Concept Architecture
Concept
Theoretically, the Accelerated Failure Time (AFT) Model is a parametric survival model that describes the effect of explanatory variables by directly accelerating or decelerating the time until an event occurs. Rather than assuming proportional hazards, the model assumes that covariates act multiplicatively on survival time, extending or shortening the expected time to the event. The concept is founded on parametric survival analysis and exists to model time-to-event outcomes when proportional hazards assumptions are inappropriate or when direct interpretation of survival time is preferred.
Mathematically, the Accelerated Failure Time Model expresses the logarithm of survival time as a linear function of covariates together with a stochastic error term whose distribution determines the specific parametric model. Common error distributions include Weibull, log-normal, log-logistic and exponential. The exponentiated regression coefficients are interpreted as time ratios, indicating the proportional acceleration or deceleration of survival time associated with each covariate.
In practice, Accelerated Failure Time Models are estimated using maximum likelihood estimation applied to censored survival data. They are widely used in oncology, chronic disease modelling and health economic evaluation to extrapolate survival beyond observed follow-up, estimate treatment effects on survival duration and generate inputs for cost-effectiveness models.
Purpose
Used to model survival times directly, estimate treatment effects on time to event, extrapolate survival beyond observed follow-up and support health economic and survival analyses when proportional hazards assumptions are not appropriate.
Mathematical Formulae
Primary Formula
ln(T) = ?? + ??X? + ??X? + ? + ??X? + ��
where:
- T = survival time
- ?? = intercept
- ?? ? ?? = regression coefficients
- X? ? X? = explanatory variables
- � = scale parameter
- � = random error term
Supporting Formulae
T = exp(?? + ?X + ��)
Time Ratio = exp(?)
Likelihood:
L = ? f(t?�X?)?? ? S(t?�X?)????
where:
- f(t) = probability density function
- S(t) = survival function
- � = event indicator
Related Mathematical Methods
- Parametric Survival Model
- Weibull Model
- Log-Normal Model
- Log-Logistic Model
- Exponential Model
- Maximum Likelihood Estimation
- Survival Function
- Hazard Function
- Cox Proportional Hazards Model
Example
An Accelerated Failure Time Model estimates a treatment coefficient of ? = 0.30.
Time Ratio:
Time Ratio = exp(0.30) = 1.35
Patients receiving the treatment are expected to survive approximately 35% longer than comparable patients receiving the reference treatment, assuming the fitted AFT model is appropriate.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| LN | =LN(B2) | Transforms observed survival times for Accelerated Failure Time regression. |
| EXP | =EXP(C2) | Converts regression coefficients into time ratios. |
| EXP | =EXP(B2+C2*D2) | Estimates survival time from fitted model coefficients. |
| LINEST | =LINEST(LN(B2:B101),C2:E101,TRUE,TRUE) | Estimates coefficients for simplified log-linear AFT analyses when appropriate. |
VBA (Optional)
A VBA macro can automate Accelerated Failure Time model prediction, survival extrapolation and generation of survival estimates for health economic models.
Sources
- Kalbfleisch JD, Prentice RL. The Statistical Analysis of Failure Time Data. 2nd ed.
- Kleinbaum DG, Klein M. Survival Analysis: A Self-Learning Text. 4th ed.
- Collett D. Modelling Survival Data in Medical Research. 4th ed.
- Royston P, Lambert PC. Flexible Parametric Survival Analysis Using Stata: Beyond the Cox Model.
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation.
Related Concepts (2)
Library
Publications
1
NICE DSU Technical Support Document 14: Survival analysis for economic evaluations alongside clinical trials – extrapolation with patient-level data — Nicholas R. Latimer, TSD 14 ed., 2013 (NICE Decision Support Unit (University of Sheffield))
The reference guidance on survival analysis for economic evaluation: fitting standard parametric models (exponential, Weibull, Gompertz, log-logistic, log-normal) to censored trial data and extrapolating to estimate lifetime survival benefit, with a process guide for model selection and justification.
Frequently Asked Questions (6)
What is an accelerated failure time model?
A class of survival models assuming covariates act multiplicatively on time itself, speeding or slowing progression to an event, unlike a Cox model.
Source: Kalbfleisch & Prentice 2002
How does an accelerated failure time model describe a covariate's effect?
An accelerated failure time model describes how a covariate speeds up or slows down the progression to an event by acting directly on the time scale itself. Rather than scaling the instantaneous hazard as a Cox model does, it stretches or compresses survival time, so a treatment might, say, double the expected time to relapse. This gives an intuitive interpretation in terms of time gained or lost, and suits data where the proportional-hazards assumption fails. Acting on time rather than hazard is its distinguishing feature. Collett (2015) describes this model.
Source: Collett 2015
How does an accelerated failure time model work?
An accelerated failure time model works by expressing the logarithm of the survival time as a function of covariates plus an error term, so that covariates multiply the time scale, accelerating or decelerating the event. A covariate with an accelerating effect shortens the expected time to the event, and a decelerating one lengthens it. The model typically assumes a particular distribution for the survival times. So an accelerated failure time model works by directly modelling survival time on covariates, with the covariates scaling the time to the event, which gives an interpretation in terms of how much a factor speeds up or slows down the progression to the event.
Source: Kalbfleisch & Prentice 2002
How does an accelerated failure time model differ from a Cox model?
An accelerated failure time model assumes covariates act multiplicatively on the survival time, speeding or slowing the event, while a Cox proportional hazards model assumes covariates act multiplicatively on the hazard, the instantaneous event rate. The accelerated failure time model usually specifies a distribution for the survival times and yields time ratios, whereas the Cox model is semi-parametric and yields hazard ratios. So the two differ in what the covariates act on and in their parameterisation, with the accelerated failure time model giving effects on the time scale and the Cox model on the hazard, offering an alternative when its assumptions or interpretation are preferred.
Source: Cox 1972
When is an accelerated failure time model used?
An accelerated failure time model is used when an interpretation of covariate effects in terms of accelerating or decelerating the time to an event is preferred, or when the proportional hazards assumption of a Cox model does not hold, since the accelerated failure time model does not require it. It is useful where an appropriate distribution for the survival times can be assumed. So an accelerated failure time model is used as an alternative to the Cox model in survival analysis, particularly when a time-scale interpretation is wanted or proportional hazards fails, providing effects expressed as how much a factor lengthens or shortens survival time.
Source: Kalbfleisch & Prentice 2002
What are the assumptions of an accelerated failure time model?
An accelerated failure time model assumes that covariates act multiplicatively on the time scale, so their effect is to accelerate or decelerate survival uniformly, and it usually assumes a specific parametric distribution for the survival times, such as a Weibull or log-normal. The choice of distribution affects the results, so it must fit the data. So an accelerated failure time model rests on the time-scale multiplicative effect of covariates and on an appropriate distributional assumption, and its validity depends on these holding, which means the fit of the assumed distribution is checked, distinguishing it from the semi-parametric Cox model that avoids specifying the baseline distribution.
Source: Kalbfleisch & Prentice 2002
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 10 Dec 2025
Content version: 1.0.0
Canonical Identity
- Term code
- HE-ES-SA-001
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