# Accelerated failure time model: Wikis

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# Encyclopedia

In the statistical area of survival analysis, an accelerated failure time model (AFT model) is a parametric model that provides an alternative to the commonly-used proportional hazards models. Whereas a proportional hazards model assumes that the effect of a covariate is to multiply the hazard by some constant, an AFT model assumes that the effect of a covariate is to multiply the predicted event time by some constant. AFT models can be therefore be framed as linear models for the logarithm of the survival time.

## Comparison with proportional hazard models

The biggest difference is that AFT models are always fully parametric, i.e. a probability distribution must be specified, as there is no known equivalent of Cox's semi-parametric proportional hazards model. The choice of origin from which to measure time at risk is important in all parametric survival models.

Unlike proportional hazards models, the regression parameter estimates from AFT models are robust to the presence of unmeasured confounders. They are also less affected by the choice of probability distribution.

The results of AFT models are easily interpreted. For example, the results of a clinical trial with mortality as the endpoint could be interpreted as a certain percentage increase in future life expectancy on the new treatment compared to the control. So a patient could be informed that he would be expected to live (say) 15% longer if he took the new treatment. Hazard ratios can prove harder to explain in layman's terms.

More probability distributions can be used in AFT models than parametric proportional hazard models, including distributions that have unimodal hazard functions.

### Distributions used in AFT models

To be used in an AFT model, a distribution must have a parameterisation that includes a scale parameter. The logarithm of the scale parameter is then modelled as a linear function of the covariates.

The log-logistic distribution provides the most commonly-used AFT model. Unlike the Weibull distribution, it can exhibit a non-monotonic hazard function which increases at early times and decreases at later times. It is similar in shape to the log-normal distribution but its cumulative distribution function has a simple closed form, which becomes important computationally when fitting data with censoring.

The Weibull distribution (including the exponential distribution as a special case) can be parameterised as either a proportional hazards model or an AFT model, and is the only family of distributions to have this property. The results of fitting a Weibull model can therefore be interpreted in either framework.

Other distributions suitable for AFT models include the log-normal, gamma and inverse Gaussian distributions, although they are less popular than the log-logistic, partly as their cumulative distribution functions do not have a closed form.

## References

1. ^ Lambert, Philippe; Collett, Dave; Kimber, Alan; Johnson, Rachel (2004), "Parametric accelerated failure time models with random effects and an application to kidney transplant survival", Statistics in Medicine 23: 3177–3192, doi:10.1002/sim.1876
2. ^ Kay, Richard; Kinnersley, Nelson (2002), "On the use of the accelerated failure time model as an alternative to the proportional hazards model in the treatment of time to event data: A case study in influenza" ( – Scholar search), Drug Information Journal 36: 571–579
• Collett, D. (2003), Modelling Survival Data in Medical Research (2nd ed.), CRC press, ISBN 1584883251

### Books

• Cox, David Roxbee; Oakes, D. (1984), Analysis of Survival Data, CRC Press, ISBN 041224490X
• Marubini, Ettore; Valsecchi, Maria Grazia (1995), Analysing Survival Data from Clinical Trials and Observational Studies, Wiley, ISBN 0470093412