Properties of Model II--Type Analysis of Variance Tests, A: Optimum Nature of the $F$-Test for Model II in the Balanced Case

Abstract

A distribution analogous to the canonical distribution used in testing the general linear hypothesis is developed for Model II analysis of variance for balanced classifications. As in the case of Model I analysis of variance, this standard distribution exhibits the sums of squares going into the analysis of variance table. By use of the standard form it is also shown that (i) all exact $F$-tests used in testing hypotheses based on balanced multiple classifications determine uniformly most powerful (u.m.p.) similar regions although they are not likelihood ratio (L.R.) tests, but (ii) in the balanced one-way classification, for all practical purposes, the test is an L.R. test, and is u.m.p. invariant. An exact $F$-test exists when we have a sum of squares, $S_1$ distributed as $(k + \sigma^2_0)$ times a chi-square variate, where $k > 0$, independently of $S_2$, which is distributed as $k$ times a chi-square variate. The test is then to reject the hypothesis that $\sigma^2_0 = 0$ whenever $S_1/S_2$ is greater than some suitably chosen number, $c$. As a corollary to property (i) it is shown that "of all invariant tests of $\sigma^2_0 = 0$ against $\sigma^2_0 > 0$ whose power is a function of $\sigma^2_0/(k + \sigma^2_0)$ only, the test $S_1/S_2 > c$ is most powerful, providing $S_1$ and $S_2$, as defined above can be found."