bone and joint strength 22,24 . Proportionate to body mass, more elongate lower legs are typically found among smaller-bodied theropods 11,16,25 . This general rule holds true in comparisons between both closely and disparately related taxa and throughout the ontogeny of a single species-with smaller juveniles having proportionately more elongate lower-legs than larger adults ( Fig. 1) 11,16 .
In this paper, we do not estimate absolute or relative maximum running speeds, but rather quantify and compare theropod limb proportions to assess their relative cursoriality 11,26 . It is the aim of this study to simply evaluate how strongly the morphology of a particular theropod species has been evolutionarily modified in favor of enhanced cursorial ability, regardless of whether or not the end result of that modification is a maximum running speed that is greater or less than that of other species. In this way, we infer phylogenetic differences in cursoriality, which most logically reflects differential selective pressures for rapid or efficient locomotion. To this end, the allometric effects of body mass on theropod limb proportions is a serious potentially confounding problem. Because allometric restrictions and cursorial adaptive pressures are effectively competing forces that exert opposite influences on the limb morphology of any particular theropod, the one has the potential to obscure the other 11 . For instance, a theropod lineage that is exposed to adaptive pressures favoring cursoriality may show relatively unaltered or even diminished lower limb proportions if, at the same time, that lineage undergoes an increase in body mass and allometric pressures match or exceed selective pressures favoring cursoriality. Similarly, a theropod lineage that undergoes a reduction in absolute body mass may show greater proportional limb elongation, not because the lineage has been exposed to greater selective pressures that favor cursoriality but simply because allometric pressures have been relieved. Carrano 11 termed this concealing allometric effect on limb proportions "muting" and "enhancing". This problem has limited large-scale studies of theropod cursorial evolution, and an approach is offered here by which allometric effects on theropod limb proportions can first be identified and then factored out, giving a true measure of the extent to which hindlimb proportions have been modified in favor of cursoriality.

Material and Methods
Composition of the initial dataset. Hindlimb length measurements were taken from an initial set of 53 theropod specimens (Table 1). In instances when multiple valid specimens were available for a particular species, only a large specimen was selected, with the aim of restricting the initial dataset to ontogenetically mature individuals. Femur length and metatarsal III length were measured from the proximal-most to the distal-most extent of both bones. It was not possible to identify the distal-most extent of the tibia in all taxa, because the distal end is frequently obscured by fusion with the astragalus and calcaneum. For this reason, "tibia length" for all taxa is actually a measure of the combined proximodistal length of the tibia and the astragalus/calcaneum, when all bones are held in tight articulation. The term "lower leg-length" is here used to refer to the combined proximodistal length of the tibia (plus the astragalus and calcaneum) and metatarsal III. Several previous studies of limb proportions have found that, within the lower leg, it is simply the proportional length of the metatarsals that typically indicates cursorial morphology, and that the length of the tibia is frequently irrelevant. We have chosen to combine both lengths into a single measure in the interest of inclusivity and because tibia length has been recognized as allometrically variable among some theropods 16 .
In the interest of consistency, all measurements were taken directly by one of either of the two authors (not pulled from previously reported data within the literature). All measurements were made directly from the fossils or, in cases when quality was judged equivalent, from casts (no measurements were made indirectly from photographs or illustrations). Standard tape measures were used for large specimens and digital calipers were used for small specimens.
The goal of this study is only to consider limb proportions in carnivorous theropods, and several large, but presumed herbivorous or largely-herbivorous, theropod groups are notably omitted from the initial set of specimens (e.g. "elaphrosaurs", ornithomimosaurs, oviraptorosaurs, and therizinosaurs). There are three major reasons for not including possible herbivorous taxa. First, the ecological differences between carnivores and herbivores undoubtedly imposes different pressures on locomotor evolution. Cursorial morphology in large herbivores frequently relates more to endurance and the ability to continuously forage across expansive home ranges. Second, many herbivorous theropods have unique foot and limb morphologies. For instance, advanced therizinosauroids have four weight-bearing toes 27 , advanced ornithomimosaurs lack a hallux 28 , and some ornithomimosaurs have broad short ungulas 29 . Key to this study is the overall conservative morphology of most carnivorous theropod limbs, and limiting the considered taxa to carnivores, therefore, removes a substantial source of potential variation from the dataset. Finally, it has been argued that the limb proportions of herbivorous theropods have a different (nearly isometric) ontogeny 30 , and this difference would confound the later consideration of ontogenetic variation.
