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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 therefore be framed as linear models for the logarithm of the survival time.

## Comparison with proportional hazard models

Unlike proportional hazards models, in which Cox's semi-parametric proportional hazards model is more widely used than parametric models, AFT models are predominately fully parametric i.e. a probability distribution is specified. (Buckley and James[1] proposed a semi-parametric AFT but its use is relatively uncommon in applied research; in a 1992 paper, Wei[2] pointed out that the Buckley–James model has no theoretical justification and lacks robustness, and reviewed alternatives.)

Unlike proportional hazards models, the regression parameter estimates from AFT models are robust to omitted covariates. They are also less affected by the choice of probability distribution.[3][4]

The results of AFT models are easily interpreted.[5] 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. Finally, the generalized gamma distribution is a three-parameter distribution that includes the Weibull, log-normal and gamma distributions as special cases.

## References

1. ^ Buckley, Jonathan; James, Ian (1979), "Linear regression with censored data", Biometrika 66 (3): 429–436, doi:10.1093/biomet/66.3.429, JSTOR 2335161
2. ^ Wei, L. J. (1992). "The accelerated failure time model: A useful alternative to the cox regression model in survival analysis". Statistics in Medicine 11 (14-15): 1871–1879. doi:10.1002/sim.4780111409. PMID 1480879. edit
3. ^ 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 (20): 3177–3192, doi:10.1002/sim.1876, PMID 15449337
4. ^ Keiding, N.; Andersen, P. K.; Klein, J. P. (1997). "The Role of Frailty Models and Accelerated Failure Time Models in Describing Heterogeneity Due to Omitted Covariates". Statistics in Medicine 16 (1-3): 215–224. doi:10.1002/(SICI)1097-0258(19970130)16:2<215::AID-SIM481>3.0.CO;2-J. PMID 9004393.
5. ^ 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", Drug Information Journal 36 (3): 571–579
• Collett, D. (2003), Modelling Survival Data in Medical Research (2nd ed.), CRC press, ISBN 1-58488-325-1

### Books

• Cox, David Roxbee; Oakes, D. (1984), Analysis of Survival Data, CRC Press, ISBN 0-412-24490-X
• Marubini, Ettore; Valsecchi, Maria Grazia (1995), Analysing Survival Data from Clinical Trials and Observational Studies, Wiley, ISBN 0-470-09341-2
• Martinussen, Torben; Scheike, Thomas (2006), Dynamic Regression Models for Survival Data, Springer, ISBN 0-387-20274-9
• Bagdonavicius, Vilijandas; Nikulin, Mikhail (2002), Accelerated Life Models. Modeling and Statistical Analysis, Chapman&Hall/CRC, ISBN 1-58488-186-0

Original courtesy of Wikipedia: http://en.wikipedia.org/wiki/Accelerated_failure_time_model — Please support Wikipedia.
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