A simple procedure to estimate the optimal sample size in case of conjunctive coprimary endpoints

For clinical studies in which two coprimary endpoints are necessary for assuring efficacy of the treatment of interest, it is important to determine the minimal sample size needed to attain a certain conjunctive power (i.e., power to reject false null hypothesis for both endpoints). The traditional method of assigning the square root of the targeted overall power to each of the two hypothesis tests is optimal only when the standardized treatment effect sizes of the two endpoints are equal. In spite of this limitation the square root method is applied routinely, resulting in frequent overestimation of the overall sample size. A new, iterative method is presented to find the two individual power values for the two endpoints so that the targeted overall power is attained with the smallest possible overall sample size. The principle is to assign more power to the endpoint for which a larger standardized effect size is likely to occur based on prior information. The main assumption of the new method is the independence of endpoints. However, this is not a serious limitation in case of type II error, thus the method yields a good approximation even if the condition of independence is not fulfilled. The advantages of the new method are (a) a considerable saving (up to 24% in our examples) in the overall sample size, (b) the flexibility as it can be applied to any combination of endpoint types (e.g., normally distributed + binomial, survival + binomial, etc.) and (c) easy to program.