Improving Estimates of Fixed Effects in a Mixed Linear Model

Abstract

Best linear unbiased estimation, or equivalently maximum likelihood under normality, is the method most frequently used in animal breeding for estimation of fixed effects. James and Stein found that maximum likelihood is inadmissible under the mean squared error criterion and proposed a nonlinear, biased estimator that has smaller mean squared error than maximum likelihood throughout the parameter space in normal linear models with more than two uniquely estimable ftxed effects. In this paper, several biased estimators of ftxed effects in the mixed linear model are considered. Dispersion parameters are assumed to be known. An estimator that minimizes mean squared error in a certain class is derived. Because this estimator regresses the maximum likelihood estimates toward zero, an alternative estimator that "shrinks" these estimates to their mean value is also considered. The two estimators require knowledge of the true values of the ftxed effects, so approximations to these are presented, including an extension of the James and Stein estimator to the mixed model. These estimators were compared with maximum likelihood in a simulation study involving a balanced group (ftxed) plus sire (random) model. The James and Stein and "minimum mean squared error" estimators gave estimates of group effects with slightly smaller mean squared error than maximum likelihood. Improvement was mini-