Biology Reference
In-Depth Information
McArdle and Anderson then discuss pseudo F-tests of the form:
ð
SS
H
Þ=ð
ÞÞ=ð
ð
SS
error
Þ=ð
ÞÞ
F
5
tr
J
2
1
tr
n
2
J
(9.42)
where the trace is simply the sum along the diagonal of the matrix. The pseudo F-test thus
does not require a matrix inversion. The calculation of
the SSCP is based on the
partitioning:
tr
ð
SS
Total
Þ
5
tr
ð
SS
H
Þ
1
tr
ð
SS
error
Þ
(9.43)
Y
0
Y
,
HY
and
H
is
X
(
X
0
X
)
2
1
X
0
where
SS
total
5
Y
(
I
H
), where Y
pred
5
XB
ε5
2
5
so that
SS
under the hypothesis is
Y
0
Y
2ε
0
ε
SS
H
5
(9.44)
and the F-ratio is formed from the trace of the SSCP matrices of the J groups and of error
sum of squares (for n specimens and J levels). In more complex factorial designs, the terms
in the F-ratio would be the same as those of the univariate F, but would use the trace of
the related SSCP instead of the univariate SS. We can therefore consider how to carry out
the partitioning and formation of F-tests based on the outer product matrix
YY
0
, which
may be derived from the matrix of all pairwise interspecimen distances. In studies of
shape, that distance will typically be the Procrustes distance, although the methods are
more widely applicable to any distance metric.
Expressing GLM Models in Terms of Distance Matrices A and B for Which We
Can Compute Both AB and BA
tr
ð
AB
Þ
5
tr
ð
BA
Þ
(9.45)
then
Y
0
Y
YY
0
Þ
tr
ð
Þ
5
tr
ð
(9.46)
and we can partition tr(
YY
0
)as
YY
0
Þ
5
ð
ε
=
ε
0
Þ
tr
ð
tr
ð
Y
pred
Y
pred
0
Þ
1
tr
(9.47)
H)
0
, we can partition
YY
0
by
simply using
YY
0
and
H
. We don't need to know
Y
, in fact, we need only
YY
0
, which can
be obtained from the matrix of all pairwise inter-specimen distances (
D
).
Given
D
we can compute
A
, whose elements are
Noting that
Y
pred
Y
pred
0
5
H(YY
0
)H
0
and
εε
0
5
(I
H)YY
0
(I
2
2
1/2d
i
2
, and then we can calcu-
α
ij
52
late Gower's centered matrix
G
from
A
II
0
Þ
II
0
Þ
G
5
ð
I
2
ð
1
=
n
Þ
A
ð
I
2
ð
1
=
n
Þ
(9.48)
and
YY
0
5
G
. This means that if we have any matrix of interspecimen distances, we can
write it in squared, centered form, and use it in the same manner as an SSCP matrix, i.e.
as a way of describing the variance
covariance structure (the SSCP matrix is simply a
multiple of the variance
covariance matrix). Using the distance matrix and the related
