*In recognition of R.A. Fisher’s birthday tomorrow, I will post several entries on him. I find this (1934) paper to be intriguing –immediately before the conflicts with Neyman and Pearson erupted. It represents essentially the last time he could take their work at face value, without the professional animosities that almost entirely caused, rather than being caused by, the apparent philosophical disagreements and name-calling everyone focuses on. 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 very similar place (no pun intended) while starting from different origins. I quote just the most relevant portions…the full article is linked below. I’d blogged it earlier here. You may find some gems in it.*

**‘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 H_{0} is more powerful than any other equivalent test, with regard to an alternative hypothesis H_{1}, when it rejects H_{0} in a set of samples having an assigned aggregate frequency ε when H_{0} is true, and the greatest possible aggregate frequency when H_{1} is true.

If any group of samples can be found within the region of rejection whose probability of occurrence on the hypothesis H_{1} is less than that of any other group of samples outside the region, but is not less on the hypothesis H_{0}, then the test can evidently be made more powerful by substituting the one group for the other. Continue reading