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 Hirano et al
(2001), Yau and Little (2001) and in particular Chib (2007) and Chib and
Jacobi (2007a, 2007b), 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 which was set-up to test the
efficacy of a job training program on subsequent mental health outcomes.