How not to answer interdisciplinary "Why?" questions.

The book under review tries to link the economic concept of "reward," or, more accurately, "capture rate," to the experimental literature of various neuroscientific quantities dealing with motor control. But this reviewer argues that such a linkage requires a richer language of quantification than the book actually affords: a language not just of "greater" or "less," but of how much greater or less. Without such a methodology, the arguments here cannot be persuasive.

Correlation of the human pubic symphysis surface with age-at-death: a novel quantitative method based on a bandpass filter.

Age-at-death estimation from skeletal remains typically utilizes the roughness of pubic symphysis articular surfaces. This study presents a new quantitative method adapting a tool from geometric morphometrics, bandpass filtering of partial warp bending energy to extract only age-related changes of the surfaces. The study sample consisted of 440 surface-scanned symphyseal pubic bones from men between 14 and 82 years of age, which were landmarked with 102 fixed and surface semilandmarks. From the original sample, 371 specimens within Procrustes distance of 0.05 of the side-specific average were selected. For this subsample, age was correlated with total bending energy (calculated as summed squared partial warps amplitudes) for a wide range of plausible bandpass filters. For our subsample's 188 right-side surfaces, the correlation between age and bandpass filtered versions of bending energy peaks relatively sharply at r = -0.648 for ages up through 49 years against the first seven partial warp amplitudes only. The finding for left symphyses is similar. The results demonstrate that below the age 50, the symphyseal surface form changes most systematically related to age may be best detected by a lowpass-filtered version of bending energy: signals at the largest geometric scales of roughness rather than its full spectrum. Combining this method with information from other skeletal features could further improve age-at-death estimation based on the symphyseal pubic surface.

Morphometric Variation at Different Spatial Scales: Coordination and Compensation in the Emergence of Organismal Form.

It is a classic aim of quantitative and evolutionary biology to infer genetic architecture and potential evolutionary responses to selection from the variance-covariance structure of measured traits. But a meaningful genetic or developmental interpretation of raw covariances is difficult, and classic concepts of morphological integration do not directly apply to modern morphometric data. Here, we present a new morphometric strategy based on the comparison of morphological variation across different spatial scales. If anatomical elements vary completely independently, then their variance accumulates at larger scales or for structures composed of multiple elements: morphological variance would be a power function of spatial scale. Deviations from this pattern of "variational self-similarity" (serving as a null model of completely uncoordinated growth) indicate genetic or developmental coregulation of anatomical components. We present biometric strategies and R scripts for identifying patterns of coordination and compensation in the size and shape of composite anatomical structures. In an application to human cranial variation, we found that coordinated variation and positive correlations are prevalent for the size of cranial components, whereas their shape was dominated by compensatory variation, leading to strong canalization of cranial shape at larger scales. We propose that mechanically induced bone formation and remodeling are key mechanisms underlying compensatory variation in cranial shape. Such epigenetic coordination and compensation of growth are indispensable for stable, canalized development and may also foster the evolvability of complex anatomical structures by preserving spatial and functional integrity during genetic responses to selection.[Cranial shape; developmental canalization; evolvability; morphological integration; morphometrics; phenotypic variation; self-similarity.].

Anatomical similarity between the Sost-knockout mouse and sclerosteosis in humans.

Sclerosteosis, a rare autosomal recessive genetic disorder caused by a mutation of the Sost gene, manifests in the facial skeleton by gigantism, facial distortion, mandibular prognathism, cranial nerve palsy, and, in extreme cases, compression of the medulla oblongata. Mice lacking sclerostin reflect some symptoms of sclerosteosis, but this is the first report of the effect on the facial skeleton. We used geometric morphometrics (GMM) to analyze the deformations of the murine facial skeleton from the wild-type to the Sost gene knockout. Landmark coordinates were obtained by surface reconstructions from micro-computed tomography. Centroid size, principal component scores in shape space and form space, and asymmetry were computed by the standard GMM formulas, and dental and skeletal jaw lengths were examined as ratios. We show here that, compared to wild type controls, mice lacking Sost have larger centroid size (effect size, p-value: 4.59, <.001), higher mean asymmetry (1.14, .065), dental and skeletal mandibular prognathism (1.36, .010 and 5.92, <.001), a smaller foramen magnum (-1.71, .015), and calvaria that are more highly curved (form space p = 4.09, .002; shape space p = 12.82, .002). These features of mice lacking sclerostin largely correspond to the changes of the facial skeleton observed in sclerosteosis. This alignment further supports claims that the Sost gene plays a fundamental role in bony facial development in rodents and humans alike.

