Accelerated Failure-Time Models › Summary

Accelerated failure-time models describe how covariates multiply the event time. On the log scale, the model is a linear regression with a distributional error term, and exponentiating a coefficient gives a time ratio that applies to every quantile of the event-time distribution. This makes AFT models particularly interpretable when the question is about lengthening or shortening time to an event.

Censoring enters through the likelihood in a natural way. An observed event contributes a density, while a censored observation contributes a survival probability. Transforming the standardized error to event time gives the event log-density terms $-\log y_i-\log\sigma$: the log-time-to-time transformation supplies $-\log y_i$, while standardizing log time supplies $-\log\sigma$. The Weibull and log-normal models are the two standard choices: the Weibull model has a monotone hazard and a special proportional hazards representation, while the log-normal model can represent non-monotone hazards but does not generally satisfy proportional hazards.

The practical lesson is to define the parameterization before interpreting signs, to check the distributional assumption against nonparametric survival curves and residuals, and to compare AFT models with Cox models using predictive performance rather than mismatched likelihood criteria. When the parametric assumption is credible, the fitted AFT parameters determine time ratios, survival curves, and event-time quantiles. When it is not, the model can still be useful as a smoothed summary, but its tail predictions should be treated with caution.