Quadratic Trends: A Morphometric Tool Both Old and New
Authors: Bookstein Fred L. 
Published: February 2024

The original exposition of the method of “Cartesian transformations” in D’Arcy Thompson’s On Growth and Form (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 anatomist’s ultimate goal of inferring biologically 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 the landmark locations and the distinct anatomical components they span. This paper explores a new 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. Those have been replaced by a close reading of the regression coefficients, accompanied by several new diagrams, of which the most striking is a novel biometric ellipse, the circuit of the trend’s second-order directional derivatives around the data plane. These new trend coordinates, directly visualizable in their own coordinate plane, do not conduce to any of the usual Procrustes or thin-plate summaries. The geometry and algebra of the 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 group’s data set of midsagittal cranial landmarks over most of the orders of the mammals. Each analysis yields intriguing 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 straitjacket of Procrustes coordinates and thin-plate splines that together so severely constrain the conversion of landmark locations into biological understanding. This restoration of a quantitative diagrammatic style for reporting effects across regions and gradient directions has the potential to enrich landmark-driven comparisons over either developmental or phylogenetic time. Extension of the paper’s quadratic methods to the next polynomial degree, cubics, probably won’t prove generally useful; but close attention to local deviations from globally fitted quadratic trends, however, might. Ultimately there will have to emerge a methodology of landmark configurations, not merely landmark locations.