Statistical Methods for Comparison of Forms - An Invariant Approach to Statistical Analysis of Shapes - page 125

research scientists and statisticians, the lack of a common
coordinate system has severe implications when comparing
forms. This is because methods that are coordinate system
based require that a particular coordinate system be adopted,
and this choice can profoundly affect analytical results. It is
important that these issues and their implications be exam-
ined and understood thoroughly.
4.3 Comparing two forms: introduction to the problem
When comparing samples of biological forms, within-population vari-
ability cannot be ignored. However, for instructional purposes we begin
our discussion by comparing two forms rather than two samples of
forms. We do this to clarify the act of form comparison and to isolate
and observe the effect of the lack of a common coordinate system on the
comparison of forms. The consequence of within-population variability
for the comparison of forms is discussed later in this chapter.
Suppose we are studying two groups of organisms (e.g., two differ-
ent species, sexes, or age groups) by choosing a representative form
from each of the groups. Typically, the first question posed is: are the
two forms different? It is important to have quantitative data that sub-
stantiate the answer to this question, especially when form differences
are subtle. However, many times the differences are substantial and
the answer is obviously affirmative. The more important question then
becomes: in what aspects do the two forms differ from one another? The
answer to this question requires identifying those loci where differ-
ences are profound and those where differences are minimal. We refer
to this process as the “localization of form differences.” Information
pertaining to the location of component parts that are most or least dif-
ferent between specimens is essential to the investigator's discovery of
the mechanisms (e.g., physiological, genetic, developmental, pathologi-
cal, environmental, evolutionary) that underlie the form difference or
the form change.
Before delving into the comparison of complex biological forms, let
us first consider a simple triangle in two dimensions with three land-
marks. We will conduct a simple experiment using three circular
transparencies with different diameters as the media on which data
are recorded. Start by drawing the three points of a triangle on one of
the transparencies using a red pen. Number the points from 1 to 3 and