Cure Fraction

Cure Fraction in Survival Extrapolation and Health Technology Assessment

In a health technology assessment (HTA) model, the cure fraction is the share of patients whose excess mortality from the disease ends, and so the height at which relative survival levels off. A cured patient is still mortal, so after the cure point the model applies the death rates of the general population, and every additional cured patient adds close to a full remaining lifetime to the life years and QALYs. In one colon cancer case study, models fitted to the same 24 months of data put the cure fraction anywhere from 0% to 62%. Trials usually stop before the plateau is clear, so the fraction is among the least certain inputs in an appraisal and has to be judged for plausibility. It is estimated inside a cure model, mixture or non-mixture, together with survival for the uncured.

What the cured proportion refers to

The idea is old. Boag's 1949 paper set out maximum likelihood estimates of the proportion of patients cured by cancer therapy. In population-based cancer studies, Lambert and colleagues define cure as the point at which the mortality rate in the diseased group returns to the level expected in the general population, and describe the cure fraction as a useful measure for monitoring trends in survival from curable disease.

Cure in this sense is statistical. TSD 21, the NICE Decision Support Unit's guide to flexible survival methods, stresses that cure is defined at the population level, as the cause-specific hazard falling to zero, and not for any individual patient. Latimer and Rutherford make the same point for mixture models, in which each patient carries a probability of being cured and nobody is assigned to a group.

The quantity also depends on the endpoint. For overall survival, Latimer and Rutherford define the cure fraction as the proportion of patients who will not die from their disease; for progression-free survival it is the proportion who will neither progress nor die. Othus and colleagues note that progression-free survival counts extra events, so a fraction estimated on it may understate a true cure fraction for survival, and its use as a surrogate needs validation in the same disease and treatment setting.

Why the cure fraction is higher than survival at the cure point

The fraction also depends on the framework in which the model is fitted. Latimer and Rutherford favour fitting cure models in a relative survival framework, which needs no cause-of-death data and in which the all-cause hazard is split into the general population hazard and the excess hazard due to the disease.

$$R(t) = \frac{S(t)}{S^{*}(t)}$$

where $R(t)$ is relative survival at time $t$, $S(t)$ is all-cause survival in the patient group and $S^{*}(t)$ is expected survival in a matched general population, taken from life tables by age, sex and calendar year. Relative survival levels off when the excess hazard reaches zero, and the cure fraction $\pi$ is the height of that plateau.

Latimer and Rutherford spell out what this means. The cure fraction is the proportion alive in a world where patients could die only from the disease of interest, so it is always higher than all-cause survival at the cure timepoint, because some patients will already have died of other causes. At the cure timepoint $t_c$ all-cause survival is therefore $S^{*}(t_c)\,\pi$, and it keeps falling afterwards as background mortality takes its course.

The cure timepoint is a separate quantity from the cure fraction. Latimer and Rutherford explain that a mixture cure model places no constraint on when the last uncured patient has died, which can be early, late or never, and TSD 21 notes that many standard cure models assume the fraction is reached only at infinity. A flexible non-mixture cure model instead fixes the time at which the excess hazard returns to zero through its boundary knot.

How a cure fraction is estimated

Felizzi and colleagues, in a tutorial on mixture cure models in oncology, describe two routes. In the uninformed route the fraction is a parameter estimated from the trial data alongside the survival of the uncured, and the result is the best estimate given what is usually short-term follow-up. In the informed route the fraction is an input set equal to an external estimate, from expert opinion or from long-term real-world data on the same cancer or class of drug, sometimes after fitting a separate model to registry data.

In the uninformed route the estimate depends heavily on the distribution chosen for the uncured. TSD 21 warns that different standard distributions can give very different cure fractions and asks that the choice rest on biological plausibility. In Latimer and Rutherford's explanation, a log-logistic distribution can represent a falling hazard and long-term survivors among the uncured, so it tends to estimate a low fraction, while a Weibull distribution with a rising hazard cannot, so it may estimate a high one, even when the two models give similar overall survival curves. On simulated data from the BRIM-3 melanoma trial, Felizzi and colleagues obtained cure fractions from 13.3% with an exponential distribution to 18.1% with Weibull and Gompertz distributions.

