Sample Size Estimation in Single-Arm Clinical Trials with Multiple Testing Under Frequentist and Bayesian Approaches

Abstract

In this paper, we study one-sided multiple testing problems for normal and binomial distributions. We use order statistics to test the null hypothesis {all H _i0 are true}. This approach allows us to uniformly address frequentist and Bayesian multiple testing models. To calculate order statistics, we use confidence limits. In frequentist models, we apply an adjustment to the confidence limits that is equivalent to the Bonferroni adjustment of p-values. In the Bayesian case, we adjust the credible limits following a concept of reconciliation between the Bayesian posterior probability and the frequentist p-value. We also study the quantitative relationship between the number of tests and the sample size of a clinical trial. If the number of tests is very large, we suggest using asymptotic order statistics. We study the performance of these statistics. The asymptotic order statistics for the normal distribution are used to extend the results for independent observations to dependent observations.

Key Words:

• Bonferroni adjustment
• Confidence limits
• Credible limits
• Multiple testing
• Order statistics

ACKNOWLEDGMENTS

The authors thank Dr. Estelle Russek-Cohen for numerous helpful comments and suggestions. They also thank an anonymous reviewer for helpful comments.