Improving Estimates of Fixed Effects in a Mixed Linear Model

¹ Department of Animal Sciences, University of Illinois, Urbana 61801

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 fixed effects. In this paper, several biased estimators of fixed 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 fixed 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 (fixed) 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 minimal when the true group effects were far from zero. However, shrinkage of group effects toward their mean value substantially reduced mean squared error of group estimates. Genetic trend (the regression of group effects on time) was considerably underestimated using this type of shrinkage. Care must be exercised when these "improved" statistics are used for estimating some functions of fixed effects.

Key Words: mixed models • fixed effects • biased estimation

Submitted on November 29, 1990
Accepted on March 21, 1991

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