A note on identification of causal effects in cluster randomized trials with post-randomization selection bias

Cluster randomized trials are commonly used for evaluating treatment effects of a cluster-level intervention. Because subjects are often recruited after cluster randomization, a main complication is the potential for selection bias due to the intervention affecting which subjects are identified as being eligible for receiving the actual treatment. In the presence of such post-randomization selection bias, direct comparison of the observed outcomes among eligible subjects between arms does not yield a valid causal effect estimate. In this note, we define average and quantile causal estimands using principal stratification in the context of post-randomization selection bias, and provide nonparametric identification assumptions and formulas for these estimands.