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
[email protected]
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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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Theodore G. Karrison
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verswlr, survival curves, log-rank test, nonproportional hazards, versatile tests, power
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