Statistical Methods for Comparison of Forms - An Invariant Approach to Statistical Analysis of Shapes - page 123

from the things around it. Consciously or unconsciously, we compare
one object to others to try to explain or justify its appearance. When
observing the form of any biological organism, two key questions come
to mind. First, why do organisms look the way they do? And second, why
do organisms look different from one another? Members of biological
groups are often distinguished from one another on the basis of appear-
ance or morphology. Morphologists have always suspected that form
follows function to some degree (Cuvier, 1828; Hildebrand et al., 1985;
Radinsky, 1987) but have sought explanations for form using not only
biomechanical reasoning and functional models but also phylogenetic
information, optimization criteria, consideration of sexual dimorphism,
developmental programs, genetic factors, evolutionary trends, and
mathematical models. To understand the relationship between form
and any of these explanatory variables, a method is needed that enables
both a precise definition of form and the quantitative comparison of
forms, including the ability to “localize” form differences.
In practice, there are several limiting factors that make something
as seemingly simple as comparing forms extraordinarily difficult. In
the following chapter we discuss the implications of these limiting fac-
tors for some of the more popular methods of superimposition and
deformation by examining several Procrustean approaches (e.g., Rohlf
and Slice, 1990; Goodall, 1991; Dryden and Mardia, 1998) and the use
of thin plate splines (Bookstein, 1989; Bookstein, 1991). The problem of
invariance discussed previously in the context of describing form is
also encountered when comparing forms. We show that the shortcom-
ings of superimposition and deformation approaches stem from their
failure to satisfy the principle of invariance. EDMA is shown to satis-
fy the principle of invariance. Sections 4.2 through 4.6 are based on
ideas originally presented in Lele (1999).
4.2 Limiting factors in morphometrics
The first limiting factor of any morphometric analysis concerns the
type of data chosen for analysis. With the choice of landmark coordi-
nate data comes the acknowledgment that landmarks can never
entirely represent a biological form or the form of the population that
the specimen represents. This issue was more fully discussed in
Chapter 2 . The decision to use landmark data to represent a form is a
primary limitation that is accepted and understood. Additional limit-
ing factors faced by morphometricians are divided in to three major