Estimation and expected sample size in Simon's two-stage designs that stop as early as possible

In early phase clinical studies in oncology, Simon's two-stage designs are widely used. The trial design could be made more efficient by stopping early in the second stage when the required number of responses is reached, or when it has become clear that this target can no longer be met (a form of non-stochastic curtailment). Early stopping, however, will affect proper estimation of the response rate. We propose a uniformly minimum-variance unbiased estimator (UMVUE) for the response rate in this setting. The estimator is proven to be UMVUE using the Rao-Blackwell theorem. We evaluate the estimator's properties in terms of bias and mean squared error, both analytically and via simulations. We derive confidence intervals based on sample space orderings, and assess the coverage. For various design options, we evaluate the reduction in expected sample size as a function of the true response rate. Our method provides a solution for estimating response rates in case of a non-stochastic curtailment Simon's two-stage design.