Confidence intervals for risk difference of two independent negative binomial proportions using method of variance estimates recovery

Risk difference (RD) is widely used to assess the impact of exposure in biomedical and epidemiological studies. When studying rare events, the negative binomial distribution is often preferred over the binomial distribution to ensure a sufficient number of cases of interest. This study introduces a family of new confidence interval (CI) estimators for the RD between two independent negative binomial proportions, using the method of variance estimates recovery (MOVER). These methods apply various CIs, including exact, score, fiducial, Bayesian and a modified Jeffreys interval. The modified Jeffreys approach employs a pair of proportion-specific prior parameters to derive confidence limits for the RD. The evaluation results indicate that the modified Jeffreys and fiducial methods maintain coverage probabilities closest to the nominal level while producing interval widths comparable to those of the other methods, particularly in small-sample setting. A real-data examples further illustrate the practical advantages of the proposed methods.