Two‐Stage Drop‐the‐Losers Design for the Selection of Effective Treatments and Estimating Their Average Worth

In multi-arm clinical trials, several new treatments are often evaluated concurrently to identify the best and confirm their superiority over a control. In this paper, we propose a framework that introduces an intermediate stage aimed at assessing the collective efficacy of treatments retained after initial screening. Estimating the average effect of the selected treatments provides an interpretable measure of their collective potential and serves as a data-driven criterion for deciding whether to continue or terminate the trial. Consider <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics><mml:mrow><mml:mi>k</mml:mi> <mml:mspace></mml:mspace> <mml:mo>(</mml:mo> <mml:mo>≥</mml:mo> <mml:mn>2</mml:mn> <mml:mo>)</mml:mo></mml:mrow> <mml:annotation>$$ k\kern0.3em \left(\ge 2\right) $$</mml:annotation></mml:semantics> </mml:math> experimental treatments whose effects are described by independent Gaussian responses with unknown means and a common variance. For the purpose of selecting the effective treatments (drugs) and estimating their average worth, we employ a two-stage drop-the-losers design (DLD). To get an idea about the structure of an optimal estimator, we first assume that the common variance is known. In the first stage of the design, data is collected to select a subset of experimental treatments so that the probability of including the best treatment is at least a prespecified level <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics> <mml:mrow> <mml:msup><mml:mrow><mml:mi>P</mml:mi></mml:mrow> <mml:mrow><mml:mo>∗</mml:mo></mml:mrow> </mml:msup> </mml:mrow> <mml:annotation>$$ {P}^{\ast } $$</mml:annotation></mml:semantics> </mml:math> . This selection rule ensures that inferior treatments are eliminated while maintaining a minimum confidence that the best treatment remains among those advanced. Given this requirement, the design either advances all selected treatments to the next stage or stops for futility. The treatment(s) selected in the subset then proceed to the second stage for estimating their collective effectiveness through point estimation of their average worth, defined as the arithmetic average of their mean effects. Since the bias of estimators is crucial in clinical studies, we derive the uniformly minimum variance conditionally unbiased estimator (UMVCUE) of the worth of the selected treatments, conditioned on the indices of treatments selected at the first stage. The mean squared error and bias performances of the UMVCUE are compared with the naive estimator (maximum likelihood estimator) via a simulation study. For the unknown variance scenario, we propose a plug-in estimator based on the structure of the UMVCUE derived for the known variance case and study its performance through simulations. A real-life data example is also provided to illustrate an application of our findings.