Statistical Methods for Comparison of Forms - An Invariant Approach to Statistical Analysis of Shapes - page 165

4.10 Methods for exploring the form difference matrix
In the previous section, we showed how to test the null hypothesis of
equality of shapes. Testing for the equality of shapes or forms is most
often a formality; one almost required in published analyses, but one
that may not provide information relevant to the problem under study.
We have found in our experience that even when the null hypothesis of
equality of overall shape or form cannot be rejected, there may be
regions or loci where the forms are different. On the other hand, sam-
ples of forms that are shown to be statistically different may share
features that are similar. What is needed is a method that localizes
form difference to those parts of the objects that contribute signifi-
cantly to the observed form difference, or that differentiate seemingly
similar forms at very specific loci. This activity falls under the heading
of data exploration . Calculating confidence intervals for elements of
the form difference matrix is one way of uncovering those differences.
In this section, we address the activity of data exploration . This
includes methods for identifying those regions that contribute sub-
stantially to the observed form or shape difference and the ranking of
parts of the object in terms of relative contribution to the observed dif-
ference.
When we specify the location of differences between forms under
study, we say that the difference is “local” to a particular linear distance,
to a particular landmark, or to a set of landmarks. We use the term
“influential” to describe landmarks whose relative ranking is high in
terms of the contribution to the difference in form or difference in
shape. We remind the reader that the relative locations of landmarks
are simply a manifestation of underlying biological processes. The land-
marks are not “causing” the form difference or “influencing” the change
in form. The difference occurs due to biological processes that affect the
phenotype. Moreover, the difference noted local to a landmark might
reflect differences local to a larger region that includes that particular
landmark exclusively or several related landmarks. With this under-
standing, we will refer to “influential” landmarks, and designate
differences that are “local” to particular regions.
4.10.1 Detection of influential landmarks
Consider the form difference matrix FDM ( B,A ). Recall that if an entry
in the form difference matrix is close to 1, it signifies that little differ-
ence exists local to the relative positions of the corresponding pair of