Saddlepoint Inference for a Proportional Reversed-Hazard Rank Test with Interval-Censored Survival Data

Interval censoring commonly arises in clinical trials, screening studies, and longitudinal medical investigations in which event status is assessed only at scheduled examination times. Rank-based procedures provide flexible tools for comparing interval-censored (IC) event-time distributions, but inference is usually based on first-order normal approximations that may be inaccurate in small or moderately sized samples and under substantial censoring. We develop a saddlepoint approximation (SPA) to the conditional permutation distribution of a linear rank statistic derived from the proportional reversed-hazard model with IC data. Conditional on the observed group size, the permutation distribution is represented through a bivariate cumulant generating function, and Skovgaard’s approximation is used to obtain computationally efficient tail probabilities without exhaustive permutation enumeration. The finite-sample performance of the proposed method is evaluated under log-normal, Weibull, and Gompertz event-time distributions and under monitoring schemes producing predominantly left, interval, or right-censored observations. Monte Carlo (MC) permutation p-values based on 106 random permutations are used as a numerical benchmark (not the exact permutation distribution). Across the evaluated simulation scenarios, the SPA generally produces p-values that are closer to the MC permutation benchmark than those obtained from the standard normal approximation (NA). Applications to lung tumor, HIV drug-resistance, and breast-cosmesis data illustrate the relevance of the method to biomedical event-time studies. The proposed approximation provides an accurate and computationally efficient approach to rank-based inference for IC medical data.