A statistical method for biological shape comparisons.

Biological shape data combine two sorts of information: genometric location and biological homology. The link between these two is the landmark, a set of biologically corresponding points located in all the forms of a series. Analysis of shapes as configurations of landmarks proceeds most effectively by D' Arcy Thompson's (1917, 1942) method of transformations. In this tradition, shape change is not the arithmetical difference of distances, angles, and ratios measured on shapes separately; rather, it is a mensurand in its own right, the geometrical deformation taking the Cartesian coordinate space of one form onto that of another in accord with biological homology. Such a deformation may be displayed, continuously or discretely, as a symmetric tensor field of principal crosses, pairs of directions which start and finish at 90 degrees. At any point, along one of these directions the dilatation, or dimensionless rate of change of length, is largest; along the other, smallest. These are the principal dilatations. All conventional scalar descriptions of the shape change may be deduced from the tensor presentation. This article reviews the statistical description of mean biological form-changes viewed in this way, and introduces linearized significance tests for mean shape change and mean size change. The tests, which make no reference to any set of size or shape variables, are based in a convenient parametric distribution for shape changes in triangles of landmarks. On a suitable null hypothesis, the ratio of the difference of the principal dilatations of the mean deformation to the sum of their standard errors goes as square root of pi/2 approximately 0.88 times an approximate t-ratio for shape change; the sum of the deviations of those dilatations from unity, divided by the sum of their standard errors, goes as square root of 2/2 approximately 0.71 times an approximate t-ratio for size change. The difference between these two constants reflects the divergent selection bias of the pair of principal dilatations. In matched studies, the difference of the principal dilatations of the mean change, divided by the mean difference of the principal dilatations case by case, is square root of 2/pi N times an approximate X2 for shape difference. Examples are presented that test cephalometric data in three scientific contexts: comparison of groups of biological shapes observed once each; characterization of mean difference in a matched design (namely, mean growth in one group observed twice); and description of mean differences in growth between groups of shapes each observed twice.