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Existing methods for estimating the mean outcome under a given sequential treatment rule often rely on intention‐to‐treat analyses, which estimate the effect of following a certain treatment rule 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 potentially differential compliance behavior. These are particularly problematic in settings with a high level of non‐compliance, such as substance use disorder studies. 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, instead of the actual effect of the treatment. We obviate this challenge by defining the target parameter as the mean outcome under a dynamic treatment regime conditional on a potential compliance stratum. We propose a flexible non‐parametric Bayesian approach based on principal stratification, which consists of a Gaussian copula model for the joint distribution of the potential compliances, and a Dirichlet process mixture model for the treatment sequence specific outcomes. We conduct extensive simulation studies which highlight the utility of our approach in the context of multi‐stage randomized trials. We show robustness of our estimator to non‐linear and non‐Gaussian settings as well.

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.

In this paper, we consider Bayesian inference on a type of multivariate median and the multivariate quantile functionals of a joint distribution using a Dirichlet process prior. Since, unlike univariate quantiles, the exact posterior distribution of multivariate median and multivariate quantiles are not obtainable explicitly, we study these distributions asymptotically. We derive a Bernstein-von Mises theorem for the multivariate 1-median with respect to a general p-norm, which in particular shows that its posterior concentrates around its true value at the n −1/2 -rate and its credible sets have asymptotically correct frequentist coverages. In particular, asymptotic normality results for the empirical multivariate median with a general p-norm is also derived in the course of the proof, which extends the results from the case p = 2 in the literature to a general p. The technique involves approximating the posterior Dirichlet process by a Bayesian bootstrap process and deriving a conditional Donsker theorem. We also obtain analogous results for an affine equivariant version of the multivariate 1-median based on an adaptive transformation and re-transformation technique. The results are extended to a joint distribution of multivariate quantiles. The accuracy

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