Relationship between Survival Distributions and Log Linear Regression in Accelerated Failure Time models

Question

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?

Answer 1

This adapts the very useful and concise course notes on parametric survival by Germán Rodríguez, Section 2.2, to your formulation of the accelerated failure time model.

In your form, $\log(T) = (\theta\cdot X) + \sigma\epsilon$, the baseline survival curve, the probability that an individual survives beyond time $t$ at the reference levels of covariates (that is, with $\theta\cdot X=0$) is:

$$ S_0(t) = \Pr\{T_0 > t\}= \Pr\{\epsilon>\log(t)/\sigma \},$$

where the baseline time scale is defined with respect to the random term, $T_0= \exp\{\sigma \epsilon \}$. For a set of non-baseline covariate values $x$, the corresponding random variable $T$ is thus distributed as $T_0 e^{\theta\cdot X}$. The corresponding survival curve conditional on those covariate values is:

$$S(t, x) = \Pr\{T >t|x \} =\Pr\{T_o e^{\theta\cdot X} >t \}=\Pr\{T_0 > t e^{-\theta\cdot X} \} = S_0(t e^{-\theta\cdot X}),$$

illustrating the effective time compression or expansion by a factor of $e^{-\theta\cdot X}$ as a function of covariate values.