In the setting of multi-armed trials, adaptive designs are a popular way to increase estimation efficiency, identify optimal treatments, or maximize rewards to individuals. Recent work has considered the case of estimating the effects of K active treatments, relative to a control arm, in a sequential trial. Several papers have proposed sequential versions of the classical Neyman allocation scheme to assign treatments as individuals arrive, typically with the goal of using Horvitz-Thompson-style estimators to obtain causal estimates at the end of the trial. However, this approach may be inefficient in that it fails to borrow information across the treatment arms. In this paper, we consider adaptivity when the final causal estimation is obtained using a Stein-like shrinkage estimator for heteroscedastic data. Such an estimator shares information across treatment effect estimates, providing provable reductions in expected squared error loss relative to estimating each causal effect in isolation. Moreover, we show that the expected loss of the shrinkage estimator takes the form of a Gaussian quadratic form, allowing it to be computed efficiently using numerical integration. This result paves the way for sequential adaptivity, allowing treatments to be assigned to minimize the shrinker loss. Through simulations, we demonstrate that this approach can yield meaningful reductions in estimation error. We also characterize how our adaptive algorithm assigns treatments differently than would a sequential Neyman allocation.

翻译：在多臂试验的背景下，自适应设计是提高估计效率、识别最优治疗方案或最大化个体奖励的常用方法。近期研究考虑了在序贯试验中估计K种活性治疗方案相对于对照组效应的情况。多篇论文提出了经典Neyman分配方案的序贯版本，用于在个体入组时分配治疗方案，其目标通常是采用Horvitz-Thompson型估计量在试验结束时获得因果估计。然而，这种方法可能存在效率不足的问题，因为它未能跨治疗臂共享信息。本文探讨了当最终因果估计采用适用于异方差数据的类Stein收缩估计量时的自适应策略。此类估计量能够在治疗效果估计间共享信息，相对于孤立估计每个因果效应，可在期望平方误差损失方面提供可证明的降低。此外，我们证明了收缩估计量的期望损失具有高斯二次型的形式，这使得通过数值积分能高效计算该损失。这一结果为序贯自适应铺平了道路，允许通过分配治疗方案来最小化收缩损失。通过模拟实验，我们证明该方法能显著降低估计误差。我们还刻画了自适应算法在治疗方案分配方面与序贯Neyman分配方案的差异特征。