Optimal Biomarker-stratified Design and Adaptive Design in Mixture Distributions

In the first part of this dissertation, an optimal design for precision medicine is introduced. Different patients could respond differently to the same drug, and the ineffective rate of a particular drug could be quite high. Because of this, precision medicine has become more and more popular in recent years, which could potentially help patients choose the right medicine and help pharmaceutical companies increase the success rate of clinical trials. Conventional clinical trials lack the ability of incorporating the patient's heterogeneity, thus many new designs for precision medicine have been proposed, among which biomarker-stratified design is quite popular. Optimality of biomarker-stratified design will be presented in the first part of this dissertation. More specifically, the optimal design is first derived under the scenario that the parameters are taken to be fixed values. Then the uncertainty of the parameters are taken into account, and optimal design is derived when the parameters are within a given interval. The performance of the optimal design is demonstrated through the relative efficiency to the balanced design. In the second part of this dissertation, adaptive design with mixture normal distribution is introduced. Interim analysis is becoming more and more desirable in clinical trials. It allows early termination when there is a significant difference between two treatments. All the available methods are considering normal distribution with common variance but different means. However, in some cases, especially when patients are receiving some standard care, the common variance assumption may not be true. Thus in the second part of this dissertation, a mixture normal distribution for the responses in the control group is considered. And the method proposed will first work on one-stage design to find the required total sample size based on some criteria and then uses the information from one-stage design to obtain the sample size and cutting point for two-stage design which can satisfy both requirements for type I error and power simultaneously.