MEASURING SHAPE DISPARITY

Geometric morphometric methods offer powerful tools for analyzing shape. They also offer a natural metric for measuring shape disparity: the Procrustes distance. The Procrustes distance is the Euclidean distance between shapes superimposed so as to minimize the sum of squared distances between homologous landmarks. Any complete set of geometric shape descriptors, such as partial warps (including the uniform component) or Procrustes shape coordinates (so long as obtained by the Generalized Least Squares Procrustes superimposition) can be used to calculate shape disparity. The same results are guaranteed because the sum of the variances of these descriptors equals the squared Procrustes Distance between shapes. This invariance to the choice of descriptors is one important characteristic of geometric analyses, differentiating geometric methods from those based on traditional morphometric data. Geometric analyses of disparity are also unaffected by body size, making them particularly useful for comparing levels of disparity over the course of ontogeny. We present formulae for measuring disparity, exemplify analyses (both geometric and traditional) using ontogenetic data from trilobites, and demonstrate software for calculating these measures and for testing hypotheses about the ontogenetic basis of disparity.