HUAYI Clinical ResearchJiaxing
Limit Theorems on Deviation Probabilities with Applications in Two-Armed Clinical Trials
I consider design problems of two-armed clinical trials conducted in two stages. The goal of the design is to maximize the expected treatment mean over all patients by properly choosing the number of patients in the first stage. Using a Bayesian approach, the treatment means of each arm are considered random. When the responses from both arms are Bernoulli and one arm has known success rate, Berry and Pearson (1985) conjectured that the optimal first-stage sample size should be in the order of square root of N, provided that the prior on the unknown success rate is beta-distributed, where N is the total number of patients in the trial or in consideration. Cheng (1992) proved this claim for the case in which the known success rate is a rational number. In this dissertation, I give a first-order approximation of the optimal sample size in the first stage under a more general condition. I also obtain the asymptotically optimal sample sizes in the first stage when both arms are unknown. The coefficients of the approximations are explicitly given. These asymptotic results are derived using the limit theorems on deviation probabilities established in this dissertation. The concept of deviation probabilities is an extension of the classical large deviations to the Bayesian context. For many applica tions in Bayesian analysis, an expression which estimates a limiting behavior can be decomposed into two parts: the limit of the expression and a deviation probability. Limit theorems on deviation probabilities provide a way of estimating these expressions analytically. Finding optimal assignment procedures in two-armed clinical trials is an application of these limit theorems. Some other applications of the limit theorems are also provided.