Statistical Methods for Comparison of Forms - An Invariant Approach to Statistical Analysis of Shapes - page 124

classes introduced below. These include restricted information, biolog-
ical variability, and lack of a common coordinate system.

1. Incomplete information: Landmark data used to study

form can be restricted in two ways.
(a) Sample size: In practice, it may be difficult to get a large
number of observations pertaining to the problem under
study. For example, the number and condition of available fos-
sil specimens may be limited, or the disease being
investigated may be too rare to obtain many cases. In certain
cases, the “normal” sample used for comparison can also be
limited in size.
(b) Number and choice of landmarks: The form of biolog-
ical objects is extremely complex. Our approach to the study
of form involves choosing biologically relevant loci to serve as
landmarks. There may be situations in which one region or
portion of a form contains more closely spaced landmarks
than another portion of the same form, or where certain
regions are without landmarks. This results in unequal repre-
sentation of aspects of a single form. One important criterion
for choosing landmarks is that they reflect positions repre-
sentative of underlying processes. This and issues of
repeatability guide our decision whether to include any spe-
cific landmark in the analysis. Consideration of both of these
concerns places limits on the number and nature of land-
marks available for analysis.

2. Variability: Variability is ubiquitous in nature and is of

great interest to biologists, but it is also our nemesis. In the
previous chapter we discussed the presence of variability
among biological organisms, the need to correctly account for
variability, and ways to model within-population variability.
We noted that the nuisance parameters of translation, rota-
tion, and reflection contribute to the difficulty of correctly
estimating variability. These same nuisance parameters con-
tribute to the difficulties in comparing samples of forms.
These issues are explored more fully in this chapter.

3. Lack of a common coordinate system: The presence of

nuisance parameters is linked to the lack of a common coor-
dinate system. The effect of these nuisance parameters on the
estimation of mean form and variability were documented in
the previous chapter. Though often unappreciated by basic