Comparing the asymptotic power of exact tests in 2 x 2 tables

A 2 x 2 table may arise from three types of sampling, depending on the number of previously fixed marginals, and may yield three possible, differing, probabilistic models. From the unconditional point of view each model requires a specific solution but, within each model, the calculation time increases as the test procedure chosen is more powerful and, between the models, the calculation time decreases in the number of marginals fixed. Moreover, each model yields a test which is generally more powerful than the test of any other model with a larger number of marginals fixed. The condition under which a less powerful test, of the same or a different model, can substitute a more powerful test with a loss of power lower than 2% is determined. It is concluded that the Fisher exact test can be used as an approximation to Barnard's exact test for a table with 0 or I fixed marginals, when the sample size is greater than or equal to 100 or when the smaller sample size is greater than or equal to 80, respectively. Similarly, Barnard's exact test for a table with I fixed marginal can be used as an approximation of the same test for a table with 0 fixed marginals, when the sample size is greater than or equal to 50. (C) 2003 Published by Elsevier B.V.