Intensity Adjusted Log-Rank Test
Wu Y and Huag X
Published on: 2025-10-09
Abstract
The log-rank test is a widely used nonparametric method for comparing survival distributions between groups. However, its power diminishes when the proportional hazards condition does not hold, which is common in practice. To address this limitation, we propose the intensity-adjusted log-rank test-a novel extension that adapts to non-proportional hazards while maintaining the simplicity and enhancing the interpretability of the original test. Our approach incorporates time-varying adjustment factors that automatically account for changing hazard ratios, thereby eliminating the need (and the associated bias) for weight specification required by weighted log-rank tests. Simulation studies and a data example demonstrate that the intensity-adjusted log-rank test achieves significant power gains over the standard log-rank test (up to a 55% improvement) under non-proportional hazards, while preserving Type I error control. We introduce a general class of log-rank type tests that includes both the standard log-rank test and our proposed method as specific cases. This new method provides researchers with a flexible, robust, and more powerful tool for survival comparisons when hazards are non-proportional, effectively addressing key limitations of traditional log-rank and weighted log-rank tests.
Keywords
Log-rank test; Intensity adjusted; Non-proportional hazard; Power enhancement; Type I error controlIntroduction
The log-rank test has long served as the fundamental method for comparing survival distributions between groups since its introduction by Nathan Mantel in 1966 [1] and subsequent refinement by Richard and Julian Peto [2]. As a nonparametric method that elegantly handles right-censored data, it has become essential in clinical trials and medical research. However, its reliance on the proportional hazards assumption represents an important limitation- when hazard functions diverge over time or survival curves cross, the test may fail to detect meaningful differences or produce misleading conclusions. The difference between two survival curves can arise from two key factors: a discrepancy in the total number of observed events, and/or a time-varying intensity of event occurrence. While the log-rank test is a powerful statistical test under the condition of proportional hazards, its power diminishes when hazard ratios are time-dependent.
To address this fundamental limitation by accounting for differences in both event counts and time-varying intensity, we propose the intensity-adjusted log-rank test, a novel modification that maintains the simplicity and interpretability of the traditional log-rank test while adapting to non-proportional hazards scenarios. The intensity-adjusted log-rank test incorporates time-varying functions that automatically adjust for changing hazard ratios over time, providing robust performance across various patterns of survival difference. Unlike existing weighted methods that require strong assumptions or weight pre-specification, our approach preserves the intuitive framework of the standard log-rank test while overcoming its most significant constraint.
Some approaches, such as weighted log-rank tests [3] and the MaxCombo test, [4,5] require pre-specified weights, which can present certain considerations in survival analysis. The process of selecting appropriate weights generally involves assumptions and justifications that may be challenging to substantiate in practice, especially when prior knowledge is limited. This level of subjectivity can be a point of discussion in regulatory review, as agencies like the FDA may request clear reasoning for methodological choices that could influence trial outcomes.