Statistical Models for Landmark Coordinate Data - An Invariant Approach to Statistical Analysis of Shapes - page 92

Variances calculated for right and left asterion (sutural intersec-
tions) and for the frontal bosses (fuzzy landmarks) are similar and
relatively large. The variance for bregma (3), lambda (4), and the pari-
etal bosses (7 and 8) (fuzzy landmarks) are similar to one another and
relatively small. This example demonstrates that the increased mea-
surement error in locating fuzzy landmarks (Valeri et al., 1998) may
not always result in fuzzy landmarks displaying greater measures of
variance relative to other landmark types.
Markedly high variances for any landmark is a condition that
needs to be closely studied in any morphometric analysis, especially
when landmarks with large variances are located close to one another
on a form. When EDMA is applied and the model-based bootstrapping
method is chosen for statistical testing, hypothetical forms will be gen-
erated from the mean form and variance-covariance matrix. When the
variability around a landmark is large and landmarks are placed rela-
tively close to one another ( Figure 3.5 ), it is possible that two
landmarks can change relative positions within a form created for the
testing procedure. The data shown in Figure 3.5 could produce a simu-
lated form with landmarks swapping relative locations, resulting in an
error that would prohibit statistical testing.
3.9 Some comments on EDMA
vs. other morphometric methods
Reflection Invariance. Earlier we defined form as “that characteristic
which remains invariant under translation, rotation, or reflection of
the object.” In some of the approaches to the analysis of landmark coor-
dinate data, reflection invariance is not included in the definition of
form (e.g., Goodall, 1991). However inclusion of reflection may sub-
stantially simplify statistical and mathematical analysis and may be
advantageous in certain research situations. Reflection invariance
does not compromise biological understanding or intuition, because
any scientist who has collected data knows those data, as well as the
data he or she would ideally like to have for analysis. Consider the fol-
lowing example. A paleontologist is trying to understand the evolution
of a species from fossilized teeth. She is collecting landmark coordinate
data from cusp tips and other landmarks identified on the occlusal sur-
face of teeth. Different tooth types (incisors, canines, premolars, and
molars) cannot be combined to increase the sample size for certain
poorly represented species. However, if in some individuals only a right