[S] Krantz/Oehlschlaegel discussion on ordering multinomial populatio ns--

Gunter, Bert (bert_gunter@merck.com)
Tue, 17 Feb 1998 09:14:20 -0500

1. This is not S-plus, but statistics. Hope it's OK. I enjoyed the
Kr/Oehl discussion anyway.

2. Disclaimer: I'm no expert on any of this. The views expressed here
are my own and not those of my employer or my colleagues.

3. But ... assuming that the choice of numbers is fixed, this is a
multinomial selection/ranking problem, one approach to which is
described in the book: SELECTING AND ORDERING POPULATIONS by Gibbons,
Olkin, and Sobel (Wiley, 1977). There is also a large literature on
selection and ranking problems (with which I am not familiar), but this
is the only book on it that I know of (others, anyone??). The approach
to these problems differs from the hypothesis testing paradigm, although
it is still frequentist (i.e., non-Bayesian). The basic ideas is:

1. Model: sample from multinomial with k categories and probabilities
P1,P2, ... Pk.

2. Start by ASSUMING one category is highest. Problem is to correctly
determine which one it is (not to estimate probabilities). Note: No null
of equality in any form.

3. Given a selection procedure, R, wish to make sure prob(CS|R) is
"large" (CS = Correct Selection), provided multinomial probabilities do
not fall into an "indifference zone" of near equality. You get to
specify the IZ and "large" as parameters of the procedure.

Indifference zones are based on some kind of distance measure over the
parameter space, Chapter 6 of the book uses the ratio of the largest to
next to largest Pi's and develops things from there. As you all can
read, too, enough said.

Bert Gunter
Biometrics Research
Merck Research Labs
P.O. Box 2000
Rahway, NJ 07065-0900

"The business of the statistician is to catalyze the
scientific learning process." George E. P. Box
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