Signature
HR = exp(-beta_1 / sigma)
| Inputs | Definition | Unit |
|---|---|---|
beta_1 | Treatment coefficient from the Weibull accelerated failure time model, whose exponent is the time ratio | log time ratio, no unit |
sigma | Scale of the error term in the log-linear model, equal to 1 divided by the Weibull shape parameter | no unit, greater than zero |
HR | Constant hazard ratio of treatment to reference implied by the Weibull fit | 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
Hazard ratio from Weibull accelerated failure time output
Excel combines the named cells Coef and Sigma. The second formula gives the same result from the time ratio and the shape parameter, Shape, equal to 1 divided by Sigma.
=EXP(-Coef/Sigma); =TimeRatio^(-Shape)
Assumptions
Weibull distribution with a common shape
Both groups follow a Weibull distribution with the same shape parameter. The relationship does not apply to log-logistic, log-normal or generalised gamma models, which do not produce a single hazard ratio.
Worked examples
Time ratio of 1.5 with sigma of 0.8
A coefficient of 0.4055, a time ratio of about 1.5, and a scale parameter of 0.8, a Weibull shape of 1.25, imply a hazard ratio of about 0.6024. A time ratio above 1 corresponds to a hazard ratio below 1.
beta_1 = 0.4055; sigma = 0.8; HR = 0.6024
Exponential special case
With sigma equal to 1 the Weibull reduces to the exponential, and the hazard ratio is the reciprocal of the time ratio, about 0.667 for a time ratio of 1.5.
beta_1 = 0.4055; sigma = 1; HR = 0.667
Common errors
Converting a time ratio to a hazard ratio without sigma
Taking the hazard ratio as 1 divided by the time ratio, about 0.667 in the first example, ignores the shape of the Weibull and is correct only for the exponential. The value with sigma of 0.8 is about 0.6024.
Mixing parameterisations from different software
Some programs report the log-linear scale sigma and others the Weibull shape, which is its reciprocal, and some report proportional hazards rather than accelerated failure time coefficients. Using a shape where sigma is expected, or a proportional hazards coefficient as beta_1, gives a wrong hazard ratio.
Sources
Weibull regression in proportional and accelerated forms
Zhang Z. Parametric regression model for survival data: Weibull regression model as an example. Annals of Translational Medicine. 2016;4(24):484. Section on the Weibull regression model (proportional hazards and accelerated failure time forms, with the hazard ratio coefficient equal to minus beta_1 divided by sigma).
Weibull as both proportional and accelerated
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.
Canonical Identity
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