Bayesian Framework for Causal Inference with Principal Stratification and Clusters

In observational studies, principal stratification is a well-established method in causal analysis to adjust the treatment effect estimation for post-treatment variables. However, this inference could be challenging when the data have a clustering structure, which is pervasive in observational studies. Adding to the issues is the fact that often the variables associated with the clusters are only recorded as the cluster label due to a budget constraint or measuring difficulties. Furthermore, the true nature of the relationship between these cluster level variables and the outcome may be unclear. Although accommodating this clustering structure via random effects based on the cluster label can address the bias issues, estimating the model is inevitably tedious and overfitting can occur with principal stratification and clustering. In this article, we propose a comprehensive framework for estimating a treatment effect when both post-treatment variable and clustering exist in a data set. Specifically, following the idea of principal stratification, we define the clustering structure as random effects with a spike and slab prior in a Bayesian hierarchical model. As a result, a parsimonious model which only contains clusters with significant effects on the outcome can be obtained without much computational cost. We demonstrate the desirable features of the proposed method with two real data sets, one about academic performance and the other about infant birth weight. To further examine the empirical performance of the proposed method, simulations with data generating mechanisms similar to our data applications, and other four hypothetical data sets are conducted.