Biometrics, biomathematics and the morphometric synthesis

Abstract

At the core of contemporary morphometrics--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 of landmark 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's shape 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 the thin-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.

An alkaloid fraction extracted from the cactusTrichocereus terscheckiiaffects fitness in the cactophilic flyDrosophila buzzatii(Diptera: Drosophilidae)

“…Wing length is a good proxy of body size known to be involved in a trade-off with developmental time in D. buzzatii (Fanara et al, 1999;Fernández Iriarte & Hasson, 2000;Cortese et al, 2002). Wing morphology was analysed by separating size from shape variation using geometric morphometric techniques (Bookstein, 1996). As a measure of wing size, we calculated the centroid size of each individual configuration of landmarks (the square root of the sum of the squared distances of each landmark to the centroid of the configuration in arbitrary units; Dryden & Mardia, 1998).…”

“…Wing shape variation was investigated using the Procrustes technique. Shape coordinates were computed using a least-squares Procrustes superimposition method, where all wings are superimposed for the examination of differences in the position of landmarks once original variations in size, position, and orientation of the wings have been eliminated (Bookstein, 1996;Dryden & Mardia, 1998). This procedure created new shape variables (Procrustes coordinates) and eliminated four degrees of freedom, resulting in 16 shape-space dimensions (see Klingenberg, McIntyre & Zaklan, 1998).…”

Second to fourth digit ratio and face shape

“…The mathematical theory and biological application of geometric morphometrics are well understood (Bookstein 1991(Bookstein , 1996Marcus et al 1996;Dryden & Mardia 1998;O'Higgins 2000;Slice 2005), and its statistical properties have been proven superior to those of distance or angle-based methods (Rohlf 2000a(Rohlf ,b, 2003. The first step in the geometric morphometric analysis is the so-called Procrustes superimposition of the raw landmark coordinates.…”

A morphometric study on morphological plasticity of shell form in crevice-dwelling Pterioida (Bivalvia)

“…The past decade has seen the rise of the new 'geometric' morphometrics (Bookstein, 1991(Bookstein, , 1993(Bookstein, , 1996aRohlf & Marcus, 1993;Marcus & Corti, 1996;Lestrel, 1997;Dryden & Mardia, 1998;Kendall et al, 1999). This technique is based on Cartesian coordinate data in two or three dimensions and is here used to analyse variation of shell form in crevice-dwelling pterioids.…”