The model family matters as much. In Latimer and Rutherford's case study of 440 patients with regional colon cancer, about 28% of whom were alive at 10 years, with the data censored at 24 months, mixture models estimated cure fractions of 0% to 21% and non-mixture models 48% to 62%. Survival at 10 years ranged from 6% to 21% and from 30% to 39% respectively, against 28% observed.

Treatment effects on the fraction need a scale. Othus and colleagues note that a mixture cure model compares cure proportions on the odds ratio scale, usually through a logistic link, while a non-mixture model uses the hazard ratio scale.

How the estimate moves as data mature

A cure fraction is estimated from the tail of the curve, where few patients remain at risk. TSD 21 lists among the limitations of cure models that reliable estimation needs sufficient numbers at risk in the tail, which randomised trials often lack, and that the Akaike information criterion and BIC give most weight to early follow-up and very little to the tail. Othus and colleagues add that an apparent plateau in a Kaplan-Meier curve can come from heavy censoring and small numbers, that adequate follow-up may be under five years for a high-mortality cancer while 20 years may not be enough for a low-mortality one, and that no formal test of adequacy exists.

Re-estimation with longer follow-up shows how far the fraction can move. Othus and colleagues refitted Weibull mixture cure models to six trials from the SWOG Leukemia Committee using 2019 data. In one trial the cured proportion fell from 66% to under 1%, and in another from 32% to under 1%, while in a third it rose from 31% to 43%; they described the resulting bias in mean survival as unpredictable.

Judging whether a cure fraction is plausible in HTA

NICE's manual for technology evaluation (PMG36) names cure approaches among the flexible methods for extrapolating time-to-event data and asks that the clinical plausibility of the underlying hazard functions be assessed routinely (section 4.6.25). Section 4.6.27 expects all model parameter values to be clinically plausible, and section 4.6.28 points to expert elicitation where plausibility is hard to define, as with innovative technologies.

TSD 21 is more specific about the cure fraction. Its model-specific recommendations (section 7.2) describe the estimate as prone to high uncertainty, particularly in small samples, and recommend that a submission justify the credibility of the fraction by comparing it with response rates, by examining earlier-phase trials with longer follow-up, and by using biological evidence on the mechanism of action and the nature of the disease. They also recommend presenting the predicted hazard and survival functions for both the cured and the uncured, and, where the existence of cure is uncertain, preferring flexible parametric models with background mortality.

Assumptions about the cured matter as well. Latimer and Rutherford note that a standardised mortality ratio can be applied to general population rates when the disease leaves a lasting excess risk, which matters most when the fraction is large. In their case study, inflating the ratio improved some models only by offsetting an overestimated cure fraction, so they prefer models that are credible under a realistic ratio. Their overall advice is to judge models by the survival proportions they predict over time rather than by the cure fraction itself, and Felizzi and colleagues recommend sensitivity analysis across a range of cure fractions.

Worked example with two treatments and an odds ratio for cure

All numbers in this example are illustrative. A comparator has a cure fraction of $\pi_0 = 0.25$ for overall survival in a relative survival framework, and a new treatment multiplies the odds of cure by 2. By year 5 the excess hazard has fallen to zero and no uncured patient remains alive.

$$\pi_1 = \frac{OR \cdot \pi_0/(1-\pi_0)}{1 + OR \cdot \pi_0/(1-\pi_0)} = \frac{2 \times 1/3}{1 + 2 \times 1/3} = \frac{2/3}{5/3} = 0.40$$

where $\pi_1$ is the cure fraction with the new treatment, $\pi_0$ the cure fraction with the comparator and $OR$ the odds ratio for cure.

1. Convert the fractions to survival. Matched general population survival is $S^{*}(5) = 0.92$ and $S^{*}(15) = 0.66$. All-cause survival at the cure point is 0.92 × 0.25 = 0.23 with the comparator and 0.92 × 0.40 = 0.368 with the new treatment. At 15 years it is 0.66 × 0.25 = 0.165 and 0.66 × 0.40 = 0.264, although the relative survival plateaus stay at 0.25 and 0.40.

