We investigate experimental design for randomized controlled trials (RCTs) with both equal and unequal treatment-control assignment probabilities. Our work makes progress on the connection between the distributional discrepancy minimization (DDM) problem introduced by Harshaw et al. (2024) and the design of RCTs. We make two main contributions: First, we prove that approximating the optimal solution of the DDM problem within even a certain constant error is NP-hard. Second, we introduce a new Multiplicative Weights Update (MWU) algorithm for the DDM problem, which improves the Gram-Schmidt walk algorithm used by Harshaw et al. (2024) when assignment probabilities are unequal. Building on the framework of Harshaw et al. (2024) and our MWU algorithm, we then develop the MWU design, which reduces the worst-case mean-squared error in estimating the average treatment effect. Finally, we present a comprehensive simulation study comparing our design with commonly used designs.

翻译：本研究探讨了在随机对照试验中，当处理组与对照组分配概率相等及不相等时的实验设计问题。我们的工作推进了Harshaw等人（2024）提出的分布差异最小化问题与随机对照试验设计之间关联的研究。我们主要做出两方面贡献：首先，我们证明了即使在一定常数误差范围内逼近DDM问题的最优解也是NP难问题。其次，我们针对DDM问题提出了一种新的乘性权重更新算法，该算法在分配概率不相等时改进了Harshaw等人（2024）所使用的Gram-Schmidt游走算法。基于Harshaw等人（2024）的理论框架及我们提出的MWU算法，我们进一步开发了MWU设计方法，该方法能降低估计平均处理效应时最坏情况下的均方误差。最后，我们通过系统的模拟研究，将我们提出的设计与常用设计方法进行了全面比较。