Bayesian noninferiority test for 2 binomial probabilities as the extension of Fisher exact test

• Published in: Statistics in medicine Vol. 36; no. 30; pp. 4789 - 4803
• Main Authors: Doi, Masaaki, Takahashi, Fumihiro, Kawasaki, Yohei
• Format: Journal Article
• Language: English
• Published: England Wiley Subscription Services, Inc 30.12.2017
• Subjects: Bayes Theorem; Bayesian analysis; Bayesian noninferiority test; binomial probability; Biostatistics - methods; Clinical Trials as Topic - statistics & numerical data; Computer Simulation; conditional power prior; Fisher exact test; historical data; HIV Infections - drug therapy; Humans; Models, Statistical; Monte Carlo Method; posterior probability; Probability; Sample Size; conditional power prior; historical data; binomial probability; posterior probability; Bayesian noninferiority test; Fisher exact test
• ISSN: 0277-6715; 1097-0258
• DOI: 10.1002/sim.7495

Noninferiority trials have recently gained importance for the clinical trials of drugs and medical devices. In these trials, most statistical methods have been used from a frequentist perspective, and historical data have been used only for the specification of the noninferiority margin Δ>0. In contrast, Bayesian methods, which have been studied recently are advantageous in that they can use historical data to specify prior distributions and are expected to enable more efficient decision making than frequentist methods by borrowing information from historical trials. In the case of noninferiority trials for response probabilities π1,π2, Bayesian methods evaluate the posterior probability of H1:π1>π2−Δ being true. To numerically calculate such posterior probability, complicated Appell hypergeometric function or approximation methods are used. Further, the theoretical relationship between Bayesian and frequentist methods is unclear. In this work, we give the exact expression of the posterior probability of the noninferiority under some mild conditions and propose the Bayesian noninferiority test framework which can flexibly incorporate historical data by using the conditional power prior. Further, we show the relationship between Bayesian posterior probability and the P value of the Fisher exact test. From this relationship, our method can be interpreted as the Bayesian noninferior extension of the Fisher exact test, and we can treat superiority and noninferiority in the same framework. Our method is illustrated through Monte Carlo simulations to evaluate the operating characteristics, the application to the real HIV clinical trial data, and the sample size calculation using historical data.