Following Bender, Augustin, and Blettner
(2003) and Leemis (1987),
simulation of survival times is possible if there is function that
invert the cumulative hazard (\(H^{-1}\)), Random survival times for a
baseline distribution can be generated from an uniform distribution
between 0-1 \(U\) as: \[ T = H^{-1}(-log(U)) \] For a survival
distribution object, this can be accomplished with the function
`rsurv(s_object, n)`

which will generate `n`

number of random draws from the distribution `s_object`

. All
objects of the s_distribution family implements a function that inverts
the survival time with the function `invCum_Hfx()`

The function `ggplot_survival_random()`

helps to graph
Kaplan-Meier graphs and cumulative hazard of simulated times from the
distribution

Survival times with hazard proportional to the baseline hazard can be simulated \[ T = H^{-1}\left(\frac{-log(U)}{HR}\right) \] where \(HR\) is a hazard ratio.

The function `rsurv_hr(s_object, hr)`

can generate random
number with hazards proportionals to the baseline hazard. The function
produce as many numbers as the length of the hr vector. for example:

```
s_obj <- s_exponential(fail = 0.4, t = 2)
group <- c(rep(0,500), rep(1,500))
hr_vector <- c(rep(1,500),rep(2,500))
times <- rsurvhr(s_obj, hr_vector)
plot(survfit(Surv(times)~group), xlim=c(0,5))
```

The function `ggplot_survival_hr()`

can plot simulated data
under proportional hazard assumption.

Survival times with accelerated failure time to the baseline hazard can be simulated \[ T = \frac{H^{-1}(-log(U))}{AFT}\] where \(AFT\) is a acceleration factor, meaning for example an AFT of 2 have events two times quicker than the baseline

The function `rsurv_aft(s_object, aft)`

can generate
random numbers accelerated by an AFT factor. The function produce as
many numbers as the length of the aft vector. for example:

```
s_obj <- s_lognormal(scale = 2, shape = 0.5)
ggplot_survival_aft(s_obj, aft = 2, nsim = 10, subjects = 1000, timeto = 5)
```

In this example, the scale parameter of the Log-Normal distribution represents the mean time and it this simulation and accelerated factor of 2 move the average median from 2 to 1

If the proportional hazard and the accelerated failure is combined
and accelerated hazard time is generated. This can be accomplished with
the function `rsurvah()`

function and the
`ggplot_random_ah()`

functions

Bender, R., Thomas Augustin, and Maria Blettner. 2003. “Generating
Survival Times to Simulate Cox Proportional Hazards Models.”
*Universitätsbibliothek Der Ludwig-Maximilians-Universität
München*. https://doi.org/10.5282/UBM/EPUB.1716.

Leemis, Lawrence M. 1987. “Variate Generation for Accelerated Life
and Proportional Hazards Models.” *Operations Research* 35
(6): 892–94.