Variation of 3D outer and inner crown morphology in modern human mandibular premolars.

OBJECTIVES: This study explores the outer and inner crown of lower third and fourth premolars (P3 , P4 ) by analyzing the morphological variation among diverse modern human groups. MATERIALS AND METHODS: We studied three-dimensional models of the outer enamel surface and the enamel-dentine junction (EDJ) from µCT datasets of 77 recent humans using both an assessment of seven nonmetric traits and a standard geometric morphometric (GM) analysis. For the latter, the dental crown was represented by four landmarks (dentine horns and fossae), 20 semilandmarks along the EDJ marginal ridge, and pseudolandmarks along the crown and cervical outlines. RESULTS: Certain discrete traits showed significantly different regional frequencies and sexual dimorphism. The GM analyses of both P3 s and P4 s showed extensive overlap in shape variation of the various populations (classification accuracy 15-69%). The first principal components explained about 40% of shape variance with a correlation between 0.59 and 0.87 of the features of P3 s and P4 s. Shape covariation between P3 s and P4 s expressed concordance of high and narrow or low and broad crowns. CONCLUSIONS: Due to marked intragroup and intergroup variation in GM analyses of lower premolars, discrete traits such as the number of lingual cusps and mesiolingual groove expression provide better geographic separation of modern human populations. The greater variability of the lingual region suggests a dominance of functional constraints over geographic provenience or sex. Additional information about functionally relevant aspects of the crown surface and odontogenetic data are needed to unravel the factors underlying dental morphology in modern humans.

Calibrating facial morphs for use as stimuli in biological studies of social perception.

Studies of human social perception become more persuasive when the behavior of raters can be separated from the variability of the stimuli they are rating. We prototype such a rigorous analysis for a set of five social ratings of faces varying by body fat percentage (BFP). 274 raters of both sexes in three age groups (adolescent, young adult, senior) rated five morphs of the same averaged facial image warped to the positions of 72 landmarks and semilandmarks predicted by linear regression on BFP at five different levels (the average, ±2 SD, ±5 SD). Each subject rated all five morphs for maturity, dominance, masculinity, attractiveness, and health. The patterns of dependence of ratings on the BFP calibration differ for the different ratings, but not substantially across the six groups of raters. This has implications for theories of social perception, specifically, the relevance of individual rater scale anchoring. The method is also highly relevant for other studies on how biological facial variation affects ratings.

A method of factor analysis for shape coordinates.

Currently the most common reporting style for a geometric morphometric (GMM) analysis of anthropological data begins with the principal components of the shape coordinates to which the original landmark data have been converted. But this focus often frustrates the organismal biologist, mainly because principal component analysis (PCA) is not aimed at scientific interpretability of the loading patterns actually uncovered. The difficulty of making biological sense of a PCA is heightened by aspects of the shape coordinate setting that further diverge from our intuitive expectations of how morphometric measurements ought to combine. More than 50 years ago one of our sister disciplines, psychometrics, managed to build an algorithmic route from principal component analysis to scientific understanding via the toolkit generally known as factor analysis. This article introduces a modification of one standard factor-analysis approach, Henry Kaiser's varimax rotation of 1958, that accommodates two of the major differences between the GMM context and the psychometric context for these approaches: the coexistence of "general" and "special" factors of form as adumbrated by Sewall Wright, and the typical loglinearity of partial warp variance as a function of bending energy. I briefly explain the history of principal components in biometrics and the contrast with factor analysis, introduce the modified varimax algorithm I am recommending, and work three examples that are reanalyses of previously published cranial data sets. A closing discussion emphasizes the desirability of superseding PCA by algorithms aimed at anthropological understanding rather than classification or ordination.

The Inappropriate Symmetries of Multivariate Statistical Analysis in Geometric Morphometrics.

