Volume 2, Issue 6, December 2017, Page: 129-140
Testing Treatment Effect in Randomized Clinical Trials with Possible Non-proportional Hazards
Belay Belete Anjullo, Department of Statistics, Arba Minch University, Arba Minch, Ethiopia
Roel Braekers, Center of Statistics, Hasselt University, Diepenbeek, Belgium
Received: Jul. 23, 2017;       Accepted: Oct. 26, 2017;       Published: Dec. 8, 2017
DOI: 10.11648/j.ijcocr.20170206.12      View  1991      Downloads  119
Many randomized clinical trials include right censored time to event data, comparing an experimental treatment with a standard treatment or placebo control. In this comparison, one tests whether the two treatments have the same survival function or equivalently the same hazard function over a given time period in order to evaluate effect of treatment. The methodological development of survival analysis for randomized clinical trials with right-censored data that have had the most profound impact are the log-rank test for comparing the equality of two or more survival distributions, and the Cox proportional hazards model for examining the covariate(s) effects on the hazard function. However, when comparing treatments in terms of their time to event distribution, there may be reason to believe that the hazard curves will cross, and in such cases standard comparison techniques could lead to misleading results [16]. Hence, in this study, the performance of new methods for testing treatment effect on randomized clinical trials when the proportional hazards assumption is not satisfied was evaluated based on simulation studies and on two real datasets. New proposed methods are based on combination of early/late treatment effects obtained from stopped/left truncated Cox or equivalently from extended Cox and the overall treatment effect from Cox proportional hazards model. These methods were compared with Cox proportional hazards model [8], pseudo values regression approach based on mean restricted survival time [1, 13] and extended Cox for the time dependent treatment effect [20]. Type I error rate and power of the proposed tests were illustrated based on simulated data under five possible treatment effect. The results of simulations and real data examples on cancer clinical trials showed that the new proposed methods performed reasonably well in case of crossing survival curves compared to Cox proportional hazards model and pseudo values regression approach based on restricted mean survival time. However, they performed about the same compared to extended Cox model. Furthermore, they performed about the same compared to Cox proportional hazards model and extended Cox under the late treatment effect. Using the proposed methods under proportional hazards alternative did not generally yield dramatic decrease in power compared to the Cox model and they allow adjusting for covariate(s).
Simulation, Stopped Cox, Kaplan-Meier Method, Cox Proportional Hazards, Pseudo Values, Regression Approach, Extended COX Model
To cite this article
Belay Belete Anjullo, Roel Braekers, Testing Treatment Effect in Randomized Clinical Trials with Possible Non-proportional Hazards, International Journal of Clinical Oncology and Cancer Research. Vol. 2, No. 6, 2017, pp. 129-140. doi: 10.11648/j.ijcocr.20170206.12
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This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Andersen, P. K, Hansen, M. G and Klein, J. P. (2004). Regression analysis of restricted mean survival time based on pseudo-observations. Lifetime Data Analysis; 10:335-350.
Andersen, P. K, Klein, J. P and Rosthoj, S. (2003). Generalized linear models for correlated pseudo-observations with applications to multi-state models. Biometrika; 90:15–27.
Byar, D. P. (1984). The Veterans Administration study of chemoprophylaxis for recurrent stage Ibladder tumors: comparisons of placebo, pyridoxine and topical thiotepa. In Bladder Tumors and Other Topics in Urological Oncology, (Edited by m. Pavone-Macaluso, P. H. Smith and F. Edsmyr). Plenum, New York, 363-370.
Burton, A, Altman, D. G, Royston, P, and Holder, R. L. (2006). The design of simulation studies in medical statistics: Wiley InterScience, Statist. Med, 25:4279–4292
Bain, L and Engelhardt, M. (1991). Statistical Analysis of Reliability and Life testing Models: Theory and Methods. Marcel Dekker, Inc., New York, 2nd edition.
Callegaro, A, Debois, M and Spiessens, B. (2014). Testing the treatment effect in randomized clinical trials with possible non-proportional hazards: Working paper: GSK Vaccines, Belgium.
Chen, P and Tsiatis, A. A. (2001). Causal inference on the difference of the restricted mean lifetime between two groups, Biometrics vol. 57 pp. 1030–1038.
