Saddlepoint inference for rank-based k-sample tests in clustered survival trials

Statistical inference for cluster randomized trials often involves complex derived endpoints, such as biomarker ratios or cumulative products, which significantly complicate the application of standard asymptotic theory. When the number of clusters is limited or intra-cluster dependence is pronounced, conventional nonparametric procedures relying on large-sample chi-squared approximations frequently break down, leading to inflated Type I error rates and inadequate confidence interval coverage. This paper develops a rigorous, unified framework to circumvent these inferential failures by constructing a rank-based k-sample test statistic specifically tailored for clustered, right-censored survival data. We derive a multivariate saddlepoint approximation for the exact permutation distribution under an urn randomization scheme, providing a solution that yields higher-order precision. This analytical strategy recovers the accuracy of computationally exhaustive Monte Carlo resampling at a fraction of the numerical cost. Furthermore, the efficiency of the saddlepoint approach enables the practical construction of nonparametric confidence intervals for relative treatment effects via test inversion. Extensive simulations demonstrate that our proposed method preserves nominal significance levels in the small-cluster regimes where standard techniques prove unreliable. The practical utility of the methodology is further evidenced through the analysis of three multi-center clinical trials-concerning leukemia, vision loss, and periodontitis-where the saddlepoint approximation provides robust and trustworthy inferential conclusions in borderline cases that would otherwise yield conflicting results.