Survival function of accumulated hazard in Cox model

Question

I have this setup:

Independent subjects following a Cox model $$\lambda_i(t,X_i) = \lambda_0(t) \exp(X_i^T \beta ) $$

the observed right censored survival data $(T_i, \delta_i, X_i)$ are i.i.d. and $T_i = min( \tilde{T}_i,C_i)$ and $\delta_i = 1(\tilde{T_i}\leq C_i)$.

$\tilde{T_i} \sim \lambda_i(t,X_i)$ and $C_i \sim \lambda_C(t,C_i$ and $\tilde{T_i} \perp C_i | X_i$. Denote the survival distribution of $\tilde{C}$ given $X_i$ as $G_C(t,X_i)$ and assume that $G_C(t,x) > \varepsilon > 0$ for all $t \leq \tau$ and all x. Assume $C_i = \min ( \tilde{C}_i,\tau)$, where $\tau$ is a fixed upper limted follow-up. Let $\Lambda_0(t) = \int_0^t \lambda_0(s)ds$

My teacher asks my in this weekly to: $\textbf{find the survival function of }$ $$ \Lambda_0(\tilde{T_i})\exp( X_i^T \beta) $$ given $X_i$.

I do not know what I am supposed to find. Apparently it is not $P(\tilde{T} > t|X)$ since this is the survival function of $\tilde{T}|X$. Or is it? Maybe someone with more experience can read of the question for me.

Answer 1

It's hard to read a teacher's mind. In my attempt to read your teacher's mind, I think that this question is designed to get you to focus on the relationship between a covariate-adjusted cumulative hazard function and a covariate-adjusted survival function. I might then have worded it: "Find the survival function corresponding to this cumulative hazard function."

If I'm correct, then you should just write the survival function $S(\tilde T|X_i)$ in terms of the cumulative hazard function $\Lambda(\tilde T|X_i) =\Lambda_0(\tilde{T_i})\exp( X_i^T \beta)$. That's an important relationship to understand. Your teacher might have something more nefarious in mind, but I doubt it.