How to generate survival data with time dependent covariates using R

Question

I want to generate survival time from a Cox proportional hazards model that contains time dependent covariate. The model is

$h(t|X_i) =h_0(t) \exp(\gamma X_i + \alpha m_{i}(t))$

where $X_i$ is generated from Binomial(1,0.5) and $m_{i}(t)=\beta_0 + \beta_1 X_{i} + \beta_2 X_{i} t$.

The true parameter values are used as $\gamma = 1.5, \beta_0 = 0, \beta_1 = -1, \beta_2 = -1.5, h_0(t) = 1$

For time-independent covariate (i.e. $h(t|X_i) =h_0(t) \exp(\gamma X_i) $ I generated as follows

#For time independent case
# h_0(t) = 1
gamma <- -1
u <- runif(n=100,min=0,max=1)
Xi <- rbinom(n=100,size=1,prob=0.5)
T <- -log(u)/exp(gamma*Xi)

Can anyone please help me to generate survival data with time-varying covariate.

Answer 1

OK from your R code you are assuming an exponential distribution (constant hazard) for your baseline hazard. Your hazard functions are therefore:

$$ h\left(t \mid X_i\right) = \begin{cases} \exp{\left(\alpha \beta_0\right)} & \text{if $X_i = 0$,} \\ \exp{\left(\gamma + \alpha\left(\beta_0+\beta_1+\beta_2 t\right)\right)} & \text{if $X_i = 1$.} \end{cases} $$

We then integrate these with respect to $t$ to get the cumulative hazard function:

$$ \begin{align} \Lambda\left(t\mid X_i\right) &= \begin{cases} t \exp{\left(\alpha \beta_0\right)} & \text{if $X_i=0$,} \\ \int_0^t{\exp{\left(\gamma + \alpha\left(\beta_0+\beta_1+\beta_2 \tau\right)\right)} \,d\tau} & \text{if $X_i=1$.} \end{cases} \\ &= \begin{cases} t \exp{\left(\alpha \beta_0\right)} & \text{if $X_i=0$,} \\ \exp{\left(\gamma + \alpha\left(\beta_0+\beta_1\right)\right)} \frac{1}{\alpha\beta_2} \left(\exp\left(\alpha\beta_2 t\right)-1\right) & \text{if $X_i=1$.} \end{cases} \end{align} $$

These then give us the survival functions:

$$ \begin{align} S\left(t\right) &= \exp{\left(-\Lambda\left(t\right)\right)} \\ &= \begin{cases} \exp{\left(-t \exp{\left(\alpha \beta_0\right)}\right)} & \text{if $X_i=0$,} \\ \exp{\left(-\exp{\left(\gamma + \alpha\left(\beta_0+\beta_1\right)\right)} \frac{1}{\alpha\beta_2} \left(\exp\left(\alpha\beta_2 t\right)-1\right)\right)} & \text{if $X_i=1$.} \end{cases} \end{align} $$

You then generate by sampling $X_i$ and $U\sim\mathrm{Uniform\left(0,1\right)}$, substituting $U$ for $S\left(t\right)$ and rearranging the appropriate formula (based on $X_i$) to simulate $t$. This should be straightforward algebra you can then code up in R but please let me know by comment if you need any further help.