Modification of Estimates of Parameters in the Construction of Genetic Selection Indices ('Bending')

A method, termed 'bending', is proposed for the modification of the $estimates of genetic (\mathbf{\hat G}) and phenotypic (\mathbf{\hat P}) covariance matrices,$ that are used in the construction of genetic selection indices for two or more traits. If P and G are estimated from the between- and within-class covariance matrices, B and W respectively, in a one-way multivariate analysis of variance, then the method consists of contracting all the eigenvalues of the matrix product W-1B towards their mean but with the corresponding eigenvectors unchanged. The usefulness of the modification procedure is investigated by Monte Carlo simulation and Taylor series approximation methods. In general, the modification procedure improves the achieved response to selection, with the amount of improvement dependent on the (unknown) population parameters and the size of sample for estimation. In practice, there is some difficulty in selecting the value of the 'bending' factor; some very simple methods of doing this are proposed. If some of the parameter estimates are known to be defective (outside their valid limits), a simple and effective method for the selection of the 'bending' factor is to contract the eigenvalues so that they are all nonnegative.