Factorial survival designs with right-censored observations are commonly inferred by Cox regression and explained by means of hazard ratios. However, in case of non-proportional hazards, their interpretation can become cumbersome; especially for clinicians. We therefore offer an alternative: median survival times are used to estimate treatment and interaction effects and null hypotheses are formulated in contrasts of their population versions. Permutation-based tests and confidence regions are proposed and shown to be asymptotically valid. Their type-1 error control and power behavior are investigated in extensive simulations, showing the new methods’ wide applicability. The latter is complemented by an illustrative data analysis.
ArborettiRFontanaRPesarinF, et al.
Nonparametric combination tests for comparing two survival curves with informative and non-informative censoring. Stat Methods Med Res2018;
12: 3739–3769.
2.
BrendelMJanssenAMayerCD, et al.
Weighted logrank permutation tests for randomly right censored life science data. Scand J Stat2015;
41: 742–761.
3.
DitzhausMPaulyM.Wild bootstrap logrank tests with broader power functions for testing superiority. Comp Statist Data Anal2019;
136: 1–11.
4.
DitzhausMFriedrichS.More powerful logrank permutation tests for two-sample survival data. J Stat Comput Simul2020;
90: 2209–2227.
5.
DitzhausMJanssenAPaulyM.Permutation inference in factorial survival designs with the CASANOVA. arXiv:2004.108182020.
6.
GorfineMSchlesingerMHsuL.k-sample omnibus non-proportional hazards tests based on right-censored data.Stat Methods Med Res2020;
29: 2830–2850.
7.
EllisLMBernsteinDSVoestEE, et al.
American society of clinical oncology perspective: raising the bar for clinical trials by defining clinically meaningful outcomes.J Clin Oncol2014;
32: 1277–1280.
8.
BrücknerMBrannathW.Sequential tests for non-proportional hazards data.Lifetime Data Anal2017;
23: 339–352.
9.
KalbfleischJDPrenticeRL.Estimation of the average hazard ratio. Biometrika1981;
68: 105–112.
10.
DoblerDPaulyM.Bootstrap-and permutation-based inference for the Mann–Whitney effect for right-censored and tied data. TEST2018;
27: 639–658.
11.
DoblerDPaulyM.Factorial analyses of treatment effects under independent right-censoring.Stat Methods Med Res2020;
29: 325–343.
12.
KoziolJAJiaZ.The concordance index C and the Mann–Whitney parameter Pr(X > Y) with randomly censored data.Biom J2009;
51: 467–474.
13.
Ben-AharonOMagneziRLeshnoM, et al.
Median survival or mean survival: which measure is the most appropriate for patients, physicians, and policymakers?Oncologist2019;
24: 1469.
14.
RoystonPParmarMKB.Restricted mean survival time: an alternative to the hazard ratio for the design and analysis of randomized trials with a time-to-event outcome.BMC Med Res Methodol2013;
13: 152.
15.
BrookmeyerRCrowleyJ.A confidence interval for the median survival time. Biometrics1982;
38: 29–41.
16.
BrookmeyerRCrowleyJ.A k-sample median test for censored data. J Amer Statist Assoc1982;
77: 433–440.
17.
ChenZZhangG.Comparing survival curves based on medians.BMC Med Res Methodol2016;
16: 33.
18.
MartinussenTPipperCB.Estimation of odds of concordance based on the Aalen additive model.Lifetime Data Anal2013;
19: 100–116.
19.
GreenS. Factorial designs with time to event endpoints. In: Crowley J and Ankerst DP (eds) Handbook of statistics in clinical oncology. Boca Raton, FL: Chapman and Hall/CRC, 2005, pp.201–210.
20.
ChenYIChangYM.Covariates-dependent confidence intervals for the difference or ratio of two median survival times.Stat Med2007;
26: 2203–2213.
21.
SuJQWeiLJ.Nonparametric estimation for the difference or ratio of median failure times. Biometrics1983;
49: 603–607.
22.
DitzhausMFriedRPaulyM.QANOVA: quantile-based permutation methods for general factorial designs. arXiv:1912.09146v22019.
23.
ChungEYRomanoJP.Exact and asymptotically robust permutation tests. Ann Statist2013;
41: 484–507.
24.
JanssenAPaulsT.How do bootstrap and permutation tests work?Ann Statist2013;
3: 768–806.
25.
PaulyMBrunnerEKonietschkeF.Asymptotic permutation tests in general factorial designs. J R Stat Soc Ser B Stat Methodol2015;
77: 461–473.
SmagaŁ.Diagonal and unscaled wald-type tests in general factorial designs. Electron J Stat2017;
11: 2613–2646.
28.
SanderJM. The weak convergence of quantiles of the product-limit estimator. Standford University, Department of Statistics, Technical report number 5, 1975.
29.
McKeanJWSchraderRG.A comparision of methods for studentizing the sample mean. Commun Statist B1984;
13: 751–773.
30.
PriceRMBonettDG.Estimating the variance of the sample median. J Stat Comput Simul2001;
68: 295–305.
31.
ChakrabortiS.Large sample tests for equality of medians under unequal right-censoring. Comm Statist Theory Methods1988;
17: 4075–4084.
32.
ChakrabortiS.A class of tests for homogeneity of quantiles under unequal right-censorship. Statist Probab Lett1990;
9: 107–109.
33.
AalenO.Nonparametric inference in connection with multiple decrement models. Scand J Stat1976;
3: 15–27.
34.
HemerikJGoemanJ.Exact testing with random permutations.TEST2018;
27: 811–825.
35.
PaulyMAsendorfTKonietschkeF.Permutation-based inference for the AUC: a unified approach for continuous and discontinuous data.Biometrical J2016;
58: 1319–1337.
MichielsSPiedboisPBurdettS, et al.
Meta-analysis when only the median survival times are known: a comparison with individual patient data results.Int J Technol Assess2005;
21: 119–125.
44.
WeiYRoystonPTierneyJF, et al.
Meta-analysis of time-to-event outcomes from randomized trials using restricted mean survival time: application to individual participant data.Stat Med2015;
34: 2881–2898.
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