Biology Reference
In-Depth Information
distinction lies in how the test will use information about the connections between the
subsets (e.g. the relative sizes of the partitions and their position relative to each other). In
terms of the details of the method, the distinction lies in whether each module is superim-
posed separately or the entire configuration is superimposed only once. When each mod-
ule is superimposed separately, information about the relative sizes and positioning of the
modules is removed from the data. What matters is the covariance between shapes. When
the variation in one shape is associated with the variation in another, that will generate
covariance between the modules. In contrast, when the subsets are regarded as parts
of a whole, any variation in the relationship between that generates covariances in
relative sizes and positions is retained in the data. As Klingenberg points out, there is no
right or wrong decision about this. In the case of the mandible, we might base the
decision on considerations of function. Because the mandible is functionally a lever, it
makes sense to treat these two parts of the lever as two parts of a whole. Alternatively, we
could decide that function is immaterial and disregard the relationship between the two
parts, testing the hypothesis that dividing the mandible into front and back produces a
lower covariance between the two shapes than any alternative division of the mandible
into two parts, having the same number of landmarks within them as our hypothesized
modules.
The second decision is whether modules must be spatially continuous. What spatially
continuous means, in this context, is that a module comprises all the landmarks that are
adjacent to each other except for those on the boundaries between modules. More pre-
cisely, the definition of spatial contiguity for partitions of landmarks uses the graph theo-
retic concepts of node and edge introduced above in the context of a structural concept of
modularity. When defining contiguity of landmarks, the nodes on the graph represent the
landmarks and the edges connect them; a set of landmarks is spatially contiguous if every
landmark within the set is connected to every other, either directly (by an edge between
that pair of landmarks) or indirectly through the other landmarks in the set. It is thus pos-
sible to reach every landmark within the set by moving along the edges. This decision
about continuity also has no right or wrong answer. It is reasonable to anticipate that
developmental interactions act over spatially continuous regions, but those regions may
not remain continuous through the whole course of development
they might be inter-
rupted by morphogenetic movements, outgrowth or cell death. Also, in cases like the man-
dible, there is a landmark within the tooth-bearing region (the one on the masseteric fossa)
that is a muscle insertion site so it might plausibly be regarded as part of the muscle-
bearing region. The reason why this decision about continuity matters to the method is
that the
RV
for the hypothesis will be compared only to the random, continuous (also
called “contiguous”) modules if continuity is a requirement for modules. For purposes
of the example, we will restrict the analysis to contiguous modules.
When comparing the
RV
to all (or a subset) of possible alternatives, the comparisons
are restricted to partitions that have the same number of landmarks as in the hypothesized
modules. For example, if a hypothesis proposes that there are two modules, one having 10
landmarks and the other 15, all the alternative hypotheses will also comprise two modules,
one having 10 landmarks and the other 15. If there are few landmarks in the configuration,
it is possible to compare the
RV
for the data to all the possible alternatives. In the case of
