Generating random survival times
Simulation studies represent an important statistical tool to investigate the performance, properties and adequacy of statistical models. Here, we will see how to generate random survival times based on the most commonly used distributions:
Distribution function of the Cox model
Thanks to the Cox proportional hazard model, it is convenient to model survival times through the hazard function, with the baseline function:
The survival function of the Cox proportional hazards models given by with
And thus, the distribution function of the Cox model is
Random survival times formula
Let be a random variable with distribution function , then follows a uniform distribution on the interval , abbreviated as . Moreover, if , then , too. Thus,
Therefore, the survival time of the Cox model can be expressed as with: , and
Therefore, as long as it is possible to compute , we can generate random survival times.
and are tuning parameters.
Linear and Nonlinear hazard function
It is possible to use nonlinear hazard functions to generate random survival times such that: where is a nonlinear function.