Quadratic trends: a morphometric tool both old and new

Abstract

The original exposition of the method of "Cartesian transformations" in DArcy Thompsons great essay On Growth and Form of 1917 is still its most cited. But generations of theoretical biologists have struggled ever since to invent a biometric method aligning that approach with the comparative anatomists ultimate goal of inferring bio-logically meaningful hypotheses from empirical geometric patterns. Thirty years ago our community converged on a common data resource, samples of landmark configurations, and a currently popular biometric toolkit for this purpose, the "morphometric synthesis," that combines Procrustes shape coordinates with thin-plate spline renderings of their various multivariate statistical comparisons. But because both tools algebraically disarticulate the landmarks in the course of a linear multivariate analysis, they have no access to the actual anatomical information conveyed by the arrangements and adjacencies of these locations as they combine in pairs or higher numbers into substructures. This paper explores a geometric approach circumventing these fundamental difficulties: an explicit statistical methodology for the simplest nonlinear patterning of these comparisons at their largest scale, their fits by what Sneath (1967) called quadratic trend surfaces. After an initial quadratic regression of target configurations on a template, the proposed method ignores individual shape coordinates completely, replacing them by a close reading of the regression coefficients accompanied by several new diagrams, notably the exhaustive summary of each regression by an unfamiliar biometric ellipse, its circuit of second-order directional derivatives. These novel trend coordinates, directly visualizable in their own coordinate plane, do not reduce to any of the usual Procrustes or thin-plate summaries. The geometry and algebra of these second-derivative ellipses seem a serviceable first approximation for applications in evo-devo studies and elsewhere. Two examples are offered, one the classic growth data set of Vilmann neurocranial octagons and the other the Marcus groups data set of midsagittal cranial landmarks over most of the orders of the mammals. Each analysis yields startling new findings inaccessible to the current GMM toolkit. A closing discussion suggests a variety of ways by which innovations in this spirit might burst the current strait-jacket of Procrustes coordinates and thin-plate splines that together so severely constrain the conversion of landmark locations into understanding across our science.