This procedure computes the eigenvalues and eigenvectors of the matrix GP-1, where G is the additive genetic and P is the phenotypic covariance matrix. These eigenvalues and eigenvectors provide useful information about the inheritance of shape, such as the range of heritabilities for all possible shape variables and the shape variables that respond most or least easily to selection (Klingenberg and Leamy 2001).
In the multivariate breeders' equation, Δμ = GP-1s, the matrix GP-1 plays a central role: it transforms the selection differential s into the selection response Dm. Both the selection differential and selection response are shape differences (or can be interpreted as such): the difference in shape means before and after selection in the parental generation and the differences between the shape means in the parental and offspring generations. This transformation includes the effects of genetic constraints, such as deflections toward 'lines of least resistance' and similar phenomena.
The matrix GP-1 can be viewed as a multivariate analogue to the univariate heritability (h2 = VA / VP, where VA is the additive genetic variance and VP is the phenotypic variance; Roff 2000). The univariate heritability has a similar role in the transformation from selection differential to selection response (e.g. Falconer and Mackay 1996).
The parallel with univariate heritability should normally be made with the utmost caution in the inherently multivariate context of geometric morphometrics, because the transformation from selection differential to selection response usually involves a change of direction (e.g. Klingenberg and Monteiro 2005). A special case concerns selection differentials that are eigenvectors of GP-1. For any eigenvector a of the matrix GP-1, the equality GP-1a = λa holds, where λ is a scalar, the eigenvalue associated with a. In this special case, therefore, the selection response is simply a scaled version of the selection differential, and the univariate breeders equation holds for the shape variable defined by the eigenvector a. The eigenvalues of GP-1 therefore are the heritabilities for the set of shape variables that correspond to the directions of the eigenvectors.
The shape variables that produce the minimal and maximal selection response are also eigenvectors of GP-1. Accordingly, the lowest and highest eigenvalues are also the lowest and highest heritabilities for any shape variables.
The method is implemented in MorphoJ as described by Klingenberg and Leamy (2001). Because the phenotypic covariance matrix is computed from Procrustes coordinates, it is not of full rank, and a generalized inverse must be used. Accordingly, eigenvectors and eigenvalues are computed for the matrix GP-.
To run the analysis of GP-, the user needs to provide estimates of the additive genetic and phenotypic covariance matrices. These estimates need to be obtained outside of MorphoJ, for instance, using the 'animal model' based on Procrustes coordinates or principal component scores previously exported from MorphoJ. Such estimated covariance matrices can be imported (Import Covariance Matrix in the File menu) and, if needed, converted from the coordinate system of PC scores back to the original landmark coordinates (Convert Covariance Matrix to Landmark Coordinates in the Variation menu).
Once the genetic and phenotypic covariance matrices are available in the MorphoJ project, the user can request the analysis by selecting Analysis of GP(inv) from the Genetics menu. A dialog box like the following will appear:
The dialog box contains a text field for entering a name for the analysis, which will be shown in the Project Tree. Below that, there are two drop-down menus for selecting the genetic and phenotypic covariance matrices.
To start the analysis, click Execute. To abort the procedure, click Cancel.
The output of this procedure in the Graphics tab contains two panels. One shows the shape changes that correspond to the eigenvectors of the matrix GP-. The other panel shows a bar chart with the eigenvalues of this matrix.
The text output of the procedure in the Results tab first indicates the name of the analysis and the names of the genetic and phenotypic covariance matrices. Then the eigenvalues of the matrix GP- are listed, and finally, a table provides the coefficients of the eigenvectors.
Falconer, D. S., and T. F. C. Mackay. 1996. Introduction to quantitative genetics. Longman, Essex.
Klingenberg, C. P., and L. J. Leamy. 2001. Quantitative genetics of geometric shape in the mouse mandible. Evolution 55:2342–2352.
Klingenberg, C. P., and L. R. Monteiro. 2005. Distances and directions in multidimensional shape spaces: implications for morphometric applications. Syst. Biol. 54:678–688.
Roff, D. A. 2000. The evolution of the G matrix: selection or drift? Heredity 84:135–142.