Introduction - An Invariant Approach to Statistical Analysis of Shapes - page 2

logical form. This approach, Euclidean Distance Matrix Analysis, or
EDMA (pronounced ed·ma), has considered invariance a central prop-
erty since its inception (Lele 1991). Real biological questions and the
issues that surround them provided the motivation for the develop-
ment of this approach. We have had the advantage of both statistical
and biological perspectives from the beginning of our collaborative
work, and believe that this has resulted in a method that is not just
statistically correct, but also valuable to biologists. We have struggled
with communicating cogent concerns from two different disciplines
throughout our collaboration, and have repeatedly stumbled onto new
insights that would not have come without scientific questions moti-
vating the development of the methods. Some of these realizations
were subtle and complex, others were simpler. Since our collaborative
publications have appeared in a number of different journals over
many years, we felt it timely to bring our methods, observations, and
insights together into a presentation suited to both the biological and
statistical community.
Biologists measure forms to obtain quantitative information about
specimens. The term measurement connotes quantitative observation,
ascertainment of dimensions, and comparison with standards.
Referring to an organism as a “large dog'' is not particularly informa-
tive unless we have an idea of what constitutes “largeness,'' or have a
common frame of reference for dogs. A record of weight and standing
height of the “large dog'' adds precision to the statement and enables
explicit comparison among dogs. Measurements can be used to vali-
date relationships that are qualitatively evident, but can also be used
to discover aspects of biological phenomena that may not be discern-
able from casual observation. Whether metric differences are obvious
or obscure, quantitative data can ultimately be used to reveal infor-
mation pertaining to growth patterns, evolutionary processes, biome-
chanical design, or phylogenetic relationships. These phenomena are
not always directly observable, but may be reflected in morphology.
Most scientific studies of biological morphology focus on one of two
primary components: design and diversity (Lauder, 1982). Design
refers to the relationship between form and function, a relationship
that serves as the basis for several disciplines. Functional anatomy,
biomechanics, biochemistry, crystallography, and bioengineering are
examples of fields built around ideas pertinent to biological design. The
other component, diversity, concerns variation observed among organ-
isms. Variation is observable between and within groups of biological
organisms that currently populate this planet, but also across time in