Statistical Methods for Comparison of Forms - An Invariant Approach to Statistical Analysis of Shapes - page 166

landmarks. On the other hand, if the entry is markedly different from
1, it signifies that the relative position of the corresponding pair of
landmarks is substantially different in the two mean forms represent-
ing the two populations. When an entry is very different from 1, this
signifies that the two landmarks involved in defining this distance fig-
ure prominently in explaining the observed form difference.
The first step in exploring the form difference matrix is to arrange
the entries in increasing order. Simply looking at the extremes (mini-
mum and maximum) of such an ordered vector provides substantial
information. If one or more distances at either the maximum or the
minimum ends of the vector delineate a relevant region, this provides
information relevant to the processes that underlie the form difference.
However, analysis does not end with the simple reporting of which
ratios are larger or smaller than 1. Instead, the researcher should
attempt to relate these observations to potential underlying processes
(i.e., physiological, biomechanical, pathological, evolutionary) and for-
mulate hypotheses. The generation of new hypotheses from EDMA
results will depend upon the scientific background of the investigator,
the goals of the research, and the availability of alternate data. Once
new hypotheses are generated, one can collect an alternate data set
that bears on the new hypotheses. These data sets may or may not
include landmark coordinates. The new hypotheses are then tested
using the new data set. This process is similar to the general scientif-
ic process of hypothesis formulation followed by hypothesis testing,
modification of the hypothesis on the basis of those results, additional
data collection, and further hypothesis testing.
Two additional tools for systematic exploration of the form differ-
ence matrix are presented below. These tools can be used to gain an
understanding of the observed form difference and as an aid in refor-
mulation of hypotheses. These tools were originally presented in Lele
and Richtsmeier (1992) and Cole and Richtsmeier (1998).

4.10.2 The landmark deletion approach.

The idea behind this technique is very simple. Recall the test statistic,

max FDM ( B A , )

T , defined in EDMA-I as Remember also that

ij , ij

T

) . min FDM ( B A ,

ij , ij

if the two mean forms are very similar to each other, the value of T is
close to 1 and as the forms become more and more dissimilar, the value
of T differs increasingly from 1.