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.     

An Integrated Approach for Landmark-Based Resistant Shape Analysis in 3D

The study of shape changes in morphology has seen a significant renovation in the last 20 years, particularly as a consequence of the development of geometric morphometric methods based on Cartesian coordinates of points. In order to extract information about shape differences when Cartesian coordinates are used, it is necessary to establish a common reference frame or system for all specimens to be compared. Therefore, a central issue in coordinate-based methods is which criterion should be used to align these configurations of points, since shape differences highly depend on those alignments. This is usually accomplished by aligning the configurations in a way that the sum of squared distances between coordinates of homologous points (landmarks) is minimized: the least-squares superimposition method. However, it is widely recognized that this method has some limitations when shape differences are not homogeneous across landmarks. Here we present an integrated approach for the resistant shape comparison of 3D landmark sets. It includes a new ordinary resistant Procrustes superimposition and its corresponding generalized resistant Procrustes version. In addition, they are combined with existing resistant multivariate statistical techniques for depicting the results. We demonstrate, by using both simulated and real datasets, that resistant Procrustes better detects and measures localized shape variation whenever present in up to half but one of the landmarks. The resistant Procrustes results are highly concordant with a priori biological information, and might dramatically improve the quality of inferences on patterns of shape variation.     

MorphoJ: an integrated software package for geometric morphometrics

Increasingly, data on shape are analysed in combination with molecular genetic or ecological information, so that tools for geometric morphometric analysis are required. Morphometric studies most often use the arrangements of morphological landmarks as the data source and extract shape information from them by Procrustes superimposition. The MorphoJ software combines this approach with a wide range of methods for shape analysis in different biological contexts. The program offers an integrated and user-friendly environment for standard multivariate analyses such as principal components, discriminant analysis and multivariate regression as well as specialized applications including phylogenetics, quantitative genetics and analyses of modularity in shape data. MorphoJ is written in Java and versions for the Windows, Macintosh and Unix/Linux platforms are freely available from http://www.flywings.org.uk/MorphoJ_page.htm.     

Multivariate analyses of shape variation for genetically distinct groups

Many multivariate techniques have been proprosed for the analysis of shape variation. This article discusses several approaches in the context of examining shape similarities and differences for landmark data from two genetically distinct groups. Describing and understanding these variations will help develop insight into how genetically determined differences arise and are maintained. We discuss techniques based on principal component analyses including the use of a «sheared» component as the shape component and the use of a holistic size measure for adjustment of the original log-transformed measurements. Finally we examine a recently developed morphometric technique of analysis of triangles defined by sets of three appropriate landmarks.     

Geometric morphometrics as a tool for interpreting evolutionary transitions in the black fly wing (Diptera: Simuliidae)

A geometric morphometric analysis was conducted on wing‐vein landmarks on exemplar species of the family Simuliidae of the following genera: Parasimulium, Gymnopais, Twinnia, Helodon, Prosimulium, Greniera, Stegopterna, Tlalocomyia, Cnephia, Ectemnia, Metacnephia, Austrosimulium, and Simulium. Generalized least squares superimposition was performed on landmarks, followed by a principal component analysis on resulting Procrustes distances. Patterns of shape change along the principal component axes were visualized using the thin‐plate spline. The analysis revealed wing shape diversity through (1) the insertion points of the subcosta and R₁, resulting in the terminus of the costa exhibiting a trend towards a more apical position on the wing, and (2) the insertion point of the humeral cross vein, resulting in the anterior branch of the media exhibiting a trend toward a more basal position on the wing. Canonical variates analysis of Procrustes distances successfully assigned all exemplar species into their a priori taxonomic groupings. The diversity in wing shape reveals a trend towards decreased length of basal radial cell and increased costalization of anterior wing veins in the evolutionary transition from plesiomorphic prosimuliines to more derived simuliines. The functional significance of these evolutionary transitions is discussed. © 2013 The Linnean Society of London     

