Treatment Effect Quantiles in Stratified Randomized Experiments and Matched Observational Studies

Evaluating the treatment effects has become an important topic for many applications. However, most existing literature focuses mainly on 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 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 treatment effects, which can be more robust in the presence of extreme individual effects. Moreover, our inference for them is purely randomization-based, avoiding any distributional assumption on the units. We first consider inference in stratified randomized experiments, extending the recent work by Caughey et al. (2021). We show that calculating valid $p$-values for testing null hypotheses on quantiles of individual effects is equivalent to solving multiple-choice knapsack problems, which are generally NP-hard. We provide efficient algorithms to calculate the $p$-values exactly or slightly conservatively. 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. The proposed randomization inference and sensitivity analysis are simultaneously valid for all quantiles of individual effects, noting that the analysis for the maximum (or minimum) individual effect coincides with the conventional analysis assuming constant treatment effects. An R package has also been developed to implement the proposed methods.