Statistical Methods for Comparison of Forms - An Invariant Approach to Statistical Analysis of Shapes - page 158

er. Differences in size exist but vary among the linear distances con-
sidered.
4.9.1 The EDMA-I hypothesis test
Details of the EDMA-I procedure were introduced by Lele and
Richtsmeier (1991) and are based on the Union-Intersection principle
(Casella and Berger, 1990). We begin by presenting an intuitive
description of this testing procedure.
Assume that there is no biological variability, and that mean form
matrices are available for the two populations under study. If the two
mean forms are simply scaled versions of each other, then the form
matrices representing the mean forms will differ by a 'scaling factor,'
S . If this is the case, then the form difference matrix will consist of a
single number, a constant, S . If the form difference matrix consists only
of S and we divide the maximum entry in the form difference matrix
by the minimum entry in it, the ratio will be equal to 1. Since we are
dividing the maximum entry in the form difference matrix by the min-
imum entry, the resulting ratio can never be smaller than 1. Moreover,
as this ratio deviates increasingly from the value of 1, the implication
is that the forms under consideration are more and more different from
one another.
In reality, it would be rare to have the form matrices of two forms
differ by a constant, S . In practice, we do not have the true mean forms
but only their estimates based on the observed landmark data. Even if
the true mean forms are scaled versions of each other, their estimates
may not be. In practice then, the ratio of the maximum entry of the
form difference matrix to the minimum entry will most likely be dif-
ferent from 1. Our goal is to determine the probability of obtaining the
observed (or a larger) maximum-to-minimum ratio value due to under-
lying biological variability when the two mean forms are, in fact,
similar. This probability is known as the p-value . If the calculated p-
value is small, we claim that the observed value is unlikely under the
null hypothesis of similarity in form and reject the null hypothesis. By
the same results, we entertain the alternative hypothesis. The EDMA-
I test of the null hypothesis of similarity in forms assumes that the
variances of the two samples being considered are equal.
The null hypothesis is that the average shapes of the two samples
are the same. The steps for testing for the equality of average shapes
are as follows. Again, let A 1 , A 2 ,…, A n and B 1 , B 2 ,…, B m be the landmark
coordinate matrices for individuals from the two samples.