Biology Reference
In-Depth Information
FIGURE 6.7
Scree plot of the pro-
portion of variance described by each
PC for the squirrel jaw data set. Arrow
indicates the inflection point.
0.0012
0.0010
0.0008
0.0006
0.0004
0.0002
0.0000
0
10
20
30
Ordinal number of principal component
FIGURE 6.8
Scatter plot showing scores
on the first two PCs for the sample of 31
squirrel jaws shown in
Figure 6.6
.
0.04
0.02
0.00
0.02
0.04
0.08
0.06
0.04
0.02
0.00
0.02
PC1
eigenvalues are equal to each other. In other words, the variation described by these com-
ponents cannot be distinguished from random variation. The eigenvalues are numbered
from
Q
R
, where
Q
is a function of
P
(the total number of eigenvalues) and
R
(the number of the particular components of interest) such that
Q
1
1to
Q
1
P
R
.
Anderson
5
2
2
χ
(1958)
derived a
statistic based on the likelihood-ratio criterion to test the hypothesis
that the
Q
1 eigenvalue is not distinct from the higher numbered eigenvalues:
1
!
NR
ln
P
j
5
Q
1
1
λ
N
X
N
j
2
χ
ln
λ
j
1
(6.36)
52
R
j
5
Q
1
1
where
N
is the sample size minus one. When
N
is large, the degrees for freedom are
(
1
2
R
(
R
1)
1)
(
d.f.
2when
R
2).
In the special case where
Q
R
P
,
the test
1
2
5
5
1
5
