# Survival function of accumulated hazard in Cox model

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.

• As this seems to be a homework-type question, please add the self-study tag to the question and read the policy on self-study questions. In the meantime: do you understand the relationship between a cumulative hazard function and a survival function?
– EdM
Dec 19, 2021 at 17:04

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.
• Thanks so much for replying! I think my teacher teacher wants me to assume that $\Lambda_0$ is increasing and continuous, hence the inverse exists and is increasing, so I can say $\Lambda_0^{-1}(\Lambda_0(\tilde{T})) = \tilde{T}$ Dec 21, 2021 at 20:18
• $\Lambda_0$ is non-decreasing, not necessarily always increasing. Do you understand the relationship between a cumulative hazard function and its associated survival function? See Wikipedia or these lecture notes or this answer (in the context of a Cox model, but the relationship holds for any continuous-time survival and cumulative-hazard function).