Mean survival time for a Gamma survival function I was not able to find the formula to allow me to calculate the mean and the median survival time for a Gamma survival function.
Does anyone knows how to calculate these?
 A: In my parametrisation (take $\lambda = 1 / \theta$ to obtain the same parametrisation as in the wikipedia page), the density function is
$$
f(t) = \frac{\lambda^k t^{k-1} \exp(-\lambda t)}{\Gamma(k)} \qquad (\lambda > 0, \,k > 0)
$$
with
$$
\Gamma(k) = \int_0^\infty x^{k-1} \exp(-x) \, \textrm{d}x
$$
Thus,
$$
\textrm{E}(T) = \int_0^\infty t f(t) \, \textrm{dt} = \frac{k}{\lambda}
$$
The median survival time is found by solving $S(t) = 1 / 2$ for $t$. However, there is no explicit formula for the median survival time. Indeed, the survival function is
$$
S(t) = 1 - \frac{\int_0^{\lambda t} s^{k-1} \exp(-s)\, \textrm{d}s}{\Gamma(k)}
$$
The integral that appears in the formula for $S(t)$ is called the incomplete gamma integral (see the handbook of mathematical functions, section 6.5) which has no closed form expression. The solution to the equation $S(t) = 1 / 2$ can however be approached numerically.
A: If the survival time has a Gamma distribution, the median survival time doesn't have a simple closed form (but see below).
The mean depends on your parameterization.
If $\alpha$ is the shape parameter, and $\beta$ the scale parameter, then the mean survival is $\alpha \beta$; alternatively, if $\alpha$ is the shape parameter, and $\theta$ the rate parameter, then the mean survival is $\alpha/\theta$. 
My parameterizations are backwards to what Wikipedia has. (Or at least has now ... I believe mine are consistent with how it used to be.) 
They have $\theta$ for the scale and $\beta$ for the rate.
(Often - but not always - survival models will be parameterized in the second form, with $\lambda$ - as the rate parameter - written as a function of predictors.)
Approximation to the median survival:
Wikipedia gives an approximation to the median: $\mu \frac{3 \alpha - 0.8}{3 \alpha + 0.2}$ where $\mu$ is the mean survival time. This approximation is not suitable if $\alpha <1$. The paper for this is findable online via Google.
