Biology Reference
In-Depth Information
method allows many models to be fit to the data, with the best-fitting model being the one
that deviates least from the data, taking into account the number of parameters fixed by
the hypothesis. The number that is fixed is important because the free parameters (the
intramodular covariances) are estimated from the data; models that have relatively few
fixed parameters will always fit well. We will call this method the “Minimum deviance
method”.
The third method differs from the other two in two major respects. First, it produces a
correlation matrix rather than a covariance matrix and second, it works with distance
matrices rather than coordinates (
Monteiro et al., 2005; Monteiro and Nogueira, 2009
). The
correlations are obtained by subdividing the data into modules, then calculating the pair-
wise Procrustes distances between all individuals for each module. Then the correlation
between the distance matrices is estimated. For example, given the hypothesized two mod-
ules of the Front/Back model, we would divide the coordinates into those two subsets
and calculate the pairwise Procrustes distances between all individuals for each subset of
landmarks and then compute the matrix correlation between those two distance matrices.
In this simple case of only two modules, we could test the hypothesis that they are inde-
pendent of each other by the Mantel test. For more complex hypotheses, we could use any
of the methods conventionally used in studies of morphological integration to evaluate the
fit of the hypotheses to the correlation (or inverse) correlation matrix. We could also use
exploratory methods to select
the best-fitting model. We will call
this method the
“Distance-matrix method”.
After discussing each method in more detail, we will apply all three of them to evaluate
four hypotheses of mandibular modularity.
Minimum Intermodular Covariance Method
The test for modularity based on estimating the covariances between modules relative
to those within modules was devised by Klingenberg and implemented in his software,
MorphoJ (
Klingenberg, 2011
). The test statistic is Escoufier's
RV
coefficient (
Escoufier,
1973
). This was introduced in Chapter 7 (Partial Least Squares), but repeated here for
convenience:
the
RV
is a multivariate extension of
the ordinary univariate squared
correlation:
R
12
R
t
12
Þ
trace
ð
RV
p
trace
(12.5)
5
R
1
R
t
1
Þ
R
2
R
t
2
Þ
ð
trace
ð
The numerator is the summed squared covariances between the two sets of variables
and the denominator is the square root of the product of the summed squared variances
within each block.
RV
ranges from 0 (no covariance) to 1 (complete covariance). When we
introduced the
RV
in Chapter 7, we used it to test the hypothesis that the two blocks of
landmarks covary so we tested the null hypothesis that the observed
RV
is no higher than
expected by chance. But, as discussed above, in the context of a study of modularity, the
expectation is that the
RV
will be
lower
than expected by chance.
The details of the test depend on two decisions that you make. The first is whether
hypothesized modules should be treated as separate shapes or as parts of a whole. The
