Generating Survival Times to Simulate Cox Proportional Hazards Models

This paper discusses techniques to generate survival times for
simulation studies regarding Cox proportional hazards models. In
linear regression models, the response variable is directly
connected with the considered covariates, the regression
coefficients and the simulated random errors. Thus, the response
variable can be generated from the regression function, once the
regression coefficients and the error distribution are specified.
However, in the Cox model, which is formulated via the hazard
function, the effect of the covariates have to be translated from
the hazards to the survival times, because the usual software
packages for estimation of Cox models require the individual
survival time data. A general formula describing the relation
between the hazard and the corresponding survival time of the Cox
model is derived. It is shown how the exponential, the Weibull and
the Gompertz distribution can be used to generate appropriate
survival times for simulation studies. Additionally, the general
relation between hazard and survival time can be used to develop
own distributions for special situations and to handle flexibly
parameterized proportional hazards models. The use of other
distributions than the exponential distribution only is
indispensable to investigate the characteristics of the Cox
proportional hazards model, especially in non-standard situations,
where the partial likelihood depends on the baseline hazard.