Procrustes Shape Coordinates Research Articles

Variation in human limb joint articular morphology.

Synovial joints in human limbs strike a balance between mobility, stability, and articular fit, yet little is known about how these conflicting demands pattern intraspecific variation in articular shape. In this study, we use geometric morphometrics to establish the apportionment and magnitude of morphological variance of the articular surfaces of the human shoulder, elbow, hip, and knee. We hypothesize that variances will be comparable between articulating surfaces within a joint and will be larger in joints with smaller ranges of motion, given their plurality of functional demands. Three-dimensional landmarks were taken on the articular surfaces of the glenohumeral, humeroulnar, acetabulofemoral, and tibiofemoral joints from CT scans of 200 skeletons from the University of Tennessee Donated Skeletal Collection (84 females, 116 males). Root mean-squared distances between articulations calculated from Procrustes shape coordinates were used to determine variance distributions. We found no difference in variances for each articular surface between the sexes or between left and right articular surfaces. A high range of motion is associated with greater morphological variance; however, this pattern is largely driven by the concave articular surfaces of each joint, which consistently exhibit statistically greater variance than their convex counterparts. The striking pattern of differential variance between articulating morphologies points to potential disparities in development between them. Consistently higher variance in concave surfaces may relate to chondral modeling theory for the formation of joints. Establishing intraspecific morphological variance patterns is a first step in understanding coordinated evolution among articular features.

Reworking Geometric Morphometrics into a Methodology of Transformation Grids

AbstractToday’s typical application of geometric morphometrics to a quantitative comparison of organismal anatomies begins by standardizing samples of homologously labelled point configurations for location, orientation, and scale, and then renders the ensuing comparisons graphically by thin-plate spline as applied to group averages, principal components, regression predictions, or canonical variates. The scale-standardization step has recently come under criticism as unnecessary and indeed inappropriate, at least for growth studies. This essay argues for a similar rethinking of the centering and rotation, and then the replacement of the thin-plate spline interpolant of the resulting configurations by a different strategy that leaves unexplained residuals at every landmark individually in order to simplify the interpretation of the displayed grid as a whole, the “transformation grid” that has been highlighted as the true underlying topic ever since D’Arcy Thompson’s celebrated exposition of 1917. For analyses of comparisons involving gradients at large geometric scale, this paper argues for replacement of all three of the Procrustes conventions by a version of my two-point registration of 1986 [originally Galton’s of 1907 (Nature 76:617–618, 1907)]. The choice of the two points interacts with another non-Procrustes concern, interpretability of the grid lines of a coordinate system deformed according to a fitted polynomial trend rather than an interpolating thin-plate spline. The paper works two examples using previously published midsagittal cranial data; there result new findings pertinent to the interpretation of both of these classic data sets. A concluding discussion suggests that the current toolkit of geometric morphometrics, centered on Procrustes shape coordinates and thin-plate splines, is too restricted to suit many of the interpretive purposes of evolutionary and developmental biology.

