Biology Reference
In-Depth Information
SAs are calculated just like scores on PCs, i.e. by multiplying the data for each specimen
by the SA (i.e. taking the dot product between an SA and the data for a specimen). Scores
are calculated for each block separately.
The fraction of the total covariance of the two blocks expressed by the ith pair of singu-
lar axes is given by:
2
I
λ
(7.4)
P
P
min
2
j
1
λ
j
5
Whether a singular value is larger than we would expect from randomly related blocks
is determined by comparing the observed singular value to the distribution produced by
randomly permuting the covariance structure between blocks. In such a permutation test,
the vectors of observations, each representing a specimen in the first block, are randomly
associated with vectors of observations from the second, thereby randomizing the covari-
ance structure between blocks without altering the variance
covariance structure within
the blocks. If the observed singular value lies outside the 95% confidence interval obtained
from the permuted data sets, the observed SA is judged to be statistically significant. The
correlation between the scores on the two blocks on the ith SA is also a measure of the sta-
tistical significance of the axis, and this correlation also may be tested via a permutation
test in exactly the same manner.
Three Block PLS
It is possible to extend PLS to more than two blocks of data, such as when we have
three blocks of data
Y
1
,
Y
2
and
Y
3
, and seek the linear combinations of variables within
each block (expressed as vectors
U
1
,
U
2
,
U
3
) which produce the greatest covariation of
scores (s
1
5
Y
1
U
1
, s
2
5
Y
2
U
2
, s
3
5
Y
3
U
3
). The vectors
U
1
,
U
2
,
U
3
are the singular axes of such
a system, and s
1
, s
2
and s
3
are the scores along the singular axes. It is then possible to com-
pute the correlation between each pair of scores r
1
2
,r
1
3
and r
2
3
. As discussed by
Bookstein and colleagues (2003)
, there is no standard approach to a multiblock PLS; sev-
eral are possible based on different properties of the PLS method. Bookstein and collea-
gues follow the approach taken by
Streissguth and colleagues (1993)
, which uses an
iterative method to estimate
U
1
,
U
2
and
U
3
.
In this approach, one starts with an arbitrary choice of the first axes
U
1
,
U
2
and
U
3
,
using random values or 1/n
0.5
if there are n entries in the vector
U
. The following set of
steps (
Bookstein et al., 2003
) is then iterated:
(a) Compute the scores (s
1
, s
2
)
s
1
5
Y
1
U
1
(7.5a)
s
2
5
Y
2
U
2
(7.5b)
s
3
5
Y
3
U
3
(7.5c)
