Optimal Unconditional Asymptotic Test in 2 × 2 Multinomial Trials

Abstract

The generally most powerful unconditional exact test for testing independence in a 2 × 2 multinomial trial is Barnard's test, but at present it is impossible to apply in samples of even moderate size. Alternatively, it can be used as an asymptotic method. This paper evaluates 10 different approximations (as well as 48 others not included here) under the criterion that the optimal approximation is the one which produces a p-value that differs from the exact test by less than a given quantity. The best method is the one based on Pirie and Hamdan's chi-squared statistics – = {|N 11 N 22 − N 12 N 21| − 0.5}2(N − 1)/{N 1• N 2• N •1 N •2} – which is valid, for ordinary significances and one(two)tailed-tests, if the minimum expected quantity E is larger than or equal to 2.5 (1.5). However, when the test is valid it may fail between 1% and 5% of times (the failures are sometimes liberal and others are conservative). If one wants to avoid failure, E will have to be rather more than 9. (Present computation capacity does not allow us to determine the exact amount.)