Clustered data with censored failure times frequently arise in clinical trials and tumorigenicity studies. For such data, the common and extensively used class of two-sample tests is the weighted log-rank tests. In this article, a double saddlepoint approximation is used to calculate the p-values of the null permutation distribution of these tests. This technique is demonstrated using three real clustered data sets. Comprehensive simulation studies are conducted to appraise the efficiency of the saddlepoint approximation. This approximation is far superior to the asymptotic normal approximation. This precision allows us to determine almost exact confidence intervals for the treatment impact.
RosnerBGroveD.Use of the Mann–Whitney U-test for clustered data.Stat Med1999;
18: 1387–1400.
2.
RosnerBGlynnRJLeeML.Incorporation of clustering effects for the Wilcoxon rank sum test: a large-sample approach.Biometrics2003;
59: 1089–1098.
3.
RosnerBGlynnRJLeeMLT.Extension of the rank sum test for clustered data: two-group comparisons with group membership defined at the subunit level.Biometrics2006;
62: 1251–1259.
4.
DattaSSattenGA.Rank-sum tests for clustered data. J Am Stat Assoc2005;
100: 908–915.
5.
JungSHJeongJH.Rank tests for clustered survival data.Lifetime Data Anal2003;
9: 21–33.
6.
D’AbreraHLehmannE.Nonparametrics: statistical methods based on ranks.
San Francisco:
Holden-Day, 1975.
7.
RosenbergerWFLachinJM.Randomization in clinical trials: theory and practice.
New Jersey:
John Wiley & Sons, 2015.
8.
ButlerRW.Saddlepoint approximations with applications.
Cambridge:
Cambridge University Press, 2007.
9.
DanielsHE.Saddlepoint approximations in statistics. Ann Math Statist1954;
25: 631–650.
10.
LugannaniRRiceS.Saddle point approximation for the distribution of the sum of independent random variables. Adv Appl Probab1980;
12: 475–490.
11.
SkovgaardIM.Saddlepoint expansions for conditional distributions. J Appl Probab1987;
24: 875–887.
12.
WangS.Saddlepoint approximations for bivariate distributions. J Appl Probab1990;
27: 586–597.
13.
RobinsonJ.Saddlepoint approximations for permutation tests and confidence intervals. J R Statist Soc B1982;
44: 91–101.
14.
DavisonACHinkleyDV.Saddlepoint approximations in resampling methods. Biometrika1988;
75: 417–431.
15.
BoothJGButlerRW.Randomization distributions and saddlepoint approximations in generalized linear models. Biometrika1990;
77: 787–796.
16.
Abd-ElfattahEFButlerRW.The weighted log-rank class of permutation tests: p-values and confidence intervals using saddlepoint methods. Biometrika2007;
94: 543–551.
17.
Abd-ElfattahEFButlerRW.Log-rank permutation tests for trend: saddlepoint p-values and survival rate confidence intervals. Can J Stat2009;
37: 5–16.
18.
Abd-ElfattahEF.The weighted log-rank class under truncated binomial design: saddlepoint p-values and confidence intervals.Lifetime Data Anal2012;
18: 247–259.
19.
Abd-ElfattahEF.Saddlepoint p-values and confidence intervals for the class of linear rank tests for censored data under generalized randomized block design. Comput Stat2015;
30: 593–604.
20.
Abd-ElfattahEF.Saddlepoint p-values for a class of tests for comparing competing risks with censored data. J Appl Stat2015;
42: 1782–1791.
21.
Abd-ElfattahEF.Saddlepoint p-values for two-sample bivariate tests. J Stat Plan Infer2016;
171: 92–98.
22.
Abd-ElfattahEF.Saddlepoint p-values for group of linear rank tests. Commun Stat-Simul C2017;
46: 4274–4280.
23.
RoutledgeR.Practicing safe statistics with the mid-p. Can J Stat1994;
22: 103–110.
24.
FlemingTRHarringtonDP.A class of hypothesis tests for one and two sample censored survival data. Commun Stat-Theor M1981;
10: 763–794.
25.
MantelN.Evaluation of survival data and two new rank order statistics arising in its consideration.Cancer Chemother Rep1966;
50: 1623–1670.
26.
GehanEA.A generalized Wilcoxon test for comparing arbitrarily singly-censored samples. Biometrika1965;
52: 203–224.
27.
PollockKHWintersteinSRConroyMJ.Estimation and analysis of survival distributions for radio-tagged animals. Biometrics1989;
45: 99–109.
28.
MakuchRWEscobarMMerrillS.Algorithm AS 262: a two-sample test for incomplete multivariate data. J R Stat Soc C-Appl Stat1991;
40: 202–212.
29.
LeCTLindgrenBR.Duration of ventilating tubes: a test for comparing two clustered samples of censored data.Biometrics1996;
52: 328–334.
30.
HougaardP.Analysis of multivariate survival data.
Berlin:
Springer Science & Business Media, 2000.
31.
DuchateauLJanssenP.The frailty model.
Berlin:
Springer Science & Business Media, 2008.
32.
JohnsonRCJohnsonNL. Survival models and data analysis. Wiley Online Library, 1980.
