Estimation of denominator degrees of freedom of F-distributions for assessing Wald statistics for fixed-effect factors in unbalanced mixed models

Tests for fixed-effect factors in unbalanced mixed models have previously u sed t-tests on a contrast-by-contrast basis or Wald statistics without a un iversally accepted method of calculating the denominator degrees of freedom . This situation has arisen because the variances of different contrasts ar e differently weighted sums of the variance components with associated degr ees of freedom that are not necessarily equal. A simultaneous F-test for di fferences between all levels of a fixed-effect factor can be derived by for ming new contrasts, by rotation of the original contrasts, with variances t hat are close to being the same weighted sum of variance components. The as sociated degrees of freedom for these new contrasts are nearly equal. A sma ll simulation study shows the appropriateness of a X-2 approximation to the distribution of the weighted sums of variance components. Three simple exa mples are used to demonstrate the effects of rotation. The last of these ex amples is also used to compare the proposed simultaneous F-test with the di stribution of the Wald statistic obtained by numerical simulation. The meth od of rotations is then applied to data on the range size of mountain hares (Lepus timidus) to assess the evidence for a two-way interaction between s eason and habitat.