In today's geometric morphometrics the commonest multivariate statistical procedures, such as principal component analysis or regressions of Procrustes shape coordinates on Centroid Size, embody a tacit roster of symmetries-axioms concerning the homogeneity of the multiple spatial domains or descriptor vectors involved-that do not correspond to actual biological fact. These techniques are hence inappropriate for any application regarding which we have a-priori biological knowledge to the contrary (e.g., genetic/morphogenetic processes common to multiple landmarks, the range of normal in anatomy atlases, the consequences of growth or function for form). But nearly every morphometric investigation is motivated by prior insights of this sort. We therefore need new tools that explicitly incorporate these elements of knowledge, should they be quantitative, to break the symmetries of the classic morphometric approaches. Some of these are already available in our literature but deserve to be known more widely: deflated (spatially adaptive) reference distributions of Procrustes coordinates, Sewall Wright's century-old variant of factor analysis, the geometric algebra of importing explicit biomechanical formulas into Procrustes space. Other methods, not yet fully formulated, might involve parameterized models for strain in idealized forms under load, principled approaches to the separation of functional from Brownian aspects of shape variation over time, and, in general, a better understanding of how the formalism of landmarks interacts with the many other approaches to quantification of anatomy. To more powerfully organize inferences from the high-dimensional measurements that characterize so much of today's organismal biology, tomorrow's toolkit must rely neither on principal component analysis nor on the Procrustes distance formula, but instead on sound prior biological knowledge as expressed in formulas whose coefficients are not all the same. I describe the problems of the standard techniques, discuss several examples of the alternatives, and draw some conclusions.

Reconsidering "The inappropriateness of conventional cephalometrics".

Of all the articles on cephalometrics this journal has published over the last half-century, the one most cited across the scientific literature is the 1979 lecture "The inappropriateness of conventional cephalometrics" by Robert Moyers and me. But the durable salience of this article is perplexing, as its critique was misdirected (it should have been aimed at the craniometrics of the early twentieth century, not merely the roentgenographic extension used in the orthodontic clinic) and its proposed remedies have all failed to establish themselves as methods of any broad utility. When problems highlighted by Moyers and me have been resolved at all, the innovations that resolved them owe to tools very different from those suggested in our article and imported from fields quite a bit farther from biometrics than we expected back in 1979. One of these tools was the creation de novo of a new abstract mathematical construction, statistical shape space, in the 1980s and 1990s; another was a flexible and intuitive new graphic, the thin-plate spline, for meaningfully and suggestively visualizing a wide variety of biological findings in these spaces. On the other hand, many of the complaints Moyers and I enunciated back in 1979, especially those stemming from the disarticulation of morphometrics from the explanatory styles and purposes of clinical medicine, remain unanswered even today. The present essay, a retrospective historical meditation, reviews the context of the 1979 publication, its major themes, and its relevance today. This essay is dedicated to the memory of Robert E. Moyers on the 100th anniversary of the American Journal of Orthodontics and Dentofacial Orthopedics.

Integration, Disintegration, and Self-Similarity: Characterizing the Scales of Shape Variation in Landmark Data.

The biologist examining samples of multicellular organisms in anatomical detail must already have an intuitive concept of morphological integration. But quantifying that intuition has always been fraught with difficulties and paradoxes, especially for the anatomically labelled Cartesian coordinate data that drive today's toolkits of geometric morphometrics. Covariance analyses of interpoint distances, such as the Olson-Miller factor approach of the 1950's, cannot validly be extended to handle the spatial structure of complete morphometric descriptions; neither can analyses of shape coordinates that ignore the mean form. This paper introduces a formal parametric quantification of integration by analogy with how time series are approached in modern paleobiology. Over there, a finding of trend falls under one tail of a distribution for which stasis comprises the other tail. The null hypothesis separating these two classes of finding is the random walks, which are self-similar, meaning that they show no interpretable structure at any temporal scale. Trend and stasis are the two contrasting ways of deviating from this null. The present manuscript introduces an analogous maneuver for the spatial aspects of ontogenetic or phylogenetic organismal studies: a subspace within the space of shape covariance structures for which the standard isotropic (Procrustes) model lies at one extreme of a characteristic parameter and the strongest growth-gradient models at the other. In-between lies the suggested new construct, the spatially self-similar processes that can be generated within the standard morphometric toolkit by a startlingly simple algebraic manipulation of partial warp scores. In this view, integration and "disintegration" as in the Procrustes model are two modes of organismal variation according to which morphometric data can deviate from this common null, which, as in the temporal domain, is formally featureless, incapable of supporting any summary beyond a single parameter for amplitude. In practice the classification can proceed by examining the regression coefficient for log partial warp variance against log bending energy in the standard thin-plate spline setup. The self-similarity model, for which the regression slope is precisely [Formula: see text] corresponds well to the background against which the evolutionist's or systematist's a-priori notion of "local shape features" can be delineated. Integration as detected by the regression slope can be visualized by the first relative intrinsic warp (first relative eigenvector of the nonaffine part of a shape coordinate configuration with respect to bending energy) and may be summarized by the corresponding quadratic growth gradient. The paper begins with a seemingly innocent toy example, uncovers an unexpected invariance as an example of the general manipulation proposed, then applies the new modeling tactic to three data sets from the existing morphometric literature. Conclusions follow regarding findings and methodology alike.

