In thestatistical area ofsurvival analysis, anaccelerated failure time model (AFT model) is aparametric model that provides an alternative to the commonly usedproportional hazards models. Whereas a proportional hazards model assumes that the effect of acovariate is to multiply thehazard by some constant, an AFT model assumes that the effect of a covariate is to accelerate or decelerate the life course of a disease by some constant. There is strong basic science evidence fromC. elegans experiments by Stroustrup et al.^[1] indicating that AFT models are the correct model for biological survival processes.

In full generality, the accelerated failure time model can be specified as^[2]

${\displaystyle \lambda (t|\theta )=\theta {\lambda }_0(\theta t)}$

where${\displaystyle \theta }$ denotes the joint effect of covariates, typically${\displaystyle \theta =\exp (-[{\beta }_1{X}_1+\cdots +{\beta }_p{X}_p])}$. (Specifying the regression coefficients with a negative sign implies that high values of the covariatesincrease the survival time, but this is merely a sign convention; without a negative sign, they increase the hazard.)

This is satisfied if theprobability density function of the event is taken to be${\displaystyle f(t|\theta )=\theta f_0(\theta t)}$; it then follows for thesurvival function that${\displaystyle S(t|\theta )=S_0(\theta t)}$. From this it is easy^{[citation needed]} to see that the moderated life time${\displaystyle T}$ is distributed such that${\displaystyle T\theta }$ and the unmoderated life time${\displaystyle T_0}$ have the same distribution. Consequently,${\displaystyle \log (T)}$ can be written as

${\displaystyle \log (T)=-\log (\theta )+\log (T\theta ):=-\log (\theta )+ϵ}$

where the last term is distributed as${\displaystyle \log (T_0)}$, i.e., independently of${\displaystyle \theta }$. This reduces the accelerated failure time model toregression analysis (typically alinear model) where${\displaystyle -\log (\theta )}$ represents the fixed effects, and${\displaystyle ϵ}$ represents the noise. Different distributions of${\displaystyle ϵ}$ imply different distributions of${\displaystyle T_0}$, i.e., different baseline distributions of the survival time. Typically, in survival-analytic contexts, many of the observations are censored: we only know that${\displaystyle T_i>t_i}$, not${\displaystyle T_i=t_i}$. In fact, the former case represents survival, while the later case represents an event/death/censoring during the follow-up. These right-censored observations can pose technical challenges for estimating the model, if the distribution of${\displaystyle T_0}$ is unusual.

The interpretation of${\displaystyle \theta }$ in accelerated failure time models is straightforward:${\displaystyle \theta =2}$ means that everything in the relevant life history of an individual happens twice as fast. For example, if the model concerns the development of a tumor, it means that all of the pre-stages progress twice as fast as for the unexposed individual, implying that the expected time until a clinical disease is 0.5 of the baseline time. However, this does not mean that the hazard function${\displaystyle \lambda (t|\theta )}$ is always twice as high - that would be theproportional hazards model.

Unlike proportional hazards models, in whichCox's semi-parametric proportional hazards model is more widely used than parametric models, AFT models are predominantly fully parametric i.e. aprobability distribution is specified for${\displaystyle \log (T_0)}$. (Buckley and James^[3] proposed a semi-parametric AFT but its use is relatively uncommon in applied research; in a 1992 paper, Wei^[4] pointed out that the Buckley–James model has no theoretical justification and lacks robustness, and reviewed alternatives.) This can be a problem, if a degree of realistic detail is required for modelling the distribution of a baseline lifetime. Hence, technical developments in this direction would be highly desirable.

When a frailty term is incorporated in the survival model, the regression parameter estimates from AFT models are robust to omittedcovariates, unlike proportional hazards models. They are also less affected by the choice of probability distribution for the frailty term.^[5][6]

The results of AFT models are easily interpreted.^[7] For example, the results of aclinical trial with mortality as the endpoint could be interpreted as a certain percentage increase in futurelife 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.

Thelog-logistic distribution provides the most commonly used AFT model^{[citation needed]}. Unlike theWeibull distribution, it can exhibit a non-monotonic hazard function which increases at early times and decreases at later times. It is somewhat similar in shape to thelog-normal distribution but it has heavier tails. The log-logisticcumulative distribution function has a simpleclosed form, which becomes important computationally when fitting data withcensoring. For the censored observations one needs the survival function, which is the complement of the cumulative distribution function, i.e. one needs to be able to evaluate${\displaystyle S(t|\theta )=1-F(t|\theta )}$.

TheWeibull distribution (including theexponential 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. However, the biological applicability of this model may be limited by the fact that the hazard function is monotonic, i.e. either decreasing or increasing.

Any distribution on amultiplicatively closed group, such as thepositive real numbers, is suitable for an AFT model. Other distributions include thelog-normal,gamma,hypertabastic,Gompertz distribution, andinverse Gaussian distributions, although they are less popular than the log-logistic, partly as their cumulative distribution functions do not have a closed form. Finally, thegeneralized gamma distribution is a three-parameter distribution that includes theWeibull,log-normal andgamma distributions as special cases.

• Bradburn, MJ; Clark, TG; Love, SB; Altman, DG (2003), "Survival Analysis Part II: Multivariate data analysis - an introduction to concepts and methods",British Journal of Cancer,89 (3):431–436,doi:10.1038/sj.bjc.6601119,PMC 2394368,PMID 12888808
• Hougaard, Philip (1999), "Fundamentals of Survival Data",Biometrics,55 (1):13–22,doi:10.1111/j.0006-341X.1999.00013.x,PMID 11318147
• Collett, D. (2003),Modelling Survival Data in Medical Research (2nd ed.), CRC press,ISBN 978-1-58488-325-8
• Cox, David Roxbee; Oakes, D. (1984),Analysis of Survival Data, CRC Press,ISBN 978-0-412-24490-2
• Marubini, Ettore; Valsecchi, Maria Grazia (1995),Analysing Survival Data from Clinical Trials and Observational Studies, Wiley,ISBN 978-0-470-09341-2
• 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

• Survival analysis

• Articles with short description
• Short description matches Wikidata
• All articles with unsourced statements
• Articles with unsourced statements from June 2018
• Articles with unsourced statements from May 2023