We distinguish between two sources of uncertainty in experimental causal inference: design uncertainty, due to the treatment assignment mechanism, and sampling uncertainty, when the sample is drawn from a super-population. This distinction matters in settings with small fixed samples and heterogeneous treatment effects, as in geographical experiments. Most bootstrap procedures used by practitioners primarily estimate sampling uncertainty. Other methods for quantifying design uncertainty also fall short, because they are restricted to common designs and estimators, whereas non-standard designs and estimators are often used in these low-power regimes. We address this gap by proposing an integer programming approach, which allows us to estimate design uncertainty for any known and probabilistic assignment mechanisms, and linear-in-treatment and quadratic-in-treatment estimators. We include asymptotic validity results and demonstrate the refined confidence intervals achieved by accurately accounting for non-standard design uncertainty through simulations of geographical experiments.

翻译：在实验因果推断中，我们区分两种不确定性来源：源于处理分配机制的设计不确定性，以及样本从超总体中抽取时产生的抽样不确定性。这种区分在样本量小且固定、处理效应存在异质性的情境中尤为重要，例如地理实验。实践者使用的大多数自助法程序主要估计抽样不确定性。其他量化设计不确定性的方法也存在不足，因为它们通常局限于常见的设计和估计量，而在这些低统计功效的场景中，研究者往往采用非标准的设计和估计量。为填补这一空白，我们提出了一种整数规划方法，该方法使我们能够针对任何已知的概率性分配机制，以及关于处理的线性估计量和二次估计量，估计其设计不确定性。我们提供了渐近有效性结果，并通过地理实验的模拟，展示了通过准确考虑非标准设计不确定性所实现的更精确的置信区间。