  In observational studies, unmeasured confounders present a crucial challenge
in accurately estimating desired causal effects. To calculate the hazard ratio
(HR) in Cox proportional hazard models for time-to-event outcomes, two-stage
residual inclusion and limited information maximum likelihood are typically
employed. However, these methods are known to entail difficulty in terms of
potential bias of HR estimates and parameter identification. This study
introduces a novel nonparametric Bayesian method designed to estimate an
unbiased HR, addressing concerns that previous research methods have had. Our
proposed method consists of two phases: 1) detecting clusters based on the
likelihood of the exposure and outcome variables, and 2) estimating the hazard
ratio within each cluster. Although it is implicitly assumed that unmeasured
confounders affect outcomes through cluster effects, our algorithm is
well-suited for such data structures. The proposed Bayesian estimator has good
performance compared with some competitors.