Calculating cursorial-limb-proportion (CLP) score. To evaluate lower-leg proportions in the context of body mass requires a measureable osteological correlate of body mass. Femoral length is here used as that correlate. Femoral size has been found to be a reasonable indicator of body mass in multiple studies of both modern animals and dinosaurs 16,31,32 . Although other femoral dimensions (such as femur circumference and diameter) are slightly better size correlates than length 33 , length was selected as the measure of femur size, because femur length is seldom distorted by taphonomic factors and could be reliably measured from the largest number of specimens. By comparison, theropod femoral circumference and diameter are often impossible to reliably measure, because theropod femora are relatively thin walled and hollowed and are, therefore, prone to collapsing internally when fossilized and buried. As noted by Campione and Evans 33 , femur-length/mass scaling follows a roughly isometric pattern in modern animals, and multiple studies have shown femoral length to be among the least variable hindlimb dimensions 12,16,[34][35][36] . Femoral length does, therefore, provide a generally reliable indicator of body mass, especially when compared among members of the same taxonomic group with the same general limb forms, and the morphology of non-avian theropod hindlimbs has long been recognised as highly conservative 20 . Additionally, it should be remembered, as pointed out by Carrano 37 in his seminal work on dinosaur size evolution, that, because femoral length has an established linear relationship with body mass, femoral length serves as a valid proxy for body mass and allows the relative sizes of dinosaurs to be compared on the same scale (which is all that is needed in this study).
The initial theropod dataset was used to create a bivariate plot 38 , with femur length and lower-leg length on either axis. A simple best-fit power curve was then applied to the plot (Fig. 2). All analyses were performed using Microsoft Excel 2013. This power curve (which is generated by Equation 1: l = 4.178f 0.8371 , where l is lower-leg length and f is femur length), is here interpreted as representing the normal relationship between body mass (approximated by femur length) and lower-leg length, as established by the dataset as a whole. As such, Equation 1 offers a way of predicting the lower-leg length of a particular species based on its femur length, and comparisons between a predicted lower-leg length and its true lower-leg length offers a way to quantitatively evaluate the relative abnormality of the lower-leg length of that species, irrespective of allometry. The percentage difference by which the true lower-leg length of a particular species differs from the lower-leg length predicted for that species by Equation 1 is here reported as the "cursorial-limb-proportion" (CLP) score of that species. This approach of deriving a comparative score of how a particular taxon differs from a prediction based on the absolute size of that taxon and analysis of a size-dependent relationship seen in a large sample of taxa is commonly used in assessments of allometricly influenced traits, with perhaps the most well-known example being the derivation of encephalization quotient (EQ) scores as a way of estimating animal intelligence from relative brain size 39 . Example. To better explain how the CLP scores were derived in this study, it may be helpful to briefly consider an example. The ceratosaurian theropod Deltadromeus agilis was given its name, which means "agile delta runner", because Sereno et al. 40 interpreted its elongate hindlimbs as being highly adapted for cursoriality. To evaluate the limb proportions of D. agilis using the methods of this study, the femur length (741 mm) is input into the equation for predicted lower-leg length (Equation 1) and yields a predicted length of 1055 mm. In actuality, D. agilis has a lower-leg length of 1134 mm. So, the true lower-leg length of D. agilis differs from its predicted lower-leg length by 79 mm. Thus the inference of Sereno et al. 40 is here supported, as D. agilis is found to have a lower-leg that is 7.5% longer than would be "normal" for a theropod of its size (based on the relationship seen in the initial 53 taxa dataset) and is given a CLP score of + 7.5. Note: if D. agilis had failed to live up to its name and had been found to have a lower-leg length that was abnormally short (below the predicted length), its CLP score would be reported as a negative value.
