The accelerated failure time model (AFT) can be expressed as:
$S_{1}(t) = S_{0}(\frac{t}{\gamma})$ where $S_{0}$ is a specified baseline survival distribution and $\frac{1}{\gamma}$ is the accelerant/decelerant.
The probability of survival in group 1 ($S_{1}$) at time t is the same as the probability of survival in the baseline group ($S_{0}$) at time $\frac{t}{\gamma}$. So the baseline group is considered to age $\gamma$ as fast as group 1. $\gamma$ is modeled using the exponent of a linear combination of features ($\gamma(x)=\exp\{\theta\cdot X\}$). So $S_{1}(t) = S_{0}\left(\frac{t}{\exp\{\theta\cdot X\}}\right)$
references: 7.3.1 Accelerated Life Models, Lecture eighteen: The accelerated Failure Time (AFT) Model, lifelines:Accelerated failure time models
The accelerated failure time model (AFT) can also be expressed as a linear function of log time (log linear regression):
$\log(T) = (\theta\cdot X) + \sigma\epsilon$ where $\epsilon$ is a specified distribution of the error and $\sigma$ is a scale factor
references:Accelerated Failure Time Models pg.15, Survival Analysis with Accelerated Failure Time
What is the mathematical relationship between the two above expressions of the accelerated failure time model? From $\log(T) = (\theta\cdot X) + \sigma\epsilon$ how do you get $S_{1}(t) = S_{0}(\frac{t}{\gamma})$ or vice versa?