Dimensions of Morphological Integration

Over several generations of evolutionary and developmental biologists, ever since Olson and Miller’s pioneering work of the 1950’s, the concept of “morphological integration” as applied to Gaussian representations N(mu ,Sigma ) of morphometric data has been a focus equally of methodological innovation and methodological perplexity. Reanalysis of a century-old example from Sewall Wright shows how some fallacies of distance analysis by correlations can be avoided by careful matching of the distance rosters involved to a different multivariate approach, factor analysis. I reinterpret his example by restoring the information (means and variances) ignored by the correlation matrix, while confirming what Wright called “special size factors” by a different technique, inspection of the concentration matrix Sigma ^{-1}. In geometric morphometrics (GMM), data accrue instead as Cartesian coordinates of labelled points; nevertheless, just as in the Wright example, statistical manipulations do better when they reconsider the normalizations that went into the generation of those coordinates. Here information about both mu and Sigma , the means and the variances/covariances, can be preserved via the Boas coordinates (Procrustes shape coordinates without the size adjustment) that protect the role of size per se as an essential explanatory factor while permitting the analyst to acknowledge the realities of animal anatomy and its trajectories over time or size in the course of an analysis. A descriptive quantity for this purpose is suggested, the correlation of vectorized mu against the first eigenvector of Sigma for the Boas coordinates. The paper reanalyzes two GMM data sets from this point of view. In one, the classic Vilmann rodent neurocranial growth data, a description of integration can be aligned with the purposes of evolutionary and developmental biology by a graphical exegesis based mainly in the loadings of the first Boas principal component. There results a multiplicity of morphometric patterns, some homogeneous and others characterized by gradients. In the other, a Vienna data set comprising human midsagittal skull sections mostly sampled along curves, a further integrated feature emerges, thickening of the calvaria, that requires a reparametrization and a modified thin plate spline graphic distinct from the digitized configurations per se. This new GMM protocol fulfills the original thrust of Olson & Miller’s (Evolution 5:325–338, 1951) “rho F-groups,” the alignment of statistical and biological explanatory guidance, while respecting the enormously greater range of morphological descriptors afforded by well-designed landmark/semilandmark configurations.

Practical and technical aspects for the 3D scanning of lithic artefacts using micro-computed tomography techniques and laser light scanners for subsequent geometric morphometric analysis. Introducing the StyroStone protocol.

Here, we present a new method to scan a large number of lithic artefacts using three-dimensional scanning technology. Despite the rising use of high-resolution 3D surface scanners in archaeological sciences, no virtual studies have focused on the 3D digitization and analysis of small lithic implements such as bladelets, microblades, and microflakes. This is mostly due to difficulties in creating reliable 3D meshes of these artefacts resulting from several inherent features (i.e., size, translucency, and acute edge angles), which compromise the efficiency of structured light or laser scanners and photogrammetry. Our new protocol StyroStone addresses this problem by proposing a step-by-step procedure relying on the use of micro-computed tomographic technology, which is able to capture the 3D shape of small lithic implements in high detail. We tested a system that enables us to scan hundreds of artefacts together at once within a single scanning session lasting a few hours. As also bigger lithic artefacts (i.e., blades) are present in our sample, this protocol is complemented by a short guide on how to effectively scan such artefacts using a structured light scanner (Artec Space Spider). Furthermore, we estimate the accuracy of our scanning protocol using principal component analysis of 3D Procrustes shape coordinates on a sample of meshes of bladelets obtained with both micro-computed tomography and another scanning device (i.e., Artec Micro). A comprehensive review on the use of 3D geometric morphometrics in lithic analysis and other computer-based approaches is provided in the introductory chapter to show the advantages of improving 3D scanning protocols and increasing the digitization of our prehistoric human heritage.

Island biogeography predicts skull gigantism and shape variation in meadow voles Microtus pennsylvanicus through ecological release and allometry

Island rule describes the graded trend of gigantism in small‐bodied species to dwarfism in large‐bodied species inhabiting islands, but causal explanations remain unresolved. We used geometric morphometrics to quantify cranial morphology of 544 meadow vole Microtus pennsylvanicus samples across 11 island and 3 mainland populations from the Outer Lands of New England (Atlantic) and the Alexander Archipelago of Alaska (Pacific). We compared the thermoregulation and endurance (TRE) and ecological release (ER) hypotheses using all‐subsets linear models employing residual randomization permutation procedures (rrpp), and Akaike information criterion (AIC) for model selection. We decoupled direct and indirect effects of island variables on size using path analysis. We evaluated shape with principal components analysis (PCA) and Procrustes ANOVA on Procrustes shape coordinates, then assessed the impact of static allometry and TRE and ER variables on shape. Six Atlantic island populations exhibit significant signals of gigantism with the largest voles occurring on the smallest islands lacking predators. ER explains 63% of cranial size differences. Island area has a significant total effect on size by influencing the number of mammalian predators, resulting in a 0.011 increase in unit centroid size for a 100 km2 decrease in island area. This corresponds to a predicted 0.9% change in size for every 100 km2. Given static allometry, cranial shape does not respond to insularity independent of size. These results suggest that island rule is a latent evolutionary process whose manifestation depends on nuanced biogeographic and ecological contexts that have important conservation and taxonomic implications.

