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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>χ</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:math> with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>J</mml:mi></mml:math> degrees of freedom in the multivariate case, with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>J</mml:mi></mml:math> 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.