Properties of Model II--Type Analysis of Variance Tests, A: Optimum Nature of the $F$-Test for Model II in the Balanced Case
1. Summary. A distribution analogous to the canonical distribution used in testinig the general linear hypothes:s 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, Si distributed as (k + a') times a chi-square variate, where k > 0, independently of S2, which is distributed as k times a chi-square variate. The test is then to reject the hypothesis that go = 0 whenever S1/S2 is greater than some suitably chosen number, c. As a corollary to property (i) it is shown that "of all invariant tests of o = 0 against co > 0 whose power is a function of o/(k + To) only, the test S1/S2 > c is most powerful, providing Si and S2, as defined above can be found." 2. Notation and terminology. We use the notation po(x) for the probability density function (p.d.f.) of the vector-valued random variable, X, which depends on the vector-valued parameter 0 - Q, where S2 will always represent the unrestricted parameter space. This notation is generic so that p may not be the same density each time it appears. The difference in functional form is indicated by the change in variable. The actual form will always be clear from the context. This same generic nlotation will be used for constants; c will usually be a constant, not necessarily the same one each time it appears. It will be clear from the context when c is not a constant. The subspace of Q specified by the hypothesis being tested will be denoted by co. No confusion will be caused when dealing with the hypothesis H: 0 E w if we sometimes speak of co rather than H as the hypothesis. By a test of anl hypothesis we mean any measurable function (p(x) with the property that 0 ? (p(x) < 1. When X is observed to take on the value x one rejects H with probability (p(x).