Optimizing the allocation of units into treatment groups can help researchers improve the precision of causal estimators and decrease costs when running factorial experiments. However, existing optimal allocation results typically assume a super-population model and that the outcome data comes from a known family of distributions. Instead, we focus on randomization-based causal inference for the finite-population setting, which does not require model specifications for the data or sampling assumptions. We propose exact theoretical solutions for optimal allocation in $2^K$ factorial experiments under complete randomization with A-, D- and E-optimality criteria. We then extend this work to factorial designs with block randomization. We also derive results for optimal allocations when using cost-based constraints. To connect our theory to practice, we provide convenient integer-constrained programming solutions using a greedy optimization approach to find integer optimal allocation solutions for both complete and block randomization. The proposed methods are demonstrated using two real-life factorial experiments conducted by social scientists.

翻译：将实验单元优化分配到处理组中，有助于研究者提高因果估计量的精度并降低析因实验成本。然而，现有最优分配结果通常假设超总体模型且结果数据服从已知分布族。与此不同，我们聚焦于有限总体中基于随机化的因果推断——该方法无需对数据模型或抽样假设进行设定。我们提出了完全随机化下采用A最优性、D最优性和E最优性准则的$2^K$析因实验最优分配的精确理论解，并将该工作拓展至区组随机化析因设计。同时，我们推导了基于成本约束的最优分配结果。为连接理论与实际应用，我们采用贪心优化方法提供了便于实施的整数约束规划方案，从而为完全随机化和区组随机化场景求解整数最优分配方案。最后，通过社会科学家开展的两项真实析因实验对所提方法进行了验证。