A common problem faced in clinical studies is that of estimating the effect of the most effective (e.g. the one having the largest mean) treatment among k(≥2) available treatments. The most effective treatment is adjudged based on numerical values of some statistic corresponding to the k treatments. A proper design for such problems is the so-called \"Drop-the-Losers Design (DLD)\". We consider two treatments whose effects are described by independent Gaussian distributions having different unknown means and a common known variance. To select the more effective treatment, the two treatments are independently administered to n1 subjects each and the treatment corresponding to the larger sample mean is selected. To study the effect of the adjudged more effective treatment (i.e. estimating its mean), we consider the two-stage DLD in which n2 subjects are further administered the adjudged more effective treatment in the second stage of the design. We obtain some admissibility and minimaxity results for estimating the mean effect of the adjudged more effective treatment. The maximum likelihood estimator is shown to be minimax and admissible. We show that the uniformly minimum variance conditionally unbiased estimator (UMVCUE) of the selected treatment mean is inadmissible and obtain an improved estimator. In this process, we also derive a sufficient condition for inadmissibility of an arbitrary location and permutation equivariant estimator and provide dominating estimators in cases, where this sufficient condition is satisfied. The mean squared error and the bias performances of various competing estimators are compared via a simulation study. A real data example is also provided for illustration purpose.

Covariate adjustment via a regression approach is known to increase the precision of statistical inference when fixed trial designs are employed in randomized controlled studies. When an adaptive multi-arm design is employed with the ability to select treatments, it is unclear how covariate adjustment affects various aspects of the study. Consider the design framework that relies on pre-specified treatment selection rule(s) and a combination test approach for hypothesis testing. It is our primary goal to evaluate the impact of covariate adjustment on adaptive multi-arm designs with treatment selection. Our secondary goal is to show how the Uniformly Minimum Variance Conditionally Unbiased Estimator can be extended to account for covariate adjustment analytically. We find that adjustment with different sets of covariates can lead to different treatment selection outcomes and hence probabilities of rejecting hypotheses. Nevertheless, we do not see any negative impact on the control of the familywise error rate when covariates are included in the analysis model. When adjusting for covariates that are moderately or highly correlated with the outcome, we see various benefits to the analysis of the design. Conversely, there is negligible impact when including covariates that are uncorrelated with the outcome. Overall, pre-specification of covariate adjustment is recommended for the analysis of adaptive multi-arm design with treatment selection. Having the statistical analysis plan in place prior to the interim and final analyses is crucial, especially when a non-collapsible measure of treatment effect is considered in the trial.

The two-stage drop-the-loser design provides a framework for selecting the most promising of K experimental treatments in stage one, in order to test it against a control in a confirmatory analysis at stage two. The multistage drop-the-losers design is both a natural extension of the original two-stage design, and a special case of the more general framework of Stallard & Friede () (Stat. Med. 27, 6209-6227). It may be a useful strategy if deselecting all but the best performing treatment after one interim analysis is thought to pose an unacceptable risk of dropping the truly best treatment. However, estimation has yet to be considered for this design. Building on the work of Cohen & Sackrowitz () (Stat. Prob. Lett. 8, 273-278), we derive unbiased and near-unbiased estimates in the multistage setting. Complications caused by the multistage selection process are shown to hinder a simple identification of the multistage uniform minimum variance conditionally unbiased estimate (UMVCUE); two separate but related estimators are therefore proposed, each containing some of the UMVCUEs theoretical characteristics. For a specific example of a three-stage drop-the-losers trial, we compare their performance against several alternative estimators in terms of bias, mean squared error, confidence interval width and coverage.