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 accelerate or decelerate the life course of a disease by some constant.
There is strong basic science evidence from C. 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] where
denotes the joint effect of covariates, typically
This is satisfied if the probability density function of the event is taken to be
From this it is easy[citation needed] to see that the moderated life time
Typically, in survival-analytic contexts, many of the observations are censored: we only know that
These right-censored observations can pose technical challenges for estimating the model, if the distribution of
in accelerated failure time models is straightforward:
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.
(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.
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 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.
The log-logistic distribution provides the most commonly used AFT model[citation needed].
Unlike the Weibull 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 the log-normal distribution but it has heavier tails.
The log-logistic cumulative distribution function has a simple closed form, which becomes important computationally when fitting data with censoring.
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
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 a multiplicatively closed group, such as the positive real numbers, is suitable for an AFT model.