Statistical Methods for Comparison of Forms - An Invariant Approach to Statistical Analysis of Shapes - page 160

two populations are different from one another, the testing should be
done twice using each of the samples as the baseline in one of the anal-
yses. If one sample is particularly small, preventing its use as the
baseline sample in testing, it is important to make this clear in report-
ing the results that the test is a one-way test. The EDMA-I test also
requires the assumption that variances in the two samples are equal.
If information is available that indicates that this assumption is not
supported by the data, results of this test may be misleading.
4.9.2 A two-way test of the null hypothesis using EDMA-II
This procedure, discussed in detail by Lele and Cole (1996), is based on
ideas outlined above, but uses a different statistic for testing the null
hypothesis of similarity in shapes. An important difference between
the two procedures is that unlike EDMA-I, EDMA-II does not require
the assumption of equality of variances in the two populations. EDMA-
II is also a two-way test removing the need to choose one of the
samples as the baseline group. However, EDMA-II does require that a
scaling factor be chosen to represent the “size” variable. A single linear
distance, or a combination of all linear distances (e.g., the geometric
mean of the distances) could be used to scale the data and represent
size.
When testing the null hypothesis of similarity of shapes using the
EDMA-II procedure, it is important to realize that choice of a measure
to represent “size” has implications. Although the choice of the scaling
factor does not affect the validity of the test, it does affect the power of
the test (Lele and Cole, 1996). The logic for choosing a particular scal-
ing factor should be biological and not based simply on mathematical
convenience. Choice of the scaling factor will be influenced by the bio-
logical problem at hand. For example, the geometric mean might be a
reasonable scaling factor for studying skulls, whereas overall length
may be a more appropriate scaling factor in the study of femur mor-
phology.
When conducting the EDMA-II test, the first step is to select a scal-
ing factor. Using this scaling factor, a shape difference matrix is
calculated. If the two mean forms are similar to each other, irrespec-
tive of which scaling factor is chosen, the shape difference matrix will
consist of zeros. Thus, if the two forms are not similar, the entry that is
farthest away from zero (either a positive or negative value) provides
us with a measure of their dissimilarity.
The steps for testing for the equality of average shapes using