Post-treatment variables (PVs), such as treatment noncompliance, behavioral responses, intercurrent events, often modify the ultimate treatment effect on the primary outcome. However, existing methods provide limited tools for studying treatment effect heterogeneity with respect to PVs. Conventional heterogeneous treatment effect estimands condition on baseline covariates. However, similarly conditioning on the observed PV can induce endogenous selection bias for the treatment effect estimation. Principal stratification offers a rigorous framework for studying principal causal effects across principal strata, but principal strata are latent and their identification often requires stringent assumptions. This paper develops an assumption-lean empirical stratification framework for characterizing treatment effect heterogeneity with respect to PVs. We define empirical scores using the predicted potential PV responses based on baseline covariates, and use the empirical scores to construct empirically accessible subgroups. The resulting empirical-stratum treatment effects (ETEs) are identifiable under standard causal assumptions. We connect the proposed framework to principal stratification by showing that the average ETE recovers principal causal effects under the principal ignorability assumption, but remains informative under violations of this assumption. We further introduce projected ETE curves and develop efficient influence function-based estimators for the semiparametric inference. We illustrate the proposed framework with two real-world applications.

翻译：事后变量（如治疗不依从性、行为反应、并发事件）通常会改变治疗对主要结局的最终效果。然而，现有方法为研究基于事后变量的治疗效果异质性提供的工具十分有限。传统的异质性治疗效果估计量以基线协变量为条件进行估计。然而，类似地以观测到的事后变量为条件，会引发治疗效果估计中的内生选择偏差。主分层为研究各主层间的因果效应提供了严谨框架，但主层是潜在变量，其识别通常需要严格假设。本文发展了一种假设精简的实证分层框架，用于刻画基于事后变量的治疗效果异质性。我们利用基线协变量预测潜在事后变量响应，据此定义实证评分，并使用该评分构建可实证访问的子群。由此得到的实证层治疗效果（ETE）在标准因果假设下可识别。我们通过证明平均ETE在主层可忽略性假设下可恢复主层因果效应，但在该假设被违反时仍具有信息量，将所提框架与主分层建立联系。我们进一步引入投影ETE曲线，并开发基于高效影响函数的半参数推断估计量。通过两个实际应用案例对所提框架进行说明。