When and are assumed known, as in BLUP, then the solutions for from the MME are BLUE and tests of hypotheses that use these solutions are best. Tests involving are unnecessary because when and have been assumed to be known, then the variation due to the random factors has already been assumed to be different from zero. The general linear hypothesis procedures are employed as in the fixed effects model. The null hypothesis is or where must be an estimable function and must have full row rank. is estimable if The test statistic is with degrees of freedom, and the test is where The degrees of freedom for F are and . If and are not known, then there is no best test because BLUE of is not possible.
Valid tests exist only under certain circumstances.
If estimates of and are used to construct the MME, then the solution for is not BLUE and the resulting tests are only approximate. If the estimate of is considered to be inappropriate, then a test of can be constructed by treating as a fixed factor, assuming that is estimable in the model with as fixed. That is, and Restrictions on Fixed Effects There may be functions of that are known and this knowledge should be incorporated into the estimation process. For example, in beef cattle, male calves of a particular breed are known to weigh 25 kg more than female calves of the same breed at 200 days of age. By incorporating a difference of 25 kg between the sexes in an analysis then all other estimates of fixed and random effects would be changed accordingly and also their variances.