Signature
S_T = S_C^HR
| Inputs | Definition | Unit |
|---|---|---|
S_C | Probability that a comparator person remains free of the event up to time t, read from the baseline survival curve | probability from 0 to 1 |
HR | Hazard ratio of treatment to comparator, assumed constant from the time origin to t | ratio, no unit |
S_T | Probability that a treated person remains free of the event up to time t | probability from 0 to 1 |
|---|
Function
Relative hazard function
Maps the hazards of an event in a treatment group and a comparator group at the same time t to their ratio. Under proportional hazards the ratio is constant over time, and it can then be applied to a baseline survival curve or a baseline transition probability to obtain absolute outcomes for the treatment group.
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Implementations
Excel
Treatment survival column from a baseline column
With comparator survival in column B from row 2 and the hazard ratio in the named cell HazardRatio, the formula in C2, filled down, returns treatment survival at each time.
=B2^HazardRatio
Assumptions
Proportional hazards from the origin to t
The hazard ratio is constant over the whole period from the time origin to t, both within the trial and in any extrapolated period. When the assumption fails, NICE directs analysts to the alternative methods in technical support document 21.
Same population, time origin and event
The baseline curve describes the target population for the same event and from the same time origin as the trial that estimated the ratio, and competing events are not treated as ordinary independent censoring.
Worked examples
Two-year survival with a hazard ratio of 0.70
With comparator two-year survival of about 0.81873, from a constant hazard of 0.10 per year, and a hazard ratio of 0.70, treatment two-year survival is about 0.8694, as in the article.
S_C = 0.81873; HR = 0.70; S_T = 0.8694
Survival at the comparator median
At the comparator median, where S_C is 0.50, a hazard ratio of 0.70 leaves about 0.6156 of the treated group event-free.
S_C = 0.50; HR = 0.70; S_T = 0.6156
Common errors
Multiplying the survival probability by the hazard ratio
Multiplying 0.81873 by 0.70 gives about 0.5731, which would imply that treatment lowers survival. The ratio acts on the cumulative hazard, so it enters as a power of the survival probability.
Applying one hazard ratio after proportional hazards has failed
When hazards cross, or the treatment effect wanes or is delayed, a single ratio applied over a lifetime horizon can misstate both the size and the timing of benefit. TSD 14 advises testing the assumption and, if it fails, fitting separate parametric models of the same type to each arm or allowing the hazard ratio to vary over time.
Sources
Applying a hazard ratio to a base survival curve
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 2.9 Modelling approaches (a hazard ratio applied to a base survival curve requires the proportional hazards assumption) and section 3.2 on log-cumulative hazard plots.
NICE manual on proportional hazards in extrapolation
National Institute for Health and Care Excellence. NICE technology appraisal and highly specialised technologies guidance: the manual (PMG36). Published 31 January 2022, last updated 31 March 2026. Chapter 4 Economic evaluation, sections 4.6.22 to 4.6.25 (assessing proportional hazards, pooling hazard ratios only when the assumption holds and is clinically plausible, alternative methods from technical support document 21 when it does not, and plausibility of extrapolated hazard functions).
Survival modelling in decision models
Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation. Oxford: Oxford University Press; 2006.
Canonical Identity
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