Interpreting intercept of parametric AFT survival model (predict.survreg output) I am trying to work out under what circumstances you would use the response type in the survreg.predict function in r. I have read an excellent response by
Terry Therneau, author of the survival package, that stated:

The survreg routine assumes that log(y) ~ covariates + error. For a log-normal distribution the error is Gaussian and thus the predict(fit, type='response') will be exp(predicted mean of log time), which is not the predicted mean time. For Weibull the error dist is asymmetric so things are more muddy. Each is the MLE prediction for the subject, just not interpretable as a mean.

So what is it?
I have a textbook that erroneously calls these values the mean survival times, which I understand it is not, but why would you need the mean of log(survival time)? The exponential is not comparable to the mean(survival time), so how can you interpret the intercept of the output of the survreg model other than as the log of the scale parameter of the distribution? Does it tell us anything useful about the data from a biological perspective, rather than a mathematical one?
As an aside, can anyone help me with the r code to get the mean survival time for each group?
Thank you, and apologies for my clear mathematical ignorance.
 A: I prefer to equivalently view the AFT model in terms of a generalized linear model like logistic regression or Poisson regression.  In these models there is no "error term."  There is simply a likelihood and a parameterization of the mean that is estimated through maximum likelihood.  Most documentation describe an AFT model in terms of a log scale parameter and an "error term."  While this is technically correct, I do not find it easy to think about or work with.
The intercept in, say, a Weibull AFT model represents the log of the scale parameter of the Weibull distribution, $\text{log}(\lambda)$, for the reference class in your model (Weibull - Wikipedia).  This parameter might just as easily be referred to as a shape parameter instead of a scale parameter.  AFT models are typically parameterized in terms of shape and scale parameters instead of mean and scale.
The good news is that with skewed distributions we are most often interested in comparing percentiles between groups (like medians).  Estimates of these percentiles and their standard errors are easily requested from the procedure.  (There is no need to directly interpret the intercept term.)  If you construct a Wald interval for a percentile I suggest using a log link function.
