In the comparison of two independent binomial proportions for small or moderate sample sizes, Fisher's exact test can be conservative in the sense of its actual significance level (or size) being much less than the nominal level. Boschloo (1970) proposed a procedure, which raises the significance level used for the p-value test derived from Fisher's exact test, such that the size of the p-value test is closer to and below the nominal level. Rohmel and Mansmann (1999) used the unconditional approach, called the supremum procedure, to a valid p-value to obtain a modified p-value test, and showed that the modified p-value test is always better than the original one. In this article, we show that Rohmel and Mansmann's procedure is equivalent to Boschloo's procedure. The supremum procedure will then be applied to some well-known p-value procedures appeared in the literature. Numerical studies including two real cases are given to illustrate the advantages of the proposed modified p-value tests. Numerical results also show that the sizes of modified p-value tests are closer to nominal levels than those of the original p-value tests for many cases, especially in case of unbalanced sample sizes.
COMMUNICATIONS IN STATISTICS-SIMULATION AND COMPUTATION