The Principal Components of Adult Female Insole Shape Align Closely with Two of Its Classic Indicators.

The plantar surface of the human foot transmits the weight and dynamic force of the owner's lower limbs to the ground and the reaction forces back to the musculoskeletal system. Its anatomical variation is intensely studied in such fields as sports medicine and orthopedic dysmorphology. Yet, strangely, the shape of the insole that accommodates this surface and elastically buffers these forces is neither an aspect of the conventional anthropometrics of feet nor an informative label on the packet that markets supplementary insoles. In this paper we pursue an earlier suggestion that insole form in vertical view be quantified in terms of the shape of the foot not at the plane of support (the "footprint") but some two millimeters above that level. Using such sections extracted from laser scans of 158 feet of adult women from the University of Zagreb, in conjunction with an appropriate modification of today's standard geometric morphometrics (GMM), we find that the sectioned form can be described by its size together with two meaningful relative warps of shape. The pattern of this shape variation is not novel. It is closely aligned with two of the standard footprint measurements, the Chippaux-Smirák arch index and the Clarke arch angle, whose geometrical foci (the former in the ball of the foot, the latter in the arch) it apparently combines. Thus a strong contemporary analysis complements but does not supplant the simpler anthropometric analyses of half a century ago, with implications for applied anthropology.

Variability of Australopithecus second maxillary molars from Sterkfontein Member 4.

The question of how many Australopithecus species lived at Sterkfontein Member 4 and Makapansgat continues to be controversial inasmuch as the fossils are poorly preserved, the stratigraphy is difficult to interpret, and the cranial, dental, and postcranial remains are mostly not associated. To proceed we applied the most intensive modern methods of 3D geometric morphometrics to dental form, specifically the shapes of the upper second molars (M(2)s) in a sample combining 13 Australopithecus, 11 Paranthropus, and 23 Homo. We analyzed outer and inner crown surfaces, as well as crown and cervical outlines both separately and together, using a total of 16 landmarks, 51 curve semilandmarks, and 48 pseudolandmarks over the four structures. Outer and inner enamel surfaces are highly correlated in this dataset, while crown outline is the least informative of the four structures. Homo was easily distinguished from both Australopithecus and Paranthropus by these methods, likewise Homo sapiens from Homo neanderthalensis. There were, however, no stable classes within the Australopithecus sample or between Australopithecus and Paranthropus. Instead, there was a gradient along which Australopithecus prometheus and Australopithecus africanus lie toward the extremes, with Paranthropus overlapping both. If there are indeed different species at this site, then either their M(2) morphologies are uninformative or else the present sample is too small to make an accurate assessment. Our findings suggest that the variability of the Australopithecus specimens will be difficult to interpret authoritatively, independent of the method used.

Virtual reconstruction of modern and fossil hominoid crania: consequences of reference sample choice.

Most hominin cranial fossils are incomplete and require reconstruction prior to subsequent analyses. Missing data can be estimated by geometric morphometrics using information from complete specimens, for example, by using thin-plate splines. In this study, we estimate missing data in several virtually fragmented models of hominoid crania (Homo, Pan, Pongo) and fossil hominins (e.g., Australopithecus africanus, Homo heidelbergensis). The aim is to investigate in which way different references influence estimations of cranial shape and how this information can be employed in the reconstruction of fossils. We used a sample of 64 three-dimensional digital models of complete human, chimpanzee, and orangutan crania and a set of 758 landmarks and semilandmarks. The virtually knocked out neurocranial and facial areas that were reconstructed corresponded to those of a real case found in A.L. 444-2 (A. afarensis) cranium. Accuracy of multiple intraspecies and interspecies reconstructions was computed as the maximum square root of the mean squared difference between the original and the reconstruction (root mean square). The results show that the uncertainty in reconstructions is a function of both the geometry of the knockout area and the dissimilarity between the reference sample and the specimen(s) undergoing reconstruction. We suggest that it is possible to estimate large missing cranial areas if the shape of the reference is similar enough to the shape of the specimen reconstructed, though caution must be exercised when employing these reconstructions in subsequent analyses. We provide a potential guide for the choice of the reference by means of bending energy.

