Biology Reference
In-Depth Information
number of zeros within each model because
γ
is linearly related to the number of zeros.
γ
The residual from that regression,
*, can then be compared across models. When the
model predicts that the covariances are zero and the observed values are indeed low,
γ
0; conversely, when the model predicts that the covariances are zero but they are actu-
ally high,
*
,
0. Because this scaling step uses a regression, the method benefits from fit-
ting
many
models to the data.
When testing model(s), the null hypothesis is that the difference between the observed
and expected covariance matrices is no greater than expected by chance. A low probability
means that the model does
not
fit the data. The test is done by comparing the observed
value of
γ
*
.
* to the range of values that could be obtained when the null hypothesis is true.
To obtain the distribution of
γ
under the null hypothesis, the modeled covariance matrix
and sample size are used to parameterize a Wishart distribution, which is the distribution
of covariance matrices of a multivariate normal population (
Wishart, 1928
). The value of
γ
γ
* is calculated between each randomly drawn matrix and the expected matrix. The proba-
bility that the model differs from the data by no more than expected by chance is calcu-
lated from the proportion of cases in which the value of
* (computed by comparing the
model to randomly drawn matrices) is larger than the observed one. Thus, a p-value of 0.9
means that in 90% of the cases in which the model is compared to random matrices,
γ
γ
*is
larger than it is when the model is compared to the data.
When comparing multiple models, the best-fitting model is the one with the lowest
γ
*.
Multiple models, however, might be nearly equal in
* and all might fit well. All the
expected covariance matrices might deviate little from the observed one. Then, the prob-
lem is to decide which model fits best. In principle, this can be decided by which has the
lowest
γ
γ
γ
* or in its rank because both
depend on the sampling of the observed covariance matrix. Before deciding that one of
the models fits best, we want to be confident in the ranks of the models. To that end, we
can resample the covariance matrices and rerun the analyses for each sample, calculating
the number of runs in which the ranking is the same as we obtained for the observed
covariance matrix. As implemented in Mint, the resampling is done by jackknifing the
data, leaving out a proportion of the sample, refitting the model to the data, recalculating
γ
*, but we may not be entirely confident either in
* and the ranks of the models at each iteration.
Distance-Matrix Method
This method for analyzing modularity using correlations between pairwise Procrustes
distance matrices was introduced by Monteiro and colleagues (
Monteiro et al., 2005,
Monteiro and Nogueira, 2009
). As briefly outlined above, this method produces a correla-
tion matrix from the matrix correlations between pairwise Procrustes distance matrices.
Those Procrustes distance matrices preserve the information about the structure of varia-
tion within each module. If the analysis is done using Procrustes distances calculated sepa-
rately for each module, the only information retained is the correlations between the
shapes; any information about the relationships between the relative sizes and positions of
the modules within the whole is disregarded. That is the procedure implemented in
Coriandis (
Marquez and Knowles, 2007
). But it is possible to retain the information about
