Abstract
SUMMARY In the three stage nested design with random effects, let ?A2 be the variance of factor A (the top stage). It is often desirable to test the hypothesis HA: GA2 = 0. In the unbalanced case, the conventional F-test for HA does not in general have the expected null distribution, and the expected mean squares are not in general equal under HA . Tietjen and Moore [1968] proposed that a Satterthwaite-like procedur-e be used to construct a denominator (and D.F.) which would have, under HA , 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 investigatiing 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.
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