Signature
TR = exp(beta_1)
| Inputs | Definition | Unit |
|---|---|---|
beta_1 | Estimated coefficient for treatment, coded 1 for treatment and 0 for the reference group, in the log-linear model | log time ratio, no unit |
TR | Factor by which treatment multiplies every percentile of the time to the event | ratio, no unit |
|---|
Function
Accelerated failure time function
Models the logarithm of the time to an event as a linear function of covariates plus a scaled error term whose distribution defines the model, such as Weibull, log-logistic or log-normal. A covariate multiplies time itself, so the survival curve of the treated group is the reference curve stretched or shrunk along the time axis: S(t|x = 1) equals S_0(t / TR), where TR is the time ratio.
Implementations
Excel
Time ratio and 95% confidence limits
With the coefficient in the named cell Coef and its standard error in SE, the three formulas return the time ratio and its lower and upper 95% limits.
=EXP(Coef); =EXP(Coef-1.96*SE); =EXP(Coef+1.96*SE)
Assumptions
Constant time ratio across the survival curve
The same ratio applies at every percentile of the survival distribution. A quantile-quantile plot of the two groups' survival percentiles that follows a straight line through the origin supports the assumption.
Same distribution and scale in both groups
Treatment enters as a covariate, so both groups share the error distribution and the scale parameter sigma. If they differ, separate models of the same type are fitted to each arm and no single time ratio exists.
Worked examples
Coefficient of 0.4055
A treatment coefficient of 0.4055 gives a time ratio of about 1.50, so treatment makes survival times 50% longer, as in the article's example.
beta_1 = 0.4055; TR = 1.50
Coefficient of minus 0.2231
A negative coefficient of minus 0.2231 gives a time ratio of about 0.80, so events occur after 80% of the reference time.
beta_1 = -0.2231; TR = 0.80
Common errors
Confusing the time ratio with the acceleration factor
Some sources define the acceleration factor as the reciprocal of the time ratio, writing S(t) as S_0 of the factor times t. A published value of 1.5 on one definition is about 0.667 on the other, so the definition is checked before the estimate is used.
Reading a time ratio as a hazard ratio
A time ratio of 1.50 means longer survival times, whereas a hazard ratio of 1.50 means a higher hazard and shorter survival. The two are linked only through a fitted distribution, and for log-logistic and log-normal models no single hazard ratio exists.
Sources
Accelerated failure time models and time ratios
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. Section on accelerated failure time models (the log-linear form and exponentiated coefficients as time ratios).
Acceleration factor in NICE survival modelling
Latimer NR. NICE DSU Technical Support Document 14: Survival analysis for economic evaluations alongside clinical trials, extrapolation with patient-level data. Sheffield: Decision Support Unit, ScHARR, University of Sheffield; June 2011, last updated March 2013. Section 2.2 (the Weibull as a proportional hazards or accelerated failure time model, with the treatment effect as an acceleration factor acting multiplicatively on the time scale) and section 2.9 (log-logistic and log-normal models do not produce a single hazard ratio).
Canonical Identity
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