Evaluating the treatment effects has become an important topic for many applications. However, most existing literature focuses mainly on the average treatment effects. When the individual effects are heavy-tailed or have outlier values, not only may the average effect not be appropriate for summarizing the treatment effects, but also the conventional inference for it can be sensitive and possibly invalid due to poor large-sample approximations. In this paper we focus on quantiles of individual effects, which can be more robust measures of treatment effects in the presence of extreme individual effects. Moreover, our inference for quantiles of individual effects are purely randomization-based, which avoids any distributional assumption on the units. We first consider inference for stratified randomized experiments, extending the recent work of Caughey et al. (2021). The calculation of valid $p$-values for testing null hypotheses on quantiles of individual effects involves linear integer programming, which is generally NP hard. To overcome this issue, we propose a greedy algorithm with a certain optimal transformation, which has much lower computational cost, still leads to valid $p$-values and is less conservative than the usual relaxation by dropping the integer constraint. We then extend our approach to matched observational studies and propose sensitivity analysis to investigate to what extent our inference on quantiles of individual effects is robust to unmeasured confounding. Both the randomization inference and sensitivity analysis are simultaneously valid for all quantiles of individual effects, which are actually free lunches added to the conventional analysis assuming constant effects. Furthermore, the inference results can be easily visualized and interpreted.

翻译：评估处理效应已成为许多应用中的重要课题。然而，现有文献主要关注平均处理效应。当个体效应呈现重尾分布或存在异常值时，平均效应不仅可能不适用于概括处理效应，而且其传统推断方法可能因大样本近似效果不佳而变得敏感甚至无效。本文聚焦于个体效应的分位数，在极端个体效应存在的情况下，分位数可作为处理效应更稳健的度量。此外，我们对个体效应分位数的推断完全基于随机化，避免了对个体做出任何分布假设。我们首先考虑分层随机实验的推断，扩展了Caughey等人（2021）的近期工作。针对个体效应分位数原假设的有效p值计算涉及线性整数规划，这通常是NP困难的。为克服这一难题，我们提出了一种基于特定最优变换的贪心算法，该算法计算成本显著降低，仍能产生有效p值，且比通常通过舍弃整数约束的松弛方法更不保守。随后，我们将方法推广至匹配观察研究，并提出敏感性分析以探讨对个体效应分位数的推断在多大程度上对未测量混杂因素具有稳健性。随机化推断与敏感性分析同时适用于个体效应的所有分位数，这实际上是传统假设效应恒定的分析之外附加的“免费午餐”。此外，推断结果易于可视化和解释。