Construction of confidence intervals for risk difference with paired correlated data using saddlepoint approximation

In clinical studies involving paired organs (e.g. eyes or ears), correlated binary outcomes necessitate specialized statistical methods to account for intra-subject dependencies. This paper proposes a saddlepoint approximation (SA) method for constructing confidence intervals (CI) for risk differences in paired binary data under Donner's correlation model. Compared to conventional approaches - Wald, likelihood ratio, score, and MOVER methods - the SA method explicitly incorporates higher-order moment information, offering improved accuracy in small-sample or rare-event settings. Simulation studies evaluate empirical coverage probability (ECP) and mean interval width (MIW) across varying correlation levels. Results demonstrate that SA consistently maintains ECPs near the nominal 95% level while achieving narrower intervals than competing methods, particularly under high correlation. In contrast, likelihood and score tests exhibit undercoverage, and Wald/MOVER intervals are overly conservative. An application to otitis media trial data further validates SA's utility, yielding interpretable inferences for correlated bilateral outcomes. The proposed method bridges a critical gap in small-sample inference for risk differences, ensuring robustness without reliance on large-sample approximations.