Statistical Methods for Comparison of Forms - An Invariant Approach to Statistical Analysis of Shapes - page 152

form and shape difference as arithmetic differences between forms, as
relative differences between forms, and arithmetic differences between
scaled forms. We now discuss statistical methods that can be used to test
for differences in form and shape.
4.8 Statistical analysis of form and shape difference
using EDMA
Traditionally, the first step in statistical analysis is the testing of a
simple null hypothesis that the two forms or shapes are identical to
each other. However, an excessive emphasis on such a test may prevent
further exploration of the data and may even mislead researchers by
obscuring important local similarities or differences in forms or
shapes. For this reason, we advocate and emphasize the use of confi-
dence intervals when comparing forms. We present both approaches to
the statistical comparison of forms.
4.8.1 Confidence intervals for localized form differences
Suppose that the estimated form difference matrix determines that a
particular distance is larger in object B as compared to object A . How
do we know that this difference is significant in a statistical sense?
Basic statistics tells us that if we have a very large number of obser-
vations, or if the biological variability is very small, we can put greater
trust in the estimated form difference matrix than if we had a small
sample size and/or large variability. It is important that any estimate
of a scientifically relevant quantity carry with it a statement about its
reliability. Point estimates do not provide any information about relia-
bility of the estimate. An interval estimate incorporates reliability into
the estimate and can therefore be a powerful statistical tool (Kowalski,
1972; Reichardt and Gollob, 1997). When an interval estimate has a
certain probability of including the true value of the parameter, that
interval is called a confidence interval . We provide a brief discussion at
the end of this section concerning the interpretation of confidence
intervals for form and shape difference. For the precise meaning of con-
fidence intervals, we ask the reader to refer to a statistical text (e.g.,
Casella and Berger, 1990; see also Harlow, Mulaik, and Steiger, 1997).
The subtleties of interpretation cannot be fully covered here but it is
important that the user understand them.