  In comparative research on time-to-event data for two groups, when two
survival curves cross each other, it may be difficult to use the log-rank test
and hazard ratio (HR) to properly assess the treatment benefit. Our aim was to
identify a method for evaluating the treatment benefits for two groups in the
above situation. We quantified treatment benefits based on an intuitive measure
called the area between two survival curves (ABS), which is a robust measure of
treatment benefits in clinical trials regardless of whether the proportional
hazards assumption is violated or two survival curves cross each other.
Additionally, we propose a permutation test based on the ABS, and we evaluate
the effectiveness and reliability of this test with simulated data. The ABS
permutation test is a robust statistical inference method with an acceptable
type I error rate and superior power to detect differences in treatment
effects, especially when the proportional hazards assumption is violated. The
ABS can be used to intuitively quantify treatment differences over time and
provide reliable conclusions in complicated situations, such as crossing
survival curves.
  The R Package "ComparisonSurv" contains the proposed methods and is available
from https://CRAN.R-project.org/package=ComparisonSurv.
  Keywords: Survival analysis; Area between two survival curves; Crossing
survival curves; Treatment benefit
