Biology Reference
In-Depth Information
(b) Normalize the scores to unit variance
=
ð
ð
ÞÞ
s
1
5
s
1
standard deviation
s
1
(7.6a)
s
2
s
2
=
ð
standard deviation
ð
s
2
ÞÞ
(7.6b)
5
s
3
5
s
3
=
ð
standard deviation
ð
s
3
ÞÞ
(7.6c)
(c) Compute the correlations between the scores
r
1
2
correlation of s
1
and s
2
(7.7a)
5
r
1
3
5
correlation of s
1
and s
3
(7.7b)
r
2
3
5
correlation of s
2
and s
3
(7.7c)
(d) Update the estimates of the
U
vectors
U
1
5
Y
t
ð
r
1
2
s
2
1
r
1
3
s
3
Þ
(7.8a)
U
2
5
Y
t
ð
r
1
2
s
1
1
r
2
3
s
3
Þ
(7.8b)
U
3
5
Y
t
ð
r
1
3
s
1
1
r
2
3
s
2
Þ
(7.8c)
The sequence of steps is then repeated, using the
U
vectors at the end of each iteration
as inputs into the next. This is repeated until the changes in
U
(or correlations r) do not
change within some desired tolerance level. It is then possible to test the significance of
the observed correlations using permutations.
USING PLS TO COMPARE PATTERNS OF COVARIANCE BETWEEN
BLOCKS ACROSS GROUPS
PLS is usually used to examine patterns of covariances between blocks of variables
measured in a single sample, but it can also be used to compare those covariances
between samples, as in a comparative analysis of geographic variation. Such comparisons
rely on the same logic (and methods) used in comparative analyses of regression equa-
tions or PCs because in all of these we are asking if the biologically corresponding vec-
tors point in the same direction. To answer that question, we can compute the angle
between comparable SAs, then test it statistically (using, for example, a bootstrapping
procedure). In a similar fashion, we can also compare SAs to PCs, asking whether the
major dimension of covariance between blocks is equivalent to the dominant dimension of
variation within blocks. For example, when our data come from an ontogenetic series, the
major dimension of variance within each block is likely to be the ontogenetic vector, and
the major dimension of the covariance between blocks may be the developmental covari-
ance between the two blocks. Comparing SAs to PCs can be especially useful for under-
standing causes of variance when PLS indicates a significant relationship between
morphology and some collection of environmental variables. That same relationship
between morphology and the environment may also explain the dominant axis of mor-
phological variation.
