Accelerated Failure-Time Models

An AFT coefficient of $0.2$ corresponds to a time ratio of $\exp(0.2)\approx1.22$: a one-unit covariate change multiplies each conditional event-time quantile by that factor, with other covariates fixed. This makes the interpretation concrete, but it also states a restriction. The same factor must stretch the entire conditional distribution, rather than changing only one part of it.

Accelerated failure-time models express the logarithm of event time as a linear function of covariates plus an error term with a common scale. A positive coefficient stretches event-time quantiles to the right, while a negative coefficient compresses them. We develop parametric versions that specify the error distribution, handle censoring through a full likelihood, and compare the resulting survival curves. Cox regression instead leaves its baseline hazard unspecified; estimating that baseline still permits survival predictions without choosing a parametric lifetime family.

This chapter develops the log-time regression model, derives the censored likelihood, and explains the two choices developed here, Weibull and log-normal event times. We will see why the Weibull AFT model has a special proportional hazards representation, why that representation does not extend automatically to every AFT distribution, and how to compare fitted survival curves under competing distributional assumptions.