Introduction - An Invariant Approach to Statistical Analysis of Shapes - page 6

Statisticians must appreciate the nuances of biology and biologists
must be cognizant of the assumptions of certain methods and require-
ments made of the data.
Our collaborations have resulted in the following axioms:

1. The importance of scientific relevance. The scientific

questions being posed, in our case those pertaining to biology,
are of paramount importance. Morphometric methods are
developed as tools for answering scientific questions and have
little value if the results are not interpretable in terms that
relate directly to the scientific question posed. In biology, sta-
tistics and mathematics are means to an end. That end is to
assist the scientist in the formulation and testing of informed
hypotheses.

2. The importance of biological variability. Variability can

be extreme or subtle, but it is inherent to biological systems.
Variability lies at the root of what interests biologists.
Variability in biological systems must be modeled as correctly
and as accurately as possible. To accomplish this, variability
should be modeled stochastically rather than deterministically.

3. The importance of invariance. In any geometrically based

analysis of biological form, the choice of a coordinate system
is arbitrary. Alignment of specimens (registration and super-
imposition are terms used synonymously with alignment) is
required for many morphometric techniques and for all meth-
ods of visualization. Alignment is usually accomplished by
selecting either a subset of landmarks, a line, or a plane for
superimposition of all specimens. When alignment is required,
the scientist should be cautioned to run the same analysis try-
ing varying alignments of the same specimens. If results of the
comparison of specimens vary between the analyses done
using different alignments, then the method used for compar-
ison reflects the effects of the chosen method of alignment
rather than the biology of the organisms. The scientific infer-
ences should be invariant to such an arbitrary choice. When
scientific inferences are invariant with respect to this arbi-
trary choice of alignment or coordinate system, they are
known in the statistical literature to satisfy the Invariance
Principle (Berger, 1985). The Invariance Principle will be fol-
lowed and discussed at length in the context of this topic.