Re: 2x2 tables in epi. Why Fisher test?
Ronald Bloom wrote: In sci.stat.consult Elliot Cramer [EMAIL PROTECTED] wrote: In sci.stat.consult Ronald Bloom [EMAIL PROTECTED] wrote: Herman as usual is absolutely correct; the validity of the Fisher test is analagous to the validity of regression tests which are derived conditional on x but, since the distribution does not involve x, are valid unconditionally even if the x's are random. If I take your analogy in the direction that leads back to the Fisher test, I should be able to paraphrase the above as the validity of the [Fisher test] which [is] derived conditional on [the fixed marginals] but, since the distribution does not involve [the fixed marginals], [is] valid unconditionally even if the [marginals] are random. Please clarify what is meant by the distribution does not involve [the fixed marginals]. I am not clear on this: the Fisher test statistic (hypergeometric upper tail probability) certainly *does* depend on the fixed marginals in this case -- they appear in every term in that tail sum. Usual the assumptions for Fishers exact test are not true. What you can fix are the row margins, or column margins or grand total or Element of row i and column j. In these cases the exact Fisher test is biased. At least in Survo (may be in some other programs too) it is possible make the test also in these cases. Look at http://www.helsinki.fi/survo/q/qu1_03.html regards Juha -- Juha Puranen Department of Statistics P.O.Box 54 (Unioninkatu 37), 00014 University of Helsinki, Finland http://noppa5.pc.helsinki.fi = Instructions for joining and leaving this list and remarks about the problem of INAPPROPRIATE MESSAGES are available at http://jse.stat.ncsu.edu/ =
Re: 2x2 tables in epi. Why Fisher test?
In article 9deiug$l0h$[EMAIL PROTECTED], Ronald Bloom [EMAIL PROTECTED] wrote: Significance tests for 2x2 tables require that the single observed table be regarded as if it were, (under the null hypothesis of uniformity or independence) but a single instance drawn at random from a universe of replicates. Insofar as there are at least three well-known distinct such sample spaces that one might arguably propose as reasonable models of the universe of replicates, different probability models by which the extremity of the observed table, under the null hypothesis, do arise. I can even provide more. But from the standpoint of classical statistics, it makes little difference. From the standpoint of decision theory it does, but then one would not be doing anything like fixing a significance level in the first place. This has given rise over the years to misunderstandings between proponents of different small-sample inferential tests of signifance for 2x2 tables. But the disputes seem largely to be due to the failure of the disputants to identify precisely that particular probability setup which is correct for the particular problem at hand. At least three distinct such ways of regarding a given 2x2 table can be distinguished: 1.) both row and column marginals regarded as fixed, and under the null hypothesis of uniformity, the observed table is treated as a random sample from the finite set of permutations of all 2x2 tables satisfying that constraint. This sample-space model gives rise to the hypergeometric distribution for the probability of the observed table; thus the Fisher Exact test. The advantage of this one is that an exact test of the prescribed level can be produced. 2.) The two row (col)marginals are treated as independent; and the observed table under the null hypothesis is regarded as being the result of two independent random samples from identical binomial distributions. The significance test used in this case is identical to the elementary test for the difference between two sample proportions. This is a much more complicated testing situation than you seem to think. Because of the nuisance parameters, it is essentially impossible to come up with a natural test at the precise level, especially for small samples. 3.) Only the total cell sum T is regarded as fixed. The observed table, under the null hypothesis, is regarded as a random draw of four cell values satisfying the constraint that their total T is specified. This leads to a multinomial distribution. Each one of these probability setups 1-3 gives rise to a somewhat different small-sample inferential test. In particular, the schemes (1),(2),(3) give rise to distributions conditioned on 3, 2, and 1 fixed parameters respectively. But these parameters are unknown. Testing with nuisance parameters is very definitely not easy, and exact tests are hard to come by. Even in other types of problems, conditional tests are often used. In fact, in many practical problems, the sample size itself need not be fixed. It is not uncommon to use the number of observations as if it were a fixed sample size, and it is easy to give examples where this can be shown not to do what is wanted. Since, for large cell values, the large-sample approximations to all of these distributions (apparently?) converge to the CHi-Squared distribution, it is only in situations with small cell sizes that the controversy over choice of probability model is of practical (?) import. As long as the conditional probabilities are the same, and one uses one of the scenarios you mentioned, the distribution of the Fisher exact test given the marginals is as stated. Thus the probability that the test at a given level rejects is precisely the stated level in all of these cases, assuming that randomized testing is used. If one uses a decision approach, none of this is correct, even if the Fisher model happens to be true. -- 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, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399 [EMAIL PROTECTED] Phone: (765)494-6054 FAX: (765)494-0558 = Instructions for joining and leaving this list and remarks about the problem of INAPPROPRIATE MESSAGES are available at http://jse.stat.ncsu.edu/ =
Re: 2x2 tables in epi. Why Fisher test?
