Biology Reference
In-Depth Information
two partitions, one having
p
landmarks,
the other
m
2
p
,
the number of possible
partitions is
5
m
!
m
p
(12.6)
ð
m
p
Þ!
p
!
2
With few landmarks, exhaustive enumeration is feasible, but if there are more than a
few (e.g.
20) the number of alternatives becomes enormous. The requirement that mod-
ules be contiguous will result in far fewer than the total number of alternatives, but the
number could still be very large. Rather than use exhaustive enumeration, the alternatives
can be randomly sampled; Klingenberg recommends at least 10 000 permutations because
we are interested in the left tail of the distribution.
The permutation procedure is straightforward when the subsets are separately superim-
posed but it becomes more complex when the subsets are simultaneously superimposed.
When they are separately superimposed, the hypothesis of independence between the sub-
sets can be tested by randomly permuting the observations in the two sets of landmarks.
At each iteration of the procedure, the observations in one subset are randomly permuted
and the
RV
is calculated; its statistical significance is assessed by the proportion of the
cases in which the observed
RV
is equal to or higher than the observed one. The procedure
is more complex when the subsets are simultaneously superimposed because the test must
take into account the interdependence between partitions produced by the superimposi-
tion procedure. The procedure is thus modified to include a new Procrustes superimposi-
tion at each iteration, so the observations are randomly permuted in one of the two
subsets, then they are combined into a single configuration. It is not likely that they are
still optimally superimposed, so the superimposition is redone and the
RV
of this re-
superimposed configuration is compared to the observed one. This whole procedure
.
random permutations of one of subset followed by a re-superimposition of the data, is
done at each iteration.
To this point, we have talked about the analysis of just two modules, but the analysis is
not limited to a two-module case even though the
RV
measures the covariance between
two blocks of data.
Klingenberg (2009)
extended it to a multiblock case, introducing the
multiset
RV
M
coefficient, which is the average of all the pairwise
RV
coefficients.
X
X
k
2
1
k
2
RV
M
5
RV
ð
i
;
j
Þ
(12.7)
k
ð
k
Þ
2
1
i
1
j
i
1
5
5
1
where
k
is the number of subsets of landmarks and
RV
(
i,j
) is the
RV
coefficient for the sub-
sets
i
,
j
. Just like the pairwise
RV
, the multiset
RV
M
coefficient can be tested against the
null hypothesis that the modules are independent, providing a test of overall integration.
The test is done by computing
RV
M
for the hypothesized modules after which the land-
marks of all but one subset are randomly permuted (and re-superimposed if the analysis
is done using simultaneously superimposed landmarks).
