Exact and Approximate Tests for Unbalanced Random Effects Designs

In the three stage nested design with random effects, let $\sigma_A^2$ be the variance of factor A (the top stage). It is often desirable to test the hypothesis H$_A$ : $\sigma_A^2$ = 0. In the unbalanced case, the conventional F-test for H$_A$ does not in general have the expected null distribution, and the expected mean squares are not in general equal under H$_A$. Tietjen and Moore [1968] proposed that a Satterthwaite-like procedure be used to construct a denominator (and D.F.) which would have, under H$_A$, the same expected mean square as the numerator. It was pointed out by W. H. Kruskal, however, that such a procedure had no justification since the mean squares were not independent and did not have chi-square distributions. This paper is an attempt to revisit the question by investigating the properties of this approximation and those of the conventional F-test. It is shown that the conventional F-test does considerably better as an approximation under imbalance than does the Satterthwaite test.