Biology Reference
In-Depth Information
measuring the correlation between them. If that correlation is statistically significant it
would support the hypothesis of a common buffering mechanism. The second approach is
less straightforward because it requires comparing covariance structures and methods for
comparing covariance structures are a matter of some contention. However, the most com-
monly used method is to estimate the correlation between two matrices and to test its sta-
tistical significance by a Mantel test (
Mantel, 1967
). The matrix correlation is a standard
Pearson product-moment correlation calculated over the corresponding entries in the two
covariance matrices. Corresponding entries would be, for example, the covariance between
the
x
-coordinate of the first two landmarks in the two matrices
that covariance in one
matrix corresponds to that same covariance in the other matrix. The matrix correlation can
be calculated by hand if the matrices are small; that involves arranging the two matrices in
two column vectors (omitting the redundant elements and, if desired, the variances along
the diagonal). The two matrices should then be matched up, with corresponding entries
on each row. Once the matrices are turned into column vectors, the correlation is calcu-
lated between the two columns.
The Mantel test is the most common test of the null hypothesis that the two matrices
are no more similar than expected by chance. To determine whether the correlation is sig-
nificant, the elements in one matrix are randomly permuted and the correlation is mea-
sured between the permuted matrix and the other one, at each iteration, repeating this
procedure many times. The correlations obtained from these permutations provide the dis-
tribution of the correlations between randomly related matrices. Given this distribution,
the number of correlations that equals or exceeds the observed one can be counted.
Usually, the observed correlation is included in the count, so if you do 100 permutations
and obtain a p-value of 0.01, one value obtained by 99 random permutations plus the
observed correlation are equal to or greater than the observed one. The test needs some
modification to be used for geometric data because the standard version of the test would
permute
x
-coordinates independently of
y
-coordinates. Adapted for geometric data, the
Mantel test permutes landmarks as units (
Klingenberg and McIntyre, 1998
). The test has
also been modified to allow for comparisons between the covariance matrix of object FA
and individual variation, which, if you recall from the previous section on FA, are in dif-
ferent subspaces. The test is done by omitting the midline landmarks (and one whole side,
which is redundant), thereby limiting the analysis to the paired landmarks on a single side
(
Klingenberg et al., 2002
).
Examples: Comparing Phenotypic Variation to FA for Prairie Deer Mouse
Mandibular and Cranial Shape
We first test the hypothesis that an individual's deviation from bilateral symmetry
is correlated with its deviation from the mean shape, using the Procrustes distance from
the bilaterally symmetric shape as our measure of FA. Similarly, we use that individual's
Procrustes distance from the mean shape as our measure of its deviation from the mean.
In the case of the mandible, the correlation is a very weak 0.09, which is not statistically
significant (
p
0.38). For the cranial data, the correlation between these two measures is
a weak 0.232, which is nonetheless statistically significant (
p
5
0.012). We then take the
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