Causal effects from panel data in randomized experiments with partial compliance

We present Bayesian models for finding the longitudinal causal effects of a randomized two-arm training program when compliance with the randomized assignment is less than perfect in the training arm (but perfect in the non-training arm) for reasons that are potentially correlated with the outcomes. We deal with the latter confounding problem under the principal stratification framework of Sommer and Zeger (1991) and Frangakis and Rubin (1999), and others. Building on the Bayesian contributions of Imbens and Rubin (1997), Hirano et al. (2000), Yau and Little (2001) and in particular Chib (2007) and Chib and Jacobi (2007, 2008), we construct rich models of the potential outcome sequences (with and without random effects), show how informative priors can be reasonably formulated, and present tuned computational approaches for summarizing the posterior distribution. We also discuss the computation of the marginal likelihood for comparing various versions of our models. We find the causal effects of the observed intake from the predictive distribution of each potential outcome for compliers. These are calculated from the output of our estimation procedures. We illustrate the techniques and ideas with data from the 1994 JOBS II trial that was set up to test the efficacy of a job training program on subsequent mental health outcomes.