Today is R.A. Fisher’s birthday. I’ll post some Fisherian items this week in honor of it. This paper comes just before the conflicts with Neyman and Pearson erupted. Fisher links his tests and sufficiency, to the Neyman and Pearson lemma in terms of power. It’s as if we may see them as ending up in a similar place while starting from different origins. I quote just the most relevant portions…the full article is linked below. Happy Birthday Fisher!
“Two New Properties of Mathematical Likelihood“
by R.A. Fisher, F.R.S.
Proceedings of the Royal Society, Series A, 144: 285-307 (1934)
The property that where a sufficient statistic exists, the likelihood, apart from a factor independent of the parameter to be estimated, is a function only of the parameter and the sufficient statistic, explains the principle result obtained by Neyman and Pearson in discussing the efficacy of tests of significance. Neyman and Pearson introduce the notion that any chosen test of a hypothesis H0 is more powerful than any other equivalent test, with regard to an alternative hypothesis H1, when it rejects H0 in a set of samples having an assigned aggregate frequency ε when H0 is true, and the greatest possible aggregate frequency when H1 is true. If any group of samples can be found within the region of rejection whose probability of occurrence on the hypothesis H1 is less than that of any other group of samples outside the region, but is not less on the hypothesis H0, then the test can evidently be made more powerful by substituting the one group for the other.
Consequently, for the most powerful test possible the ratio of the probabilities of occurrence on the hypothesis H0 to that on the hypothesis H1 is less in all samples in the region of rejection than in any sample outside it. For samples involving continuous variation the region of rejection will be bounded by contours for which this ratio is constant. The regions of rejection will then be required in which the likelihood of H0 bears to the likelihood of H1, a ratio less than some fixed value defining the contour. (295)…
It is evident, at once, that such a system is only possible when the class of hypotheses considered involves only a single parameter θ, or, what come to the same thing, when all the parameters entering into the specification of the population are definite functions of one of their number. In this case, the regions defined by the uniformly most powerful test of significance are those defined by the estimate of maximum likelihood, T. For the test to be uniformly most powerful, moreover, these regions must be independent of θ showing that the statistic must be of the special type distinguished as sufficient. Such sufficient statistics have been shown to contain all the information which the sample provides relevant to the value of the appropriate parameter θ . It is inevitable therefore that if such a statistic exists it should uniquely define the contours best suited to discriminate among hypotheses differing only in respect of this parameter; and it is surprising that Neyman and Pearson should lay it down as a preliminary consideration that ‘the testng of statistical hypotheses cannot be treated as a problem in estimation.’ When tests are considered only in relation to sets of hypotheses specified by one or more variable parameters, the efficacy of the tests can be treated directly as the problem of estimation of these parameters. Regard for what has been established in that theory, apart from the light it throws on the results already obtained by their own interesting line of approach, should also aid in treating the difficulties inherent in cases in which no sufficient statistics exists. (296)
It might worth noting that Fisher (1934) is the first and last time Fisher talked about alternative hypotheses and the power of a test in favorable light. The irreparable rift between Fisher and Neyman occurred at Neyman’s second presentation to the RSS in March of 1935 on “Statistical Problems in Agricultural Experimentation” in which he criticized Fisher’s treatment of randomized block vs. Latin square methods. Neyman put forward an explicit linear model to frame his criticisms and call into question Fisher’s implicit model of Latin squares as being less efficient than that of randomized block. In his comments, Fisher was acerbic in tone, criticizing Neyman for discussing a topic `he knew very little about’, in contrast to his previous presentation to the RSS. The subsequent reply by Neyman drew the battle lines for an endless war between them on “all things statistical”, including topics they were in agreement before this episode. The first casualty of this war was their common ground on Confidence Intervals as specified in Fisher (1930) and Neyman (1934), with both authors going out of their way to exaggerate their differences.