This paper considers the problem of testing equality between two independent binomial proportions. Hwang and Yang (Statist. Sinica 11 (2001) 807) apply the Neyman-Pearson fundamental lemma and the estimated truth approach to derive optimal procedures, named expected p-values. This p-value has been shown to be identical to the mid p-value in Lancaster (J. Amer. Statist. Assoc. (1961) 223) for the one-sided test. For the two-sided test, the paper proves the usual two-sided mid p-value is identical to the expected p-value in the balanced sample case. © 2003 Elsevier B.V. All rights reserved.
Yang, M. C., Lee, D. W., & Hwang, J. T. G. (2004). The equivalence of the mid p-value and the expected p-value for testing equality of two balanced binomial proportions. Journal of Statistical Planning and Inference, 126(1), 273–280. https://doi.org/10.1016/j.jspi.2003.07.010