A Bayesian Semiparametric Approach to Intermediate Variables in Causal Inference

In causal inference studies, treatment comparisons often need to be adjusted for confounded post-treatment variables. Principal stratification (PS) is a framework to deal with such variables within the potential outcome approach to causal inference. Continuous intermediate variables introduce inferential challenges to PS analysis. Existing methods either dichotomize the intermediate variable, or assume a fully parametric model for the joint distribution of the potential intermediate variables. However, the former is subject to information loss and arbitrary choice of the cutoff point and the latter is often inadequate to represent complex distributional and clustering features. We propose a Bayesian semiparametric approach that consists of a flexible parametric model for the potential outcomes and a Bayesian nonparametric model for the potential intermediate outcomes using a Dirichlet process mixture (DPM) model. The DPM approach provides flexibility in modeling the possibly complex joint distribution of the potential intermediate outcomes and offers better interpretability of results through its clustering feature. Gibbs sampling based posterior inference is developed. We illustrate the method by two applications: one concerning partial compliance in a randomized clinical trial, and one concerning the causal mechanism between physical activity, body mass index, and cardiovascular disease in the observational Swedish National March Cohort study.