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ulations, to identify locations or regions that differ least and/or most
between two populations, to evaluate a new observation in relation to
another group, and to cluster or classify a group of individuals into
potential subgroups. The way in which K * and the eigenvalues of D
can be used to address many of these issues will be shown in subse-
quent chapters. Although nuisance parameters limit our inferences
many biologically relevant and interesting questions can be addressed
using the landmark coordinate matrix data.
Our conclusion is simple. Landmark coordinate data can be used to
obtain a coordinate system-free representation of form, the form
matrix, or FM, which is invariant to the operations of translation,
rotation and reflection. Landmark coordinate data can also be used to
obtain useful features of sample variability. These parameters can be
obtained in a statistically sensible and operationally simple fashion.
We show in Part 2 of this chapter that these estimators are statistical-
ly consistent in that as the sample size increases, the estimators
approach the true values. Moreover, these estimators are simple to
compute and have high efficiency as compared to other estimators
(e.g., maximum likelihood estimators) (Lele and McCulloch, 2000).
These topics are more fully developed and studied in Part 2 of this
chapter. In the next section, we present estimates of the mean and
variance (mean form and variance-covariance matrices) for the biolog-
ical data sets introduced in Chapter 1 .
3.8 Analysis of example data sets
In this section we provide the estimates of the mean form, both in
terms of the form matrix (FM) and the landmark coordinate matrix for
data sets introduced earlier. Recall that the coordinate matrix repre-
sentation is useful for pictorially representing the object under study
but carries with it all of the problems associated with nuisance param-
eters. We also present the variance-covariance matrices for these data
sets. Throughout this section, we only consider the model where D
I .
The algorithms used to estimate mean forms and the perturbation
covariance structure ( K * ) for each sample are presented in Part 2 of
this chapter.
3.8.1 Ts65Dn mouse mandibles
Individuals with Trisomy 21 (Ts21) or Down Syndrome (DS) express
different subsets of the phenotypes that characterize the syndrome
(e.g., Hirschprungs disease, cardiovascular anomalies, atlanto-axial
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