Population genetics and geometric morphometrics of the freshwater snail Segmentina nitida reveal cryptic sympatric species of conservation value in Europe

Segmentina nitida Müller 1774 is a rare European freshwater snail of drainage ditches and marshland, which has seen a marked decrease in range (~ 80%) over the last 100 years in the UK. This has been attributed to over-dredging of drainage ditches for land management, conversion of grazing marshes to arable farmland, as well as eutrophication. Segmentina nitida is identified as a priority species in the UK Biodiversity Action Plan (UKBAP) that recommends further research to inform reintroduction and translocation for its conservation. We used nuclear markers (microsatellites and ITS2) and a mitochondrial (COI) marker to investigate population structure in S. nitida individuals sampled from Poland, Germany, Sweden, and the UK to identify differences within and between populations. Data based on 2D landmark-based geometric morphometrics of S. nitida shells was used to determine if phenotypic variation followed genetic differentiation. Two distinct genetic lineages of S. nitida were identified in ITS and COI phylogenies as well as cluster analysis of microsatellite markers, one of these lineages was present in eastern Europe (Poland, Sweden- Lineage 2), and one in western Europe (UK, Germany- Lineage 1), with lineages co-occurring in German populations. No genetic admixture was observed in German populations containing both lineages. These two lineages were also distinct in shape, with lineage 2 individuals having significantly wider shells and taller and wider apertures than those in Lineage 1. ~ 85% of shells assigned to the predicted lineage in a discriminant analysis of Procrustes shape coordinates. We infer that S. nitida includes at least one sympatric cryptic species. We discuss the implications of these findings on the conservation status of S. nitida in the UK and Europe.

Centric Allometry: Studying Growth Using Landmark Data

The geometric morphometric (GMM) construction of Procrustes shape coordinates from a data set of homologous landmark configurations puts exact algebraic constraints on position, orientation, and geometric scale. While position as digitized is not ordinarily a biologically meaningful quantity, and orientation is relevant mainly when some organismal function interacts with a Cartesian positional gradient such as horizontality, size per se is a crucially important biometric concept, especially in contexts like growth, biomechanics, or bioenergetics. “Normalizing” or “standardizing” size (usually by dividing the square root of the summed squared distances from the centroid out of all the Cartesian coordinates specimen by specimen), while associated with the elegant symmetries of the Mardia–Dryden distribution in shape space, nevertheless can substantially impeach the validity of any organismal inferences that ensue. This paper adapts two variants of standard morphometric least-squares, principal components and uniform strains, to circumvent size standardization while still accommodating an analytic toolkit for studies of differential growth that supports landmark-by-landmark graphics and thin-plate splines. Standardization of position and orientation but not size yields the coordinates Franz Boas first discussed in 1905. In studies of growth, a first principal component of these coordinates often appears to involve most landmarks shifting almost directly away from their centroid, hence the proposed model’s name, “centric allometry.” There is also a joint standardization of shear and dilation resulting in a variant of standard GMM’s “nonaffine shape coordinates” where scale information is subsumed in the affine term. Studies of growth allometry should go better in the Boas system than in the Procrustes shape space that is the current conventional workbench for GMM analyses. I demonstrate two examples of this revised approach (one developmental, one phylogenetic) that retrieve all the findings of a conventional shape-space-based approach while focusing much more closely on the phenomenon of allometric growth per se. A three-part Appendix provides an overview of the algebra, highlighting both similarities to the Procrustes approach and contrasts with it.

