Need help to simulate Survival Times with a time-varying Covariate

Question

I would really appreciate if someone can help me figure this out as I have been working on it for few days now and I don't know what I'm missing that it's not working. Basically, my goal is to sample survival times from a weibull distribution when I have a time-varying covariate in my model. My problem is when I fit a Cox PH model to my simulated data, I can't get the coefficients I used to simulate my data.

Here is how I simulate data. Perhaps there is something fundamental missing from my reasoning as I had checked the code over and over again and there is no error with the code:

Suppose $t \sim Weibull(\tau, \lambda_i)$ where $\lambda_i = \beta_0 + \beta_1 X_i(t)$

$X_i(t)$ is my time varying covariate and is basically of the form:

$$X_i(t) = \alpha^{(0)}_i + \alpha^{(1)}_i t + \alpha^{(2)}_i t^2$$

$\alpha$ coefficients above are indexed by i as every subject has it's own trajectory for X(t)

As I said, $t \sim Weibull(\tau, \lambda)$. I use the parameterization below for weibull:

$$f(t|\tau, \lambda_i) = \tau t^{\tau - 1} exp(\lambda_i - exp(\lambda_i) t^\tau)$$

From this, I can easily get survival function as: $$S(t|\tau, \lambda_i) = 1-F(t | \tau, \lambda_i) = exp(-exp(\lambda_i) t^{\tau})$$

where: $X_i(t) = \alpha^{(0)}_i + \alpha^{(1)}_i t + \alpha^{(2)}_i t^2$

Ok, here is what I do. For every subject:

1) I randomly sample $u \sim uniform(0,1)$

2) I then set $u = S(t|\tau, \lambda_i)$ where: $$S(t|\tau, \lambda_i) = exp(-exp(\beta_0 + \beta_1 X_i(t)) t^{\tau})$$ where $$X_i(t) = \alpha^{(0)}_i + \alpha^{(1)}_i t + \alpha^{(2)}_i t^2$$

3) I then simply find the root (t) in $u = S(t|\tau, \lambda_i)$ and call it my survival time.

Am I missing something? does this make sense? If so, how come when I fit my CoxPH model, I cannot get back my true $\beta_1$ where I used to simulate data?

Answer 1

Ok, after thinking for a few days, I think I now have the answer. The process above is wrong and the reason and here is the reason:

Consider the survival function as: $$S(t | \tau, \lambda_i) = exp(-exp(\beta_0 + \beta_1 X_i(t)) t^\tau)$$

I consider $\lambda = \beta_0 + \beta_1 X_i(t)$ where $\lambda$ is the scale parameter in my representation of Weibull distribution (not be confused with hazard!). So:

$$S(t | \tau, \lambda_i) = exp(-exp(\lambda_i) t^\tau)$$

Of course, it's obvious that $\lambda_i$ is a function of time. Let's look at the derivative of S(t|...) with respect to t in order to see how S(t|...) changes over time: $$\frac{dS(t | \tau, \lambda_i)}{dt} = S(t|\tau, \lambda_i) \times (- \tau t^{\tau - 1} exp(\lambda_i) - \frac{d\lambda_i}{dt} exp(\lambda_i)t^{\tau}) $$

the issue is $\frac{d\lambda_i}{dt}$ is non-zero as $\lambda_i$ is a function of time!