Previous work



Inferring the nature of allometry from geometric data

Kim van der Linde & David Houle (2009)

Evolutionary Biology: 36(3): 311-322.

The form of an organism is the combination of its size and its shape. For a sample of forms, biologists wish to characterize both mean form and the variation in form within the sample. For geometric data, where form is characterized as the spatial locations of homologous points, the first step in analysis is usually to superimpose the forms, which requires an assumption about what measure of size is appropriate. Standard geometric morphometrics assumes that centroid size is the natural measure of size, and that variation around the mean form is isometric with size. These assumptions are motivated by geometric considerations rather than biological assumptions, and therefore strongly limit the interpretation of the resulting estimates of mean and variance in form. We illustrate these problems using allometric variation in shape. While allometric changes are readily tested for following Procrustes superimposition, the nature of the changes in shape with size cannot be recovered because of the assumption of isometry, and because both size and shape must be estimated from the same data. To ameliorate this problem, we propose that alignments based on subsets of the available data that can be assumed to be more isometric will yield superior inferences about the remaining, more allometric, variation. We propose and demonstrate two superimposition techniques based on this idea. In subset superimposition, landmarks are progressively discarded from the data used for superimposition if they result in significant decreases in the variation among the remaining landmarks. In outline superimposition, regularly distributed semi-landmarks on the continuous outline of a form are used as the basis for superimposition of the landmarks contained within it. We use simulations to show that these techniques can result in dramatic improvements in the accuracy of estimated variance-covariance matrices among landmarks when our assumptions are roughly satisfied. The pattern of variation inferred by means of our superimposition techniques can be quite different from that recovered from the standard generalized Procrustes superimposition. The pattern of shape variation in the wings of drosophilid flies appears to meet these assumptions. Adoption of superimposition procedures that incorporate biological assumptions about the nature of size and of the variation in shape can dramatically improve the ability to infer the pattern of variation in geometric morphometric data. We urge further development of techniques that allow improved inferences about patterns of variation.