An application of Edgeworth expansion in Bayesian inferences: optimal sample sizes in clinical trials

Abstract

This paper presents an asymptotic optimal solution based on an Edgeworth expansion of an utility function for the design of a two-stage clinical trial. The success rate of the control treatment is known. For an optimal design, only the treatment with unknown success rate will be observed in the first stage. One must decide how many patients should be assigned to the first stage while the total sample size of the trial is N. Information is updated at the beginning of the second stage according to Bayes’ theorem. The objective is to maximize the expected percentage of successes. The first two dominant terms of the optimal sample size for the first stage are found explicitly in terms of N and prior density function. The rates of the first and the second dominant terms are square root of N and fourth root of N, respectively. This paper suggests using these two terms as an estimate of the optimal sample size. The asymptotic bias rates of estimated optimal sample sizes and estimated optimal utilities are provided. The simulation result shows that the actual optimal sample size and the corresponding estimated size are very close, even if N is moderate. The criterion of the selection of prior distribution is addressed. The asymptotic solution is also extended to the situation when there is additional treatment cost for the patients in the first stage.