Developmental shape changes in facial morphology: Geometric morphometric analyses based on a prospective, population-based, Chinese cohort in Hong Kong.

BackgroundThorough understanding of developmental changes of human facial shape is lacking. The present study aimed to evaluate developmental shape changes of facial morphology based on a prospective, population-representative, Chinese cohort in Hong Kong.MethodsA population-representative sample of Chinese in Hong Kong was followed. Serial facial images of over 260 participants were obtained at age 12, 15, and 18 years. Facial landmarks were digitized and the corresponding coordinates were submitted for Generalized Procrustes Analysis. The resultant Procrustes shape coordinates, which captured shape information encoded by the facial landmarks, were then used for statistical shape analyses.ResultsSmall but significant developmental changes in mean facial shape were observed (p < 0.0001 for all pairwise comparisons). Significant age-related changes in the magnitude of variance of facial shape were also observed (p < 0.05). Phenotypic growth trajectories representing developmental shape changes were similar in size (p > 0.05) between sexes but differed in direction (p < 0.05) in shape space and trajectory shape (p < 0.05). The magnitude of shape differences between sexes remained constant from 12 to 18 years. Results of frontal facial shape analyses after removing the effect of allometry were similar to results obtained before removal of allometry. For lateral facial configurations, allometric trajectories among the age-by-sex groups were similar in slope (p > 0.05) but varied in directions in the multidimensional shape space.ConclusionsOur findings suggested significant age-related changes in facial shape and provided a dynamic view of developmental changes in sexual dimorphism of facial shape. Allometry contributed minimally to developmental changes in frontal facial shape. In addition, the allometric trajectories for lateral facial configurations were similar in rate of shape change but differed in their directions in shape space.

Cranial and endocranial diversity in extant and fossil atelids (Platyrrhini: Atelidae): A geometric morphometric study.

Platyrrhines constitute a diverse clade, with the modern Atelidae exhibiting the most variation in cranial and endocast morphology. The processes responsible for this diversification are not well understood. Here, we present a geometric morphometric study describing variation in cranial and endocranial shape of 14 species of Alouatta, Ateles, Brachyteles, and Lagothrix and two extinct taxa, Cartelles and Caipora. We examined cranial and endocranial shape variation among species using images reconstructed from CT scans and geometric morphometric techniques based on three-dimensional landmarks and semilandmarks. Principal components analyses were used to explore variation, including the Procrustes shape coordinates, summing the logarithm of the Centroid Size, the common allometric component, and residual shape components. Differences in endocranial shape are related to a relative increase or decrease in the volume of the neocortex region with respect to brainstem and cerebellum regions. The relative position of the brainstem varies from a posterior position in Alouatta to a more ventral position in Ateles. The shape of both the cranium and endocast of Caipora is within the observed variation of Brachyteles. Cartelles occupies the most differentiated position relative to the extant taxa, especially in regards to its endocranial shape. The pattern of variation in the extant species in endocranial shape is similar to the variation observed in previous cranial studies, with Alouatta as an outlier. The similarities between Caipora and Brachyteles were unexpected and intriguing given the frugivorous adaptations inferred from the fossil's dentition. Our study shows the importance of considering both extant and fossil species when studying diversification of complex traits.

Three-dimensional analysis of sexual dimorphism in ribcage kinematics of modern humans.