Principles and methods of geometric morphometrics

The basic concepts, notions and methods of geometric morphometrics (GM) are considered. This approach implies multivariate analysis of landmark coordinates located following certain rules on the surface of a morphological object. The aim of GM is to reveal differences between morphological objects by their shapes as such, the "size factor" being excluded. The GM is based on the concept of Kendall's space (KS) defined as a hypersphere with points distributed on its surface. These points are the shapes defined as aligned landmark configurations. KS is a non-Euclidian space, its metrics called Procrustes is defined by landmark configuration of a reference shape relative to which other shapes are aligned and compared. The differences among shapes are measured as Procrustes distances between respective points. For the linear methods of multivariate statistics to be applied to comparison of shapes, the respective points are projected onto the tangent plane (tangent space), the tangent point being defined by the reference. There are two principal methods of shape comparisons in GM: the Procrustes superimposition (a version of the least squares analysis) and thin-plate spline analysis. In the first case, Procrustes residuals are the outcome shape variables which remain after isometric alignment of the shapes being compared. Their summation over all landmarks yields Procrustes distances among these shapes. The Procrustes distances can be used in multivariate analyses just as the Euclidian distances. In the second case, the shapes are fitted to the references by stretching/compressing and shearing until complete identity of their landmark configurations. Eigenvectors of resulting bending energy matrix are defined as new shape variables, principal warps which yield another shape space with the origin defined by the reference. Projections of the shapes being compared onto principal warps yield partial warps, and their covariance matrix decomposition into eigenvectors yields relative warps which are similar to principal components (in particular, they are mutually orthogonal). Both partial and relative warps can be used in many multivariate statistic analyses as quantitative shape variables. Results of thin-plate spline analysis can be represented graphically by transformation grid which displays type, amount and localization of the shape differences. Basis rules of sample composition and landmark positioning to be used in GM are considered. At present, rigid (with minimal degrees of freedom) 2D morphological objects are most suitable for GM applications. It is important to recognize three type of real landmarks, and additionally semi-landmarks and "virtual" landmarks. Some procedures of thin-plate spline analysis are considered exemplified by some study cases, as well as applications of some standard multivariate methods to GM results. They make it possible to evaluate correlation between different shapes, as well as between a shape and some non-shape variables (linear measurements etc); to evaluate the differences among organisms by shape of a morphological structure; to identify landmarks which most accounted for both correlation and differences between the shapes. An annotated list of most popular softwares for GM is provided.     

Primate cranial diversity

Many studies in primate and human evolution focus on aspects of cranial morphology to address issues of systematics, phylogeny, and functional anatomy. However, broad analyses of cranial diversity within Primates as an Order are notably absent. In this study, we present a 3D geometric morphometric analysis of primate cranial morphology, providing a multivariate comparison of the major patterns of cranial shape change during primate evolution and quantitative assessments of cranial diversity among different clades. We digitized a set of 18 landmarks designed to capture overall cranial shape on male and female crania representing 66 genera of living primates. The landmark data were aligned using a Generalized Procrustes Analysis and then subjected to a principal components analysis to identify the major axes of cranial variation. Cranial diversity among clades was compared using multivariate measurements of variance. The first principal component axis reflects differences in cranial flexion, orbit size and orientation, and relative neurocranial volume. In general, it separates strepsirrhines from anthropoids. The second axis reflects differences in relative cranial height and snout length and primarily describes differences among anthropoids. Eulemur, Mandrillus, Pongo, and Homo are among the extremes in cranial shape. Anthropoids, catarrhines, and haplorhines show a higher variance than prosimians or strepsirrhines. Hominoids show the highest variance in cranial shape among extant primate clades, and much of this diversity is driven by the unique cranium of Homo sapiens. Am J Phys Anthropol 142:565–578, 2010. © 2010 Wiley‐Liss, Inc.     

Biometrics,biomathematics and the morphometric synthesis

At the core of contemporarymorphometrics—the quantitative study of biological shape variation—is a synthesis of two originally divergent methodological styles. One contributory tradition is the multivariate analysis of covariance matrices originally developed as biometrics and now dominant across a broad expanse of applied statistics. This approach, couched solely in the linear geometry of covariance structures, ignores biomathematical aspects of the original measurements. The other tributary emphasizes the direct visualization of changes in biological form. However, making objective the biological meaning of the features seen in those diagrams was always problematical; also, the representation of variation, as distinct from pairwise difference, proved infeasible. To combine these two variants of biomathematical modeling into a valid praxis for quantitative studies of biological shape was a goal earnestly sought though most of this century. That goal was finally achieved in the 1980s when techniques from mathematical statistics, multivariate biometrics, non-Euclidean geometry and computer graphics were combined in a coherent new system of tools for the complete regionalized quantitative analysis oflandmark points together with the biomedical images in which they are seen. In this morphometric synthesis, correspondence of landmarks (biologically labeled geometric points, like “bridge of the nose”) across specimens is taken as a biomathematical primitive. The shapes of configurations of landmarks are defined as equivalence classes with respect to the Euclidean similarity group and then represented as single points in David Kendall'sshape space, a Riemannian manifold with Procrustes distance as metric. All conventional multivariate strategies carry over to the study of shape variation and covariation when shapes are interpreted in the tangent space to the shape manifold at an average shape. For biomathematical interpretation of such analyses, one needs a basis for the tangent space compatible with the reality of local biotheoretical processes and explanations at many different geometric scales, and one needs graphics for visualizing average shape differences and other statistical contrasts there. Both of these needs are managed by thethin-plate spline, a deformation function that has an unusually helpful linear algebra. The spline also links the biometrics of landmarks to deformation analysis of the images from which the landmarks originally arose. This article reviews the history and principal tools of this synthesis in their biomathematical and biometrical context and demonstrates their usefulness in a study of focal neuroanatomical anomalies in schizophrenia.     

