Estimation Following Surrogate Based Selection in Multi-Arm Drop-The-Losers Designs

This paper studies inference for primary endpoints following surrogate-based selection in a multi-arm, two-stage drop-the-losers design, assuming bivariate normal responses with a common but unknown covariance matrix. In this framework, treatment selection at the first stage is based on the largest sample mean of the surrogate endpoint (the first component), after which additional observations are collected from the selected treatment arm to estimate the primary endpoint (the second component of the mean vector). We consider several estimators for the primary endpoint, including the natural (maximum likelihood) estimator, two plug-in estimators that account for covariance estimation, and the uniformly minimum variance conditionally unbiased estimator (UMVCUE) derived using Rao-Blackwellization. The natural estimator is shown to be minimax under the scaled squared error loss function, while the UMVCUE provides unbiased inference with only a modest loss in efficiency. Simulation studies demonstrate that for a fixed total sample size, the bias of the natural estimator increases with stronger correlation and larger first-stage sample sizes, whereas the UMVCUE exhibits stable performance across a wide range of configurations. An illustrative real-data example is also presented, highlighting the practical relevance of the proposed methodology for adaptive and sequential clinical trial designs.