Professor Birnbaum, in his paper (1976), makes a reasonable and valid distinction between 'evidence', i.e. information affecting beliefs, and 'decision', i.e. choice of course of action. He claims that the Neyman-Pearson theory could be looked at in two ways, either as a theory of inference, that is, of evidence, or as a simple kind of decision theory. In fact, he says there are really two distinct 'Neyman-Pearson Theories'. In the first he calls a conclusion obtained from the theory an 'elliptical' or 'evidential' decision, in the second a 'literal' decision. L. J. Savage and D. V. Lindley have presented arguments claiming to show that a reasonable user of the Neyman-Pearson theory should behave as if he was a Bayesian, even if she (or he) calls herself (or himself) a non-Bayesian.
[For convenience, from now on we use 'she' to mean 'she or he'.] Professor Birnbaum admits the validity of these arguments in the 'literal' decision case, but argues that they are fallacious in the 'evidential' case.
Before commenting on these points I would like to say that ever since I came to study statistics the Neyman-Pearson theory has impressed me as a very notable intellectual achievement. Since then I have also been greatly impressed by the work of L. J. Savage and D. V. Lindley and A. Birnbaum. (In fact, my first meeting with Savage was at a lecture at which he expounded an argument essentially the same as one quoted in Birnbaum's paper, and at the time it seemed striking and convincing.) I therefore argue here only apologetically and with regrets that the Neyman-Pearson theory is not a reasonable theory of inference, and only in very restricted circumstances is it at all useful as a decision theory, the Lindley-Savage argument is irrelevant, and Birnbaum's objections are based on a misunderstanding. Perhaps it might be as well to add that these rather sweeping condemnations concern the philosopher and theoretician more than the practical statistician. Thus even if (as seems unlikely) all statisticians were instantly converted to the view that significance tests are less than best possible, they would still use them in some situations for Synthese 36. (1977) 71-85. All Rights Reserved.