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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?

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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 $N(0, 1)$

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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.

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  • $\begingroup$ Do you have a reference on case studies where estimating sigma as a function of features was performing well ( compared to assuming constant scale for any X -- which seems very unrealistic especially in survival context). $\endgroup$ Commented Jan 12, 2023 at 13:17
  • $\begingroup$ @GeorgM.Goerg I'm not familiar with "studies where estimating sigma as a function of features was performing well." The standard accelerated failure time model of the OP assumes that $\sigma$ is a constant to be estimated from the data, and such survival models can work quite well. The assumption is similar to that of constant variance in ordinary least squares, as another answer shows. Modeling $\sigma$ as a function of features is allowed by the R flexsurv package, but I don't know how often or successfully it's done. $\endgroup$
    – EdM
    Commented Jan 12, 2023 at 14:41
  • $\begingroup$ Right, but similarly that constant variance assumption in OLS is not really practical/necessary. Especially for time to event the shape of the distribution is often more peaked at the start (I know pretty well that this will take about 20-30mins) vs as we approach the end of life the waiting distribution becomes more exponential like ["it could happen any moment now"]. Will check out flexsurv; I have seen it used in failure time prediction (ragulpr.github.io/2016/12/22/WTTE-RNN-Hackless-churn-modeling), but wondering if in more classic survival context it also showed up $\endgroup$ Commented Jan 13, 2023 at 2:34
  • $\begingroup$ @GeorgM.Goerg the type of behavior you describe might better be handled by a proportional-hazards (PH) model rather than the accelerated failure time model implicit in the OP. A Cox PH model makes no assumptions about baseline hazard form, just that hazard at any time is proportional to a function of covariates. $\endgroup$
    – EdM
    Commented Jan 13, 2023 at 15:53

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