Statistical Methods for the Comparison of Crossing Survival Curves

Abstract

Publisher Summary This chapter focuses on the detection of crossing-curve alternatives—the alternatives hypotheses where the two underlying survival curves cross each other at a time point other than at t = 0. For the comparison of crossing survival curves, the chapter presents three different major approaches: (1) a modified Kolmogorov–Smirnov test by Fleming et al., (2) a Levene-type test aimed directly at crossing-curve alternatives, and (3) a subclass of linear rank tests where locally most powerful linear ranks can be formed by proper choices of an optimal score function. The Kolmogorov–Smirnov test was not specifically designed to compare crossing survival curves. A Smirnov-type statistic is designed to measure the maximum distance between the estimates of two (survival) functions. The modified Kolmogorov–Smirnov test is more effective than the Tarone–Ware family when the two survival curves cross each other or when the two survival distributions differ substantially for some short range of time values but not necessarily elsewhere. The Levene-type test has a powerful tool to differentiate crossing survival curves. However, it appears that it does not have the flexibility that would allow studying short-term or long-term risks. The direct generalization of linear rank tests to the censored data situation is conceptually simple, but derivations of variance formulas are difficult, even in the cases of the linear score function and the logarithmic score function.