Adjusting Simon's optimal two-stage design for heterogeneous populations based on stratification or using historical controls

In many cancer studies, the population under consideration is highly heterogeneous in terms of clinical, demographical, and biological covariates. As the covariates substantially impact the individual prognosis, the response probabilities of patients entering the study may strongly vary. In this case, the operating characteristics of classical clinical trial designs heavily depend on the covariates of patients entering the study. Notably, both type I and type II errors can be much higher than specified. In this paper, two modifications of Simon's optimal two-stage design correcting for heterogeneous populations are derived. The first modification assumes that the patient population is divided into a finite number of subgroups, where each subgroup has a different response probability. The second approach uses a logistic regression model based on historical controls to estimate the response probabilities of patients entering the study. The performance of both approaches is demonstrated using simulation examples.