Converting log hazard function to survival probability

Question

I am currently working through a survival problem and I wanted to get some advice with how to proceed.

I wanted to estimate survival probabilities over time based on knowing only the $\log({\rm hazard})$ which takes the form of powers of $t$. For example: $\ln({\rm hazard}(t)) = -5 + 1.2t^2 + 0.5t$

I've read elsewhere that this obtains an analytically intractable integral and thus requires numerical techniques.

Essentially I am looking to obtain a formula which predicts survival over time based on this hazard. Can anyone give me a heads-up about how I proceed with this? Does anyone know of a function in R that may help with the problem?

Answer 1

Survival probability $S(t) = Pr(T > t) = exp(-\int_0^t h(u) du)$.

In R, you can use function integrate:

h <- function(t) exp(-5 + 1.2*t^2 + 0.5*t)

and the survival probability

t <- 2
exp(-integrate(h, 0, t)$value)