Mean survival time for a log-normal survival function 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:

 A: 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.

A: The R rms package can help:
require(rms)
f <- psm(Surv(dtime, event) ~ ..., dist='lognormal')
m <- Mean(f)
m   # see analytic form
m(c(.1,.2)) # evaluate mean at linear predictor values .1, .2
m(predict(f, expand.grid(age=10:20, sex=c('male','female'))))
# evaluates mean survival time at combinations of covariate values

A: In case someone really does want the mean survival time as originally asked, it's $e^{\mu+{\sigma^2\over 2}}$.  (In fact, the original poster should carefully consider whether they want the mean or the median for their use of the resulting number.  For the example given with $\sigma=1.1$, the mean is almost twice the median.)
