Improved P-Values for Testing Marginal Homogeneity in 2 × 2 Contingency Tables

Abstract

This article considers the problem of testing marginal homogeneity in a 2 × 2 contingency table. We first review some well-known conditional and unconditional p-values appeared in the statistical literature. Then we treat the p-value as the test statistic and use the unconditional approach to obtain the modified p-value, which is shown to be valid. For a given nominal level, the rejection region of the modified p-value test contains that of the original p-value test. Some nice properties of the modified p-value are given. Especially, under mild conditions the rejection region of the modified p-value test is shown to be the Barnard convex set as described by Barnard (1947). If the one-sided null hypothesis has two nuisance parameters, we show that this result can reduce the dimension of the nuisance parameter space from two to one for computing modified p-values and sizes of tests. Numerical studies including an illustrative example are given. Numerical comparisons show that the sizes of the modified p-value tests are closer to a nominal level than those of the original p-value tests for many cases, especially in the case of small to moderate sample sizes.

Keywords:

• Confidence interval p-value
• Marginal homogeneity
• McNemar's p-value
• Mid p-value
• P-value test

Mathematics Subject Classification:

• Primary 62F03
• Secondary 62P10, 62P25

Notes

Note: For each pair of p-value tests, if the modified p-value test has size closer to α, the superscript of its size is marked by a “○”. For each n, the size closest to and not greater than 0.05 has been bolded.

Note: For each pair of p-value tests, if the modified p-value test has size closer to α, the superscript of its size is marked by a “○”. For each n, the size closest to and not greater than 0.05 has been bolded.