Scale Parameter in a Log-Linear Accelerated Failure Time Model

Question

Let the logarithm of the random varible $T_i$, associated with the lifetime of the $i$th individual in a survival study, follow the disitribution

$$log(T_i) = \mu + X_i\beta + \sigma\epsilon_i$$

with $\mu$ being a constant that is equal for all individuals, $X_i$ a vector of covariate values with its coeffcient vector $\beta$, $\epsilon_i$ a random error term for individual $i$ that follows a particular probability distribution, and $\sigma$ a so called scale parameter. Since the effect of covariates is proportional with regard to the survival time, this model is called an accelerated failure time model.

I am about confused that this scale paramter $\sigma$ is included. Why not simply leave it out, so that the log-linear formulation of this accelerated failure time model looks the formulation of a linear regression model? Is this so that $T_i$ follows a known probability distribution?

Answer 1

Is this so that $T_i$ follows a known probability distribution?

In your formula, it's the choice of the functional form of $\epsilon$ that determines the distribution of event times $T$. This handout is a concise reference. From page 8:

... extreme value, generalized extreme value, normal or logistic [distributions ... lead] to Weibull, generalized gamma, log-normal or log-logistic models for T.

Then $\sigma$ in the model determines the width of the distribution. That's called a scale parameter, like the variance of a normal distribution is called a scale parameter. You estimate $\sigma$ in the modeling to give the best fit to the data, given the assumed distribution.

Answer 2

EdM answer is good, and I want to also add that the scale parameter is also in linear regression, it's just there implicitly. Often we write LR like:

$$Y_i = \mu + X_i\beta + \epsilon_i$$

where $\epsilon_i$ is distributed $N(0, \sigma^2)$.

That is equivalent to writing

$$Y_i = \mu + X_i\beta + \sigma\epsilon_i$$

where $\epsilon_i$ is distributed $N(0, 1)$