Exploring consistency in multi-specimen taxa. Because calculating the CLP score for any particular species requires femoral, tibia, and metatarsal III length measurements, CLP scores can only be calculated from specimens with relatively complete hindlimbs. This limits the number of taxa that are able to contribute to the initial dataset. It also means that, for the vast majority of species, it is only possible to base the CLP score calculation on measurements taken from a single specimen. For the sake of consistency, all species in the initial dataset are represented only by single specimens (in instances where more than one potential specimen was available, the largest or the best preserved specimen was generally chosen). This imparts a potential source of error. In the first place, the initial dataset does not consider the amount of individual variation that may be present within a species. Secondly, and potentially more seriously, some species may be represented by specimens that are ontogenetically immature.
The few theropod species for which multiple specimens with sufficiently complete hindlimbs are known offer a chance to explore both the degree of individual variation in CLP score and the effect of ontogeny. Limb measurements were taken from multiple specimens of six theropod species (Albertosaurus sarcophagus n = 4, Allosaurus fragilis n = 8, Coelophysis bauri n = 10, Gorgosaurus libratus n = 6, Herrerasaurus ischigualestensis n = 7, and Tyrannosaurus rex n = 4) ( Table 2).

Results
The CLP scores derived from the initial dataset are reported in Table 3, with all taxa arranged in rank order of CLP score, and in Fig. 3, with taxa arranged phylogenetically. The scores range from -20.4 to + 40.6, with an average of + 0.7.
The CLP scores derived from the multi-specimen dataset are presented in Table 2 and Fig. 4. The highest variation in the scores within any species was found in the four specimens of Tyrannosaurus rex-scores ranged from + 15.5 to + 9.1, a difference of 6.4. However, a series of F-tests confirm that the amount of variance seen in the Tyrannosaurus rex data is not significantly greater than that observed in any of the other multi-specimen taxa. The greatest deviation of any score from the mean score of its species was found in the Tyrannosaurus rex specimen MOR 555, which deviates from the mean by 3.2. This suggests that, when interpreting the scores derived from the initial dataset, it is reasonable to assume that the reported CLP scores may deviate from the average CLP score of that species by as much + /− 3.2.
Of the eight oldest and phylogenetically least-derived theropods included in the initial dataset, none had a positive CLP score. Guaibasaurus candelariensis, and Herrerasaurus ischigualestensis, the most basal dinosaurs included in the study, both have scores below − 10 (implicating them as among the least cursorially adapted).  These results contradict previous interpretations that some early theropods are examples of highly cursorial forms, and suggests that such interpretations were misled by the effect of allometry and the relative small size of these early theropods. Instead, the primitive theropod condition appears to have been hindlimb proportions that are relatively non-cursorial. Evidence of high cursorial limb proportions were found among deinonychosaurs, tyrannosauroids, compsognathids, and also found in the non-coelurosaurian theropods Concavenator corcovatus and Deltadromeus agilis. Note that we have treated Guaibasaurus candelariensis, and Herrerasaurus ischigualestensis as theropods. There is currently debate over whether these taxa belong within the Theropoda proper or if they fall outside   Table 3. Cursorial-limb-proportion (CLP) scores from the initial dataset.

