Principal stratification with continuous treatments and continuous post-treatment variables

Abstract

Principal stratification (PS) is a commonly used approach for understanding the mechanisms through which a treatment affects an outcome. The goal of this work is to extend the PS framework to studies with continuous treatments, which introduces a number of both challenges and opportunities in terms of defining causal effects and performing inference. This manuscript provides multiple key methodological contributions: 1) we introduce principal causal estimands for continuous treatments that provide insights into different causal mechanisms, 2) we show that nonparametric identification is possible under a principal ignorability assumption, but only under a restrictive assumption on the joint distribution of potential mediators, which can be dropped under mild parametric assumptions, 3) we utilize nonparametric Bayesian models for the joint distribution of the potential mediating variables to ensure our approach is robust to model misspecification, and 4) we provide theoretical justification for utilizing an outcome model to identify the joint distribution of the potential mediating variables, and show that this is only possible if a principal ignorability assumption is violated. 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.