To compare covariance matrices, or any other objects, an effective approach is to focus on those aspects that show the most marked differences between them. This procedure contrasts two morphometric covariance matrices by focusing on those shape variables that show the greatest differences in the amounts of variation. In other words, the analysis will identify those shape features that are much more variable in one covariance matrix than in the other.
The two covariance matrices compared by this method should concern corresponding landmarks and normally should be computed from landmark configurations that were superimposed in the same Procrustes fit (to avoid differences from different overall orientation).
This analysis is of an exploratory nature and not intended as a formal test, although it is related to Roy's greatest root, one of the standard test statistics in multivariate analysis. Some caution is advised, in particular, because the shape features with extreme ratios of variance in the two covariances may account for only minor amounts of covariance in both covariance matrices (e.g. 7 times nearly nothing may still be nearly nothing...).
This procedure contrasts two covariance matrices S1 and S2 by finding the eigenvectors of the matrix S1S-2, where S-2 is the generalized inverse of S2. The eigenvectors of this matrix that are associated with the minimal and maximal eigenvalues are those linear combinations that have most extreme ratios of variances in the two covariance matrices. Therefore, they identify the shape features that are much more variable in one sample than in the other.
To start the analysis, select Contrast Covariance Matrices 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, which will appear in the Project Tree.
The two drop-down menus are for selecting the two covariance matrices. 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).
To start the analysis, click Execute. To abort the procedure, click Cancel.
The analysis produces two panels with graphs. One is a graph showing the shape changes that correspond to the eigenvectors of the matrix S1S-2. The other is a bar chart of the eigenvalues of this matrix.
The text output in the Results window provides a range of statistics related to the eigenvalues of the matrix S1S-2. These include the eigenvalue itself, the amount of variance (in units of squared Procrustes distance) and percentage of the total variance in Matrix 1 for which the shape variable for this contrast accounts, as well as the amount and percentage of variation for which it accounts in Matrix 2. Note that these variances and percentages need not add up to the total variance in either covariance matrix or 100 %, respectively.
A table of the coefficients of the eigenvectors of the matrix S1S-2 is also provided.