A Permutation Test for a Modified Spearman’s Rank Correlation Using Martingale-Residual Ranks Under Right Censoring

This note presents a robust permutation testing approach for assessing a modification of the Spearman rank correlation coefficient between right-censored variables. The method is based on univariate martingale residuals (UMRs), which provide a principled rank-generating transformation that incorporates censoring through the Kaplan–Meier estimators while reducing to the identity transformation when no censoring is present. Under the null hypothesis \(H_0 : \rho_s = 0\), we apply the studentized permutation framework to the paired UMRs, yielding a test whose permutation distribution consistently estimates the sampling distribution of the statistic. The resulting procedure achieves asymptotically correct Type I error control, applies when one or both margins are censored, and reduces exactly to the classical Spearman permutation test in the absence of censoring. Simulation studies demonstrate strong Type I error control and competitive power across a range of censoring scenarios, and two real-data examples illustrate the method’s practical utility. Readily accessible R software implementations support the proposed test.