A revolution in morphometries

TREE vol. 8, no. 4, April 1993 A Revolution in Morphometries F. James Rohlf and Leslie F. Marcus We are now in the midst of a revolution in morphometry methodology. The new ap- proaches are more effective in capturing information about the shape of an organ- ism and result in more powerful statistical procedures for testing for differences in shape. They are also more effective in enabling a researcher to visualize differ- ences in shape and in suggesting simple traditional measurements that could be used in future studies. In this review we emphasize applications to exploratory studies in taxonomy and evolution. The field of morphometries is concerned with methods for the description and statistical analysis of shape variation within and among samples of organisms and of the analysis of shape change as a result of growth, experimental treatment or evolution. Morphometric methods are needed whenever one needs to describe and to compare shapes of organisms or of particular structures. The samples may represent geographic localities, developmental stages, gen- etic effects, environmental effects, etc. Traditional morphometries The approach now referred to as traditional morphometries12 or mul- tivariate morphometries3 is only a few decades old. It is characterized by the application of multivariate stat- istical methods to sets of variables. The variables usually correspond to various measured distances on an organism. The measurements are usually lengths and widths of struc- tures and distances between certain landmarks. Sometimes angles and ratios are used. Applications have frequently been concerned with allometry (change in shape as a function of size) and size correction (to enable the study of shape differences among samples of organ- isms adjusted to a common size). The results are mostly expressed numerically and graphically in terms of linear combinations of the measured variables. Examples of the techniques used are principal component analysis, canonical variate analysis, discriminant functions and generalized distances1,4. It is not possible to recover the shape of the original form from the lames Rohlf is at the Dept of Ecology and Evolution, State University of New York, Stony Brook, NY 11794- 5245, USA; Leslie Marcus is at the Dept of Biology, Queens College of CUNY, Flushing, NY 11367, USA. usual data matrices of distance measurements, even as an abstract representation. The overall form is neither really archived nor used in the analysis. An investigator may know, for example, that several measurements share a common landmark, but this information is not used in the multivariate analy- ses. As a result the analyses cannot be expected to be as powerful as they could be if that information were taken into account. The new morphometries The following points characterize the new approach: (1) Data are recorded to capture the geometry of the structure being studied. This is in the form of two- dimensional (2-D) or three-dimen- sional (3-D) coordinates of mor- phological landmark points. The coordinates are much more useful than traditional measurements, and, of course, the usual distance measurements can be computed from the coordinates. One can check their adequacy in covering the struc- tures of interest by a visual evalu- ation of a graphical display of the landmarks. Emphasis is given to recording homologous landmarks, since this allows a more complete biological interpretation of the results. Rather than just reporting that the shape has changed, one can report that certain structures have moved relative to others. When it is not possible to find such land- marks, one is forced to use pseudo- landmarks - points located at ends of structures, points at extremes of curvature of the outline of a struc- ture, or arbitrary points along an outline. If one is interested in the overall outline or surface of a struc- ture (or of just parts of a structure between landmarks in 2-D or a sur- face in 3-D), then this can be cap- tured by a sequence of digitized points along the outline or over a surface. Such approaches have been used for many years. (2) The geometrical relationships among the landmarks are not inherent in the raw coordinates themselves. The relationship among the points is captured by fitting an appropriate function to them in 2- or 3-D. The estimates of the parameters of the fitted function can then be used as variables in standard univariate and multivariate statistical analyses. For landmark data, Bookstein5 7 has recently proposed the use of thin-plate spline functions to fit the differences in the positions of land- marks in one organism relative to their positions in another. The term 'thin-plate spline' comes from a model of the deformation of a thin metal sheet. The use of this spline does not imply that biological tis- sue behaves like metal sheets (just as the use of Fourier analysis does not imply that outlines behave as vibrating strings). It is simply a con- venient function that is able to express the differences in two con- figurations of landmarks as a con- tinuous deformation. One can also transform arbitrary grid lines, out- line contours and any other points describing the image that are not used in the computation of the transformation. These properties enable the automatic construction of transformation grids such as those associated with D'Arcy Thompson8. An especially important feature of this transformation is that one can easily separate those changes due to differences in size, trans- lation, rotation and uniform shape change (affine transformation) and those describing purely inhomo- geneous changes (non-affine or local deformations). The purely inhomo- geneous part can be further split into principal warps - geometrically orthogonal components correspond- ing to deformations at different geometric scales (analogous to different powers in polynomial curve fitting or harmonics in Fourier analysis). A recent study suggests that the relative contributions of these components can be used as taxonomic characters9. Figure 1 shows an example of their use as variables in a multivariate analysis of 13 cranial landmarks in Talpidae (Mammalia, Insectivora)10. Figure 2 shows the landmarks used and an example of how one can visualize variation along a canonical axis by the use of thin-plate splines. The coordinates of points around an outline (or some simple trans- formation of them) are usually approximated by a weighted sum © 1993, Elsevier Science Publishers Ltd (UK) 0169-5347/93/S06.00 129 TREE vol. 8, no. 4, April 1993 of sine and cosine terms corre- sponding