  Principal stratification provides a causal inference framework that allows
adjustment for confounded post-treatment variables when comparing treatments.
Although the literature has focused mainly on binary post-treatment variables,
there is a growing interest in principal stratification involving continuous
post-treatment variables. However, characterizing the latent principal strata
with a continuous post-treatment presents a significant challenge, which is
further complicated in observational studies where the treatment is not
randomized. In this paper, we introduce the Confounders-Aware SHared atoms
BAyesian mixture (CASBAH), a novel approach for principal stratification with
continuous post-treatment variables that can be directly applied to
observational studies. CASBAH leverages a dependent Dirichlet process,
utilizing shared atoms across treatment levels, to effectively control for
measured confounders and facilitate information sharing between treatment
groups in the identification of principal strata membership. CASBAH also offers
a comprehensive quantification of uncertainty surrounding the membership of the
principal strata. Through Monte Carlo simulations, we show that the proposed
methodology has excellent performance in characterizing the latent principal
strata and estimating the effects of treatment on post-treatment variables and
outcomes. Finally, CASBAH is applied to a case study in which we estimate the
causal effects of US national air quality regulations on pollution levels and
health outcomes.
