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Generating survival times to simulate Cox proportional hazards models with time-varying covariates
Article first published online: 4 JUL 2012
Copyright © 2012 John Wiley & Sons, Ltd.
Statistics in Medicine
Volume 31, Issue 29, pages 3946–3958, 20 December 2012
How to Cite
Austin, P. C. (2012), Generating survival times to simulate Cox proportional hazards models with time-varying covariates. Statist. Med., 31: 3946–3958. doi: 10.1002/sim.5452
- Issue published online: 23 NOV 2012
- Article first published online: 4 JUL 2012
- Manuscript Accepted: 3 MAY 2012
- Manuscript Received: 4 NOV 2011
- survival analysis;
- proportional hazards model;
- time-varying covariates;
- power and sample size calculation;
- time-dependent covariate;
- exponential distribution;
- Weibull distribution;
- Gompertz distribution
Simulations and Monte Carlo methods serve an important role in modern statistical research. They allow for an examination of the performance of statistical procedures in settings in which analytic and mathematical derivations may not be feasible. A key element in any statistical simulation is the existence of an appropriate data-generating process: one must be able to simulate data from a specified statistical model. We describe data-generating processes for the Cox proportional hazards model with time-varying covariates when event times follow an exponential, Weibull, or Gompertz distribution. We consider three types of time-varying covariates: first, a dichotomous time-varying covariate that can change at most once from untreated to treated (e.g., organ transplant); second, a continuous time-varying covariate such as cumulative exposure at a constant dose to radiation or to a pharmaceutical agent used for a chronic condition; third, a dichotomous time-varying covariate with a subject being able to move repeatedly between treatment states (e.g., current compliance or use of a medication). In each setting, we derive closed-form expressions that allow one to simulate survival times so that survival times are related to a vector of fixed or time-invariant covariates and to a single time-varying covariate. We illustrate the utility of our closed-form expressions for simulating event times by using Monte Carlo simulations to estimate the statistical power to detect as statistically significant the effect of different types of binary time-varying covariates. This is compared with the statistical power to detect as statistically significant a binary time-invariant covariate. Copyright © 2012 John Wiley & Sons, Ltd.