Optimum Invariant Tests in Unbalanced Variance Components Models

Abstract

In this paper the problem of testing the hypothesis $\Delta \leqq \Delta_0$ against $\Delta > \Delta_0$, where $\Delta$ is the ratio of variances in the one-way classification of the analysis of variance with variance components, is treated. The model is not restricted to equal class frequencies. It is found that the most powerful invariant test against an alternative $\Delta_1$ depends upon $\Delta_1$, but has the property of maximising the minimum power over the set of alternatives with $\Delta \geqq \Delta_1$. The test statistic is distributed like a ratio of linear combinations of independent chi-square distributed random variables. It is shown that a statistic used by Wald [6] to derive a confidence interval for $\Delta$ gives a test that is almost equal to the most powerful invariant tests against large alternatives $\Delta_1$. For the case $\Delta_0 = 0$ it is equal to the usual test in the fixed-effects model. In the balanced case the test reduces to the usual $F$-test which Herbach [2] has proved to be both uniformly most powerful invariant and uniformly most powerful unbiased.