to various types of Fourier analysis"12. Elliptic Fourier analysis13 is now commonly used for complex shapes. Those studies in which only the harmonic ampli- tudes (the sum of the squares of the coefficients of the sine and cosine terms) are used should not be considered as examples of geometric morphometries since the ability to capture and reconstruct the original outlines is lost when the information on phase angle (the starting points for each sine and cosine function) is discarded14. (3) Rather than having to decide beforehand exactly which variables should be measured, the analyses are designed to indicate directions of maximum variation and hence may suggest which conventional vari- ables one should emphasize in ver- bal descriptions of the results. For example, Fig. 2 suggests that a character such as the ratio of dis- tances between landmarks 11, 12 and 13 would be useful. Further- more, this guides the choice of vari- ables to measure and use in future studies to efficiently capture the most important patterns of variation. (4) Displays of the results of the analyses are emphasized, using Fig. 1. Three-dimensional ordination of ten samples of moles based on a canonical variate analysis of partial warp scores to show differences in nonuniform shape (79% of the among-sample relative to within-sample variance is accounted for). A minimum spanning tree based on generalized distances has been superim- posed to show near neighbor relationships. Original data are coordinates of 13 landmarks from the skull of seven fossorial species from the family Talpidae10. Male and female samples are pooled since a MANOVA showed no significant differences between sexes nor of an interaction between sex and sample (P>0.9). Sample codes: I and 2, Talpa romana-, 3, Mogera latoucheh 4, Parascalops brewerii, 5, T. occidentalism, 6, T. stanko- vicki, 7, T. caeca-, 8-10, T. europaea. Computations and plot produced by TPSRW and NTSYSpc programs. differences or changes that can be shown on pictorial representations of the organisms studied. The dis- plays are expressed in terms of distances in the 2- or 3-D space of the organism, rather than distances in multidimensional vector spaces (although such distances are used in the computations and the tests of significance). It is easier to visu- alize and interpret the results from these displays than from tables of numerical coefficients. There are now several alternative approaches that fit our rubric 'new morphometries'. All of these can analyse shape variation in a sample of organisms or compare shape differences among two or more samples. Below, we summarize some of the characteristics of these methods and speculate on the future direction of the field. We expect and look forward to con- tinued vigorous discussions on the relative merits of the different approaches. Relative warp analysis Perhaps the most exciting new method is relative warp analysis. This analysis finds the thin-plate spline transformations that map a reference configuration of land- marks (usually the mean of a sample) onto each specimen. The par- ameters of these transformations can be used as variables in conven- tional multivariate statistical analyses (principal component analysis, canonical vector analysis, etc.). This is because the new variables are simply weighted linear combinations of the deviations of the specimens from the reference. Under the null hypothesis of no shape variation, the scatter of each specimen's landmark positions should deviate like digitizing error from the position in the reference. Thus, these new variables will have multivariate normal distributions if the deviations at each landmark are normally distributed. An important aspect of the use of such variables is that one can express the results of statistical analyses in terms of displays of thin-plate splines and show, as in Fig. 2, the effect of the first canoni- cal variate as a deformation of the average specimen - rather than having to examine lists of numeri- cal coefficients as in a traditional morphometric study. In Bookstein's original method, the principal warps corresponding to large-scale changes are given greater weights, which seems appropriate in growth studies5. These weights may be adjusted by a single parameter. Rohlf15, based on a suggestion by Bookstein7, has investigated setting this parameter to zero for taxonomic and other exploratory studies, where one has no a priori expec- tation that important deformations will occur at particular scales. Figure 3 shows an example of an analysis of allometry. Both uni- form and nonuniform components of shape were regressed on size and then visualized by transform- ing the reference configuration of landmarks (the average rat) into that predicted for the smallest and the largest rats (which match the observed very well since R2 - 0.94). One can see that most of the shape change during growth is uniform. Superimposition methods Superimposition methods (also called Procrustes methods) are based on the simple idea of over- laying the images of two or more specimens so that their homolo- gous landmarks match as closely as possible according to an optimality criterion. One then reports any dif- ference in shape in terms of re- siduals, usually shown graphically as displacement vectors at each land- mark. Several different criteria for fitting are available. The resistant fit16-18 approach usually yields a more satisfactory alignment when shape change is mostly limited to a small proportion of the landmarks. There have also been important developments in the statistical analysis of the properties of super- imposition methods19. One can also align the specimens with differ- ences due to uniform shape change (affine transformations) taken into consideration. This technique is effective in showing relative levels of variation at different landmarks. On the other hand, it is difficult to show covariation in displacements at different landmarks, which is one of the strong points of relative warp analysis. Euclidean distance matrix analysis Lele20 has proposed Euclidean distance matrix analysis, which is 130 TREE vol. 8, no. 4, April 1993 based on an examination of ratios of distances between all pairs of landmarks in two specimens. If they are all the same then the two organisms must have the same shape and the constant ratio gives the difference in size. While the technique can show the existence of statistically significant shape changes, the results are expressed in the form of lists of distances that are unusually larger or smaller in one specimen than in another. These lists are