Re: Accept H0, or fail to reject?

In article <19660104.1219506829010.JavaMail.jakarta@xxxxxxxxxxxxxxxxxxxxxx>,
Peter <peterpc5j@xxxxxxxxxxx> wrote:
I have seen you write variations on this theme several times. Now I understand that your reluctance against point null hypotheses stems from a notion of the relation between probability density and probability that got you wound up in the E(1/Y) discussion.

I do not have a degree in mathematics or statistics. I use statistics in an applied context. Now put yourself in the situation of scientific practice for a moment. You will want to defend yourself against people who attack some empirical finding of yours. My hostile opponent will not ask whether "mu _is_ 0", but whether "mu _could be_ 0", no matter how unlikely that statement is a priori. Once your opponent challenges you this way, you have to give him or her a priori as much chance to win as yourself - th

Could mu be 0? It could be close to 0, but the probability
that it is 0 is usually zero.

at is, you distort the prior towards a mixture, in a way that now the 0 value has 50% of being true, and any other value also 50%, even if the null hypothesis doesn't make much sense from a mathematical point of view.

No, there is nothing to force an even chance.

Of course we eventually can never know whether H0 is true or not, that is a misunderstanding that should be rooted out in a stat 101 course.

It should, but it will not happen until decision theory enters.
This can be done in high school, but alas many established
statisticians cannot understand the simple reasoning, and for
the textbook writers and selectors, it is worse.

But, enlighten me, what is, for you, the conceptual difference between "accept" and "fail to reject"?

Acceptance has drawbacks in cases where mu is not exactly
zero, usually growing with the absolute value of mu. So
does failing to accept; if mu is close enough to zero, it
is likely to be better to accept. This is a decision
problem, which I considered in my paper in the First
Purdue Symposium, and the book is not closed on it by
any means. The loss-prior combination does matter more
than in most situations.

And, about the idea that a point null hypothesis is almost certainly false - that would depend on the nature of your parameter, wouldn't it? If you make me guess blindfoldedly whether you are holding up a red or black card from a normal deck, H_0: prob correct=.5 is a reasonable null hypothesis, that has a realistic chance of actually being true.

This is a reasonable null hypothesis, but alas it does not
have a reasonable chance of actually being true, even if the
card is "drawn". Cards are not identical. This is, however,
a case where the loss-prior combination, NOT the prior, can
be piled up at the point null with little loss of accuracy.
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@xxxxxxxxxxxxxxx Phone: (765)494-6054 FAX: (765)494-0558
.

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