Biology Reference
In-Depth Information
want to find the
Y
1
that maximizes the ratio of between-group variance to within-group
variance. The within-group variance of
Y
1
is:
s
Y
1
within
5
A
1
S
W
A
1
(6.38)
and the between-group variance of
Y
1
is:
s
Y
1
between
5
A
1
S
B
A
1
(6.39)
The form of these expressions should be familiar from our discussion of PCA.
As before, we use the Lagrange multiplier
λ
1
to form the expression:
!
s
Y
1
between
s
Y
1
within
5
A
1
A
1
2
λ
(6.40)
1
then make the substitutions indicated by
Equations 6.38 and 6.39
to form:
!
A
1
S
B
A
1
A
1
S
W
A
1
2
A
1
A
1
Þ
2
λ
ð
1
(6.41)
1
This is the expression we will maximize relative to
A
1
, under the constraint that
A
1
A
1
5
Taking the partial derivative of this expression again yields a characteristic
equation that can be solved for the eigenvalues and corresponding eigenvectors of
S
2
W
S
B
:
Note that the matrix inversion in
Equation 6.41
does require that the pooled within group var-
iance
1
:
covariance be
of full-rank, meaning that the number of variables must equal the number of degrees of free-
dom in the system, which poses some challenges, particularly when working with semiland-
marks. The number of CVA axes appearing is also limited to a maximum of the number
of groups minus one.
covariance matrix be invertible. This, in turn, requires that the variance
Interpretation of Results
To help interpret the CVA result, and to illustrate the effect of the rescaling step on the
result, we first show PCA on the data set that will be used in the CVA example. This data
set is composed of 15 landmarks on the lower jaws of 119 squirrels from three geographic
areas (
Figure 6.15
). For each landmark, the cloud of points overlaps broadly, suggesting
similarly broad overlaps in the distribution of the whole shapes. Closer examination
shows there are slight differences in the relative positions of some landmarks
circles
predominate at one end of the clusters, triangles at the other. This is most evident for land-
mark 13, which is more anterior in the western Michigan sample and more posterior in
the southern sample. The scores on PC1 (
Figure 6.16A
) show there are differences between
the distributions of mandible shape in the three samples, and that each sample varies con-
siderably along this axis. As shown by the deformation grid (
Figure 6.16B
), that difference
primarily consists of shifts in the relative position of landmark 13, as surmised from the
superimposed shapes, but there is also expansion of the angular process (landmarks
10
12) and contraction of the space between the molars and the tips of the coronoid and
