Technical note: computing tests of fixed effects in a restricted class of mixed models

Inferences about fixed effects in mixed linear models are important in a variety of animal science studies. The statistical theory for making such inferences is well known, and if the variance components are known up to a proportionality constant, then optimal exact tests can be performed. Computing the test statistics, however, can still be problematic when the random effects have many levels. In practice, approximate tests that are easily computed but less efficient are usually employed. This article describes reduction in error sum of squares procedures for performing the exact test and for computing associated confidence intervals. By taking advantage of iterative algorithms for solving Henderson's mixed-model equations, the tests can be performed without inverting the covariance matrix or computing a generalized inverse of the mixed-model coefficient matrix. The procedures are illustrated on an animal model that has three random effects, two with 1,372 levels and one with 450 levels.