# Mean Survival Time Under Weibull Model Using survreg

Under the Weibull parametric model, we assume the survival time $$T \sim \text{Weibull}(\alpha, \lambda)$$, with density $$f_T(t) = \alpha \lambda (\lambda t)^{\alpha - 1} \exp(-(\lambda t)^\alpha)$$. Then the mean would be $$ET = \Gamma(1+1/\alpha)/\lambda$$ (from wiki, although they use $$1/\lambda$$ for the scale and $$k$$ for the shape). I'm trying to fit a Weibull model to a dataset, but I find there's a discrepancy between the estimated mean survival time from the model, and the mean I calculate using the fitted parameters:

> library(survival)
> data(kidney)
> m <- survreg(Surv(time,status)~1,data=kidney[kidney$$sex==1,],dist="weibull") > predict(m)  50.37909 # estimate from the model > summary(m) > alpha <- 1/m$$scale
> lambda <- 1/exp(m\$coef)
> gamma(1+1/alpha)/lambda
 62.0962  # my calculation using the formula for expected value


What's the reason for this difference?

> set.seed(1234)

predict.survreg does not return the expected survival time. For details see for example this answer.