Geometric estimates of heritability in biological shape

The recently developed geometric morphometrics methods represent an important contribution of statistics and geometry to the study of biological shapes. We propose simple protocols using shape distances that incorporate geometric techniques into linear quantitative genetic models that should provide insights into the contribution of genetics to shape variation in organisms. The geometric approaches use Procrustes distances in a curved shape space and distances in tangent spaces within and among families to estimate shape heritability. We illustrate the protocols with an example of wing shape variation in the honeybee, Apis mellifera. The heritability of overall shape variation was small, but some localized components depicting shape changes on distal wing regions showed medium to large heritabilities. The genetic variance-covariance matrix of the geometric shape variables was significantly correlated with the phenotypic shape variance-covariance matrix. A comparison of the results of geometric methods with the traditional multivariate analysis of interlandmark distances indicated that even with a larger dimensionality, the interlandmark distances were not as rich in shape information as the landmark coordinates. Quantitative genetics studies of shape should greatly benefit from the application of geometric methods.     

Shape Variation in the Dermatocranium of the Greater Short-Horned Lizard <Emphasis Type="Italic">Phrynosoma hernandesi</Emphasis> (Reptilia: Squamata: Phrynosomatidae)

Dermatocranial shape and horn morphology display great disparity among the species of Phrynosoma. Ontogenetic change in dermatocranial shape in a series of 79 specimens of the short-horned Phrynosoma hernandesi (54F: 25M) was examined using geometric morphometric techniques. A multivariate ANCOVA of Procrustes residuals with sex as a factor and ln(centroid size) as the covariate indicated sexual shape dimorphism. Separate multivariate regressions of Procrustes residuals on ln(centroid size) for each sex indicated that allometry accounts for ~52–54% of the total sample shape variance. Comparisons of ontogenetic shape change between sexes indicate that sexual shape dimorphism is minimal and of uncertain biological significance. Groupings of multivariate regression coefficients by magnitude and sign suggest that allometric integration of the dermatocranium is not uniform over the dermatocranium. Principal component analysis of the landmark configurations corrected for sex and allometry yields a first principal component which describes shape variance concentrated in the posterolateral and posterior regions of the dermatocranium, and again is indicative of non-uniform shape variation over the dermatocranium. Our findings for P. hernandesi indicate that the adult shape of the dermatocranium may contribute to a passive defence against predation. We hypothesize that the complexity in dermatocranial shape demonstrated here for P. hernandesi indicates parcellation of shape variance, which may contribute to explanations of the pronounced dermatocranial disparity exhibited by the species of Phrynosoma.     

Allometric and nonallometric components of Drosophila wing shape respond differently to developmental temperature

Phenotypic plasticity of wing size and shape of Drosophila simulans was analyzed across the entire range of viable developmental temperatures with Procrustes geometric morphometric method. In agreement with previous studies, size clearly decreases when temperature increases. Wing shape variation was decomposed into its allometric (24%) and nonallometric (76%) components, and both were shown to involve landmarks located throughout the entire wing blade. The allometric component basically revealed a progressive, monotonous variation along the temperature. Surprisingly, nonallometric shape changes were highly similar at both extremes of the thermal range, suggesting that stress, rather than temperature per se, is the key developmental factor affecting wing shape.     