The relation between geometric morphometrics and functional morphology, as explored by Procrustes interpretation of individual shape measures pertinent to function.

A frequent concern in today's functional morphology is the relation of a landmark configuration to some a priori index or suite of indices of function. When an index is itself a generic mathematical or biomechanical shape function of landmark locations, meaning a dimensionless expression that has a nonzero gradient everywhere in the feasible region of morphospace, the question becomes sharper: how can we exploit it as a reference direction for representations within the realm of the customary geometric morphometric (GM) analyses? This article argues that the only valid approach to this problem is geometric, not statistical: to represent any such a priori index by way of its differential (its gradient) calculated as an explicit vector in the Procrustes dual space of the complete list of landmarks whether or not involved in the formulation of the index. Interpretation of the index follows by comparing its direction after this embedding with other interesting directions in the same shape space, such as principal warps, relative warps, group mean shape contrasts, specific form factors extracted independently, or directions corresponding to other functional indices. Here, I work an artificial but realistic example of this technique in complete detail: the construction of a Procrustes shape formula exactly aligned with a specific angle among three landmarks within an arbitrary configuration of six. A closing discussion traces the spirit of this intervention to comments by W. W. Howells and C. E. Oxnard, originally intended for anthropometric contexts other than GM, on the different purposes of systematics and functional morphology.

The psychology of inherence is self-referential (and that is a good thing).

Cimpian & Salomon (C&S) appear to characterize the inherence heuristic and essentialism as unwise or childish aspects of human reasoning. But actually, these cognitive modes lie at the core of statistical analysis across all of the quantitative sciences, including the developmental cognitive psychology in which the argument here is couched. Their whole argument is as much an example of its topic as an analysis of it.

Analysis of the human female foot in two different measurement systems: from geometric morphometrics to functional morphology.

The relationship of geometric morphometrics (GMM) to functional analysis of the same morphological resources is currently a topic of active interest among functional morphologists. Although GMM is typically advertised as free of prior assumptions about shape features or morphological theories, it is common for GMM findings to be concordant with findings from studies based on a-priori lists of shape features whenever prior insights or theories have been properly accounted for in the study design. The present paper demonstrates this happy possibility by revisiting a previously published GMM analysis of footprint outlines for which there is also functionally relevant information in the form of a-pri-ori foot measurements. We show how to convert the conventional measurements into the language of shape, thereby affording two parallel statistical analyses. One is the classic multivariate analysis of "shape features", the other the equally classic GMM of semilandmark coordinates. In this example, the two data sets, analyzed by protocols that are remarkably different in both their geometry and their algebra, nevertheless result in one common biometrical summary: wearing high heels is bad for women inasmuch as it leads to the need for orthotic devices to treat the consequently flattened arch. This concordance bears implications for other branches of applied anthropology. To carry out a good biomedical analysis of applied anthropometric data it may not matter whether one uses GMM or instead an adequate assortment of conventional measurements. What matters is whether the conventional measurements have been selected in order to match the natural spectrum of functional variation.

"The map is not the territory".

Bentley et al.'s claim that their "map … captures the essence of decision making" (target article, Abstract) is deconstructed and shown to originate in a serious misunderstanding of the role of principal components and statistical graphics in the generation of pattern claims and hypotheses from profile data. Three alternative maps are offered, each with its radiation of further investigations.

Ontology of physics for biology: representing physical dependencies as a basis for biological processes.

