  Existing methods in estimating the mean outcome under a given dynamic
treatment regime rely on intention-to-treat analyses which estimate the effect
of following a certain dynamic treatment regime regardless of compliance
behavior of patients. There are two major concerns with intention-to-treat
analyses: (1) the estimated effects are often biased toward the null effect;
(2) the results are not generalizable and reproducible due to the potential
differential compliance behavior. These are particularly problematic in
settings with high level of non-compliance such as substance use disorder
treatments. Our work is motivated by the Adaptive Treatment for Alcohol and
Cocaine Dependence study (ENGAGE), which is a multi-stage trial that aimed to
construct optimal treatment strategies to engage patients in therapy. Due to
the relatively low level of compliance in this trial, intention-to-treat
analyses essentially estimate the effect of being randomized to a certain
treatment sequence which is not of interest. We fill this important gap by
defining the target parameter as the mean outcome under a dynamic treatment
regime given potential compliance strata. We propose a flexible non-parametric
Bayesian approach, which consists of a Gaussian copula model for the potential
compliances, and a Dirichlet process mixture model for the potential outcomes.
Our simulations highlight the need for and usefulness of this approach in
practice and illustrate the robustness of our estimator in non-linear and
non-Gaussian settings.
