Shrinkage estimation in two-stage adaptive designs with midtrial treatment selection

We consider the problem of estimation in adaptive two-stage designs with selection of a single treatment arm at an interim analysis. It is well known that the standard maximum-likelihood estimator of the selected treatment is biased. We prove that selection bias of the maximum-likelihood estimator is maximal when all treatment effects are equal and the most-promising treatment is selected. Furthermore, we consider shrinkage estimation as a solution for the selection bias problem. We thereby extend previous work of Hwang on Lindley's estimator for single-stage multi-armed trials with four or more treatments and post-trial treatment selection. Following Hwang's ideas, we show that a simple two-stage version of Lindley's estimator has uniformly smaller Bayes risk than the maximum-likelihood estimator when assuming an empirical Bayesian framework with independent normal priors for the group means. For designs that start with two or three treatment groups, we suggest using a two-stage version of the common estimator of the best linear unbiased predicator of the corresponding random effects model. We show by an extensive simulation study that the shrinkage estimators perform well compared with maximum-likelihood and previously suggested bias-adjusted estimators in terms of selection bias and mean squared error.