A Kernel Log-Rank Test of Independence for Right-Censored Data

We introduce a general nonparametric independence test between right-censored survival times and covariates, which may be multivariate. Our test statistic has a dual interpretation, first in terms of the supremum of a potentially infinite collection of weight-indexed log-rank tests, with weight functions belonging to a reproducing kernel Hilbert space (RKHS) of functions; and second, as the norm of the difference of embeddings of certain finite measures into the RKHS, similar to the Hilbert–Schmidt Independence Criterion (HSIC) test-statistic. We study the asymptotic properties of the test, finding sufficient conditions to ensure our test correctly rejects the null hypothesis under any alternative. The test statistic can be computed straightforwardly, and the rejection threshold is obtained via an asymptotically consistent Wild Bootstrap procedure. Extensive investigations on both simulated and real data suggest that our testing procedure generally performs better than competing approaches in detecting complex nonlinear dependence.

本文提出了一种针对右删失生存时间与协变量（可为多变量）的通用非参数独立性检验方法。本检验统计量具有双重解读：其一，可表示为一组潜在无穷多的加权对数秩检验的上确界，其中权重函数属于再生核希尔伯特空间（reproducing kernel Hilbert space, RKHS）；其二，可表示为特定有限测度嵌入至该RKHS后的差值的范数，其形式与希尔伯特-施密特独立性准则（Hilbert–Schmidt Independence Criterion, HSIC）的检验统计量类似。本文研究了该检验的渐近性质，推导得到充分条件以确保本检验在任意备择假设下均可正确拒绝原假设。该检验统计量可直接计算，拒绝阈值则通过渐近一致的野生自助法（Wild Bootstrap）流程获取。针对模拟数据与真实数据的大量实验结果表明，本检验方法在检测复杂非线性相关性方面，整体性能优于现有同类竞争方法。