Variances, Covariances, Correlations

Genetic and PE matrices

Much time was spent looking up genetic parameters, either variances and covariances or correlations. Estimates for many combinations of traits simply do not exist in the literature, especially health traits. With 220 traits, there are 24,090 possible covariances. Permanent environmental covariances and variances were made equal to one half the value of the genetic values. Several people helped to obtain values for the G matrix including Jarmila Bohmanova, Sven Konig, Jalal Fatehi, and LRS. Values for lactose, MUN, and Omega 3 fatty acids were made equal or proportional to those for other production traits.

With covariances and correlations coming from several dozen sources, the resulting matrices were going to be non positive definite, meaning they would be unsuitable for use in genetic evaluation or in a simulation program. The Schaeffer PD modification procedure was used to make the matrices positive definite.

First, the eigenvalues (D), and eigenvectors (U), are calculated, where UU'=I. There were 21 negative eigenvalues in D out of 220. Let X be the value of the smallest positive eigenvalue, and Z_{big} be the largest negative eigenvalue. For each negative eigenvalue compute a new value as X(1 - (Z_{i}*Z_{i})/(Z_{big}*Z_{big} + 1)). Now all eigenvalues are positive, they continue to get smaller than the last positive eigenvalue, and they are all greater than zero. Lastly, reform G_{new} using U(d)U' where (d) is the new set of eigenvalues. Then G_{new} is positive definite.

The variances in G_{new} are very similar to the old values, and the covariances are only slightly modified in most cases. G can be decomposed into the product of two triangular matrices for the simulation. The same procedure was applied to the PE matrix. Not all traits have PE effects associated with them, and were ignored for those traits.

Residual Matrix

For production traits, the residual variances were matrices of order 4, for the 4 traits of milk, fat, protein, and SCS. All other residual covariances were assumed to be zero. Many of the estimates found in the literature were not different from zero.

The availability of new values for these matrices should be incorporated from time to time. A database should be created so that values can be added easily, for the existing traits and new traits, and so that the covariance matrix for a block of traits can be retrieved.