BACKGROUND: In prior work, we presented the Ontology of Physics for Biology (OPB) as a computational ontology for use in the annotation and representations of biophysical knowledge encoded in repositories of physics-based biosimulation models. We introduced OPB:Physical entity and OPB:Physical property classes that extend available spatiotemporal representations of physical entities and processes to explicitly represent the thermodynamics and dynamics of physiological processes. Our utilitarian, long-term aim is to develop computational tools for creating and querying formalized physiological knowledge for use by multiscale "physiome" projects such as the EU's Virtual Physiological Human (VPH) and NIH's Virtual Physiological Rat (VPR). RESULTS: Here we describe the OPB:Physical dependency taxonomy of classes that represent of the laws of classical physics that are the "rules" by which physical properties of physical entities change during occurrences of physical processes. For example, the fluid analog of Ohm's law (as for electric currents) is used to describe how a blood flow rate depends on a blood pressure gradient. Hooke's law (as in elastic deformations of springs) is used to describe how an increase in vascular volume increases blood pressure. We classify such dependencies according to the flow, transformation, and storage of thermodynamic energy that occurs during processes governed by the dependencies. CONCLUSIONS: We have developed the OPB and annotation methods to represent the meaning-the biophysical semantics-of the mathematical statements of physiological analysis and the biophysical content of models and datasets. Here we describe and discuss our approach to an ontological representation of physical laws (as dependencies) and properties as encoded for the mathematical analysis of biophysical processes.

Infant growth patterns of the mandible in modern humans: a closer exploration of the developmental interactions between the symphyseal bone, the teeth, and the suprahyoid and tongue muscle insertion sites.

The ontogenetic development of the mental region still poses a number of unresolved questions in human growth, development and phylogeny. In our study we examine the hypotheses of DuBrul & Sicher (1954) (The Adaptive Chin. Springfield, IL: Charles) and Enlow (1990) (Facial Growth, 3rd edn. Philadelphia, PA: Saunders) to explain the presence of a prominent mental region in anatomically modern humans. In particular, we test whether the prominence of the mental region and the positioning of the teeth are both correlated with the developmental relocation of the tongue and the suprahyoid muscles inserting at the lingual side of the symphysis. Furthermore, we test whether the development of the mental region is associated with the development of the back of the vocal tract. Using geometric morphometric methods, we measured the 3D mandibular and tooth surfaces in a cross-sectional sample of 36 CT-scanned living humans, incorporating the positions of the tongue and the geniohyoid and digastric muscle insertions. The specimens' ages range from birth to the complete emergence of the deciduous dentition. We used multivariate regression and two-block partial least squares (PLS) analysis to study the covariation among the mental region, the muscle insertions, and the teeth both across and within age stages. In order to confirm our results from the 3D cross-sectional sample, and to relate them to facial growth and the position of the cervical column and the hyoid bone, we used 46 lateral radiographs of eight children from the longitudinal Denver Growth Study. The 3D analysis demonstrates that the lingual side of the lower border of the symphysis develops downwards and forwards. These shape changes are significantly correlated with the relocation of muscle insertion sites and also with the vertical reorientation of the anterior teeth prior to emergence. The 2D analysis confirms the idea that as the mental region prominence develops, the space of the laryngopharynx becomes restricted due to upper mid-face retraction and the acquisition of upright body posture. In agreement with the hypotheses of DuBrul & Sicher (1954) and Enlow (1990), our results suggest that the presence of a prominent mental region responds to the space restriction at the back of the vocal tract, and to the packaging of the tongue and suprahyoid muscles in order to preserve the functionality of the laryngopharynx during respiration, feeding and speech.

Technical note: a novel geometric morphometric approach to the study of long bone shape variation.

Procrustes-based geometric morphometrics (GM) is most often applied to problems of craniofacial shape variation. Here, we demonstrate a novel application of GM to the analysis of whole postcranial elements in a study of 77 hominoid tibiae. We focus on two novel methodological improvements to standard GM approaches: 1) landmark configurations of tibiae including 15 epiphyseal landmarks and 483 semilandmarks along articular surfaces and muscle insertions along the tibial shaft and 2) an artificial affine transformation that sets moments along the shaft equal to the sum of the moments estimated in the other two anatomical directions. Diagrams of the principal components of tibial shapes support most differences between human and non-human primates reported previously. The artificial affine transformation proposed here results in an improved clustering of the great apes that may prove useful in future discriminant or clustering studies. Since the shape variations observed may be related to different locomotor behaviors, posture, or activity patterns, we suggest that this method be used in functional analyses of tibiae or other long bones in modern populations or fossil specimens.