In sci.stat.edu Herman Rubin [EMAIL PROTECTED] wrote: Each one of these probability setups 1-3 gives rise to a somewhat different small-sample inferential test. In particular, the schemes (1),(2),(3) give rise to distributions conditioned on 3, 2, and 1 fixed parameters respectively. But these parameters are unknown. Testing with nuisance parameters is very definitely not easy, and exact tests are hard to come by. Even in other types of problems, I was not here referring to the unknown nuisance parameter (namely the unknown binomial probability). In schemes (1), (2), (3) the 3, 2, and 1 fixed conditioning parameters are, respectively: (a) two row marginals and one column marginal (b) two independent row marginals (c) the total sum of four cells. In the conditioning arguments which yield the signficance tests I alluded to above, those 3, 2, or 1 parameters are *known*. As long as the conditional probabilities are the same, which conditional probabilities are you referring to? and one uses one of the scenarios you mentioned, the distribution of the Fisher exact test given the marginals is as stated. Thus the probability that the test at a given level rejects is precisely the stated level in all of these cases, assuming that randomized testing is used. If one uses a decision approach, none of this is correct, even if the Fisher model happens to be true. I was only addressing the matter of the logical relationship between the probability model used in the significance test to the implied underlying sample space of 2x2 tables from which the observed table was drawn. It seems to me that the choice of experimental design has some bearing on the choice of such universe and I was wondering why the Fisher Universe of permutations with 4 fixed marginals is chosen as the basis for inferential tests for experimental setups in which quite plainly only *two* marginals can be regarded as fixed (e.g. case-control studies, so on). Is there a simple answer to this question? (I guess there really is not...) -- 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, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399 [EMAIL PROTECTED] Phone: (765)494-6054 FAX: (765)494-0558 = Instructions for joining and leaving this list and remarks about the problem of INAPPROPRIATE MESSAGES are available at http://jse.stat.ncsu.edu/ =
Re: 2x2 tables in epi. Why Fisher test?
- I offer a suggestion of a reference. On 10 May 2001 17:25:36 GMT, Ronald Bloom [EMAIL PROTECTED] wrote: [ snip, much detail ] It has become the custom, in epidemiological reports to use always the hypergeometric inference test -- The Fisher Exact Test -- when treating 2x2 tables arising from all manner of experimental setups -- e.g. a.) the prospective study b.) the cross-sectional study 3.) the retrospective (or case-control) study [ ... ] I don't know what you are reading, to conclude that this has become the custom. Is that a standard for some journals, now? I would have thought that the Logistic formulation was what was winning out, if anything. My stats-FAQ has mention of the discussion published in JRSS (Series B) in the1980s. Several statisticians gave ambivalent support to Fisher's test. Yates argued the logic of the exact test, and he further recommended the X2 test computed with his (1935) adjustment factor, as a very accurate estimator of Fisher's p-levels. I suppose that people who hate naked p-levels will have to hate Fisher's Exact test, since that is all it gives you. I like the conventional chisquared test for the 2x2, computed without Yates's correction -- for pragmatic reasons. Pragmatically, it produces a good imitation of what you describe, a randomization with a fixed N but not fixed margins. That is ironic, as Yates points out (cited above) because the test assumes fixed margins when you derive it. -- Rich Ulrich, [EMAIL PROTECTED] http://www.pitt.edu/~wpilib/index.html = Instructions for joining and leaving this list and remarks about the problem of INAPPROPRIATE MESSAGES are available at http://jse.stat.ncsu.edu/ =
Re: 2x2 tables in epi. Why Fisher test?
In sci.stat.consult Ronald Bloom [EMAIL PROTECTED] wrote: Herman as usual is absolutely correct; the validity of the Fisher test is analagous to the validity of regression tests which are derived conditional on x but, since the distribution does not involve x, are valid unconditionally even if the x's are random. Incidentally, if one randomizes to get an exact p value, the Fisher test is uniformly most powerful. Herman can tell us if this is for all three cases. = Instructions for joining and leaving this list and remarks about the problem of INAPPROPRIATE MESSAGES are available at http://jse.stat.ncsu.edu/ =
Re: 2x2 tables in epi. Why Fisher test?
In sci.stat.edu Ronald Bloom [EMAIL PROTECTED] wrote: It has become the custom, in epidemiological reports to use always the hypergeometric inference test -- The Fisher Exact Test -- when treating 2x2 tables arising from all manner of experimental setups -- e.g. Only for tables with small cell sizes (and for combination of multiple such tables), and only because software is freely available. I would have thought it is more likely to be seen used for large sparse 2xK tables, eg HLA literature. Its shortcomings (conservative under the other setups) are also well known (I hope!). David Duffy. = Instructions for joining and leaving this list and remarks about the problem of INAPPROPRIATE MESSAGES are available at http://jse.stat.ncsu.edu/ =