2. Count the cured. Per 1,000 patients, the new treatment cures 400 − 250 = 150 more, but only 368 − 230 = 138 more are alive at year 5, because 12 of those 150 have died of other causes.

3. Turn the fractions into life years. Othus and colleagues note that mean survival is the weighted average of mean survival in the cured and uncured groups. With an illustrative background life expectancy of 19 years for the cured and a mean of 2 years for the uncured, mean survival is 0.25 × 19 + 0.75 × 2 = 6.25 years with the comparator and 0.40 × 19 + 0.60 × 2 = 8.8 years with the new treatment, a gain of 2.55 undiscounted life years per patient, equal to 0.15 × (19 − 2).

4. Interpret. Each percentage point of cure fraction is worth 0.17 life years per patient here, so a fraction overstated by 10 points from an immature data cut would add 1.7 years. Reading the 36.8% alive at year 5 as the cure fraction would understate it by 3.2 points, while reading 40% as the share alive would overstate survival. If later data showed both arms at 0.25, the gain would disappear.

Boundaries with nearby quantities

Several quantities are easily confused with the cure fraction. The proportion alive at the last trial visit mixes cured patients, uncured patients still alive and losses to background mortality. A model can also predict long-term survivors while estimating a 0% cure fraction, because, as Latimer and Rutherford note, a distribution for the uncured with a long tail produces survivors of its own. The fraction is one output of a cure rate model, which must also specify survival for the uncured.

Limitations and misreadings of reported cure fractions

Reported fractions are not comparable across models, because each is a function of the framework and the distribution used, and Latimer and Rutherford advise that different models should be expected to give different values. A fraction fitted where cure is not a reasonable assumption flatters the treatment; TSD 21's simulations found substantial bias in mean overall survival in those circumstances. The SWOG re-analyses show that an early fraction can be revised in either direction. A reported fraction also says nothing on its own about the quality of life or costs of cured patients, which the model has to supply separately.

Sources

• Boag JW. Maximum likelihood estimates of the proportion of patients cured by cancer therapy. Journal of the Royal Statistical Society Series B (Methodological). 1949;11(1):15-44. https://doi.org/10.1111/j.2517-6161.1949.tb00020.x
• Felizzi F, Paracha N, Pöhlmann J, Ray J. Mixture cure models in oncology: a tutorial and practical guidance. PharmacoEconomics Open. 2021;5(2):143-155. https://doi.org/10.1007/s41669-021-00260-z
• Lambert PC, Thompson JR, Weston CL, Dickman PW. Estimating and modeling the cure fraction in population-based cancer survival analysis. Biostatistics. 2007;8(3):576-594. https://doi.org/10.1093/biostatistics/kxl030
• Latimer NR, Rutherford MJ. Mixture and non-mixture cure models for health technology assessment: what you need to know. PharmacoEconomics. 2024;42(10):1073-1090. https://doi.org/10.1007/s40273-024-01406-7
• National Institute for Health and Care Excellence. NICE technology appraisal and highly specialised technologies guidance: the manual (PMG36). 2022, last updated 13 October 2026. Sections 4.6.25, 4.6.27 and 4.6.28. https://www.nice.org.uk/process/pmg36/chapter/economic-evaluation-2
• Othus M, Barlogie B, LeBlanc ML, Crowley JJ. Cure models as a useful statistical tool for analyzing survival. Clinical Cancer Research. 2012;18(14):3731-3736. https://doi.org/10.1158/1078-0432.CCR-11-2859
• Othus M, Bansal A, Koepl L, Wagner S, Ramsey S. Accounting for cured patients in cost-effectiveness analysis. Value in Health. 2017;20(4):705-709. https://doi.org/10.1016/j.jval.2016.04.011
• Othus M, Bansal A, Erba H, Ramsey S. Bias in mean survival from fitting cure models with limited follow-up. Value in Health. 2020;23(8):1034-1039. https://doi.org/10.1016/j.jval.2020.02.015
• 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). Sections 3.6, 5.3.4 and 7.2. https://doi.org/10.15131/shef.data.33849118