Studentized permutation method for comparing two restricted mean survival times with small sample from randomized trials

Abstract

Recent observations, especially in cancer immunotherapy clinical trials with time-to-event outcomes, show that the commonly used proportional hazard assumption is often not justifiable, hampering an appropriate analysis of the data by hazard ratios. An attractive alternative advocated is given by the restricted mean survival time (RMST), which does not rely on any model assumption and can always be interpreted intuitively. Since methods for the RMST based on asymptotic theory suffer from inflated type-I error under small sample sizes, a permutation test was proposed recently leading to more convincing results in simulations. However, classical permutation strategies require an exchangeable data setup between comparison groups which may be limiting in practice. Besides, it is not possible to invert related testing procedures to obtain valid confidence intervals, which can provide more in-depth information. In this paper, we address these limitations by proposing a studentized permutation test as well as respective permutation-based confidence intervals. In an extensive simulation study, we demonstrate the advantage of our new method, especially in situations with relatively small sample sizes and unbalanced groups. Finally, we illustrate the application of the proposed method by re-analyzing data from a recent lung cancer clinical trial.

Keywords: hazard ratio; permutation methods; restricted mean survival time; survival analysis; time-to-event outcomes.

FIGURE 1

The survival functions of the 9 different settings from Section 3 under the null hypothesis as well as under the alternative ${\mu }_2=1.5+{\mu }_1$ (see A and B) and the survival functions for the three different censoring scenarios (see C)

FIGURE 2

Coverage in % (nominal level $\alpha =5\%$) of the confidence intervals based on the asymptotic approximation (Asy), the empirical likelihood ratio (ELR) and the studentized permutation approach with the regenerating strategy (Perm I) as well as with the horizontal extension of the permuted Kaplan–Meier curves (Perm II), respectively. The dashed, horizontal lines represent the binomial 95%-confidence interval [94.4%, 95.6%]

FIGURE 3

Kaplan–Meier curves of the reconstructed data