Identification and estimation of causal effects in the presence of confounded principal strata

Principal stratification has become a popular tool to address a broad class of causal inference questions, particularly in dealing with non-compliance and truncation by death problems. The causal effects within principal strata, which are determined by joint potential values of the intermediate variable, also known as the principal causal effects, are often of interest in these studies. The analysis of principal causal effects from observational studies mostly relies on the ignorability assumption of treatment assignment, which requires practitioners to accurately measure as many covariates as possible so that all potential sources of confounders are captured. However, in practice, collecting all potential confounding factors can be challenging and costly, rendering the ignorability assumption questionable. In this paper, we consider the identification and estimation of causal effects when treatment and principal stratification are confounded by unmeasured confounding. Specifically, we establish the nonparametric identification of principal causal effects using a pair of negative controls to mitigate unmeasured confounding, requiring they have no direct effect on the outcome variable. We also provide an estimation method for principal causal effects. Extensive simulations and a leukemia study are employed for illustration.