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Small proportions

=======================Werner Besenfelder, 22 May 1997==========ssc From: Werner Besenfelder <wbesenfe@mstore.rz.uni-ulm.de> Subject: Re: Estimation with small proportions Message-ID: <Pine.OSF.3.96.970522084739.19466B-100000@pythia.rz.uni-ulm.de> > >Most methods I have seen for testing similarity of two (unpaired) proportions >(e.g. for equivalence) assume Normal approximations and will thus extend to >estimating confidence intervals for their difference. What if the proportions >are likely to be small enough to doubt the Normality assumption - is there a >method for calculating confidence intervals analagous to using Fisher's exact >test in a hypothesis testing approach? > >Any guidance and/or references would be appreciated. Try Joachim Roehmel. (1996). "Precision intervals for estimates of the difference in success rates for binary random variables based on the permutation principle." Biometrical Journal. 38:977-993. Werner Besenfelder *--------

Comparing small proportions (2)

=======================Rich Ulrich, 6 May 1997==========ssc Subject: Re: diff between binom proportions Message-ID: <5knufc$g7l@usenet.srv.cis.pitt.edu> The story up to now: between EC <<Elliot Cramer (cramer@email.unc.edu)>> and RU << Richard F Ulrich (wpilib+@pitt.edu)>> (me) ==============>start of previous EC: : : The p value for Fisher's exact test does NOT depend on the unknown : : : parameter (prob in the null case); it depends only upon the data. Also : : : FET does not assume fixed marginals; it is CONDITIONAL on the marginals : : : and is thus valid unconditionally RU: : -- The way I read it, the FET does, of course, assume fixed : : marginals. If I do a randomization without fixed marginals, I get : : a rather different outcome - which would not happen if the Exact Test : : applied without fixed marginals. EC: That is true but irrelevant. If the test ASSUMED fixed marginals, it : would provide a valid p value only if the EXPERIMENTER fixes the : marginals. In fact it provides valid p values regardless of what : marginals (including none) are fixed. It's hard to think of a situation : where the experimenter could fix all marginals. Fisher clearly intended : it to apply to any situation. I think that I'm in good company. <==============end of previous [ going sentence by sentence... ] Elliot, I am not sure what you mean by a 'valid p' -- but the only time that I am *sure* that Fisher's applies is, yes, only when "the Experimenter fixes the marginals". I have seen other arguments about including some other conditions, but I have never been convinced. If 'valid' includes, 'okay if it is on the conservative side', then we have no argument, but I know you are arguing the broader case than that, which I still do not understand. I still prefer to start from an imagined randomization, and not from the outcome of the occasion. I do see that one COULD start from the outcome, but not why one would insist on doing so. Where does the Experimenter fix the marginals? - 'median split' for the outcome (for two fixed groups) is the simple case, and it is the circumstance when I insist on Fisher's Test, or Yates' correction. I was not aware of Fisher's 'intention' in describing the test; nor am I aware of the status of tests at the time Fisher first described it. I *do* remember reading that 2x2 testing was messed up for a decade (to about 1920) because Pearson insisted that Pearson's contingency chisquared for the 2x2 must have 3 degrees of freedom, not one; and he was such an arrogant bully that no one dared to publicize his error. - I suggest that Fisher's intention at the time might not be entirely important, today. The company of other statisticians: Yates argued in the JRSS (Journal of the Royal Statistical Society, Series B, about 1982 or 3) that marginal totals should be treated as fixed, and almost all testing (but not quite *all*, I think) should use either Fisher's test, or the the Yates' correction which he (Yates) had devised 50 years earlier. He was fairly adamant about treating the marginals as fixed, almost always. Comments of 5 or 6 other statisticians were printed, and I don't remember that any of them were 100% with him, or 100% against him - but there were several different points at issue. I did not feel like I had to give up *my* interpretation of the test; but, if they just insist on a bit of good company, neither would anyone else. [ "Journal of the Royal Statistical Society" references added, May, 2001. Fishers vs 2x2 Pearson. ] Yates, et al. JRSS Series A (1984) 147:426-463. Shuster. JRSS Series A (1985) 148:317-327. Upton. JRSS Series A (1992) 155:395-402. In the 1984 article, Upton leant strongly against using Fishers' test. In this article, he announces own conversion, crediting the arguments of Barnard. * * * * * * * * * * * * * * * * * * * * * * * * * * * * *

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