Quantifying Individual Risk for Binary Outcome
This paper addresses the challenge of quantifying individual-level treatment risk in binary-outcome settings. We propose the Fraction of Negative Average (FNA) metric—the proportion of individuals whose outcomes deteriorate upon treatment—thereby complementing the Conditional Average Treatment Effect (CATE), which captures only subgroup-level averages and obscures individual harm. Under the ignorability assumption, we introduce the Pearson correlation coefficient between potential outcomes as a sensitivity parameter and derive tight, feasible theoretical bounds for FNA—substantially improving upon the classical Fréchet–Hoeffding bounds. We establish an analytical relationship among FNA, CATE, and the correlation coefficient, revealing the counterintuitive phenomenon that positive CATE can coexist with substantial individual harm. We further propose principled guidelines for selecting plausible correlation ranges and develop a nonparametric estimator for FNA that is consistent and asymptotically normal.