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Practical Tests of 2 × 2 Contingency Tables

Douglas Liddell
Journal of the Royal Statistical Society. Series D (The Statistician)
Vol. 25, No. 4 (Dec., 1976), pp. 295-304
Published by: Wiley for the Royal Statistical Society
DOI: 10.2307/2988087
Stable URL: http://www.jstor.org/stable/2988087
Page Count: 10
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Practical Tests of 2 × 2 Contingency Tables
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Abstract

Although a 2 × 2 contingency table can have both, one or no margins fixed, methods of analysis are asymptotically identical, and the "conditional exact" test, when joined with randomization, is UMPU (universally most powerful, unbiased) in each case. However, randomization, which can be absurd as well as arbitrary, is quite impractical. Without randomization, and when at least one margin is free, the conditional exact test is highly conservative and of low power, and indeed irrelevant. For the comparative trial (one margin fixed), a test based on the maximum likelihood estimator of the single nuisance parameter has certain desirable properties, including having greater power than the conditional exact test without randomization. True P-values can be obtained from a Fortran program; approximate values from the χ2 statistic, adjusted rather less conservatively than with Yates correction. With neither margin fixed, the χ2 statistic does not require adjustment.

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