Reconciling AFT Model Output With AFT Theory (Survival Analysis) TLDR: I have spent quite a lot of time trying to reconcile the output of AFT regression libraries with the expected outputs from the theory. Any input would be greatly appreciated.
Suppose you are using a Weibull AFT (Accelerated Failure Time) Regression Model i.e. given survival time $T$, suppose we have observed values of covariates $x_{i1}, ..., x_{ip}$ and possibly censored survival time $t_i$, then the Weibull AFT model is:
$$ log(t_i) = \beta_0 + \beta_1 x_{i1} + ... + \beta_p x_{ip}+ \sigma \epsilon_i  $$
For $i = 1, 2, ..., n$ samples. Where $\epsilon_i$ are IID according to a Gumbel (Extreme Value Type 1) distribution, $\sigma$ is a scale parameter etc.
Then it follows that the pdf for $T$ is given by:

Where the PDF of the Weibull distribution is given as:

I.e. if we let shape $\gamma = \frac{1}{\sigma}$ and scale $\rho = exp(x\beta)$ then we see $T \sim W(exp(x\beta), \frac{1}{\sigma})$.
My question is about the results from AFT libraries in R (e.g. aftreg or flexsurv) - the results are similar from both libraries. An example model fit output is given below - in this example you have two features in the model (TDC1 is time dependent covariate 1 coefficient and TDC2 is time dependent covariate 2 coefficient):
I.e. there are four parameters produced by the model: one for each of the input variables and a log(scale) and log(shape) parameter (baseline parameters apparently).
My questions are:

*

*Why is there no estimate for $\sigma $ (only the baseline shape)?

*Is it possible to reconstruct the pdf for $T$ (as above) using the four parameters estimated i.e. covariate coefficients and the baseline scale and shape?

See below for output from eha package (aftreg):


The shape parameter corresponds to the parameter you have identified as $\sigma$ (after skimming the flexsurv paper for 2 min)
