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.     

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.     

Evolution of shape in oligocene and mioceneNotocarinovalva (Ostracoda, Crustacea): A multivariate statistical study

Largely due to L. F. Bookstein, geometric morphometrics has been developed as an amalgamation of techniques drawn from mathematical statistics, non-Euclidean geometry and computer graphics, applied to labelled points (landmarks) and the biological images upon which they are registered. A tool of fundamental importance is the method of interpolation known as the thin-plate spline. Bookstein's sample-oriented procedure of relative warps, applied to data composed of coordinate pairs observed on eight landmarks on three samples of two species of fossil marine ostracods (bivalved microcrustacean),Notocarinovalva airella andN. yulecartensis, separated in time, is used to obtain weight matrices, which, when appropriately partitioned, constitute the familiar data matrices of multivariate statistical analysis. Standard multivariate statistical analysis of samples may be carried out in the tangent space to shape space at the Procrustes average shape. Linear discriminant function scores were used for assessing, approximately, the evolutionary relevance of shape change and change in size (based on standard distance measures) in the species from the upper and lower Oligocene and lower middle Miocene of Victoria, Australia. The findings indicate that a small, though consistent, change in non-uniform shape of the carapace has taken place. The analysis indicates that the evolution in shape of the carapace could have been caused by random genetic drift in moderately large populations, tempered by weak selection.     

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.     

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.     

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.     

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. Am J Phys Anthropol, 2012. © 2012 Wiley Periodicals, Inc.     

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.     

Variation in scale shape among alternative sympatric phenotypes of Arctic charr Salvelinus alpinus from two lakes in Scotland

Landmark‐based geometric morphometric analysis was used to detect differences in scale shape between ecologically distinct phenotypes of Arctic charr Salvelinus alpinus coexisting in the same lake. Relative warp analysis and standard multivariate analyses of the partial warps, obtained after a Procrustes superimposition, showed that scale landmarks were efficient in discriminating among two closely related alternative phenotypes within each of the two lakes. In Loch Tay, S. alpinus exhibited a bimodal body size‐frequency distribution among sexually mature fish, whereas in Loch Awe, S. alpinus are unimodal in body size but segregated into two distinct spawning phenotypes. In both lakes, alternative phenotypes showed significant differences in foraging ecology, habitat use and life history. It is probable that differences in scale shape reflect differences in ecology of these forms.     

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.     

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.     

Do dung beetles show interrelated evolutionary trends in wing morphology,flight biomechanics and habitat preference?

In flying organisms, wing shape and biomechanical properties are recognized as key traits related to dispersal, foraging behavior, sexual selection and habitat preferences. To determine if differences in dung beetle wing shape and flight biomechanics are consistent with habitat preferences in a phylogenetic context, we examined how wing morphology varied in a set of 18 Mozambique forest and grassland dung beetle (Scarabaeinae) species, representing nine genera and six tribes. Geometric morphometric measurements were taken of entire wings, as well as two additional shape characters comprising the RA4 and CuA to J regions of veins. Ordination (Principal Components Analysis and Canonical Variate Analysis) of landmark data revealed three different trends in wing shape related to expansion or contraction in external wing margins. These trends were consistent with published dung beetle phylogenies and a phylogenetic reconstruction of ancestral morphological changes using parsimony analysis of wing landmark configurations. Analysis of variance showed that the Procrustes distances between wing shapes were significantly correlated to species identity (~?48% of variance), wing size (~?27%), habitat (~?11%) and two of the three, tested, biomechanical variables (wing loading, wing aspect ratio: ~?1%). However, while a phylogenetic generalized least squares analysis confirmed a strongly significant phylogenetic signal for wing shape, it found no significant effect of any other variable. Therefore, wing shape evolution in dung beetles appears to have been phylogenetically constrained and habitat may constitute only a weak selective pressure for changes in wing shape.     

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.     

Getting into Shape: An Empirical Comparison of Traditional Truss-Based Morphometric Methods with a Newer Geometric Method Applied to New World Cichlids

Body shape is a difficult, but important, trait to quantify. Researchers have traditionally used multivariate analysis of several linear measures ('trusses') across the body form to quantify shape. Newer geometric morphometric methods claim to better estimate shape because they analyze the geometry among the locations of all landmarks simultaneously rather than the linear distances between pairs of landmarks. We tested this claim by comparing the results of several traditional morphometric analyses against a newer geometric analysis involving thin-plate splines (TPS), all applied to a common data set of morphologically variable new world cichlids Amphilophus citrinellus and A. zaliosus. The TPS method yielded slightly stronger evidence of morphological differences among forms, although traditional methods also distinguished the two species. Perhaps our most important result was the idiosyncratic interpretation of shape variation among the traditional truss-based methods, whereas the generation of deformation grids using the TPS approach yielded clear and visually interpretable figures. Our results indicate that geometric morphometrics can be a more effective way to analyze and interpret body form, but also that traditional methods can be relied upon to provide statistical evidence of shape differences, although not necessarily accurate information about the nature of variation in shape.