  Motivated by a potential-outcomes perspective, the idea of principal
stratification has been widely recognized for its relevance in settings
susceptible to posttreatment selection bias such as randomized clinical trials
where treatment received can differ from treatment assigned. In one such
setting, we address subtleties involved in inference for causal effects when
using a key covariate to predict membership in latent principal strata. We show
that when treatment received can differ from treatment assigned in both study
arms, incorporating a stratum-predictive covariate can make estimates of the
"complier average causal effect" (CACE) derive from observations in the two
treatment arms with different covariate distributions. Adopting a Bayesian
perspective and using Markov chain Monte Carlo for computation, we develop
posterior checks that characterize the extent to which incorporating the
pretreatment covariate endangers estimation of the CACE. We apply the method to
analyze a clinical trial comparing two treatments for jaw fractures in which
the study protocol allowed surgeons to overrule both possible randomized
treatment assignments based on their clinical judgment and the data contained a
key covariate (injury severity) predictive of treatment received.
