The accelerated failure time (AFT) model is a different type of model that may be used for the analysis of survival time data. For a group of patients with covariates (x1, x2, … xp), the model is written mathematically as
where S0(t) is the baseline survivor function and ϕ is an ‘acceleration factor’ that depends on the covariates according to the formula
The principle here is that the effect of a covariate is to stretch or shrink the survival curve along the time axis by a constant relative amount ϕ. demonstrates this for the case of a single covariate (x1) with two levels, for example, x1=0 for a placebo group and x1=1 for a new treatment group. The survival probabilities, S(t), for the placebo and new treatment groups are S0(t) and S0(ϕt), respectively. The proportion of patients who are event-free in the placebo group at any time point t1 is the same as the proportion of those who are event-free in the new treatment group at a time t2=ϕt1. shows the cases where ϕ>1 and ϕ<1, which represent situations where the length of survival is increased and decreased in the new treatment group compared with the placebo, respectively.
The AFT model is commonly rewritten as being log-linear with respect to time, giving
is a measure of (residual) variability in the survival times. Thus, the survival times can be seen to be multiplied by a constant effect under this model specification, and the exponentiated coefficients, exp(bi
), are referred to as time ratios
. A time ratio above 1 for the covariate implies that this ‘slows down’, or prolongs the time to the event, while a time ratio below 1 indicates that an earlier event is more likely.
When the survival times follow a Weibull distribution, it can be shown that the AFT and PH models are the same. However, the AFT family of models differs crucially from the PH model types in terms of their interpretation of effect sizes as time ratios as opposed to hazard ratios.
The survival times are usually assumed to follow a specific distributional form in the AFT framework. Distributions such as the Log-Normal, Log-Logistic, Generalised Gamma
may be used to represent such survival data. Alternative methods include the method of Buckley and James (1979)
, which is discussed by Stare et al (2000)
, and semiparametric AFT models, in which the baseline survivor function is estimated nonparametrically (see Wei, 1992
, for an overview), but have not yet been widely implemented in statistical software.
As with the PH approach, other quantities such as projected survival probabilities may be derived. Also in keeping with PH models is the fact that AFT models make assumptions; the appropriate choice of statistical distribution needs to be made, and also the covariate effects are assumed to be constant and multiplicative on the timescale, that is, that the covariate impacts on survival by a constant factor.
Parametric AFT models fitted to the lung cancer trial data
We use the non-small cell lung cancer dataset to illustrate the AFT model, focusing on the relapse-free survival (i.e., the time from diagnosis to the reappearance of cancer, with patients censored at time of death if no recurrence had appeared). Again, we present both the univariate and multivariate effect sizes in . The specific comparison of interest was the effect of adjuvant (platinum-based) chemotherapy and radiotherapy compared with radiotherapy alone. The unadjusted treatment effect may be summarised by a time ratio of 1.91 (95% CI: 1.21–3.01; P=0.005), which, having allowed for other covariates increased slightly to 2.05. Therefore, we can conclude that the time to recurrence was significantly prolonged (approximately doubled) among patents given adjuvant chemotherapy in comparison with those who were not.
Time ratios from the generalised gamma AFT model for the lung cancer trial
Again, we can derive model-based predictions: overall, patients allocated to receive adjuvant chemotherapy had a predicted median survival time of approximately 16 months, as opposed to 8 months among those treated with radiotherapy alone. Other factors are also significant and would influence these times, but these are of less importance in the context of the comparative trial. We will return to this example in the next paper of this series.