Statistical Models for Landmark Coordinate Data - An Invariant Approach to Statistical Analysis of Shapes - page 85

Table 3.2a. Mean form of the aneuploid Ts65Dn mandibles using the land-
mark subset (Coordinates corresponding to the mean form matrix)
Landmark (number) X Y Z

Landmark 1 1.504 2.670 -0.269

Landmark 2 4.370 -2.801 -0.009

Landmark 3 3.891 1.588 0.286

Landmark 5 -0.588 -1.783 -0.152

Landmark 7 -6.005 -0.718 0.104

Landmark 10 -3.172 1.044 0.041

Table 3.2b. Mean form matrix of the aneuploid Ts65Dn mandible using the
landmark subset (mm)
1

2 3 5 7 10

Landmark 1 0.000 6.182 2.678 4.922 8.247 4.961

Landmark 2 6.182 0.000 4.425 5.064 10.583 8.467

Landmark 3 2.678 4.425 0.000 5.623 10.162 7.088

Landmark 5 4.922 5.064 5.623 0.000 5.527 3.835

Landmark 7 8.247 10.583 10.162 5.527 0.000 3.336

Landmark 10 4.961 8.467 7.088 3.835 3.336 0.000

K * of the aneuploid Ts65Dn
mandibles using the landmark subset ( mm 2 )
1

Table 3.2c. Variance-covariance matrix

2 3 5 7 10

1 0.01633 -0.01579 0.00444 0.02117 -0.02009 -0.00606

2 -0.01579 0.02376 -0.00804 0.00017 0.00428 -0.00438

3 0.00444 -0.00804 0.00286 -0.00329 0.00111 0.00292

5 0.02117 0.00017 -0.00329 0.07947 -0.06613 -0.0314

7 -0.02009 0.00428 0.00111 -0.06613 0.05744 0.02339

10 -0.00606 -0.00438 0.00292 -0.0314 0.02339 0.01553

When we compare these estimates with those based on all eleven land-
marks ( Tables 3.1a to 3.1c ) , we notice that corresponding elements of
the mean form expressed as a form matrix are identical (within
numerical accuracy) to those of the form matrix estimated using the
full landmark set. However, landmark coordinate locations in X , Y , Z
space are quite different between the estimates. This occurs because