Statistical Models for Landmark Coordinate Data - An Invariant Approach to Statistical Analysis of Shapes - page 81

Table 3.1d. Mean form of the euploid Ts65Dn mandible ( mm )
(Coordinates corresponding to the mean form matrix)
Landmark (number) X Y Z

coronoid process (1) 3.138 2.903 0.394

mandibular angle (2) 5.414 -3.428 -0.031

anterior aspect of condyle (3) 4.714 1.591 -0.103

posterior aspect of condyle (4) 6.277 0.691 -0.505

high point on mandibular body (5) 0.070 -1.834 0.451

posterior aspect at base of alveolar thickening (6) -1.932 -2.341 -0.197

posterior aspect of incisor at alveolus (7) -5.408 -0.845 -0.126

anterior aspect of incisor at alveolus (8) -5.915 1.492 -0.469

body of alveolar ridge (9) -3.162 -0.206 0.297

anterior aspect of M1 on alveolus (10) -2.472 0.999 0.074

intersection of coronoid process and alveolar rim (11) -0.723 0.978 0.215

There is a direct correspondence between the coordinate data present-
ed for the mean forms and the mean form matrices. Each cell of the
form matrix represents a distance in three-dimensional space that
does not require a coordinate system. For example, the cell that con-
tains the number 6.741 in the mean form matrix of the euploid
mandibles ( Table 3.1e ) represents the distance between landmarks 1
and 2, a distance that was calculated directly from the landmark coor-
dinate data, but that could be measured on a mouse mandible using
calipers (though perhaps with less precision). The landmark coordi-
nates calculated for the mean form and the mean form matrix are
equivalent expressions; given one, you can derive the other. However,
to express the mean form in terms of landmark coordinates, a coordi-
nate system must be chosen. The mean form matrix does not require
that such a choice be made.
When we compare

K * for the two samples, the landmarks with
high levels of variation are different for the two samples. Landmarks
3,6,9,10 and 11 show low variance in the euploid sample, while land-
marks 10 and 11 appear to be the least variable landmarks in the
aneuploid sample. Also note that the aneuploid Ts65Dn mouse
mandible is more variable overall. Finally, both mandibles show land-
marks 5 and 8 to be highly variable.
The variance-covariance matrix provides an indication of variabili-
ty local to each landmark (variances along the diagonal) and of the
association of variability measures between pairs of points (covari-
ances on the off-diagonals), but nuisance parameters prevent us from