Differentially Private Confidence Intervals for Relative Risk and Odds Ratio

Abstract

Ratio statistics, such as relative risk and odds ratio, are central tools for comparing outcome probabilities across populations, particularly in clinical and epidemiological studies. However, when the underlying counts are released under differential privacy, the resulting ratio estimators can exhibit nontrivial bias and variance, and principled methods for statistical inference in this setting remain limited. In this work, we study estimation and inference for ratio statistics computed from counts perturbed by additive Laplace or Gaussian noise. We show that the natural plug-in estimators based on privatized counts are consistent, and we derive confidence intervals that explicitly account for both sampling variability and privacy-induced noise, making our approach appealing for moderate sample sizes. Our analysis provides transparent variance decompositions that clarify when privacy meaningfully affects inference. Empirical results on simulated and real datasets demonstrate that, for moderate privacy levels and sample sizes, the proposed confidence intervals closely match their non-private counterparts, while properly reflecting additional uncertainty in small-sample regimes. Our results provide both theoretical guarantees and practical guidance for conducting statistical inference with ratio statistics under differential privacy.