Size and Shape Spaces for Landmark Data in Two Dimensions

Biometric studies of the forms of organisms usually consider size and shape variations in the geometric configuration of landmarks, points that correspond biologically from form to form. The size variables may be usefully considered the linear vector space spanned by the set of all distances between pairs of landmarks. The shape of a single triangle $\Delta ABC$ of landmarks may be reduced to a single pair of shape coordinates locating the vertex $C$ in the coordinate system with landmark $A$ sent to (0,0) and landmark $B$ to (1,0). A useful space of shape variables is the span of all such shape coordinate pairs for various triples of landmarks. On a convenient null model of identical circular normal perturbations at each landmark independently, one size variable $S$, which may be taken as the mean square of all the interlandmark distances, has covariance zero with every shape variable. Then associations between shape and size may be tested by the $F$ ratio for multiple regression of $S$ on any basis for shape space. For a single triangle of landmarks, the existence of any mean difference or mean change in shape may be tested by Hotelling's $T^2$ applied to any pair of shape coordinates for that triangle. When such a difference is statistically significant, it may be interpreted as the ratio of a pair of size variables measured along directions at an angle averaging $90^\circ$ in the samples of forms. One size variable will bear the greatest mean rate or ratio of change between the forms, the other the least. Analysis of configurations of more than three landmarks reduces to consideration of size variables involving at most three landmarks. These techniques are demonstrated in a study of the growth of the head in 62 normal Ann Arbor youth. Each comparison of interest is summarized in its own orthogonal coordinate system, the biorthogonal grid pair.