Estimation After Selection in Seamless Phase II/III Drop-the-Losers Trials with Time-to-Event Endpoints

Seamless Phase II/III clinical trials are increasingly used in modern drug development, supported by regulatory guidance from the U.S. Food and Drug Administration (FDA) emphasizing efficiency and valid post-selection inference. A common approach is the drop-the-losers (DTL) design, where inferior treatments are eliminated at an interim stage and only the most promising arm proceeds to confirmation. However, such data-driven selection introduces bias in estimating the selected treatment effect. While post-selection inference is well studied under normality, many clinically relevant endpoints, particularly time-to-event outcomes, are positively skewed and better modeled using a gamma distribution. In this paper, we develop a unified estimation framework for two-stage DTL designs with gamma-distributed endpoints. Assuming a known common shape parameter and unknown scale parameters, we derive the uniformly minimum variance conditionally unbiased estimator (UMVCUE) of the selected mean. Under a scaled squared error loss, we establish the minimaxity of the generalized Bayes estimator and provide conditions for inadmissibility of scale-equivariant estimators.To ensure practical utility, we extend the framework to accommodate relaxed foundational assumptions. The generalized Bayes estimator maintains its superior risk profile even when the shape parameter is unknown and estimated from data. Furthermore, the proposed estimators demonstrate robustness under model misspecification (Weibull, log-normal, piecewise exponential) and adapt to unequal shape parameters across arms. Crucially, we expand the design for right-censored data using conditional maximum likelihood and MCMC-driven Bayesian approaches. Supported by extensive simulations, the methodology is illustrated using uncensored leukemia and right-censored lung cancer datasets.