Biology Reference
In-Depth Information
FIGURE 12.15
Logic of testing for low
RV
. Two developmental modules are
shown, each enclosed in a rectangle, with
the boundary between them represented by
a black line. There are extensive interac-
tions within each module, represented by
the arrows connecting the landmarks. Few
interactions occur between modules. The
covariance between the two modules will
be lowest if the two are separated along the
black line, due to the few interactions
between landmarks across that line. In con-
trast, the covariance between the two mod-
ules will be higher if the two are separated
along the dotted gray line, due to the many
interactions between the landmarks across
that line.
One method for testing the hypothesis of modularity is to assess the covariance between
the two modules relative to the covariance within them (
Klingenberg, 2009
). If the hypoth-
esized boundary between the modules is correctly positioned, the covariance between the
two blocks should be lower than the covariance obtained by any alternative partitioning,
subject to the constraint that the alternative also contains two blocks having the same
number of landmarks as in the proposed modules. The logic of the test is depicted in
Figure 12.15
(after
Klingenberg, 2008
). The black line separates the two actual modules,
one containing five landmarks, the other six. The covariances within each of the two sub-
sets are high, as evident from the many arrows depicting interactions between them.
Interactions between the two modules are few, so the covariance between them is low.
The gray line separates the two subsets of landmarks that are hypothesized to be modules,
one containing five landmarks, the other six. But the line separates landmarks belonging
to the same actual modules. As a result, the covariances between the hypothesized mod-
ules will be high, far higher than the covariance that would be obtained by partitioning
the landmarks along the black line. Because the covariance between modules should be
lower than the covariance between randomly partitioned subsets of landmarks, the testing
procedure determines whether the covariance between the hypothesized modules is signif-
icantly
lower
than expected by chance. This method, like the other two, has no name, so
we will call it the “Minimum intermodular covariance method”.
The second method tests a hypothesis of modularity by producing the covariance
matrix predicted by the hypothesis and assessing the goodness of fit (
Marquez, 2008
). The
covariance matrix predicted by the model is estimated by making the modules statistically
independent of each other
they are placed into orthogonal subspaces. These intermodu-
lar covariances are fixed but the within-module covariances are estimated from the data.
Having produced the covariance matrix predicted by the model, it can then be compared
to the observed covariance matrix. The null hypothesis is that the difference between the
observed and expected covariance matrices is no greater than expected by chance. This is
tested by comparing the difference between observed and expected matrices to the range
of values for the difference that could be obtained when the null hypothesis is true. The