Discussion and Additional Analyses
Tyrannosauroids. One clear result from the initial dataset CLP score calculations is that tyrannosauroids have high CLP scores. Among the sampled tyrannosauroids, the basal taxa Dilong paradoxus, Dryptosaurus aquilunguis, Guanlong wucaii, and Yutyrannus huali have the lowest scores, while the more advanced tyrannosauroids Alectrosaurus olsoni and Appalachiosaurus montgomeriensis and all tyrannosaurs have much higher scores. This confirms previous assessments that tyrannosauroids are characterised by proportionately elongate hindlimbs and that lower-leg length became more exaggerated in later and more advanced forms 11,12,20 . The development of high CLP scores in derived tyrannosauroids is consistent with the evolution of an arctometatarsus. The arctometatarsus is a modified metatarsal form that has been linked to fast linear locomotion 12,47 and enhanced agility [48][49][50] . That tyrannosauroids have exceptionally elongate lower-legs is a factor that may modify how the results of this study should be interpreted. Because of their more recent heritage and the resulting high abundance of more complete specimens, tyrannosauroids make a large contribution to the initial dataset (eleven taxa, accounting for more than 20%). In particular, tyrannosauroids are disproportionately represented among the extremely large theropods in the initial dataset (tyrannosauroids account for seven of the thirteen theropods with femur length greater than 750 mm and four of the six theropods with femur length greater than 1000 mm). This high concentration of proportionately long legged but extremely large theropods may have skewed the dataset and had undue influence on the derivation of the predicted lower-leg length equation. To test this possibility, the tyrannosauroids data was separated from the initial dataset and the resulting two new datasets were subjective to an analysis of covariance (ANCOVA) using R statistical software. The result suggests that the tyrannosauroid data does have a significant influence (F = 21.06, p > 0.001). This indicates that special caution is warranted when interpreting the scores of species found by this study to have negative or "abnormally" low CLP scores. In fact, the proportions of these species may actually be closer to the norm or even above it, but have received a negative score because they are being considered within a dataset that has a high concentration of the extremely leggy tyrannosauroids. In particular, the low scores reported for other large theropods should be interpreted cautiously, and the method and approach here outlined will benefit from the future addition of more large non-tyrannosauroid taxa, but, at present, sufficiently complete specimens from such taxa are lacking.
Deinonychosaurs. Gatesy and coauthors 18,[51][52][53][54] observed that a major change in hindlimb locomotive style occurred during the evolution of birds and their close relatives: the size and importance of the caudofemoral musculature was greatly reduced and the importance of knee flexion increased, while the importance of femoral retraction decreased. Although it was originally hypothesised that this change in locomotive musculature and emphasis occurred gradually across the whole of the theropod lineage, caudofemoral musculature remained important and unreduced in many coelurosaurian groups 55,56 . However, even the most basal deinonychosaurs show evidence of substantial caudofemoral reduction and are inferred to have begun the corresponding change in locomotor style 57 . This means that the CLP scores calculated for deinonychosaurs should be interpreted with special caution. As seen in modern birds, greater emphasis on knee flexion requires, if stride length and speed are not to be diminished, greater elongation of the lower leg and concomitantly reduced femora 53,58 . Because increased emphasis on knee flexion generally requires proportionately longer metatarsals and shorter femora to still accomplish high speed running and because deinonychosaurs are classically regarded as among the more cursorily adapted theropods, deinonychosaurs would be doubly expected to have high CLP scores. However, this expectation is only partially met. All four troodontid species were found to have high CLP scores (ranging from + 4.5 to + 40.6 -the highest score of any of the considered taxa), and a Grubbs' test found Sinornithoides youngi, which has the highest CLP score in the dataset (40.6) to be a significant outlier (Z = 3.337, critical Z = 3.151). This suggests that at least some troodontids had adapted avian-like limb proportions. However, the CLP scores of the eight dromaeosaurs were decidedly mixed (ranging from −20.4 to +18.5). Of these, Mahakala omnogovi, Microraptor gui, and Saurornitholestes langstoni have extremely high scores, while the CLP scores of the other five dromaeosaurs are all negative.
That a majority of the considered dromaeosaurs were found to have low scores indicates dromaeosaurs, as a group, did not undergo strong adaptive limb specialization for high-speed running. Indeed, given the reduced caudofemoral musculature of dromaeosaurs and the higher CLP scores of most compsognathids and tyrannosauroids (both more basal coelurosaurian groups), the opposite seems true: a majority of dromaeosaurs appear to have undergone a de-emphasis on cursorial limb proportions and to have been exposed to strong selective pressures favoring reduced running ability. These results are largely consistent with those of Carrano (1990) 11 . As in tyrannosauroids, the deinonychosaurs CLP scores are consistent with the presence/absence of an arctometatarsus: the consistently high scoring troodontids possess an arctometatarsus, while the dromaeosaurs do not.
