Signature
S = S_star * (pi + (1 - pi) * S_u)
| Inputs | Definition | Unit |
|---|---|---|
S_star | Expected survival at time t of the general population matched for age, sex and calendar year, from life tables | probability from 0 to 1 |
pi | Proportion of patients who will not die from their disease | proportion from 0 to 1 |
S_u | Survival at time t of uncured patients from the disease-related (excess) hazard alone, so that an uncured patient's all-cause survival is S_star times S_u | probability from 0 to 1 |
S | Proportion of the cohort alive at time t | probability from 0 to 1 |
|---|
Function
Cure rate model survival and lifetime mean survival with a cure fraction
Maps a cure fraction, the expected survival of the matched general population and the survival of uncured patients to all-cause survival over time and to the lifetime mean survival and QALYs that drive an appraisal. Cured patients face background mortality only; uncured patients also carry an excess hazard from the disease. The mixture form weights the two groups directly, the non-mixture form bounds the excess cumulative hazard so that relative survival falls to the cure fraction, and mean survival is the cure-fraction weighted average of the two group means. Reused, not repeated here: the all-cause hazard as background plus excess hazard HE-FM-BTH-005, the continuous discount rate log(1 plus d) HE-FM-CONT-003, the per-cycle probability from a hazard HE-FM-TP-001 and the restricted mean from a Kaplan-Meier curve HE-FM-ADMC-003. Notation follows the Cure Rate Model article.
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Implementations
Excel
Mixture cure all-cause survival from named cells
With BgSurv, CureFrac and UncuredSurv named for one time point, the formula returns all-cause survival, held in AllCauseSurv. Filled down a column of times, it gives the whole curve.
=BgSurv*(CureFrac+(1-CureFrac)*UncuredSurv)
Assumptions
Population-level cure with zero excess hazard in the mixture cure model
Cure means that the excess (cause-specific) hazard of the cured group is zero, so cured patients die at general population rates; the model does not identify which patients are cured. Because a survival function must reach zero, the model is fitted to relative or cause-specific survival, not to all-cause survival alone.
Expected survival from matched life tables in the mixture cure model
S_star comes from life tables for the right calendar year, stratified by age and sex. If cured patients keep an excess risk, a standardised mortality ratio can be applied to the general population mortality rates behind S_star.
Mature data and a justified uncured distribution for a mixture cure fit
TSD 21 lists three assumptions: the data are sufficient to estimate a cure fraction reliably, a cure fraction exists and cure is reasonable at a given time point, and the distribution chosen for the uncured is appropriate. Different uncured distributions fitted to the same data can give very different values of pi.
Worked examples
Mixture cure survival at 5 years with a cure fraction of 0.25
With the article's illustrative constant hazards, background survival at 5 years is exp(minus 0.04 times 5), about 0.8187, and survival of the uncured from an excess hazard of 0.46 a year is exp(minus 2.3), about 0.1003. A cure fraction of 0.25 gives all-cause survival of about 0.2662 (computed here for illustration).
S_star = 0.818731; pi = 0.25; S_u = 0.100259; S = 0.266247
Mixture cure survival at 5 years with a cure fraction of 0.10
The same inputs with the comparator's cure fraction of 0.10 give about 0.1557, so the extra 15 percentage points of cure already add about 0.11 to survival at 5 years (computed here for illustration).
S_star = 0.818731; pi = 0.10; S_u = 0.100259; S = 0.155750
Mixture cure survival at 20 years once the uncured have almost all died
By 20 years survival of the uncured is about 0.0001, so relative survival is close to the cure fraction of 0.25, but all-cause survival is only about 0.1124 because cured patients also die of other causes; background survival is about 0.4493 (computed here for illustration).
S_star = 0.449329; pi = 0.25; S_u = 0.000101; S = 0.112366
Common errors
Fitting a mixture cure model to all-cause survival with no background mortality
Without S_star the cured group never dies, so all-cause survival stays at pi for ever and mean survival grows with the model horizon. TSD 21 notes that the event cannot be all-cause mortality because the survival function must reach zero; the model belongs in a relative or cause-specific survival framework.
Reading the cure fraction off the all-cause survival curve
In the relative survival form the cure fraction is higher than all-cause survival at the cure time, because some patients die of other causes first. In the illustrative example all-cause survival at 20 years is about 0.112 while the cure fraction is 0.25, so taking the level at which the all-cause curve flattens as pi understates the cure.
Sources
Mixture cure model in the relative survival setting in DSU TSD 21
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 (updated March 2022). Section 3.6.2, equation 11, the mixture cure model with expected survival incorporated, S(t) = S_star(t)(pi plus (1 minus pi)S_u(t)), and the three assumptions of cure models; section 3.6.1 on cure defined at population level and on fitting to cause-specific or relative survival because the event cannot be all-cause mortality.
Cure models fitted in an excess mortality framework for HTA
Latimer NR, Rutherford MJ. Mixture and non-mixture cure models for health technology assessment. PharmacoEconomics. 2024;42(10):1073-1090. Sections on frameworks for cure models and on mixture cure models: the all-cause hazard is split into the general population hazard and the excess hazard, all-cause survival is relative survival multiplied by expected survival from life tables for the right calendar year by age and sex, and the cure fraction is always higher than all-cause survival at the cure timepoint.
Canonical Identity
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