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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.

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