Best Linear Unbiased Prediction Using Relationship Matrices Derived from Selected Base Populations

Abstract

Most applications of mixed model equations for computing best linear unbiased prediction of additive genetic value invoke a numerator relationship matrix derived from a base population of unselected animals. In contrast, there could be situations in which one or more base populations have been selected with unknown means and the remaining animals in the sample are descendants of these selected animals. The numerator relationship matrix is altered from what it would have been had there been no selection, and it can be singular. Methods for using such matrices in mixed model equations are described. The resulting predictions of breeding values must be interpreted relative to breeding values of base populations. Means of selected base populations are treated as group constants in the solution. Another possibility is that the type of selection on base animals is such that no distributional property can be invoked. Consequently, their breeding values are regarded as fixed. Their descendants, however, have random breeding values with numerator relationships altered. Then the mixed model equations are derived from an incidence matrix in which the ancestors’ fixed breeding values appear in the model for their descendants.