Sexual dimorphism is an important biological factor underlying morphological variation in the human skeleton. Previous research found sex-related differences in the static ribcage, with males having more horizontally oriented ribs and a wider lower ribcage than females. Furthermore, a recent study found sex-related differences in the kinematics of the human lungs, with cranio-caudal movements of the caudal part of the lungs accounting for most of the differences between sexes. However, these movements cannot be quantified in the skeletal ribcage, so we do not know if the differences observed in the lungs are also reflected in sex differences in the motion of the skeletal thorax. To address this issue, we quantified the morphological variation of 42 contemporary human ribcages (sex-balanced) in both maximal inspiration and expiration using 526 landmarks and semilandmarks. Thoracic centroid size differences between sexes were assessed using a t test, and shape differences were assessed using Procrustes shape coordinates, through mean comparisons and dummy regressions of shape on kinematic status. A principal components analysis was used to explore the full range of morphological variation. Our results show significant size differences between males and females both in inspiration and expiration (p < .01) as well as significant shape differences, with males deforming more than females during inspiration, especially in the mediolateral dimension of the lower ribcage. Finally, dummy regressions of shape on kinematic status showed a small but statistically significant difference in vectors of breathing kinematics between males and females (14.78°; p < .01). We support that sex-related differences in skeletal ribcage kinematics are discernible, even when soft tissues are not analyzed. We hypothesize that this differential breathing pattern is primarily a result of more pronounced diaphragmatic breathing in males, which might relate to differences in body composition, metabolism, and ultimately greater oxygen demand in males compared to females. Future research should further explore the links between ribcage morphological variation and basal metabolic rate.

Functional aspects of metatarsal head shape in humans, apes, and Old World monkeys

Modern human metatarsal heads are typically described as “dorsally domed,” mediolaterally wide, and dorsally flat. Despite the apparent functional importance of these features in forefoot stability during bipedalism, the distinctiveness of this morphology has not been quantitatively evaluated within a broad comparative framework. In order to use these features to reconstruct fossil hominin locomotor behaviors with any confidence, their connection to human bipedalism should be validated through a comparative analysis of other primates with different locomotor behaviors and foot postures, including species with biomechanical demands potentially similar to those of bipedalism (e.g., terrestrial digitigrady). This study explores shape variation in the distal metatarsus among humans and other extant catarrhines using three-dimensional geometric morphometrics (3DGM). Shape differences among species in metatarsal head morphology are well captured by the first two principal components of Procrustes shape coordinates, and these two components summarize most of the variance related to “dorsal doming” and “dorsal expansion.” Multivariate statistical tests reveal significant differences among clades in overall shape, and humans are reliably distinguishable from other species by aspects of shape related to a greater degree of dorsal doming. Within quadrupeds, terrestrial species also trend toward more domed metatarsal heads, but not to the extent seen in humans. Certain aspects of distal metatarsus shape are likely related to habitual dorsiflexion of the metatarsophalangeal joints, but the total morphological pattern seen in humans is distinct. These comparative results indicate that this geometric morphometric approach is useful to characterize the complexity of metatarsal head morphology and will help clarify its relationship with function in fossil primates, including early hominins.

Evaluating developmental shape changes in Homo antecessor subadult facial morphology

The fossil ATD6-69 from Atapuerca, Spain, dated to ca. 900 ka (thousands of years ago) has been suggested to mark the earliest appearance of modern human facial features. However, this specimen is a subadult and the interpretation of its morphology remains controversial, because it is unclear how developmental shape changes would affect the features that link ATD6-69 to modern humans. Here we analyze ATD6-69 in an evolutionary and developmental context. Our modern human sample comprises cross-sectional growth series from four populations. The fossil sample covers human specimens from the Pleistocene to the Upper Paleolithic, and includes several subadult Early Pleistocene humans and Neanderthals. We digitized landmarks and semilandmarks on surface and CT scans and analyzed the Procrustes shape coordinates using multivariate statistics. Ontogenetic allometric trajectories and developmental simulations were employed in order to identify growth patterns and to visualize potential adult shapes of ATD6-69. We show that facial differences between modern and archaic humans are not exclusively allometric. We find that while postnatal growth further accentuates the differences in facial features between Neanderthals and modern humans, those features that have been suggested to link ATD6-69's morphology to modern humans would not have been significantly altered in the course of subsequent development. In particular, the infraorbital depression on this specimen would have persisted into adulthood. However, many of the facial features that ATD6-69 shares with modern humans can be considered to be part of a generalized pattern of facial architecture. Our results present a complex picture regarding the polarity of facial features and demonstrate that some modern human-like facial morphology is intermittently present in Middle Pleistocene humans. We suggest that some of the facial features that characterize recent modern humans may have developed multiple times in human evolution.

