Statistical Models for Landmark Coordinate Data - An Invariant Approach to Statistical Analysis of Shapes - page 69

(d) Correlated landmarks with correlations between the axes.
When we are fortunate in having even more information per-
taining to the variation patterns of the landmarks, it may be
possible to add this information to the model. Assume that we
are collecting landmark data on femora of human male adoles-
cent individuals. We know that these teenagers are experiencing
a pronounced growth spurt affecting most parts of the body and
that the magnitude and the exact chronological timing of the
growth spurt vary between individuals. The covariance struc-
ture for correlated landmarks ( Figure 3.1d ) allows us to assume
that landmarks located on the shaft of the femur will have sim-
ilar covariance structures with correlation between the X- and
Y-axes. In the same model, we can specify an alternate covari-
ance structure for landmarks on the joint surfaces that allows
for a different correlation between the X- and Y-axes.
At this point, we remind the reader of an important caveat for any
statistical or mathematical model. Models are only approximations
that should fit the data as closely as possible. They cannot reflect real-
ity perfectly. This is the reason that the scientist's knowledge of the
data under study should be used when choosing a model. If the statis-
tical model deviates substantially from reality, any statistical analysis
based on that model is of limited utility. This is one reason to strive for
procedures that are valid under a variety of models (Lele, 1991; Lele
and Richtsmeier, 1991).
3.3 Effect of nuisance parameters
Let us return to the transparency experiment where the ten trans-
parencies and the original template are lying in their original
orientation on the table. Now let us suppose that while searching for
the reprint of an important article on covariance structures, the entire
pile of transparencies is knocked to the floor. In the act of retrieving
them from the floor, the red template transparency slips behind a file
cabinet, where it is lost forever. The remaining black pen transparen-
cies are placed back onto the desk and an attempt is made to restore
order to the pile by straightening them (translating and rotating). The
rotation and translation of each transparency away from its original
orientation (the path each transparency took as it flew to the floor, was
retrieved and then straightened within the stack) is unknown and