Designing clinical trials for the comparison of single and multiple quantiles with right-censored data

Based on the test for equality of quantiles originally introduced by Kosorok (1999), we propose new power formulas for the comparison of one quantile between two treatment groups, as well as for the comparison of a collection of quantiles. Under the null hypothesis of equality of quantiles, the test statistic follows asymptotically a normal distribution in the univariate case and a chi-squared with J degrees of freedom in the multivariate case, with J the number of quantiles compared. The variance of the test statistic depends on the estimation of the probability density function of the distribution of failure times at the quantile being tested. In order to apply the test on real data, we propose to estimate this quantity using a resampling-based method, as an alternative to Kosorok’s original kernel density estimator. The whole procedure provides a practical tool for designing and analyzing data arising from clinical trials using quantiles of survival as an endpoint. Simulation studies are performed to show the appropriateness of the power formulas. We illustrate the proposed test in a phase III randomized clinical trial where the proportional hazards assumption between treatment arms does not hold.

💡 Research Summary

This paper extends the non‑parametric two‑sample test for equality of survival quantiles originally introduced by Kosorok (1999) to a fully operational tool for clinical trial design and analysis with right‑censored data. The authors first derive explicit asymptotic power formulas for both the univariate case (comparing a single quantile) and the multivariate case (comparing a vector of J pre‑specified quantiles). Under the null hypothesis of equal quantiles, the test statistic converges to a standard normal distribution for a single quantile and to a chi‑square distribution with J degrees of freedom for multiple quantiles. Under a fixed alternative where the true difference in quantiles is Δ (or a vector Δj), the statistic is asymptotically normal (or a non‑central chi‑square) with non‑centrality parameters that depend on the sample size n, the effect size Δ, and a variance term σ². The variance term itself involves the density of the failure‑time distribution evaluated at the quantile of interest, the cumulative hazard, and the censoring distribution.

A major methodological contribution is the replacement of Kosorok’s original kernel density estimator with a resampling‑based estimator inspired by Lin et al. (2011). The new estimator generates multiple realizations of a centered Gaussian variable with a variance tuned by a grid‑search, then fits a least‑squares model to obtain a direct estimate of the density at the target quantile. This approach avoids bandwidth selection, reduces mean‑squared error, and provides a consistent estimate needed for σ².

The paper presents the derived power formulas in closed form, enabling straightforward calculation of the required sample size for a desired power (1‑β) at a given significance level α, assuming known survival and censoring distributions. The authors illustrate the method with two simulation scenarios: (1) proportional hazards where both arms follow exponential distributions, and (2) non‑proportional hazards with a piecewise‑exponential experimental arm that exhibits a delayed treatment effect. For each scenario, analytical expressions for the quantiles, the variance components ϕk, and the overall σ² are provided, allowing exact power computation without Monte‑Carlo simulation.

To demonstrate real‑world applicability, the authors apply the test to data from the OAK phase‑III lung cancer trial, a setting where the proportional hazards assumption is violated due to immunotherapy’s delayed effect. They compare both the median survival time and the 75th percentile between control and experimental arms, finding statistically significant differences for both quantiles. This showcases the method’s ability to capture treatment benefits that traditional hazard‑ratio analyses might miss.

All computational procedures are implemented in R and made publicly available on GitHub, including functions for Kaplan‑Meier estimation, the resampling density estimator, power and sample‑size calculations, and the final test statistic. The supplemental material provides additional technical proofs, algorithmic details, and extended simulation results.

In summary, the paper delivers a comprehensive statistical framework for quantile‑based comparison of survival curves with censored data: it supplies explicit power and sample‑size formulas, introduces a more accurate and user‑friendly density estimator, validates the approach through extensive simulations, and proves its practical value on an actual immuno‑oncology trial. This work fills a critical gap in the design of clinical trials where proportional hazards do not hold, offering clinicians and statisticians a robust alternative metric—differences in survival quantiles—to quantify treatment effects.