not very easy to visualize in terms of changes in shape of organisms. The fact that the method is coordinate-free, and therefore invariant to translation and rotation since it operates only on distance, has been given great emphasis2'. However, statistics based on thin-plate splines, Procrustes analyses, shape coordinates, etc., all have this same property, so the supposed advantage of coordinate- free methods is unclear. The concern seems to be that there is a possibility of bias due to the use of any particular coordinate system - even if it is used only for displaying the results of a statisti- cal analysis. Finite element scaling analysis Finite element scaling analy- sis22,23 is another useful method. In this technique the landmarks are connected so as to form small closed regions (often triangles in 2-D and tetrahedra in 3-D). Each region is then compared to its cor- responding region in a second organism by computing the affine transformations that map one into the other. The results are usually displayed in terms of strain crosses that indicate the directions of the principal strains and their magnitudes within each element, or are averaged in some way and then shown as strains at each landmark. This shows how two organisms differ in shape. However, a problem with this approach is that the division of the organism into regions is not unique. While different divisions are mathematically equivalent for showing overall differences in shape, the strain crosses can be rather different. This means that one can obtain different values for summary statistics over the regions, such that the average angle of the principal strain or the variance in its angle depends upon how the organism is divided into elements. Thus, it is difficult to rec- ognize global effects such as uni- form shape change. Bookstein7 describes these and other prob- lems that limit the usefulness of this technique for the statistical analysis of shape. Problems to be solved Traditional morphometric methods employing multivariate analyses of selected distance measurements are not 'wrong'. There is nothing wrong with multivariate methodologies as such (in fact they are essential for the statistical analyses of the variables generated by the new methods); the approach is just not nearly as (d) ib 1 1! 12 .....1-31 ♦......^ -• 3 2 •• • 4> Fig. 2. Visualization of the component of nonuniform shape change corresponding to the first canonical variate shown in Fig. 1. (a) shows the locations of the landmarks on a dorsal view of the skull, (b) and (d) correspond to an extrapolation beyond the left and right ends of the first canonical vector axis (this exaggeration was necessary in order to make the differences visible), (c) shows the average pos- itions of the landmarks with vectors pointing in the direction of positive changes along the axis. Plots produced using the TPSREGR program. 131 TREE vol. 8, no. 4, April 1993 (b) (c) powerful as it could be, since it is not able to take into account the geometrical relationships among the measurements - that pairs of measurements were made from a common landmark, or that a set of three measurements correspond to a triangle of landmarks on the organism. There are limitations and prob- lems to be solved in the new morphometries: (1) There is a need for an appro- priate distance measure for summa- rizing differences between organ- isms, and for evaluating characters so that phenetic or cladistic taxo- nomic relationships can be esti- mated. Sneath and Sokal2' point out that the available distances have standard errors that are a function of the number of variables analysed - not just of sample size as for tra- ditional statistics. Bookstein7 is much less optimistic about obtain- ing sufficiently large numbers of Fig. 3. Regression of both uniform and nonuniform com- ponents of shape on size. Rat calvarial growth data (ages 7-150 days, digitized by M. Moss from roentgenograms by H. Vilmann, published and illustrated Bookstein7). tal, Ibl and (cl correspond to predicted shapes for rats at sizes corresponding to the youngest, the average and the oldest rats, respectively. The vectors in |b) show the direction of shape change from the average to the largest rats. Computations and plots produced using the 1PSRECR program. landmarks since their variation is not independent. Another problem is that there does not seem to be a unique way of defining morpho- metric distance and thus there is a degree of arbitrariness in any such results reported. (2) Different interpolating func- tions can yield different numerical results and pictorial displays since they give different weight to data points. Superimposition analy- sis1718 uses an identity function which weights all points equally - but that is also an arbitrary choice. It is important to point out that many methods agree on the nature of the differences among a set of shapes, but not with respect to the relative amount of difference between different shapes. (3) Much more work is needed on methods to capture surfaces and the texture of both outlines and surfaces - especially in combi- nation with landmark data. (4) An important potential area of application is in taxonomy and phylogenetic inference, where characters are currently coded in roughly ordered or unordered cat- egories. But the problem is that there are different metrics for measur- ing the distance between different forms and it is unclear how to choose among them or how to generate the kinds of characters desired for phylogenetic inference methods. The statistics of shape distances (e.g. Procrustes distance) is a surprisingly complex subject"26. One cannot assume they can safely be used as measures of taxonomic distance. Theoretical research in this area is very active at present. As in any scientific revolution, one cannot expect unanimity of opinion as to whose method is most effective. There is, however, a consensus among most workers that it is important to take ge- ometry into account21. Acknowledgements We gratefully acknowledge the extensive and helpful comments by F.L. Bookstein and R. Reyment on a draft of this article and the use of Sidney Horenstein's scanner to capture the image in Fig. 2a. This work was supported, in part, by a grant to F|R (BSR-89-18630) from the Systematic Biology Program of the National Science Foundation. This paper is contribution num- ber 837 from Graduate Studies in Ecology and Evolution, State University of New York at Stony Brook. References 1 Reyment, R.A. (1991) Multidimensional Paleobiology, Pergamon Press 2 Marcus, L.F. 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