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Inferring the nature of allometry from geometric data
Kim van der Linde & David Houle (2009)
Evolutionary Biology: 36(3): 311322.
Abstract
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 semilandmarks 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 variancecovariance
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.
