Accelerated failure time model

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 accelerate or decelerate the life course of a disease by some constant. This is especially appealing in a technical context where the 'disease' is a result of some mechanical process with a known sequence of intermediary stages.

Model specification

In full generality, the accelerated failure time model can be specified as

\lambda (t|\theta )=\theta \lambda _{0}(\theta t)

where $\theta$ denotes the joint effect of covariates, typically $\theta =\exp(\beta _{1}X_{1}+\cdots +\beta _{p}X_{p})$ .

This is satisfied, if the probability density function of the event is taken to be $f(t|\theta )=\theta f_{0}(\theta t)$ , from which is follows for the survival function that $S(t|\theta )=S(\theta t)$ . From this it is easy to see that the moderated life time $T$ is distributed such that $T/\theta$ and the unmoderated life time $T_{0}$ have the same distribution. Consequently, $log(T)$ can be written as

log(T)=log(\theta )+log(T/\theta )

where the last term is distributed as $log(T_{0})$ , i.e. independently of $\theta$ . This reduces the accelerated failure time model into a linear model, $log(\theta )$ representing the fixed effects, and $log(T/\theta )$ representing the noise. Different distributional forms of $log(T/\theta )$ imply different distributional forms of $T_{0}$ , i.e. different baseline distributions of the survival time. It is typical of survival-analytic contexts, that many of the observations are censored, i.e. we only know that $T_{i}>t_{i}$ , not $T_{i}=t_{i}$ . In fact, the former case represents survival over the follow-up, while the later case represents event/death/censoring. These right-censored observations can pose technical challenges for estimating the model, if the distribution of $T_{0}$ is unusual.

Statistical issues

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.

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.

Connection with multistate models

If an event occurs by a constant rate $\lambda _{0}(t)=c$ , it is easy to see that multiplying $c$ by $\theta$ results in a model that fulfills both proportional hazards and accelerated failure time assumptions. The accelerated failure time model turns out to be the proper generalization of this model for a multistate (time-homogenous) Markov chain. Namely, consider that it is possible for a system to be in any of states $1,\ldots ,n$ at any moment $t>0$ , and let us assume that the transition between these states occurs according to a constant set of rates $q_{ij}$ . (The collection of these rates is known as the transition matrix $Q$ , also sometimes known as the infinitesimal generator^[6]). Let us then assume that $Q$ is multiplied by $\theta$ , i.e. all of transition rates are multiplied by the same constant. It is easy to show using standard results from the theory of Markov chains^[7] that the instantaneous rate of arriving in any one state follows an accelerated failure time model.

The biological interpretation of this is such that if a disease develops as a result of a known sequence of states $1,\ldots ,n$ , and the effect of risk factors accelerates this process linearly, the overall risk of disease per time unit follows an accelerated failure time model, not e.g. a proportional hazards model. Of course, it may be biologically more realistic to assume that $\theta$ scales each of the rates $q_{ij}$ differentially. The resulting model in this case is a hybrid (i.e. neither of the commonly used types), and it is difficult to say anything concerning its properties in full generality.

Producing gamma distribution

One may produce the gamma distribution as a special case of the previous Markov chain model. Let us assume that a disease develops such that an individual progresses from state $1$ (complete health) to state $n$ (clinical condition or death) always in the numerical order, and each of the transitions occurs in the baseline case by a constant rate $\lambda$ . Then it follows that the waiting time in each of the $n-1$ pre-stages follows an exponential distribution with mean $1/\lambda$ . In this case, it is known that the total waiting time, i.e. total time until arrival to $n$ , has a gamma distribution. Gamma distribution is a two-parameter family that readily applies to parametric survival analysis; it is possible to estimate both $\lambda$ , $n$ and $\theta$ from empirical data.

References

^ Buckley, Jonathan; James, Ian (1979), "Linear regression with censored data", Biometrika, 66 (3): 429–436, doi:10.1093/biomet/66.3.429, JSTOR 2335161
^ Attention: This template ({{cite doi}}) is deprecated. To cite the publication identified by doi:10.1002/sim.4780111409, please use {{cite journal}} (if it was published in a bona fide academic journal, otherwise {{cite report}} with |doi=10.1002/sim.4780111409 instead.
^ 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
^ Template:Cite doi/10.1002.2F.28SICI.291097-0258.2819970130.2916:2.3C215::AID-SIM481.3E3.0.CO.3B2-J
^ 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
^ Norris, J.R. (1997). Markov Chains. Cambridge: Cambridge University Press.
^ Norris, J.R. (1997). Markov Chains. Cambridge: Cambridge University Press.