Exploiting multiple outcomes in Bayesian inference for causal effects with intermediate variables

Confounded post-treatment variables are often present in intervention studies and need to be adjusted for drawing valid inferences on the causal effects of interest, which are generally defined in terms of those post-treatment variables. Principal Stratification (PS) is a framework to deal with such intermediate variables. Due to the latent nature of principal strata, strong structural assumptions are often invoked to sharpen inference. As an alternative, distributional assumptions may be invoked using a model-based PS approach. These usually lead to weakly identified models, weakly in the sense that the likelihood function has substantial regions of flatness. Information on multiple outcomes is routinely collected in practice, but is rarely used to improve inference. This article develops a Bayesian approach to exploit multivariate outcomes to sharpen inferences in weakly identified PS models. Simulation studies are performed to illustrate the potential gains in estimation of causal effects from jointly modeling more than one outcome. This approach can also be used to assess plausibility of structural assumptions and sensitivity to deviations from these structural assumptions. Two model checking procedures via posterior predictive checks are also discussed. The method is applied to evaluate the causal effect of a job training program on trainees' depression. [Discussion Paper 2012-15, Department of Statistical Science, Duke University]