In causal inference studies, interest often lies in understanding the mechanisms through which a treatment affects an outcome. One approach is principal stratification (PS), which introduces well-defined causal effects in the presence of confounded post-treatment variables, or mediators, and clearly defines the assumptions for identification and estimation of those effects. The goal of this paper is to extend the PS framework to studies with continuous treatments and continuous post-treatment variables, which introduces a number of unique challenges both in terms of defining causal effects and performing inference. This manuscript provides three key methodological contributions: 1) we introduce novel principal estimands for continuous treatments that provide valuable insights into different causal mechanisms, 2) we utilize Bayesian nonparametric approaches to model the joint distribution of the potential mediating variables based on both Gaussian processes and Dirichlet process mixtures to ensure our approach is robust to model misspecification, and 3) we provide theoretical and numerical justification for utilizing a model for the potential outcomes to identify the joint distribution of the potential mediating variables. Lastly, we apply our methodology to a novel study of the relationship between the economy and arrest rates, and how this is potentially mediated by police capacity.

翻译：在因果推断研究中，核心目标常在于理解处理变量影响结果变量的作用机制。主分层方法（PS）作为一种有效分析框架，可在存在混杂事后变量（即中介变量）时定义清晰的因果效应，并明确阐明识别与估计这些效应所需的假设条件。本文旨在将主分层框架拓展至连续处理变量与连续事后变量的研究场景，这为定义因果效应及开展统计推断引入了若干独特挑战。本研究提出三项关键方法论贡献：1) 针对连续处理变量定义新颖的主层估计量，为不同因果机制提供重要洞见；2) 利用基于高斯过程和狄利克雷过程混合模型的贝叶斯非参数方法，对潜在中介变量的联合分布进行建模，确保方法对模型误设具有稳健性；3) 从理论与数值角度论证采用潜在结果模型识别潜在中介变量联合分布的合理性。最后，我们将所提方法论应用于一项关于经济与逮捕率关系及其通过警察能力产生中介效应机制的前沿研究。