Quantifying Individual Risk for Binary Outcomes

Understanding treatment effect heterogeneity is crucial for reliable decision-making in treatment evaluation and selection. The conditional average treatment effect (CATE) is widely used to capture treatment effect heterogeneity induced by observed covariates and to design individualized treatment policies. However, it is an average metric within subpopulations, which prevents it from revealing individual risk, potentially leading to misleading results. This article fills this gap by examining individual risk for binary outcomes, specifically focusing on the fraction negatively affected (FNA), a metric that quantifies the percentage of individuals experiencing worse outcomes under treatment compared with control. Even under the strong ignorability assumption, FNA is still unidentifiable, and the existing Fr\'{e}chet--Hoeffding bounds are often too wide and attainable only under extreme data-generating processes. By invoking mild conditions on the value range of the Pearson correlation coefficient between potential outcomes, we obtain improved bounds compared with the Fr\'{e}chet--Hoeffding bounds. We show that paradoxically, even with a positive CATE, the lower bound on FNA can be positive, i.e., in the best-case scenario, many individuals will be harmed if they receive treatment. Additionally, we establish a nonparametric sensitivity analysis framework for FNA using the Pearson correlation coefficient as the sensitivity parameter. Furthermore, we propose nonparametric estimators for the refined FNA bounds and prove their consistency and asymptotic normality. We use simulation to evaluate the performance of the proposed estimators and apply the method to a canonical observational study.