Approximation for response adaptive designs using Stein's method

Abstract

Stein's method introduced by Charles Stein (1972) is a powerful tool in distributional approximation, especially in classes of random variables that are stochastically dependent. In recent years, researchers have concentrated more on adaptive designs. For the response adaptive randomization procedures, the patient's allocation depends on the aggregated information that is acquired from the responses of the previously treated patients. This design uses the information of patients' responses to modify treatment allocation in order to assign more patients to a successful treatment, thus introduce dependent structure in the data. In this thesis we investigate the use of Stein's method in statistical inference for response adaptive design. We have acquired asymptotic normality of the maximum likelihood estimators for treatment effects by deriving an upper bound for these estimators using Stein's method. We examine the performance of three types of response adaptive designs under various success probabilities through simulation studies. Since adaptive designs generate a dependent sequence of random variables that are not exchangeable, we present the advantage of using bootstrap re-sampling in adaptive designs and the efficiency of this method. We compare bootstrap confidence intervals with the asymptotic confidence interval under different success rates of three allocation methods. Also, we discuss the normal approximation based on the Wald's statistic in the numerical studies.