Likelihood-based instrumental variable methods for Cox proportional hazards model

The Cox proportional hazards model is widely used for analyzing time-to-event outcomes while adjusting for covariates. However, obtaining a consistent estimator can be challenging if certain covariates remain unobserved. Instrumental variable (IV) methods are frequently employed to address unmeasured covariates; however, standard IV methods cannot be applied to time-to-event outcomes without modification. In this study, we propose extending IV methods capable of handling binary, multivalue, or continuous treatments, as well as non-binary and multiple IVs, which has not been achieved by previous methods. Our approach leverages the frailty model to directly capture the variability of unmeasured covariates and allows for flexible treatment models. The proposed method outperforms previous approaches, as demonstrated by theoretical considerations and numerical experiments. Furthermore, we performed a real data analysis using health insurance claims data from Hiroshima Prefecture, Japan. The proposed method is expected to address the issue of unmeasured covariates in time-to-event outcomes across a wider range of scenarios. Additionally, it can serve as a type of sensitivity analysis to assess the influence of unmeasured covariates.