Sample size for simultaneous testing of rate differences in non-inferiority trials with multiple endpoints

This paper discusses sample size estimation for several simultaneous tests of rate differences in non-inferiority trials. Formulae corresponding to the commonly used methods in a single endpoint study, which were proposed by [Blackwelder, W.C., 1982. Proving the null hypothesis in clinical trials. Controlled Clinical Trials 3, 345-353] and [Farrington, C.P., Manning, G., 1990. Test statistics and sample size formulae for comparative binomial trials with null hypothesis on non-zero risk difference or non-unity relative risk. Statistics in Medicine 9, 1447-1454] are presented. The focus of this study is a non-inferiority trial with multiple co-primary endpoints. The sample size for simultaneous testing is not only determined by Type I and II error rates, but also by several other factors including the effect sizes of each endpoint, the number of endpoints and the correlation among test statistics in each endpoint. To demonstrate the sample sizes required to achieve various levels of statistical significance under different correlations, this manuscript presents plots of the distribution of P-values as a function of sample size. These plots are obtained simultaneously from each endpoint under the alternative hypothesis. The results are applied to the design of a non-inferiority trial.