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the relative sizes and positions of the modules by not superimposing the modules sepa-
rately after subdividing the subsets of landmarks. Rather than computing Procrustes dis-
tances, we can instead calculate Euclidean distances between each pair of individuals for
each module (implemented in an R script in the workbook accompanying this text). Given
those distance matrices, we can then calculate the correlation between each pair, which
tells us whether variation in shape shows the same pattern for both modules. For example,
if individuals who are most different from each other in the shape of one module are also
most different in the shape of the other module, and those who are most similar to each
other in the shape of one module are also most similar to each other in the shape of the
other module, the correlation between the modules will be high. Should individuals who
are most similar to each other in the shape of one module be the
least
similar to each other
in shape of the other, the correlation will be high but negative. The correlation will be near
zero when similarities among individuals in shape of one module do not predict similari-
ties in shape of another module.
An important difference between this method and the other two that were introduced
above is the treatment of correlations
within
modules. Using the present method, those
intramodular correlations are not assessed
only the correlations between modules enter
into the analysis. To overcome that limitation, each putative module can be subdivided
into two or more parts, and the correlations between the parts of a module can then be
assessed relative to the correlations between modules (
Zelditch et al., 2008, 2009
).
However, if those correlations are included in the analysis, the strength of the intramodu-
lar correlations is not taken into account when analyzing the correlations between mod-
ules. In effect, the hypothesized intramodular correlations are treated no differently than
the hypothesized intermodular correlations.
Once we have the correlation matrix we can analyze it by any of the methods con-
ventionally used for testing hypotheses of morphological integration and modularity
(
Cheverud, 1982; Cowley & Atchley, 1990; Cheverud, 1995; Herrera et al., 2002; Young &
Hallgr´msson, 2005
). One method for assessing a hypothesis of modularity is to predict
that the correlations between modules are zero, allowing the intramodular correlations to
be estimated from the data. The question is whether this model fits the data. We can com-
pare this to a model that fits the data perfectly and contains no fixed values. This model
is usually termed the “saturated model”, which can be represented as a graph in which
the nodes are the subsets of landmarks and the edges between them are the correlations
between the subsets and all the nodes are connected to all others (
Figure 12.16
).
Figure 12.16
shows the saturated model for 12 subsets because the hypothesized modules
were divided into two parts, whenever possible. Our objective is to reproduce the
observed correlation matrix using as few edges as possible. To assess the fit of the model
that includes only some of the edges relative to this saturated model, we use a measure
of the deviance (
D
) between the models (
Box, 1949; McCullagh & Nelder, 1989
).
D
is
2times
2
the log-likelihood ratio of the model being tested compared to the saturated model:
jθ
jθ
s
(12.9)
where
θ
o
are the fitted parameters of the model being tested and
θ
s
are the parameters of
the “saturated” model, the one shown in
Figure 12.16
.
D
is approximately distributed as a
chi-square with degrees of freedom equal to the difference in the number of parameters in
D
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