Inferring the Nature of Allometry from Geometric Data

The form of an organism is the combination of its size and its shape. For a sample of forms, biologists wish to characterize both mean form and the variation in form. For geometric data, where form is characterized as the spatial locations of homologous points, the first step in analysis superimposes the forms, which requires an assumption about what measure of size is appropriate. Geometric morphometrics adopts centroid size as the natural measure of size, and assumes that variation around the mean form is isometric with size. These assumptions limit the interpretation of the resulting estimates of mean and variance in form. We illustrate these problems using allometric variation in shape. We show that superimposition based on subsets of relatively isometric points can yield superior inferences about the overall pattern of variation. We propose and demonstrate two superimposition techniques based on this idea. In subset superimposition, landmarks are progressively discarded from the data used for superimposition if they result in significant decreases in the variation among the remaining landmarks. In outline superimposition, regularly distributed pseudolandmarks on the continuous outline of a form are used as the basis for superimposition of the landmarks contained within it. Simulations show that these techniques can result in dramatic improvements in the accuracy of estimated variance-covariance matrices among landmarks when our assumptions are roughly satisfied. The pattern of variation inferred by means of our superimposition techniques can be quite different from that recovered from full generalized Procrustes superimposition. The pattern of shape variation in the wings of drosophilid flies appears to meet these assumptions. Adoption of superimposition procedures that incorporate biological assumptions about the nature of size and of the variation in shape can dramatically improve the ability to infer the pattern of variation in geometric morphometric data.     

Morpho-Functional Analysis Using Procrustes Superimposition by Static Reference

In conventional geometric morphometric analyses of limb long bones, differences in the evolutionary capacity of articular surfaces and non-articular structures often remain unrecognised. It can be shown that areas of high spatial variance dominate shape data, which is problematic for the functional interpretation of limb long bone shape. We herein introduce Procrustes superimposition by static reference (PSSR), a novel analysis strategy that aims to facilitate morpho-functional inference. This procedure exploits the spatial constraint of some reference structures (in our case, articular surfaces) for the superimposition of other subareas (e.g. muscle attachment sites) in relation to that static reference. PSSR allows for the transformation of raw scan data, enabling researchers to extract geometric models of two- and three-dimensional substructures that cannot effectively be integrated with landmarks. As we demonstrate by a simple model analysis for one muscle attachment site, this procedure can yield measures of direct functional relevance. Multivariate analysis of an extensive set of subareas indicates how this type of data relates to conventional shape coordinates. The shape evolution of xenarthran humeri, which has previously been subject to a detailed study (Milne et al., J Zool 278(1):48–56, 2009), serves as a test case. The concept of a variance-based separation of landmark subsets expands mathematical methods by incorporating knowledge about evolutionary constraints. PSSR could therefore find application far beyond the intuitive case study of long bone shape.     

Bias and error in estimates of mean shape in geometric morphometrics

Sampling experiments were performed to investigate mean square error and bias in estimates of mean shape produced by different geometric morphometric methods. The experiments use the isotropic error model, which assumes equal and independent variation at each landmark. The case of three landmarks in the plane (i.e., triangles) was emphasized because it could be investigated systematically and the results displayed on the printed page. The amount of error in the estimates was displayed as RMSE surfaces over the space of all possible configurations of three landmarks. Patterns of bias were shown as vector fields over this same space. Experiments were also performed using particular combinations of four or more landmarks in both two and three dimensions.It was found that the generalized Procrustes analysis method produced estimates with the least error and no pattern of bias. Averages of Bookstein shape coordinates performed well if the longest edge was used as the baseline. The method of moments (Stoyan, 1990, Model. Biomet. J. 32, 843) used in EDMA (Lele, 1993, Math. Geol. 25, 573) exhibits larger errors. When variation is not small, it also shows a pattern of bias for isosceles triangles with one side much shorter than the other two and for triangles whose vertices are approximately collinear causing them to resemble their own reflections. Similar problems were found for the log-distance method of Rao and Suryawanshi (1996, Proc. Nat. Acad. Sci. 95, 4121). These results and their implications for the application of different geometric morphometric methods are discussed.     

The problem of assessing landmark error in geometric morphometrics: theory, methods, and modifications

Geometric morphometric methods rely on the accurate identification and quantification of landmarks on biological specimens. As in any empirical analysis, the assessment of inter- and intra-observer error is desirable. A review of methods currently being employed to assess measurement error in geometric morphometrics was conducted and three general approaches to the problem were identified. One such approach employs Generalized Procrustes Analysis to superimpose repeatedly digitized landmark configurations, thereby establishing whether repeat measures fall within an acceptable range of variation. The potential problem of this error assessment method (the "Pinocchio effect") is demonstrated and its effect on error studies discussed. An alternative approach involves employing Euclidean distances between the configuration centroid and repeat measures of a landmark to assess the relative repeatability of individual landmarks. This method is also potentially problematic as the inherent geometric properties of the specimen can result in misleading estimates of measurement error. A third approach involved the repeated digitization of landmarks with the specimen held in a constant orientation to assess individual landmark precision. This latter approach is an ideal method for assessing individual landmark precision, but is restrictive in that it does not allow for the incorporation of instrumentally defined or Type III landmarks. Hence, a revised method for assessing landmark error is proposed and described with the aid of worked empirical examples.     

