Leveraging density ratio models in a binary instrumental variable inference with a binary outcome: A retrospective approach

Conditional Local Risk Ratio (CLRR) is a widely used metric for assessing heterogeneous treatment effects of binary outcomes in randomized clinical trials involving noncompliance. Existing methods, such as moment-based and likelihood-based approaches, often overlook the inherent mixture structure in data, necessitate stringent parametric assumptions, or yield estimates with implausible values. In this paper, we introduce a novel semiparametric likelihood-based (SPL) method for estimating CLRR. Our method requires only three parametric model assumptions, significantly fewer than the six models needed by existing likelihood-based methods, thereby reducing model complexity and enhancing robustness. This simplicity also results in fewer unknown parameters, further boosting computational efficiency. Unlike moment-based methods, our SPL method fully exploits the mixture structure of the observed data and the principal strata framework. Additionally, our method ensures that the final CLRR estimate always fall within a valid range. We establish the asymptotic normality of our estimator and demonstrate its superiority over existing methods through numerical simulations. We further apply our method to analyze the Oregon Health Insurance Experiment dataset, providing valuable insights into the heterogeneous effects of Medicaid on both physical and mental health.