Comparing endocranial ontogenetic trajectories in extant great and lesser apes

Differences in adult morphology arise as a result of modifications to ancestral developmental patterns. Studies of evolutionary changes to brain development in primates can provide important clues about the evolution of human cognition, but are hindered by the lack of preserved neural tissue in the fossil record. As brain structure is mirrored in the endocranium, we have used endocasts (virtual casts of the endocranial cavity) to capture brain shape in a geometric morphometrics analysis. Here, we compare the postnatal developmental changes of the endocasts of modern humans (n=93), chimpanzees (n=61), gorillas (n=64), orangutans (n=72) and hylobatids (n=21) to assess evolutionary changes in the tempo and mode of brain development. We tested whether ontogenetic trajectories based on Procrustes shape coordinates and centroid sizes overlapped in shape‐space and form‐space and differed in their direction. In the subspace of the first three principal components, explaining &gt;;71% of the variance in shape‐space, our results show a conserved pattern of endocranial shape development in hominids. Hominids and hylobatids have similar ontogenetic trajectories in shape‐space, yet have distinct shapes at all stages of development. These findings indicate that differences in ontogenetic shape changes are observable from the first stage of postnatal development, suggesting that divergences from an ancestral pattern of primate development occur at an earlier prenatal stage. This research was supported by the Max Planck Society and by the Natural Sciences and Engineering Research Council of Canada (NS).

A modified Procrustes analysis for bilaterally symmetrical outlines, with an application to microevolution inBaculites

In quantitative paleobiology, one common format for image-derived information is the closed two-dimensional outline curve. Before the end of the last century a great variety of morphometric tools were on offer to deal with this data type, including diverse analyses of distance from a central point, “eigenshape analysis” of a carefully normalized tangent angle function, and Procrustes analysis of configurations of sliding semilandmarks. For the special case ofoutline forms that are nominally bilaterally symmetric, this paper offers a hybrid approach that fuses the Procrustes toolkit with a variant of a much older method, analysis of radial distance according to angles out of a center. The synthesis represents each outline as a regularly resampled data set of radially aligned shape coordinates analogous to bilateral Procrustes shape coordinates. When the center is iteratively located in one particular way, this representation has the same Procrustes distance statistics as a conventional radial representation, while the formalism of shape coordinates permits symmetrization, localizability, and thin-plate spline visualizations. We explain the method and sketch some of its advantages and limitations in the course of an example involving 99Baculites inornatusoutlines spanning 90 m of deposits from a high-precision measured section in the Campanian-age Rosario Formation cropping out at Punta San Jose, Baja California. This sequence shows substantial changes of pole curvature over time, changes that may involve a punctuation event regarding hydrostatics, hydrodynamics, or defense against predators. A concluding comment deals with implications of the developments here forBaculitesmicroevolution and for morphometrics.

Detection of non-affine shape outliers for matched-pair shape data

ABSTRACT Cleft lip/palate (CLP) is a relatively common birth defect so dis- figuring that nowadays it is almost always corrected surgically as early as possible. The postnatal surgical correction does not, however, result in a normally growing upper jaw, but instead, owing to scar tissue, one that grows abnormally. It is important to decide if a clinical treatment group is homogeneous. The example involves data from digitally processed lateral X-ray films of 48 boys who have complete unilateral CLP but no other malformation. 22 landmarks were represented by their Procrustes shape coordinates, principal components of matched- pair differences were examined, and the distribution of the 48 shape changes was studied for outliers in the affine and non-affine subspaces of the full Procrustes shape and form space. To separate outliers from inliers we use bagplots. There are no outliers apparent in the affine subspace. In the non-affine subspaces, we found no outliers in the subspace of bending patterns at large scale but four out- liers in the subspace of local changes at small scale. Almost the same outliers were found by form-space PCA. These latter are associated with possible creases of the corresponding thin-plate splines. In those cases we can use the same spline formalism to relax the outlying form to an inlier by optimal relaxation along the curve d´ecolletage that weighs bending energy against Procrustes distance and stop relaxation on the fence. These maneuvers suggest a possibly novel and interesting fusion of the Procrustes-spline toolkit with outlier detection. They also have practical implications for craniofacial management of CLP follow-up as well as suggestive implications for outlier detection in applied craniometrics and anthropometrics more generally.

