<- file stat 2x2.html -> Here is a note with an on-line reference, and then another with further comments.

2x2 tables

=======================Roger L Berger, 18 Jul 1995==========ssc Subject: Re: appropriate test statistic for 2x2 contingency tables Message-ID: <5qnvu7$2ap@uni00nw.unity.ncsu.edu> Reply-To: berger@eos.ncsu.edu (Roger L Berger) Mike Vagell asked which test to use to analyze a 2X2 table. I agree with David Smith that it is better to use an exact test rather than an approximate test. Fisher's Exact test is an exact conditional test. I would prefer an exact unconditional test. Six exact unconditional tests are available at http://www.stat.ncsu.edu/ click on Exact Unconditional Tests Choosing "Fishers Exact-Boschloo" for the test statistic and "NO" for the confidence interval method will always give a test that is uniformly more powerful than Fisher's Exact (conditional) Test. That is, this test yields a valid p-value that is always smaller than from Fisher's Exact (conditional) Test. In this case, this exact unconditional test yields a p-value of .0028. And this is a valid, exact p-value, not an approximation. (I assumed the two sample sizes 10 and 16 were fixed in this table, for this calculation.) No single test is always (for all sample sizes) the best for this problem. But the references on the above web page suggest that using the "Fishers Exact-Boschloo" test statistic and "YES" to the confidence interval method often yields a nearly optimal test. Roger * * * * * * * * * * * * * * * * * * * * * * * * * * * * *

2x2 table, comments

=======================John Vokey, 19 Jul 1997==========ssc Message-ID: <33D138CC.B8A22F58@uleth.ca> Subject: Re: appropriate test statistic for 2x2 contingency tables The replies to Mike Vagell's question to this point imply that the distinction between the various tests was either a matter of "exact" vs. "approximation" or less vs. more power-- exact and more powerful therefore preferred. But the principle distinction between the tests (aside from the "correction for continuity" that is a side issue) is the underlying statistical model: random sampling from a limited partitioning of N discrete values (Fisher's fixed marginals of the "exact" test) or random sampling from the complete set of all possible partitions of N discrete values (which can also have an "exact" test, but is usually handled by the Pearson chi-square because of the vast number of possible samples). [The Yates' correction for continuity concerns only the problem that chi-square is a continuous distribution, but counts are discrete.] The appropriate answer to Mike's question then is: it depends. Which probablity are you interested in, the probablity of observing your particular 2x2 cell values given only the marginals actually obtained, or the probability of observing your particular cell values given the possibility of other marginal values (including one marginal fixed, or both sampled)? _All_ are legitimate tests, depending on the question Mike wants answered. Only _after_ Mike has decided which probability is of interest do the secondary issues of "exact" vs. "approximate" and power come to the fore. And only Mike can answer that question. Thus, there is and can be no such thing as a universally best test for the 2x2 contingency table. -- Dr. John R. Vokey, Department of Psychology and Neuroscience mailto:vokey@uleth.ca http://www.uleth.ca/~vokey * * * * * * * * * * * * * * * * * * * * * * * * * * * * *

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