On the use of shape spaces to compare morphometric methods

Abstract

Several methods have been proposed to use differences in configurations of landmark points to measure the amount of shape difference between two structures. Shape difference coeffi- cients ignore differences in thc configurations that could be due to the effects of translation, rotation, and scale. One way to understand the differences between these methods is to compare the multidi- mensional shape spaces corresponding to each coefficient. This paper compares Kendall's shape space, Kendall tangent space, the shape spaces implied by EDMA-I and EDMA-I1 test statistics, the shape space of log size-scaled inter-landmark distances, and the shape space implied by differences in an- gles of lines connecting pairs of landmarks. The case of three points in the plane (Le., landmarks at the vertices of a triangle) is given special emphasis because the various shape spaces can be illus- trated in just 2 or 3 dimensions. The results of simulalions are shown both for random samples of all possible triangles as well as for normally distributed independent variation at each landmark. Gener- alizations to studies of more than three landmarks are suggested. It is shown that methods other than those based on Procrustes distances strongly constrain the possible results obtained by ordination analyses, can give misleading results when used in studies of growth and evolutionary trajectories. Kup~or-ds: Kendall shape space; tangent space: EDMA; inter-landmark distanccs; multivariate analysis: morphometrics: Procrustes distance; thin-plate spline.