Middle Pleistocene human facial morphology in an evolutionary and developmental context

Neanderthals and modern humans exhibit distinct facial architectures. The patterning of facial morphology of their predecessors, the Middle Pleistocene humans, is more mosaic showing a mix of archaic and modern morphologies. Significant changes in facial size and robusticity occurred throughout Pleistocene human evolution, resulting in temporal trends in both facial reduction and enlargement. However, the allometric patterning in facial morphology in archaic humans is not well understood. This study explores temporal trends in facial morphology in order to gain a clearer understanding of the polarity of features, and describes the allometric patterning of facial shape.The modern human sample comprises cross-sectional growth series of four morphologically distinct human populations. The fossil sample covers specimens from the Middle Pleistocene to the Upper Paleolithic. We digitized landmarks and semilandmarks on surface and computed tomography scans and analyzed the Procrustes shape coordinates. Principal component analyses were performed, and Procrustes distances were used to identify phenetic similarities between fossil hominins. In order to explore the influence of size on facial features, allometric trajectories were calculated for fossil and modern human groups, and developmental simulations were performed.We show that facial features can be used to separate Pleistocene humans into temporal clusters. The distinctly modern human pattern of facial morphology is already present around 170 ka. Species- and population-specific facial features develop before two years of age, and several of the large-scale facial differences between Neanderthals and Middle Pleistocene humans are due to scaling along a shared allometric trajectory. These features include aspects of the frontal bone, browridge morphology, nasal aperture size and facial prognathism. Infraorbital surface topography and orientation of the midface in the European Middle Pleistocene hominins is intermediate between the African Middle Pleistocene and Neanderthal condition. This could suggest that the European Middle Pleistocene hominins display incipient Neanderthal features.

Linear Discrimination, Ordination, and the Visualization of Selection Gradients in Modern Morphometrics

Linear discriminant analysis (LDA) is a multivariate classification technique frequently applied to morphometric data in various biomedical disciplines. Canonical variate analysis (CVA), the generalization of LDA for multiple groups, is often used in the exploratory style of an ordination technique (a low-dimensional representation of the data). In the rare case when all groups have the same covariance matrix, maximum likelihood classification can be based on these linear functions. Both LDA and CVA require full-rank covariance matrices, which is usually not the case in modern morphometrics. When the number of variables is close to the number of individuals, groups appear separated in a CVA plot even if they are samples from the same population. Hence, reliable classification and assessment of group separation require many more organisms than variables. A simple alternative to CVA is the projection of the data onto the principal components of the group averages (between-group PCA). In contrast to CVA, these axes are orthogonal and can be computed even when the data are not of full rank, such as for Procrustes shape coordinates arising in samples of any size, and when covariance matrices are heterogeneous. In evolutionary quantitative genetics, the selection gradient is identical to the coefficient vector of a linear discriminant function between the populations before vs. after selection. When the measured variables are Procrustes shape coordinates, discriminant functions and selection gradients are vectors in shape space and can be visualized as shape deformations. Except for applications in quantitative genetics and in classification, however, discriminant functions typically offer no interpretation as biological factors.

