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In causal inference studies, interest often lies in understanding themechanisms through which a treatment affects an outcome. One approach isprincipal stratification (PS), which introduces well-defined causal effects inthe presence of confounded post-treatment variables, or mediators, and clearlydefines the assumptions for identification and estimation of those effects. Thegoal of this paper is to extend the PS framework to studies with continuoustreatments and continuous post-treatment variables, which introduces a numberof unique challenges both in terms of defining causal effects and performinginference. This manuscript provides three key methodological contributions: 1)we introduce novel principal estimands for continuous treatments that providevaluable insights into different causal mechanisms, 2) we utilize Bayesiannonparametric approaches to model the joint distribution of the potentialmediating variables based on both Gaussian processes and Dirichlet processmixtures to ensure our approach is robust to model misspecification, and 3) weprovide theoretical and numerical justification for utilizing a model for thepotential outcomes to identify the joint distribution of the potentialmediating variables. Lastly, we apply our methodology to a novel study of therelationship between the economy and arrest rates, and how this is potentiallymediated by police capacity.

Principal stratification with continuous treatments and continuous post-treatment variables