Cox, D. R. (1972). Regression models and life-tables. Journal of the Royal Statistical Society, Series B, 34 (2), 187-220.
Fisher, R. A. (1925). Statistical Methods for Research Workers. Oliver and Boyd (Edinburgh). ISBN 0-05-002170-2.
Gillen, D. L and Emerson, S. S. (2005). A note on P-Values under Group Sequential Testing and Non proportional Hazards: Biometrics 61, 546-551, DOI: 10.1111/j.1541-0420.2005.040342.x
Greenwood, M. ( 1926 ) The natural duration of cancer, in Reports on Public Health and Medical Subjects, vol. 33, Her Majesty ’ s Stationary Office, London, pp. 1–26.
Kaplan, E. L and Meier, P. (1958). Nonparametric estimation from incomplete observations. Journal of the American Statistical Association, 53, 457–481.
Klein, J. P, Logan, B. R and Harhoff, M and Andersens, P. K. (2007). Analyzing survival curves at a fixed point in time. Statistics in Medicine, 26, 4505-4519.
Klein, J. P. and Moeschberger, M. L. (1997). Survival Analysis: Techniques for Censored and Truncated Data, New York: Springer-Verlag.
Liang, K. Y and Zeger, S. L. (1986). Longitudinal data analysis using generalized linear models. Biometrika, 73, 13-22.
Logan, B. R, Klein, J. P and Zhang, M. J. (2008). Comparing Treatments in the Presence of Crossing Survival Curves: An Application to Bone Marrow Transplantation: Biometrics: 733–740. doi:10.1111/j.1541-0420.2007.00975.x.
Mac Kenzie, G and Ha, Li Do. (2007). Modelling Survival Data with Crossing Hazards. IWSM.
Mantel, N. (1966). Evaluation of survival data and two new rank order statistics arising in its consideration. Cancer Chemotherapy Reports 50 (3): 163–70. PMID 5910392: can be accessed: http://en.wikipedia.org/wiki/Log-rank_test
Oquigley, J and Pessione, F. (1991). The problem of a covariate time qualitative interaction in a survival study. Biometrics; 47:101–115.
Putter, H, Sasako, M, Hartgrink, H. H, van de, Velde C. J and van Houwelingen, J. C. (2005). Long-term survival with non-proportional hazards: results from the Dutch Gastric Cancer Trial. Stat Med, 24, 2807-2821.
Royston, P. and Parmar, M. K. B. (2011). The use of restricted mean survival time to estimate the treatment effect in randomized clinical trials when the proportional hazards assumption is in doubt. Statist. Med., 30, 2409-2421.
Schemper, M., Wakounig, S., and Heinze, G. (2009). The estimation of average hazard ratios by weighted Cox regression. Statist. Med., 28, 2473-2489.
Sheldon, E. H. (2006). Choosing the Cut Point for a Restricted Mean in Survival Analysis, a Data Driven Method, PhD dissertation at Virginia Commonwealth University.
Spiessens, B. and Debois, M. (2010). Adjusted significance levels for subgroup analysis in clinical trials. Cont Clin Trials, 31, 647-656.
Stablein, D. M., Carter, W. H., and Novak, J. W. (1981). Analysis of survival data with non proportional hazards functions. Controlled Clinical Trials, 2, 149-159.
Yang, S and Zhao, Y. (2007). Testing treatment effect by combining weighted log-rank tests and using empirical likelihood: Science Direct: Statistics & Probability Letters 77: 1385–1393.
Van Houwelingen, H. C, and Putter, H. (2014). Comparison of stopped Cox regression with direct methods such as pseudo-values and binomial regression. Lifetime Data Anal, DOI 10.1007/s10985-014-9299-3.
Zhang, M. J and Klein, J. P. (1998). Confidence Bands for the Difference of Two Survival Curves Under Proportional hazards Model: Technical Report 29, Medical College of Wisconsin
Zhou, M. (2006). Log-rank Test: When does it Fail and how to fix it: University of Kentucky, Department of Statistics Tech Report: Submitted/under revision: http://www.ms.uky.edu/~mai/research/LogRank2006.pdf accessed on July 25, 2014.
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