On confidence intervals for the hazard ratio in randomized clinical trials.
The log-rank test is widely used to compare two survival distributions in a randomized clinical trial, while partial likelihood (Cox, 1975) is the method of choice for making inference about the hazard ratio under the Cox (1972) proportional hazards model. The Wald 95% confidence interval of the hazard ratio may include the null value of 1 when the p-value of the log-rank test is less than 0.05. Peto et al. (1977) provided an estimator for the hazard ratio based on the log-rank statistic; the corresponding 95% confidence interval excludes the null value of 1 if and only if the p-value of the log-rank test is less than 0.05. However, Peto's estimator is not consistent, and the corresponding confidence interval does not have correct coverage probability. In this article, we construct the confidence interval by inverting the score test under the (possibly stratified) Cox model, and we modify the variance estimator such that the resulting score test for the null hypothesis of no treatment difference is identical to the log-rank test in the possible presence of ties. Like Peto's method, the proposed confidence interval excludes the null value if and only if the log-rank test is significant. Unlike Peto's method, however, this interval has correct coverage probability. An added benefit of the proposed confidence interval is that it tends to be more accurate and narrower than the Wald confidence interval. We demonstrate the advantages of the proposed method through extensive simulation studies and a colon cancer study.
Department of Biostatistics, University of North Carolina, Chapel Hill, North Carolina 27599, U.S.A.
Dan-Yu
DY
Lin
Boehringer Ingelheim Investment Co., Ltd., 1601 Nanjing Road West, Shanghai 200040, P.R. China.
Luyan
L
Dai
Boehringer Ingelheim Investment Co., Ltd., 1601 Nanjing Road West, Shanghai 200040, P.R. China.
Gang
G
Cheng
Boehringer Ingelheim Pharma GmbH & Co. KG, Birkendorfer Strasse 65, 88397 Biberach an der Riss, Germany.
Martin Oliver
MO
Sailer
D003110:Colonic Neoplasms; D003198:Computer Simulation; D016001:Confidence Intervals; D006801:Humans; D011336:Probability; D016016:Proportional Hazards Models; D016032:Randomized Controlled Trials as Topic / Q000706:statistics & numerical data*
Censoring; Cox model; Log-rank test; Partial likelihood; Peto's method; Proportional hazards; Score test