Sammendrag
To test for association in an unordered rxc table, it is common to choose Pearson's chi square, Likelihood ratio (LR), or Fischer's exact test. These three tests are asymptotically equivalent. But which test is preferable in small samples? The research in this area is scant. It has focused primarily on the question of which of the three asymptotic tests matches its exact counterpart best, see Metha & Patel (2001) and references therein. Today, algorithms for computing exact p-values for the test statistics are available in the software packages like StatXact and SPSS. One should compute the exact p-values when available for small and moderate sample sizes, instead of using the asymptotic values based on the chi square approximation of the test statistics. These exact p-values are computed conditionally given the row and column sums of the table, as for the Fisher's exact test. Even for 2x2 tables, which is the focus of the present presentation, the resulting p-values for a given data set can differ substantially. Consider the following example: Group A with 51 trials, 39 successes and 12 failures versus group B with 8 trials, 3 successes and 5 failures. The exact p-values (two-sided) for Pearson, LR and Fischer are 0.037, 0.090, and 0.037. The asymptotic p-values, which are not recommended used here, equal 0.024, 0.032 and 0.030. The exact p-values for a one-sided test all equal 0.037. These one-sided tests are equivalent in 2x2 tables (Davis, 1986). A golden rule is to choose the testmethod with highest power, before we look at the data. For example, with 51 and 8 trials in each group, hypothesized success probabilities 0.765 and 0.375, and a=0.05, the exact power of the three tests are 58.4%, 47.7% and 58.4%. However, the obtainedsignificance levels may differ. For example, a common success probability of 0.712 gives obtained significance levels of 3.3%, 2.2% and 3.3%. So a fair comparison of the tests should not only look at power. One way to obtain a fair comparison is to consider randomized tests. The presentation will show results of exact power calculations for the three tests, for small samples with equal and unequal sample sizes, both for unrandomized and randomized tests. Some differences in power are found, but in most cases the differences are small. No systematic pattern is found that could indicate areas where one of these exact tests is superior or inferior to the other. Davis, L. J. (1986). Exact Tests for 2x2 Contingency Tables. The American Statistician, May 1986, Vol 40, No 2, pp 139-141. Metha, C. and Patel, N. (2001). StatXact 5 User Manual. CYTEL Software Corporation
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