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The Stata Journal
Volume 16 Number 3: pp. 678-690



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Versatile tests for comparing survival curves based on weighted log-rank statistics

Theodore G. Karrison
Department of Public Health Sciences
University of Chicago
Chicago, IL
tkarrison@health.bsd.uchicago.edu
Abstract.  The log-rank test is perhaps the most commonly used nonparametric method for comparing two survival curves and yields maximum power under proportional hazards alternatives. While the assumption of proportional hazards is often reasonable, it need not hold. Several authors have therefore developed versatile tests using combinations of weighted log-rank statistics that are more sensitive to nonproportional hazards. Fleming and Harrington (1991, Counting Processes and Survival Analysis, Wiley) consider the family of (G^{ ho}) statistics and their supremum versions, while Lee (1996, Biometrics 52: 721–725) and Lee (2007, Computational Statistics and Data Analysis 51: 6557–6564) propose tests based on the more extended (G^{ ho,gamma}), family. In this article, I consider (Z_m = mathrm{max}(|Z_1|, |Z_2|, |Z_3|)), where (Z_1), (Z_2), and (Z_3) are z statistics obtained from (G^{0,0}), (G^{1,0}), and (G^{0,1}) tests, respectively. (G^{0,0}) corresponds to the log-rank test, while (G^{1,0}) and (G^{0,1}) are more sensitive to early and late-difference alternatives. I conduct a simulation study to compare the performance of (Z_m) with the log-rank test, the more optimally weighted test, and Lee’s (2007) tests, under the null hypothesis, proportional hazards, early difference, and late-difference alternatives. Results indicate that the method based on (Z_m) maintains the type I error rate, provides increased power relative to the log-rank test under early difference and late-difference alternatives, and entails only a small to moderate power loss compared with the more optimally chosen test. I apply the procedure to two datasets reported in the literature, both of which exhibit nonproportional hazards. Versatile tests such as (Z_m) may be useful in clinical trial settings where there is concern that the treatment effect may not conform to the proportional hazards assumption. I also describe the syntax for a Stata command, verswlr, to implement the method.
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