Matrix correlation is a widespread procedure for making a quick overall comparison of two matrices. In geometric morphometrics, these matrices are usually covariance matrices (exclusively so in MorphoJ).
The two covariance matrices for which a matrix correlation is computed should concern corresponding sets of landmarks and normally should be derived from landmark configurations superimposed in the same Procrustes fit (to avoid differences from different overall orientation).
To test the association statistically, it is usual to use a matrix permutation test (e.g. Cheverud et al. 1989). This test evaluates the matrix correlation against the null hypothesis that there is no relationship whatsoever between the matrices. Note that a significant matrix correlation on its own does not imply that the matrices are proportional; it only tells that they are not totally dissimilar.
In geometric morphometrics, some adjustments to the computation of matrix correlation and to the procedure for the matrix permutation test are necessary to take into account the fact that the x, y and possibly z coordinates of landmarks are interdependent, as well as the possible object symmetry of the configuration (Klingenberg and McIntyre 1998; Klingenberg et al. 2002). MorphoJ makes these adjustments automatically.
The origin from landmark coordinates imposes a special structure on the covariance matrices used in geometric morphometrics, which particularly affect the way matrix correlations are computed. This results in some differences to the use of matrix correlation in different contexts.
A first difference is that geometric morphometrics consistently uses covariance matrices, and not correlation matrices. Because the computation of correlation matrices involves a differen scaling of each variable, this would result in a different scaling of each landmark coordinate and a distortion of the geometry of the landmark configuration. Other differences arise from this use of covariance matrices.
Analyses on matrix correlation between correlation matrices (e.g. Cheverud et al. 1989) have excluded the diagonal entries of the matrices because they are always fixed at the value 1.0, and therefore would tend to inflate the matrix correlation. Because matrix correlations in geometric morphometrics are computed from covariance matrices, the diagonal elements do carry real information, and there is a good case for including them in the analysis (Klingenberg & McIntyre 1998).
Because the coordinates of each landmark are clearly associated, the decision to include or exclude diagonal elements of a covariance matrix concerns the entire blocks of the variances and covariances of the coordinates at each landmark (Klingenberg et al. 2002). If these blocks are removed, the comparison of covariance matrices is limited to the covariances between different landmarks.
The choice of whether to include or exclude the diagonal blocks of the covariance matrix needs to made from case to case. The two ways to compute matrix correlations include different information and may therefore lead to different results. For instance, because variances and covariances at a given landmark are often higher than the covariances between landmarks, the matrix correlations including the diagonal blocks are often higher than those excluding them. Investigators may also decide to use both options and to compare the results.
The answer is usually more clear-cut for the question of whether it is more sensible to carry out the premutation test by landmarks or by separate coordinates (Klingenberg & McIntyre 1998). Because the coordinates of each landmark clearly are not independent of each other, it normally makes sense to maintain these associations of coordinates during the permutation procedure, and to permute the landmarks and not the individual coordinates in the covariance matrix.
Object symmetry can be a major complicating factor for computing matrix correlations, praticularly for the comparisons between covariance matrices for the symmetric and asymmetry components (Klingenberg et al. 2002).
The set of landmarks: For computing matrix correlations between the symmetry and asymmetry components of some configuration, only the paired landmarks are used, whereas the landmarks of the midline or median plane are ignored. The landmarks of the midline are limited to move within the median axis of plane for the symmetric component, and in the direction perpendicular to the median axis or plane for the asymmetry component (Klingenberg et al. 2002). Therefore, the covariances between the symmetry and asymmetry components is bound to be zero for these landmarks, and they can be excluded from the analysis without affecting the results.
Moreover, for the paired landmarks, only one landmark of each pair is included in comparisons of the symmetry and asymmetry components.
The matrix permutation test: The permutation test for the matrix correlation between covariance matrices for the symmetry and asymmetry components of variation in a landmark configuration uses a one-sided test for the magnitude of the absolute matrix correlation.
To compute a matrix correlation and the associated test, select Matrix Correlation from the Variation menu. A dialog box like the following will appear.
The text field at the top of the dialog box is for entering a name for the analysis that will appear in the Project Tree.
The two drop-down menus are for selecting the two covariance matrices. Users should select the first matrix first, because the first drop-down menu lists all the covariance matrices in the Project Tree. The second drop-down menu lists only those covariance matrices that are compatible with the covariance matrix selected as Matrix 1 (same number of landmarks and dimensions). If no such covariance matrices are available in the project, a notice will appear in the dialog box.
Below the drop-down menus, there is a button for selecting whether the diagonal blocks (variances and covariances of coordinates at each landmark) are to be included in the analysis.
For the matrix permutation test, the user can specify the number of permutation rounds and whether landmarks or individual coordinates are to be permuted. (Normally, permuting landmarks is preferable; see above.)
To start the analysis, click Execute. To abort the procedure, click Cancel.
The only graphical output from this procedure is a scatter plot of the corresponding elements of the two covariance matrices.
The text output provides the names of the two covariance matrices being compared and the matrix correlation between them, including an indication of whether the diagonal blocks were included. The information about the matrix permutation test includes the number of permutations, whether the landmarks or coordinates wre permuted, and the P-value achieved in the test.
Cheverud, J. M., G. P. Wagner, and M. M. Dow. 1989. Methods for the comparative analysis of variation patterns. Systematic Zoology 38:201–213.
Klingenberg, C. P., and G. S. McIntyre. 1998. Geometric morphometrics of developmental instability: analyzing patterns of fluctuating asymmetry with Procrustes methods. Evolution 52:1363–1375.
Klingenberg, C. P., M. Barluenga, and A. Meyer. 2002. Shape analysis of symmetric structures: quantifying variation among individuals and asymmetry. Evolution 56:1909–1920.