Empirical Bayes estimation of the selected treatment mean for two-stage drop-the-loser trials: a meta-analytic approach

Point estimation for the selected treatment in a two-stage drop-the-loser trial is not straightforward because a substantial bias can be induced in the standard maximum likelihood estimate (MLE) through the first stage selection process. Research has generally focused on alternative estimation strategies that apply a bias correction to the MLE; however, such estimators can have a large mean squared error. Carreras and Brannath (Stat. Med. 32:1677-90) have recently proposed using a special form of shrinkage estimation in this context. Given certain assumptions, their estimator is shown to dominate the MLE in terms of mean squared error loss, which provides a very powerful argument for its use in practice. In this paper, we suggest the use of a more general form of shrinkage estimation in drop-the-loser trials that has parallels with model fitting in the area of meta-analysis. Several estimators are identified and are shown to perform favourably to Carreras and Brannath's original estimator and the MLE. However, they necessitate either explicit estimation of an additional parameter measuring the heterogeneity between treatment effects or a quite unnatural prior distribution for the treatment effects that can only be specified after the first stage data has been observed. Shrinkage methods are a powerful tool for accurately quantifying treatment effects in multi-arm clinical trials, and further research is needed to understand how to maximise their utility.