Accelerated Failure Time Model

Accelerated Failure Time Model

Most survival analysis in clinical trials reports a hazard ratio, but many health economic models need something different, namely a way to extrapolate survival beyond the end of a trial, where the proportional hazards assumption may not hold. The accelerated failure time model offers an alternative description of a treatment effect, as a speeding up or slowing down of time to the event, and several of the distributions most often used for extrapolation belong to it.

How the model works

An accelerated failure time model treats the logarithm of survival time as the outcome and models it as a linear function of the covariates plus an error term. Bradburn and colleagues describe the principle as a covariate stretching or shrinking the survival curve along the time axis by a constant relative amount.

$$\log T = \beta_0 + \beta_1 x + \sigma \varepsilon$$

where $T$ is the time to the event, $x$ is a covariate such as treatment group, $\beta_0$ and $\beta_1$ are coefficients, $\sigma$ is a scale parameter and $\varepsilon$ is an error term whose distribution defines the particular model. In this form the exponentiated coefficient is the time ratio.

$$\text{TR} = e^{\beta_1}, \qquad S(t \mid x = 1) = S_0\!\left(\frac{t}{\text{TR}}\right)$$

where TR is the time ratio, $S_0$ is the survival function for the reference group and $S(t \mid x = 1)$ is survival in the treated group. A time ratio above 1 prolongs the time to the event, and a time ratio below 1 shortens it. NICE Decision Support Unit Technical Support Document 14 describes the treatment effect in these models as an acceleration factor that acts multiplicatively on the time scale. Some sources define the acceleration factor as the reciprocal of this time ratio, so the direction of the effect should be checked before a published estimate is used. The model assumes that the time ratio is the same at every point on the survival curve, which can be checked with a quantile-quantile plot of the two groups' survival percentiles.

Accelerated failure time and proportional hazards

The two families describe a treatment effect in different ways. In a proportional hazards model, such as the Cox proportional hazards model, a covariate multiplies the hazard by a constant hazard ratio at every time point. In an accelerated failure time model, a covariate multiplies time itself, so the whole survival experience happens faster or slower.

Technical Support Document 14, Bradburn and colleagues, and Carroll together classify the common distributions as shown below.

The Weibull, which includes the exponential as a special case, is the only distribution that is both proportional and accelerated, which is why it can report either a hazard ratio or a time ratio. The document notes that log-logistic and log-normal models do not produce a single hazard ratio, so the proportional hazards assumption does not hold for them, although the treatment effect can still be estimated as an acceleration factor.

Worked example of what a time ratio means

The figures below are illustrative. In a trial, the median survival of the control group is 12 months, and an accelerated failure time model estimates a time ratio of 1.5 for the new treatment.

$$\text{median}_{\text{treated}} = 1.5 \times 12 = 18$$

where the result is the median survival of the treated group in months. The same time ratio applies to every point on the survival curve, so the time by which a quarter of the treated group has died is 1.5 times the corresponding time in the control group. In the same way, $S(18 \mid x = 1) = S_0(18/1.5) = S_0(12) = 0.5$. The mean survival time is also multiplied by the time ratio, although a restricted mean over a fixed horizon is not. A decision-maker can read the result as treatment making the typical patient's survival 50 per cent longer, which is often easier to explain than a ratio of hazards.

Choosing a model for extrapolation

In economic evaluation, the choice of survival model matters most when trial data must be extrapolated over a lifetime horizon. Technical Support Document 14 advises that, when patient-level data are available, analysts should test the proportional hazards assumption rather than rely on it, and that if it does not seem appropriate, it is likely to be most sensible to fit separate parametric models of the same type to each treatment arm.

The document recommends considering the exponential, Weibull, Gompertz, log-logistic, log-normal and generalised gamma models, and comparing log-cumulative hazard plots, quantile-quantile plots or suitable residual plots to guide the initial selection. Statistical fit can be compared with measures such as the Akaike information criterion. The document defines an appropriate model by two criteria, whether a model fits the observed data, and whether its extrapolated portion is clinically and biologically plausible. Models that meet only one of these criteria are likely to be inappropriate.

What the time ratio can and cannot tell a decision-maker

Bradburn and colleagues describe the time ratio as arguably more interpretable than a ratio of two hazards, and Zhang notes that although hazard ratios are more familiar to clinicians, the time ratio may be easier to understand. For a parametric survival model used in a cost-effectiveness analysis, however, the output that matters most is the full extrapolated survival curve, since the area under it determines the life-years gained and, with utility values, the QALYs gained.

A time ratio also depends on the chosen distribution. The same trial data can give very different long-term survival under Weibull and log-logistic models even when both fit the observed period well, because the log-logistic often produces a long tail, so the choice of model can change the cost-effectiveness result. Technical Support Document 14 therefore recommends sensitivity analysis using alternative plausible survival models when carrying out extrapolation.

Common errors with time ratios and extrapolation

A frequent error is to assume proportional hazards by default and report a single hazard ratio from a model where the assumption does not hold. Checking the assumption against the data avoids this. When it fails, Technical Support Document 14 favours fitting separate models of the same type to each arm, or allowing the hazard ratio to vary over time. An accelerated failure time model with treatment as a covariate is an option only if its own assumption holds, which requires a constant time ratio across the whole curve, with the same shape parameter in both arms. Another is to confuse the two parameterisations of the Weibull. The same fitted model gives a hazard ratio in one form and a time ratio in the other, and the two numbers are not interchangeable. For the Weibull, they are linked by $\log \text{HR} = -\beta_1/\sigma$, so a time ratio above 1 corresponds to a hazard ratio below 1.

A third is to choose the model that fits the trial best without asking whether its long-term predictions make sense. A curve that fits well over two years of follow-up can imply implausible survival over thirty years, and the extrapolated part usually drives the economic result.

Sources

• Latimer NR. NICE DSU Technical Support Document 14: Survival analysis for economic evaluations alongside clinical trials, extrapolation with patient-level data. Report by the Decision Support Unit. June 2011, last updated March 2013.
• Bradburn MJ, Clark TG, Love SB, Altman DG. Survival analysis part II: multivariate data analysis, an introduction to concepts and methods. British Journal of Cancer. 2003;89(3):431-436.
• Bradburn MJ, Clark TG, Love SB, Altman DG. Survival analysis part III: multivariate data analysis, choosing a model and assessing its adequacy and fit. British Journal of Cancer. 2003;89(4):605-611.
• Zhang Z. Parametric regression model for survival data: Weibull regression model as an example. Annals of Translational Medicine. 2016;4(24):484.
• Carroll KJ. On the use and utility of the Weibull model in the analysis of survival data. Controlled Clinical Trials. 2003;24(6):682-701.