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matrix. The constraint specifies that the parameters corresponding to
translation and affine deformation are such that
and then
U
-1
(
M
2
1
t
T
M
1
A
)
N
.
Depending on the choice of the weight matrix, one obtains different
solutions.
An important point to make is that this choice is external to the
group structure. Fixing this constraint is analogous to fixing a linear
constraint among the
α
i
in the model for a one-way ANOVA:
E
[
y
ij
]
α
i
. For example, Arnold (1981) considers the constraint
One can also use the constraint
α
0
or any number of other
1
constraints. For the sake of estimation, such an arbitrary constraint
may be necessary, however when it comes to scientific and statistical
inference, such a constraint should have no effect. For example, in the
analysis of variance, inferences are based only on estimable functions
of the
α
i
's, which are invariant to the choice of the constraint.
Analogous to the linear models situation, the principle of invari-
ance suggests that such external constraints should have no effect on
the statistical inference when the deformation approach is used to
define form difference. Unfortunately the thin-plate methodology does
not satisfy the principle of invariance. Such lack of invariance is also
unappealing from a scientific point of view. Those inferences that vary
depending on the choice of the external constraints are, generally, sci-
entifically unacceptable. In
Part 1
,
Section 5
of this chapter, we have
illustrated the scientific consequences of this lack of invariance of the
thin-plate splines methodology.
4.15 Superimposition methods for comparison of forms
We now examine the superimposition method for comparing two forms.
For details on this method, we refer the reader to Siegel and Benson
(1982), Rohlf and Slice (1990) and Lele (1991). This method of form dif-
ference analysis has been in use in morphometrics since the beginning
of this century (Cole, 1996) and has been recently proposed by Goodall
(1991) and Rohlf and Slice (1990). Dryden and Mardia (1998) provide
a book length treatment of this approach. Small (1996, page 35) gives
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