One-tailed asymptotic inferences for the difference of proportions: Analysis of 97 methods of inference

Two-tailed asymptotic inferences for the difference d = p₂ - p₁ with independent proportions have been widely studied in the literature. Nevertheless, the case of one tail has received less attention, despite its great practical importance (superiority studies and noninferiority studies). This paper assesses 97 methods to make these inferences (test and confidence intervals [CIs]), although it also alludes to many others. The conclusions obtained are (1) the optimal method in general (and particularly for errors α = 1% and 5%) is based on arcsine transformation, with the maximum likelihood estimator restricted to the null hypothesis and increasing the successes and failures by 3/8; (2) the optimal method for α = 10% is a modification of the classic model of Peskun; (3) a more simple and acceptable option for large sample sizes and values of d not near to ±1 is the classic method of Peskun; and (4) in the particular case of the superiority and inferiority tests, the optimal method is the classic Wald method (with continuity correction) when the successes and failures are increased by one. We additionally select the optimal methods to make compatible the conclusions of the homogeneity test and the CI for d, both for one tail and for two (methods which are related to arcsine transformation and the Wald method).