Ontogenetic variation and controversial taxa. Among the multi-specimen dataset were individuals of the same species that differed from each other substantially in terms of femur length, and, therefore, assumed body mass. Although absolute size is not always an indicator of relative age, it is reasonable to assume that many of the smaller specimens probably represent younger individuals. For instance, the largest of the Coelophysis bauri specimens (CMNH 10971a) has a femur that is more than 53% longer than the smallest (AMNH 7246), and the largest of the Herrerasaurus ischigualestensis specimens (PVL 2566) has a femur that is more than 58% longer than the smallest (MACN 18.060). Four of the tyrannosaur specimens are known juveniles (the Albertosaurus sarcophagus specimen NMC 11315, the Gorgosaurus libratus specimens AMNH 5423 and FMNH PR 2211, and the Tyrannosaurus rex specimen LACM 23845). Of these, FMNH PR 2211 has a femur length that is less than half that of NMC 11593 (implying an order of magnitude difference in likely bodyweight). Yet, the CLP scores calculated for even these exceptionally large and small specimens do not strongly vary from each other and fall within or near the range of scores calculated from other more moderately sized members of the same species (Fig. 4). The CLP score of a particular theropod species, therefore, appears to neither increase nor decrease with mass and age. This indicates that the widely documented changes in theropod hindlimb proportions over ontogeny can be largely explained by factors simply relating to growth in body mass.
Aside from indicating a generalized growth pattern across theropods, the recognition that, with respect simply to body mass, theropod hindlimb proportions follow roughly the same trend ontogenetically as interspecifically has several implications. First, it suggests that even if some specimens included in the initial dataset are immature, the CLP scores derived for those specimens are not likely to be misrepresentative. Second, it means that comparing CLP scores offers a potential independent method for assessing the validity of novel taxa erected based on immature or age-indeterminate specimens, which are suspected of belonging to pre-existing species, particularly when differences in limb proportions are hypothesised to be discriminating characters.
For example, the controversial Late Cretaceous tyrannosaur Nanotyrannus lancensis has been interpreted by some as a separate genus [59][60][61][62] and by others as a junior synonym of Tyrannosaurus rex 16,63,64 . Arguments that favor the synonymy of N. lancensis and T. rex center on the interpretation of the various traits that appear to differ between N. lancensis and T. rex (which include various proportions of the skull, a small foramen in the quadratojugal, braincase morphology, tooth counts, and the form of the glenoid) as being ontogenetically dependent and indicating that all alleged N. lancensis specimens are, in actuality, immature specimens of T. rex. There has been much discussion over the legitimacy of N. lancensis within the literature, but there will be no attempt here to summarize the points and counterpoints of, and to, the various arguments made by both camps (instead, readers are directed to consult Carr 63 and Larson 59 ). These arguments include a variety of anatomical proportions and characters, with Carr 64 emphasising synapomorphies and N. lancensis diagnoses generally focusing on differences between it and T. rex. The elongate hindlimb proportions of specimens referred to N. lancensis have been specifically implicated in the debate as a trait that distinguishes N. lancensis and, alternatively, as a trait that can be explained away as simply reflecting immaturity.
The allegedly elongate hindlimbs of Nanotyrannus lancensis also have potential paleoecological implications. Excluding N. lancensis, Tyrannosaurus rex is the only large carnivorous theropod known from the uppermost Maastrichtian beds of North America. This implies a lower diversity in large predators than is seen in most other well sampled dinosaur faunas. Furthermore, it has been postulated that the more elongate hindlimbs of N. lancensis reflect a form of predatory ecological niche partitioning between N. lancensis and T. rex. Bakker 65 has suggested that the two tyrannosaurs are analogous to modern lions and cheetahs, with the smaller, more gracile, and longer legged N. lancensis being adapted for high-speed running. Such a comparison implies that N. lancensis should have proportionately more elongate hindlimbs and should, therefore, be predicted to have a much higher CLP score than T. rex.
