Signature
H_t = lambda * t^gamma; logH = log(lambda) + gamma * log(t)
| Inputs | Definition | Unit |
|---|---|---|
lambda | Scale parameter, above zero, in the parameterisation S(t) = exp(minus lambda t^gamma) used in TSD 21; equal to H at t = 1 | per time unit raised to gamma |
t | Time from the origin of the fitted curve, above zero | years or the time unit used in fitting |
gamma | Shape parameter, above zero; the gradient of log H against log t | none |
H_t | Cumulative hazard of the Weibull curve at time t | none |
|---|---|---|
logH | Natural logarithm of H_t, the vertical axis of the log-cumulative hazard plot | none |
Function
Cumulative hazard function linking the hazard, survival and cycle probabilities
Maps a hazard function over follow-up to its running total from the time origin, the cumulative hazard H(t). Survival is the exponential of minus H(t), so H(t) can be read off a survival curve and turned back into survival, and the rise in H(t) across a model cycle gives the probability of the event in that cycle for people event free at its start. On the log scale a Weibull cumulative hazard is a straight line in log time, which is the basis of the log-cumulative hazard plot. The notation follows the Cumulative Hazard article.
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Implementations
Excel
Weibull cumulative hazard and its log from named parameters
With the parameters in cells named WeibullScale and WeibullShape and the time in TimeYears, the formulas return the cumulative hazard and its natural log.
=WeibullScale*TimeYears^WeibullShape; =LN(WeibullScale)+WeibullShape*LN(TimeYears)
Assumptions
Weibull parameterisation stated before transfer to a model
The formula uses lambda t^gamma, as in TSD 21. Software and tutorials also write (lambda t)^gamma, as in the DARTH tutorial, so the scale parameter differs between sources for the same curve and must be converted before use.
Straight line on the log scale judged within follow-up only
A straight log-cumulative hazard plot supports the Weibull or exponential shape within the observed data. TSD 14 notes that such checks address internal validity, not whether the extrapolated part of the curve is plausible.
Worked examples
Weibull cumulative hazard of 0.8 at year 4 with gamma of 1.5
With lambda of 0.10 and gamma of 1.5, H(4) = 0.10 x 8 = 0.8 and its natural log is about minus 0.2231, as in the article.
lambda = 0.1; gamma = 1.5; t = 4; H_t = 0.8; logH = -0.2231
Weibull log cumulative hazard at year 1 equals log lambda
At t of 1, H equals lambda, 0.1, and its log is about minus 2.3026, the intercept of the line. The gradient between years 1 and 4 is (minus 0.2231 plus 2.3026) divided by 1.3863, which is 1.5, the shape parameter.
lambda = 0.1; gamma = 1.5; t = 1; H_t = 0.1; logH = -2.3026
Exponential case with gamma of 1 on the log-cumulative hazard scale
With gamma of 1 the curve is exponential: H(4) = 0.4, log H about minus 0.9163, and the line has gradient 1 (computed here for illustration).
lambda = 0.1; gamma = 1; t = 4; H_t = 0.4; logH = -0.9163
Common errors
Reading a gradient below 1 as a rising hazard
A log-cumulative hazard line steeper than 1 means a rising hazard and one flatter than 1 a falling hazard. TSD 14's illustrative example read a gradient below 1 after about five weeks as making an exponential model unlikely for that arm.
Taking a straight log-cumulative hazard plot as proof of the extrapolation
A good straight-line fit within follow-up says nothing about the tail. Flexible models also become linear in log time beyond the final knot, so their extrapolation resembles a Weibull whether or not that is plausible.
Sources
Weibull log cumulative hazard as a straight line in log time
Rutherford MJ, Lambert PC, Sweeting MJ, Pennington B, Crowther MJ, Abrams KR, Latimer NR. NICE DSU Technical Support Document 21: Flexible methods for survival analysis. Sheffield: Decision Support Unit, ScHARR, University of Sheffield; 2020. Section 3.2.2, equation 1: transforming S(t) = exp(minus lambda t^gamma) gives log H(t) = log(lambda) + gamma log(t), a linear function of log time with intercept log(lambda) and gradient gamma; section 4.3.1: beyond the final knot a flexible parametric model on this scale has a log cumulative hazard linear in log time, similar to a Weibull.
Log-cumulative hazard plots for choosing parametric survival models
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 3.2 (plots of log of minus log survival against log time test the Weibull and exponential distributions and show whether proportional hazards is reasonable; a gradient below 1 makes an exponential model unlikely), section 3.5 (internal, not external, validity) and section 6.1 (model selection steps).
Weibull cumulative hazard written as (lambda t)^gamma in the DARTH tutorial
Alarid-Escudero F, Krijkamp E, Enns EA, Yang A, Hunink MGM, Pechlivanoglou P, Jalal H. A tutorial on time-dependent cohort state-transition models in R using a cost-effectiveness analysis example. Medical Decision Making. 2023;43(1):21-41. Section 4.2, equation 2: H(tau) = (lambda tau)^gamma, the alternative parameterisation.
Canonical Identity
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