Signature
S = S_star * pi^F
| Inputs | Definition | Unit |
|---|---|---|
S_star | Expected survival at time t of the matched general population, from life tables | probability from 0 to 1 |
pi | Proportion of patients who will not die from their disease, the asymptote of relative survival | proportion from 0 to 1 |
F | Value at time t of the cumulative distribution function F_z(t), for example a Weibull or a restricted cubic spline form; 0 at time zero and approaching 1 as time tends to infinity | 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
Non-mixture cure all-cause survival from named cells
With BgSurv, CureFrac and CdfValue named for one time point, the formula returns all-cause survival, held in NmcSurv.
=BgSurv*CureFrac^CdfValue
Assumptions
No separate cured and uncured groups in the non-mixture cure model
The cohort is not split into cured and uncured groups, so there is no survival model specific to uncured patients. The distribution behind F_z must be flexible enough to follow the cohort's survival as it approaches the cure fraction.
Cure time fixed by the final boundary knot in a flexible non-mixture model
When F_z is defined with restricted cubic splines, the analyst fixes the time at which the excess hazard is taken to reach zero through the final boundary knot; from then on the all-cause hazard is other-cause mortality alone.
Worked examples
Non-mixture cure survival halfway along the distribution function
With a cure fraction of 0.25, F_z(t) of 0.5 and background survival of 0.8187, relative survival is 0.25 to the power 0.5, which is 0.5, and all-cause survival is about 0.4094 (computed here for illustration).
S_star = 0.818731; pi = 0.25; F = 0.5; S = 0.4093655
Non-mixture cure survival once the distribution function has reached 1
When F_z(t) has reached 1, relative survival equals the cure fraction and all-cause survival is S_star times pi, about 0.1123 with background survival of 0.4493, the level the mixture model approaches once the uncured have died (computed here for illustration).
S_star = 0.449329; pi = 0.25; F = 1; S = 0.112332
Common errors
Putting a survival function in the exponent of the non-mixture cure model
F_z(t) must be a cumulative distribution function. Its complement, a survival function, makes relative survival start at pi and rise to 1: with pi = 0.25 the model would give relative survival of 0.25 at time zero instead of 1.
Setting the non-mixture cure time earlier than the data support
Fixing the final boundary knot early forces the excess hazard down too soon. Latimer and Rutherford report that a 5-year boundary knot gave hazards that fell steeply at around 2 years, so the knot position needs clinical justification and a check of the fitted hazard.
Sources
Non-mixture cure model and its relative survival extension 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 12, the non-mixture model S(t) = pi to the power F_z(t) with F_z a cumulative distribution function, its extension to the relative survival setting by incorporating expected survival, and the cure time fixed through the final boundary knot of a flexible parametric cure model.
Non-mixture cure models in an HTA tutorial
Latimer NR, Rutherford MJ. Mixture and non-mixture cure models for health technology assessment. PharmacoEconomics. 2024;42(10):1073-1090. Section on non-mixture cure models (no split into cured and uncured groups, no survival model specific to uncured patients) and discussion of choosing boundary knots (a 5-year boundary knot gave hazards that fell steeply at around 2 years).
Canonical Identity
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