The exact unconditional z-pooled test for equality of two binomial probabilities: optimal choice of the Berger and Boos confidence coefficient
Exact unconditional tests for comparing two binomial probabilities are generally more powerful than conditional tests like Fisher's exact test. Their power can be further increased by the Berger and Boos confidence interval method, where a p-value is found by restricting the common binomial probability under H-0 to a 1 - gamma confidence interval. We studied the average test power for the exact unconditional z-pooled test for a wide range of cases with balanced and unbalanced sample sizes, and significance levels 0.05 and 0.01. The detailed results are available online on the web. Among the values 10(-3), 10(-4), ... , 10(-10), the value gamma = 10(-4) gave the highest power, or close to the highest power, in all the cases we looked at, and can be given as a general recommendation as an optimal gamma.
power exact test 2 x 2 table pearson's chi-squared small sample