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nonidentifiabilty of the true common coordinate system in which forms
can be compared. Any coordinate system can be chosen, but that choice
has implications for statistical results and scientific interpretations.
Only those inferences that are invariant to such an arbitrary choice
are acceptable.
There is an exception to the strict application of this principle. The
application of the principle of invariance may be relaxed if, under all
possible choices of the external constraints in the superimposition
approach or the deformation approach, the scientific conclusions drawn
are identical (or at least similar enough to make no practical differ-
ence). In this case, the scientific results will be effectively invariant to
the external constraints. To determine whether noninvariant descrip-
tors of form change based on superimposition and deformation produce
“scientifically invariant” answers, all noninvariant descriptors must be
applied. This requires a great deal of work. Beyond this consideration,
the following simple example is given to show that scientific invariance
does not necessarily hold in real biological situations.
The method of Roentgenographic cephalometry (RCM) was intro-
duced earlier in this chapter. A cephalometric radiograph is an x-ray
produced under controlled conditions that allows for correction of dis-
tortion and enlargement, both of which occur during exposure of the
film, by an x-ray beam. The data analyzed can consist of two-dimen-
sional coordinates of landmarks identified on the cephalometric
radiographs. Clinicians and dentists compare x-rays of skulls using
this method. The method is often applied to compare the x-ray of a nor-
mal child with that of a child with a disorder, or to compare x-rays of a
child taken at different points during growth (see Figure 4.1 ). The goal
is to be able to describe and quantify differences between forms or sam-
ples of forms with reference to specific biological landmarks.
Apert syndrome is a genetic disorder that affects the bones of the
skull. The genetic mutations responsible for this syndrome are known
(Reardon and Winter et al., 1994; Park and Meyers et al., 1995; Wilkie
and Slaney et al., 1995; Park and Theda et al., 1995c; Oldridge and
Zackie et al., 1999). A method like RCM is applied to x-ray or comput-
ed tomography data to understand the craniofacial phenotypic aspects
of this syndrome for the purpose of planning possible surgical proce-
dures, planning the timing of these procedures, and inferring the
biological processes that are responsible for the obvious difference in
craniofacial form.
Let us again take the tracings of two cephalometric radiographs, one
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