New insights into the relationship between suture closure and craniofacial dysmorphology in sagittal nonsyndromic craniosynostosis

Premature closure of the sagittal suture occurs as an isolated (nonsyndromic) birth defect or as a syndromic anomaly in combination with other congenital dysmorphologies. The genetic causes of sagittal nonsyndromic craniosynostosis (NSC) remain unknown. Although variation of the dysmorphic (scaphocephaly) skull shape of sagittal NSC cases has been acknowledged, this variation has not been quantitatively studied three-dimensionally (3D). We have analyzed the computed tomography skull images of 43 infants (aged 0.9-9 months) with sagittal NSC using anatomical landmarks and semilandmarks to quantify and characterize the within-sample phenotypic variation. Suture closure patterns were defined by dividing the sagittal suture into three sections (anterior, central, posterior) and coding each section as 'closed' or 'fused'. Principal components analysis of the Procrustes shape coordinates representing the skull shape of 43 cases of NSC did not separate individuals by sex, chronological age, or dental stages of the deciduous maxillary first molar. However, analysis of suture closure pattern allowed separation of these data. The central section of the sagittal suture appears to be the first to fuse. Then, at least two different developmental paths towards complete fusion of the sagittal suture exist; either the anterior section or the posterior section is the second to fuse. Results indicate that according to the sequence of sagittal suture closure patterns, different craniofacial complex shapes are observed. The relationship between craniofacial shape and suture closure indicates not only which suture fused prematurely (in our case the sagittal suture), but also the pattern of the suture closure. Whether these patterns indicate differences in etiology cannot be determined with our data and requires analysis of longitudinal data, most appropriately of animal models where prenatal conditions can be monitored.

Sexual dimorphism and population divergence in the Lake Tanganyika cichlid fish genus Tropheus

BackgroundWith about 120 colour morphs currently assigned to six nominal species, the genus Tropheus is an ideal model to study evolutionary divergence of populations in allopatry. The morphology of Tropheus has been described as relatively static, but reproductive constraints are sexually dimorphic due to mouthbrooding in females. We analysed phenotypic variation in six populations of T. moorii and one population of T. polli using geometric morphometrics to assess morphological differences among sexes in relation to the differentiation of populations and species.ResultsThe mean shapes differed significantly between sexes, populations, and species even though within-sex variation exceeded the divergence among populations. The first principal component of Procrustes shape coordinates revealed differences between populations and species in mouth position and ventral head shape. The second principal component reflected sex-specific shape differences, mainly comprising a relatively larger female viscerocranium and, in particular, a larger buccal area. While shape variation between populations and between sexes was primarily located in the cranial region, within-sex variation was relatively uniform across all landmarks.ConclusionsDeviations of the between-population and between-sex pattern of shape variation from that within sex indicate that the differences in head shape likely result from both adaptations to female mouthbrooding and population-specific foraging strategies.

Advances in Geometric Morphometrics

Geometric morphometrics is the statistical analysis of form based on Cartesian landmark coordinates. After separating shape from overall size, position, and orientation of the landmark configurations, the resulting Procrustes shape coordinates can be used for statistical analysis. Kendall shape space, the mathematical space induced by the shape coordinates, is a metric space that can be approximated locally by a Euclidean tangent space. Thus, notions of distance (similarity) between shapes or of the length and direction of developmental and evolutionary trajectories can be meaningfully assessed in this space. Results of statistical techniques that preserve these convenient properties—such as principal component analysis, multivariate regression, or partial least squares analysis—can be visualized as actual shapes or shape deformations. The Procrustes distance between a shape and its relabeled reflection is a measure of bilateral asymmetry. Shape space can be extended to form space by augmenting the shape coordinates with the natural logarithm of Centroid Size, a measure of size in geometric morphometrics that is uncorrelated with shape for small isotropic landmark variation. The thin-plate spline interpolation function is the standard tool to compute deformation grids and 3D visualizations. It is also central to the estimation of missing landmarks and to the semilandmark algorithm, which permits to include outlines and surfaces in geometric morphometric analysis. The powerful visualization tools of geometric morphometrics and the typically large amount of shape variables give rise to a specific exploratory style of analysis, allowing the identification and quantification of previously unknown shape features.