Biology Reference
In-Depth Information
Figure 3.2. Left mouse hemi-mandible showing 11 landmarks used in analysis.
obtaining a valid estimate of the exact magnitudes of variability local
to each landmark (see Result 2 , page 77). Whether a measure of co-
variability is positive or negative is not meaningful in our use of the
variance-covariance matrices. Even though we cannot treat the diago-
nal values from the variance-covariance matrix as proper estimates of
variability values, our estimates of the mean form and the variance-
covariance matrix can be used to obtain a visual representation of how
the location of landmarks varies across a sample of specimens. We do
this by generating hypothetical forms using parameters computed
from the original data. The resulting data can be used to provide a
visual impression of how variability is distributed across the sample of
specimens.
As an example, the mean form matrix and variance-covariance
matrix for the normal and Ts65Dn mandibles were used to generate
hypothetical populations of 50 specimens for each sample ( Figure 3.3 ).
To generate data like those represented in Figure 3.3 , a mean form
matrix is estimated following algorithm 1 ( Chapter 3, Part 2 ) and is
transformed back to landmark coordinates using an algorithm 1. The
landmark coordinates of the mean form are used as a template, just as
the original transparency was used to generate triangles in our pre-
ceding exercise with transparencies. However, instead of randomly
placing points around each of the given landmarks as we did with the
transparencies, the points are placed according to the Gaussian distri-
bution with the estimated variance-covariance matrix. The
hypothetical data displayed in Figure 3.3 (Richtsmeier et al., 2000) are
constrained by the variances calculated local to each landmark and by
the covariances that represent the relationships among variances for
all landmarks. The diagram drawn of the hypothetical data ( Figure
3.3 ) enables visualization of differences in variances local to specific
landmarks within a sample. The true sample size, of course, remains
small. Remember that the choice of coordinate system into which to
 
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