A New Method for Landmark-Based Studies of the Dynamic Stability of Growth, with Implications for Evolutionary Analyses

Dimensions of Morphological Integration

Over several generations of evolutionary and developmental biologists, ever since Olson and Miller’s pioneering work of the 1950’s, the concept of “morphological integration” as applied to Gaussian representations $$N(\mu ,\Sigma )$$ N ( μ , Σ ) of morphometric data has been a focus equally of methodological innovation and methodological perplexity. Reanalysis of a century-old example from Sewall Wright shows how some fallacies of distance analysis by correlations can be avoided by careful matching of the distance rosters involved to a different multivariate approach, factor analysis. I reinterpret his example by restoring the information (means and variances) ignored by the correlation matrix, while confirming what Wright called “special size factors” by a different technique, inspection of the concentration matrix $$\Sigma ^{-1}.$$ Σ - 1 . In geometric morphometrics (GMM), data accrue instead as Cartesian coordinates of labelled points; nevertheless, just as in the Wright example, statistical manipulations do better when they reconsider the normalizations that went into the generation of those coordinates. Here information about both $$\mu $$ μ and $$\Sigma ,$$ Σ , the means and the variances/covariances, can be preserved via the Boas coordinates (Procrustes shape coordinates without the size adjustment) that protect the role of size per se as an essential explanatory factor while permitting the analyst to acknowledge the realities of animal anatomy and its trajectories over time or size in the course of an analysis. A descriptive quantity for this purpose is suggested, the correlation of vectorized $$\mu $$ μ against the first eigenvector of $$\Sigma $$ Σ for the Boas coordinates. The paper reanalyzes two GMM data sets from this point of view. In one, the classic Vilmann rodent neurocranial growth data, a description of integration can be aligned with the purposes of evolutionary and developmental biology by a graphical exegesis based mainly in the loadings of the first Boas principal component. There results a multiplicity of morphometric patterns, some homogeneous and others characterized by gradients. In the other, a Vienna data set comprising human midsagittal skull sections mostly sampled along curves, a further integrated feature emerges, thickening of the calvaria, that requires a reparametrization and a modified thin plate spline graphic distinct from the digitized configurations per se. This new GMM protocol fulfills the original thrust of Olson & Miller’s (Evolution 5:325–338, 1951) “$$\rho $$ ρ F-groups,” the alignment of statistical and biological explanatory guidance, while respecting the enormously greater range of morphological descriptors afforded by well-designed landmark/semilandmark configurations.