Geometric morphometric analysis of mandibular shape diversity in Pan

The aim of this research is to determine whether geometric morphometric (GM) techniques can provide insights into how the shape of the mandibular corpus differs between bonobos and chimpanzees and to explore the potential implications of those results for our understanding of hominin evolution. We focused on this region of the mandible because of the relative frequency with which it has been recovered in the hominin fossil record. In addition, no previous study had explored in-depth three-dimensional (3D) mandibular corpus shape differences between adults of the two Pan species using geometric morphometrics. GM methods enable researchers to quantitatively analyze and visualize 3D shape changes in skeletal elements and provide an important compliment to traditional two-dimensional analyses.Eighteen mandibular landmarks were collected using a Microscribe 3DX portable digitizer. Specimen configurations were superimposed using Generalized Procrustes analysis and the projections of the fitted coordinates to tangent space were analyzed using multivariate statistics. The size-adjusted corpus shapes of Pan paniscus and Pan troglodytes could be assigned to species with approximately 93% accuracy and the Procrustes distance between the two species was significant. Analyses of the residuals from a multivariate linear regression of the data on centroid size suggested that much of the shape difference between the species is size-related. Chimpanzee subspecies and a small sample of Australopithecus specimens could be correctly identified to taxon, at best, only 75% of the time, although the Procrustes distances between these taxa were significant. The shape of the mandibular symphysis was identified as especially useful in differentiating Pan species from one another. This suggests that this region of the mandible has the potential to be informative for taxonomic analyses of fossil hominoids, including hominins. The results also have implications for phylogenetic hypotheses of hominoid evolution.     

GEOMETRIC MORPHOMETRICS OF DEVELOPMENTAL INSTABILITY: ANALYZING PATTERNS OF FLUCTUATING ASYMMETRY WITH PROCRUSTES METHODS

Although fluctuating asymmetry has become popular as a measure of developmental instability, few studies have examined its developmental basis. We propose an approach to investigate the role of development for morphological asymmetry by means of morphometric methods. Our approach combines geometric morphometrics with the two-way ANOVA customary for conventional analyses of fluctuating asymmetry and can discover localized features of shape variation by examining the patterns of covariance among landmarks. This approach extends the notion of form used in studies of fluctuating asymmetry from collections of distances between morphological landmarks to an explicitly geometric concept of shape characterized by the configuration of landmarks. We demonstrate this approach with a study of asymmetry in the wings of tsetse flies (Glossina palpalis gambiensis). The analysis revealed significant fluctuating and directional asymmetry for shape as well as ample shape variation among individuals and between the offspring of young and old females. The morphological landmarks differed markedly in their degree of variability but multivariate patterns of landmark covariation identified by principal component analysis were generally similar between fluctuating asymmetry (within-individual variability) and variation among individuals. Therefore there is no evidence that special developmental processes control fluctuating asymmetry. We relate some of the morphometric patterns to processes known to be involved in the development of fly wings.     

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.     

A geometric morphometric approach to the analysis of skull shape in Triassic dicynodonts (Therapsida,Anomodontia) from South America

Dicynodont therapsids were a major component of the Permo-Triassic terrestrial ecosystems across Pangea and have been regarded as specialized herbivores. In South America, the group was represented by several taxa of the clade Kannemeyeriiformes spanning from the Middle to the Late Triassic. In order to evaluate if cranial differences among taxa are potentially related to differences in feeding function, we performed a geometric morphometric analysis on 28 South American dicynodont crania. We digitized 19 cranial landmarks and conducted generalized Procrustes analysis, principal component analysis (PCA), principal component analysis between groups (bg-PCA), and a branch weighted squared-change parsimony approach. Phylogenetic inertia was not a significant driver of cranial shape evolution in the group, whereas PCA and bg-PCA support that major morphological shape differences are concentrated in the preorbital region (relative length of the snout and width of the caniniform process), in the position of quadrate condyle in relation to the caniniform process, and in the increase in the intertemporal surface area. In this context, tusked Dinodontosaurus, “Kannemeyeria,” and Vinceria have relatively smaller adductor attachment areas and input moment arm than younger taxa lacking tusks, such as Ischigualastia, Stahleckeria, and Jachaleria. Differences in cranial morphology in later dicynodonts reflect modifications in feeding mechanics, probably due to changes in food resources (vegetation) in their habitats toward the end of the Triassic.