Biology Reference
In-Depth Information
fitted to the correlation matrix, the models are termed “covariance graphical models” and
the search for models that reproduce the correlation matrix is termed “covariance model
selection”. Using the inverse correlation matrix has the advantage that it allows us to test
for the conditional evolvability of modules; the use of covariance graphical models for
studies of morphological integration and modularity was recommended by
Magwene
(2001)
. But, for purposes of comparison with the other two methods, when we apply this
method to our data, below, we will fit the models to the correlation matrix and do an
exploratory analysis using covariance model selection.
An alternative method for testing hypotheses of integration and modularity is to
construct the expected correlation matrix by predicting correlations of either zero or one
between traits according to whether they belong to different or the same module (
Wagner,
1988; Kingsolver & Wiernasz, 1991; Cheverud, 1995; Hallgr´msson et al., 2004; Young, 2004
).
We do not anticipate that correlations will typically be either zero or one, but the compari-
son between the observed and expected matrices is done by computing the matrix correla-
tion between them (which is tested by a Mantel test). The matrix correlation will be high
if the matrices are proportional to each other so it is the pattern of relatively high versus
relatively low correlations that is tested. In the case of the Front/Back model, we would
predict the correlation matrix shown in
Table 12.4
. When comparing the hypothesized to
observed matrices by the Mantel test; we would randomly permute the rows and columns
in one matrix and compute the correlation between matrices at each for each permutation.
TABLE 12.4
Correlation Matrix Among Six Partitions of the Mandible Predicted by the Front/Back
Hypothesis
IncD IncP Molar RamusD RamusP CorD CorP CondD CondP Cond AngD AngP
IncD
1
1
1
1
0
0
0
0
0
0
0
0
IndP
1
1
1
1
0
0
0
0
0
0
0
0
Molar
1
1
1
1
0
0
0
0
0
0
0
0
RamusD 1
1
1
1
0
0
0
0
0
0
0
0
RamusP
0
0
0
0
1
1
1
1
1
1
1
1
CorD
0
0
0
0
1
1
1
1
1
1
1
1
CorP
0
0
0
0
1
1
1
1
1
1
1
1
CondD
0
0
0
0
1
1
1
1
1
1
1
1
CondP
0
0
0
0
1
1
1
1
1
1
1
1
Cond
0
0
0
0
1
1
1
1
1
1
1
1
AngD
0
0
0
0
1
1
1
1
1
1
1
1
AngP
0
0
0
0
1
1
1
1
1
1
1
1
IncD: distal incisor partition; IncP: proximal incisor partition; Molar: molar incisor partition; RamusD: distal horizontal ramus
partition; RamusP: proximal horizontal ramus partition; CorD: istal coronoid process partition; CorP: proximal coronoid process
partition; CondD: distal condyloid process partition; CondP: proximal condyloid process parition; Cond: condyle; AngD: distal
angular process partition; AngP: proximal partition of the angular process. See
Figure 12.20
.