To test this prediction, leg measurements were taken from two specimens that have been referred to Nanotyrannus lancensis. The first of these specimens is BMRP 2002.4.1 ("Jane"). The second is BHI-6437, a 3-D digital specimen produced through photogrammetry 60  rex. Moreover, the scores of BMRP 2002.4.1 and BHI-6437 exceed the range of scores established by any of the other tyrannosauroids, including similarly sized and immature Albertosaurus and Gorgosaurus specimens (Fig. 4). The prediction that the alleged Nanotyrannus lancensis specimens should show limb proportions indicative of high cursorial adaptation is, therefore, met. Indeed, the two N. lancensis scores exceed those of any other non-avialan theropod included in the initial dataset, making N. lancensis arguably the most cursorily adapted of all non-avialan carnivorous theropods.
Nevertheless, some caution is warranted in the interpretation of the exceptionally high CLP scores here reported for the two alleged Nanotyrannus lancensis specimens. Although the score of BMRP 2002.4.1 and BHI-6437 exceed those of adult Tyrannosaurus rex specimens and those of juvenile specimens of other tyrannosaurs, there is, as yet, no clearly identified juvenile T. rex specimen with limb proportions different from the referred N. lancensis specimens. It could, therefore, still be argued that T. rex and N. lancensis are synonymous, and that these results simply show that juvenile T. rex possessed abnormally cursorial limb proportions that became altered over ontogeny. This is an interpretation with its own significant ecological implications -perhaps for ontogenetic diet shifts and adult vs. juvenile niche partitioning. However, such an interpretation is an argument for a special case, because, in at least two other genera of large-bodied tyrannosaurs (Albertosaurus and Gorgosaurus), it is known that no similar changes in limb proportions occur. Establishing growth series for other large tyrannosaurs, including the closely related Tarbosaurus, will help further address this challenge.

Conclusion.
Accounting for the influence of allometry permits cursorial hindlimb proportions to be scored across all carnivorous theropods, regardless of body mass. CLP scores are generally low among early primitive theropods but are high in more derived forms, including both the small-bodied compsognathids and the large-bodied tyrannosaurs. This supports previous arguments that coelurosaurs are characterised by highly cursorial limb proportions 11,12 and supports more general inferences of increased relative limb elongation throughout the evolutionary history of predatory theropods 66 . However, dromaeosaurs constitute an exception, as several dromaeosaur taxa appear to have strongly reduced cursorial limb proportions.
That the same allometric correcting method derived from interspecific comparisons also appears effective in intraspecific ontogenetic comparisons, indicates that much of the ontogenetic limb variation previously reported within different theropod taxa can be explained in terms of simple allometry. Although in the tyrannosaurs Albertosaurus and Gorgosaurus, small bodied juveniles were found to fall within the same CLP-score range as large-bodied adults, CLP scores calculated for the tyrannosaur Nanotyrannus fell well outside the range of scores calculated for Tyrannosaurus. This result suggests that the proportionately elongate lower legs of Nanotyrannus are not allometrically equivalent to those of Tyrannosaurus and are, therefore, a legitimate character to cite as a morphological discriminator between the two. This illustrates how CLP scores may in future studies be used in taxonomic assessments of juvenile specimens. Unlike many previous attempts at estimating maximum running speeds in theropods and other approaches to allometric assessments of cursorial adaptations, the method here outlined is simple and Equation 1 can easily be applied to other theropod taxa. This method offers a way to quantify the degree of hindlimb elongation in descriptions of new theropod taxa, such that cursoriality can be quantitatively and more accurately assessed. Hopefully this method will be refined and utilized as new specimens become available, and the methodology will be applied to other dinosaur groups.  Table 4. Limb measurements and CLP scores from the Nanotyrannus lancensis dataset (all measurements in mm).