Statistical Methods for Comparison of Forms - An Invariant Approach to Statistical Analysis of Shapes - page 169

landmark that reduces the value of T to the next lowest value is the
second-most influential landmark, and so on. Thus, one can rank the
landmarks according to their influence in explaining the form differ-
ence. If testing for similarity in shapes is of concern, the p-value can be
estimated and checked for each value of T ( - i) noting those landmarks
that make a substantial difference in the level of significance.
Instead of, or in addition to, working through this process one land-
mark at a time, groups of landmarks can be identified and deleted as
a set. The object could be divided into regions or features, each defined
by a set of landmarks, and the sets can be deleted one at a time. Parts
of the object can then be ranked according to their role in influencing
the form difference. Those parts of the object that reduce the value of
T substantially when deleted are highly influential.
In summary, if a particular part of the object is responsible for a
majority of the form difference, deletion of that part of the object
should result in an improved match of the forms defined by the
remaining landmarks. Using these observations, the influence of par-
ticular landmarks or sets of landmarks can be ranked by observing the
reduction in T caused by deleting that particular set: the larger the
reduction, the larger the influence.
Two concerns should be noted when applying this logic and this
approach. First, remember that the FDM is based on the comparison of
mean form matrices and that these are only estimates of the mean
form, and not the true mean forms. The various values of T calculated
during the landmark deletion process are not directly comparable due
to sampling variability. We need a way to calibrate the T values so that
sampling variability is taken into account. The p -values can be used
towards this purpose. After deletion of a set of landmarks, a substan-
tial increase in the p -value indicates substantial influence of the
deleted landmarks. Recall that as T gets closer to 1, the corresponding
p -value increases. Thus, any reduction in the value of T corresponds to
an increase in the p -value.
Second, some consideration should be given to the number of land-
marks deleted. To this end, common sense is our only guide. For
example, if deletion of a single landmark or relatively small set of land-
marks achieves a reduction in T (or increase in the p-value) similar to
that achieved by the deletion of a larger set, the smaller set is more
influential. That is, the smaller set accounts for a “statistically compara-
ble difference,” but because this difference is confined to a smaller area,
the locus of the difference is more identifiable and hypotheses regarding
processes responsible for the differences may be more specific.