Concept Architecture
Concept
Theoretically, the Cure Fraction Model is a class of survival models that assumes a proportion of individuals are permanently free from the event of interest, while the remaining individuals continue to experience disease-related risk. It is founded on survival analysis and mixture modelling and exists to represent long-term survival in populations where a genuine cure is possible. In health economics, cure fraction models are widely used to estimate lifetime survival for interventions that produce durable treatment responses, particularly in oncology.
Mathematically, cure fraction models partition the study population into cured and uncured subgroups. Overall survival is expressed as a weighted combination of the cured proportion and the survival distribution of uncured individuals. Model parameters are estimated simultaneously using maximum likelihood methods, allowing estimation of both the cure fraction and the underlying survival distribution.
In practice, cure fraction models are estimated from long-term follow-up data obtained from clinical trials, registries and observational studies. They are increasingly used in health technology assessment and cost-effectiveness analysis to extrapolate survival beyond observed follow-up, estimate lifetime QALYs and project long-term healthcare costs.
Purpose
Used to estimate long-term survival when a proportion of patients are effectively cured, improve survival extrapolation and generate lifetime estimates for health economic evaluation and health technology assessment.
Mathematical Formulae
Primary Formula
S(t) = � + (1 ? �) ? S?(t)
where:
- � = cure fraction
- S(t) = overall survival
- S?(t) = survival function for uncured individuals
Supporting Formulae
Long-term survival:
lim??� S(t) = �
Logistic model for the cure fraction:
� = exp(X?) � (1 + exp(X?))
Likelihood function:
L = ?S(t?)???? ? f(t?)??
where:
- �? = event indicator
- f(t?) = probability density function
- S(t?) = survival function
Related Mathematical Methods
- Mixture cure models
- Non-mixture cure models
- Parametric survival modelling
- Maximum likelihood estimation
- Relative survival modelling
- Flexible parametric survival models
Example
A clinical trial evaluating an immunotherapy estimates that 40% of patients achieve long-term remission. A cure fraction model estimates � = 0.40, while the remaining 60% follow a Weibull survival distribution. The model predicts long-term survival, lifetime QALYs and healthcare costs by combining these two components, providing inputs for cost-effectiveness analysis.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| Logistic Function | =EXP(B2)/(1+EXP(B2)) | Estimate the cure fraction from model coefficients. |
| Survival Function | =C2+(1-C2)*D2 | Calculate overall survival using a mixture cure model. |
| EXP | =EXP(-(A2/$B$1)^$C$1) | Calculate Weibull survival for the uncured subgroup. |
VBA (Optional)
Automate estimation of cure fraction model outputs and generate lifetime survival projections for health economic decision models.
Sources
- Boag JW. Maximum Likelihood Estimates of the Proportion of Patients Cured by Cancer Therapy. Journal of the Royal Statistical Society Series B. 1949;11(1):15?53.
- Berkson J, Gage RP. Survival Curve for Cancer Patients Following Treatment. Journal of the American Statistical Association. 1952;47:501?515.
- Lambert PC, Thompson JR, Weston CL, Dickman PW. Estimating and Modelling the Cure Fraction in Population-Based Cancer Survival Analysis. Biostatistics.
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation. Oxford University Press.
- NICE. Health Technology Evaluation Manual.
Related Concepts (2)
Library
Tools & Resources
1
survHE — Survival Analysis for Health Economic Evaluation (R package) — Gianluca Baio, R package ed., 2023 (CRAN)
An R package for fitting and comparing parametric survival models for health economic evaluation, including Bayesian estimation, and for extrapolating time-to-event data to inform cost-effectiveness models.
Software (R package)View source →
Frequently Asked Questions (6)
What is a cure fraction model?
A survival model separating a population into a cured subgroup facing only background mortality and an uncured subgroup remaining at risk of the disease event.
Source: Boag 1949
How does a cure fraction model combine two survival processes?
A cure fraction model represents a population as a mixture of two groups with different futures. The cured group is modelled as facing only the background mortality of the general population, since their disease-specific risk has gone, while the uncured group is modelled with a survival distribution reflecting continuing disease risk on top of that background. The overall survival curve is a weighted blend of the two, with the weight being the cure fraction. This structure lets long-term survival level off realistically. Othus and colleagues (2012) describe it.
Source: Othus et al. 2012
How does a cure fraction model work?
A cure fraction model works by treating the population as a mixture: a cured fraction, whose survival is governed only by background mortality, and an uncured fraction, whose disease-specific survival follows a parametric distribution. The model estimates the proportion cured and the survival of the uncured. Overall survival combines the two, so that after the uncured have mostly experienced the event, survival plateaus at the cured fraction, subject to background mortality, reproducing the long-term plateau seen in curable conditions.
Source: Latimer 2013
Why include background mortality in a cure fraction model?
Background mortality is included so that the cured subgroup, though free of disease-specific risk, still faces the general population's risk of death, keeping their survival plausible over long horizons rather than implying they never die. The cured are modelled as subject only to background mortality, while the uncured face both background and excess disease hazard. This ensures the model represents cure as freedom from the disease event, not from all mortality, giving realistic long-term survival for the cured fraction.
Source: Latimer 2013
When is a cure fraction model appropriate?
A cure fraction model is appropriate when evidence suggests a proportion of patients are effectively cured, so that disease-specific survival plateaus above zero and a subgroup experiences no further disease events, a pattern seen in some cancers with long-term survivors. A visible plateau in the survival curve and long follow-up support its use. Where all patients remain at risk of the disease event, a standard survival model without a cure fraction is used instead, so the cure model is reserved for genuinely curable conditions.
Source: Boag 1949
What are the challenges of cure fraction models?
Cure fraction models require long follow-up to distinguish genuinely cured patients from those with slow progression, since a plateau must be observed, and short trials may leave the cured fraction uncertain. The estimate depends on the survival distribution assumed for the uncured and on background mortality, and the plausibility of cure may be clinically contested. Because the cure fraction strongly affects long-term and mean survival, its uncertainty is important, so evidence of cure and sensitivity to the assumption are examined carefully.
Source: Boag 1949
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 20 Oct 2025
Content version: 1.0.0
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