# Hazard function for shifted survival time

I have a hazard function $\lambda$ for survival time T, is there a way to find the hazard function $\lambda_\alpha$ for T + $\alpha$, where alpha is some integer.

I'm thinking $$S_\alpha(t)=P(T+\alpha > t)=P(T>t-\alpha)=\exp(-\int_0^{t-\alpha}\lambda(s)ds)$$$$=\exp(-\int_0^{t}\lambda(s)ds+\int_{t-\alpha}^{t}\lambda(s)ds)$$

and I'm not really sure how to continue.

• Are you saying that you know $\lambda(s)$ only for $0 < s \leq T$ or maybe just $\lambda(T)$, and from this you want to deduce the value of $\lambda(T+\alpha)$? Or that you know the survival function $S(t)$ for all $t$ and from this and the knowledge of the value of $\lambda(T)$, you want to deduce the value of the hazard $\lambda(T+\alpha)$? The former question is unsolvable and the latter is trivial. – Dilip Sarwate Jan 16 '17 at 14:13
• It's definitely not the former. But I'm not looking for a specific value either. The way I interpret it is that I need to find the formula for the hazard that describe this new survival time (T+alpha) that has a different distribution. – ChuckP Jan 16 '17 at 14:28

The hazard function $\lambda(t)$ is related to the survival function $S(t)$ as \begin{align}\lambda(t) &= \frac{-\frac{\mathrm d(St)}{\mathrm dt}}{S(t)}\\ S(t) &= \exp\left(-\int_0^t\lambda(s)\,\mathrm ds\right)\end{align} and so \begin{align}S_\alpha(T) &= S(T+\alpha)\\ &= \exp\left(-\int_0^{T+\alpha}\lambda(s)\,\mathrm ds\right) \\&= \exp\left(-\int_0^{T}\lambda(s)\,\mathrm ds - \int_T^{T+\alpha}\lambda(s)\,\mathrm ds\right)\\ &= S(T)\cdot \exp\left(-\int_T^{T+\alpha}\lambda(s)\,\mathrm ds\right)\end{align} where you calculated everything except the very last step in writing your question. Unfortunately, no further simplification is possible without more specific information about the hazard function. But, since we do know that $\lambda(t) \geq 0$ for all $t$, we do get the cold comfort of knowing that $S_\alpha(T)$ is a nonincreasing function of $\alpha$.
• If the original hazard is of the simple additive form: $X^T(t)\beta(t)$, how does one obtain the alpha hazard? I tried to take -log to obtain the cumulative hazard, but not sure how to differentiate the latter expression. – ChuckP Jan 16 '17 at 15:14