On a Heuristic Method of Test Construction and its use in Multivariate Analysis

In this paper two closely related heuristic principles of test construction (to be explained in Section 3), called Type I and Type II methods, of which Type II is identified with the usual likelihood ratio method, are noticed as underlying most of the classical tests of hypotheses, simple or composite, on means of univariate normal populations, and on total or partial correlations or regressions in the case of multinormal variates. In these situations the two methods are found to lead to identical tests having properties which happen to be very good in certain cases and moderately good in others. For certain types of composite hypotheses an extension is then made of the Type I method which is applied to construct tests of three different classes of hypotheses on multinormal populations (so as to cover, between them, a very large area of multivariate analysis), yielding in each case a test in general different from the corresponding and current likelihood ratio test. In each case, however, the two tests happen to come out identical for some degenerate "degrees of freedom." In contrast to the likelihood ratio test it is found that in these cases, for general "degrees of freedom," the corresponding Type I test is much easier to use on small samples, because of the relatively greater simplicity of the corresponding small sample distribution problem under the null hypothesis. In each case a lower bound of the power function of the Type I test is also given (against all relevant alternatives), anything like which, so far as the author is aware, would be far more difficult to obtain for the Type II tests in these situations. In this paper the general approach to the two methods is entirely of a heuristic nature except that, under fairly wide conditions, a lower bound to the power functions for each of the two types of tests is indicated to be readily available, which, however, is much too crude or wide a bound in general.