A Simulation Study to Evaluate the Performance of Extended Cox model in Testing Treatment Effect with Possible Non-proportional Hazards

Belay Belete Anjullo

Abstract


Background: Many randomized clinical studies with long-term follow-up regularly measure time-to-event outcomes, such as survival time to compare 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. 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.  It was shown that extended Cox model performed well to test effect of treatment in case of crossing survival curves at 20% and 40% administratively censored time. However, most of the studies did not assess the statistical properties of this approach under high censoring rates.

Objective: In this paper, the main objective of the study was to evaluate whether the power and type I error rate of extended Cox model was robust with variations in censoring rates under five possible treatment effects based on data from simulations.

Methodology: a simulation study was designed to compare the performance of the extended Cox and the data are replicated 1000 times to estimate type I error rate and empirical power of the test.

Results: type I error rates for the test from the model approached the nominal level of 0.05 at 10% and 30% of censoring rates regardless of sample size per treatment group considered in the study, however, type I error rates are slightly inflated at 70% of censoring rate. Moreover, with respect to powers of the test, extended Cox model performed reasonably well with powers of test above 80% in case of crossing survival curves under censoring rates of 10% and 30% to test treatment effect without including baseline covariates in the model. Although 70% censoring rate appeared to have influence on the power of test, test from extended Cox model exhibits moderate power in this situation, particularly for sample of size 200 per treatment group. Furthermore, extended Cox model with heavy side function performed well under the early treatment effect in which hazards is expected to crosses. Using the extended Cox model with baseline covariate in case of crossing survival curves did not generally yield dramatic decrease in power.

Conclusions: 70% censoring rate appeared to have influence on the power of test and type I error rate although test exhibits moderate power for sample of size 200 per treatment group.

Keywords: Simulation, Type I error rate, Cox Proportional Hazards, Power of the test, Extended Cox Model


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References


Fisher, R. A. (1925). Statistical Methods for Research Workers. Oliver and

Boyd (Edinburgh). ISBN 0-05-002170-2.

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.

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-

2007.00975.x.

Cox, D. R. (1972). Regression models and life-tables. Journal of the Royal

Statistical Society, Series B, 34 (2), 187-220.

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,

-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,

-2489.

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.

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.

Machin D, Cheung YB, Parmar MK..,(2006). Survival analysis a practical

approach, 2nd edition, West Sussex: John Wiley and Sons Ltd.

Leung KM, Elashoff RM, Afifi AA.(1997). Censoring issues in survival

analysis. Annu Rev Public Health. 1997;18:83–104.

Andrea Callegaro & Bart Spiessens (2017) Testing Treatment Effect in

Randomized Clinical Trials With PossibleNonproportional

Hazards,Statistics in Biopharmaceutical Research, 9:2, 204-

,DOI:10.1080/19466315.2016.1257436.

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.

Lee, S. H. (2007), “On the Versatility of the Combination of the Weighted

Log-Rank Statistics,” Computational Statistics & Data Analysis,

,6557–6564.

Campbell, H., and Dean, C. B., (2013). The Consequences of Proportional

Hazards Based Model Selection, Statistics in Medicine, 33, 1042–

Li,H.,Han,D.,Hou, Y.,Chen,H., andChen, Z. (2015). Statistical Inference

Methods for Two Crossing Survival Curves: A Comparison of

Methods, PLoS One, 10, e0116774.

Pettitt, A. N. and Daud, I. Bin (1990). Investigating time dependence in

Cox's proportional hazards model, Applied statistics 39, 313-329.

Hartina Husain , Sri Astuti Thamrin, Sulaiha Tahir , Ahmad Mukhlisin,

Mirna Apriani M. (2018). The Application of Extended Cox

Proportional Hazard Method for Estimating Survival Time of Breast

Cancer; Journal of Physics: Conf. Series 979 (2018) 012087 doi

:10.1088/1742-6596/979/1/012087.

Gillen, D. L and Emerson, S. S. (2005). A note on P-Values under Group

Sequential Testing and Non proportional Hazards: Biometrics 61, 546-

, DOI: 10.1111/j.1541-0420.2005.040342.x.

Liang, K.Y and Zeger, S.L. (1986). Longitudinal data analysis using

generalized linear models. Biometrika, 73, 13-22.

Fisher, L. D. ve Lin D. Y. (1999). Time-dependent covariates in the Cox

proportional hazards regression model,Annual Review of Public Health

, 145-157.


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