> Fourth, contrary to Jens Oehlschlaegel, I don't expect that Bayesians
> will do much laughing about this problem. Formulating a sensible
> hierarchical prior leading to a good Bayesian analysis for this
> problem strikes me as an interesting challenge.
Dear Dave Krantz,
Let me explain my comment "bayesian laughter". I share your statement,
that probably a bayesian analysis is an interesting alternative here.
As a psychologist, my methodological training focused on hypotheses
testing, this is rejecting sharp null hypotheses as described in my post.
In the psychological arena, a lot of people (unfortunately) don't like
(and don't teach) bayesian approaches (at least in Germany), because of
the need to choose arbitrary prior distributions. They think that this
makes an analysis less objective, since the results do not only depend on
the objective data but also on "subjective" priors. I know a guy, who
ruined his career by trying to convince the psychology scientific
community in Germany about the usefulness of bayesian approaches.
However, I have always considered the non-bayesian approach to be a
special case of a bayesian approach, so I find it impossible to be
non-bayesian, and I regret not having received proper bayesian training.
In the present case, I demonstrated the use of a sharp null hypothesis,
which is in a way even more "subjective" than choosing a bayesian prior,
since it puts probability weight *one* on the sharp null, as the basis for
the analytic decision, whereas the bayesian approach assumes a prior
probability *distribution*. So my current understanding is that using the
sharp null
(1) makes *more* prior assumptions than a bayesian approach
(2) feeds the illusion to be less subjective
which gives rise to "bayesian laughter", which is indeed a laughing
awareness about the choosen approach and my own restrictions as a
not-yet-bayesian.
By the way, the possibility you are pointing out in
> Third, the strategy to use the binomial with p=.10 as the null
> hypothesis obviously won't do--what would happen if 200 out of
> 1000 people chose 7 and the other 800 chose 3? One could safely
> reject p=.10 for 7, but one could scarcely conclude that 7 was the
> most popular choice.
has been indicated in my post in
> of course the above conclusion p(7)>0.1 did not rule out p(1)>0.2
> i.e. it could still p(7)<=p(1)
Of course, because of this, your answer has been hitting the question much
better than mine. Thank you very much for presenting the likelihood ratio
approach.
Best regards
Jens Oehlschlaegel
-- Jens Oehlschlaegel-Akiyoshi Psychologist/Statistician Project TR-EAT + COST Action B6 F.rankfurt oehl@psyres-stuttgart.de A.ttention +49 711 6781-408 (phone) I.nventory +49 711 6876902 (fax) R .-----. / ----- \ Center for Psychotherapy Research | | 0 0 | | Christian-Belser-Strasse 79a | | ? | | D-70597 Stuttgart Germany \ ----- / -------------------------------------------------- '-----' - (general disclaimer) it's better
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