I've found plenty of formulas showing how to find the mean survival time for an exponential or Weibull distribution, but I'm having considerably less luck for log-normal survival functions.
Given the following survival function:
$$S(t) = 1 – \phi \left[ {{{\ln (t) – \mu } \over \sigma }} \right]$$
How does one find the mean survival time. As I understand it, $\sigma$ is the estimated scale parameter, and that exp($\beta$) from a parametric survival model is $\mu$. While I think I can manipulate it symbolically to get t all by itself after setting S(t) = 0.5, what's especially stumping me is how to handle $\phi$ in something like R when it actually comes down to inputting all the estimates and obtaining a mean time.
Thus far, I've been generating the survival function (and associated curves), like so:
beta0 <- 2.00
beta1 <- 0.80
scale <- 1.10
exposure <- c(0, 1)
t <- seq(0, 180)
linmod <- beta0 + (beta1 * exposure)
names(linmod) <- c("unexposed", "exposed")
## Generate s(t) from lognormal AFT model
s0.lnorm <- 1 - pnorm((log(t) - linmod["unexposed"]) / scale)
s1.lnorm <- 1 - pnorm((log(t) - linmod["exposed"]) / scale)
## Plot survival
plot(t,s0.lnorm,type="l",lwd=2,ylim=c(0,1),xlab="Time",ylab="Proportion Surviving")
lines(t,s1.lnorm,col="blue",lwd=2)
Which yields the following:
Best Answer
The median survival time, $t_{\textrm{med}}$, is the solution of $S(t) = \frac{1}{2}$; in this case, $t_{\textrm{med}} = \exp(\mu)$. This is because $\Phi(0) = \frac{1}{2}$ when $\Phi$ denotes the cumulative distribution function of a standard normal random variable.
When $\mu = 3$, the median survival time is around $20.1$ as depicted in the picture below.