Cure-Rate Model

Definition

A cure-rate (or surviving-fraction) model is a survival model for populations in which a fraction of subjects will never experience the event of interest — they are "cured" (or immune), so their event time is effectively infinite. Such models decompose the population into a cured subpopulation and a susceptible subpopulation, and are widely used in oncology (e.g. cancer relapse), where a nonzero long-run survival plateau is observed.

Key Ideas

Standard mixture model (Berkson-Gage 1952): the population survivor function is $S_{\text{pop}}(t) = \pi + (1-\pi)S^*(t)$ , where $\pi$ is the cure fraction and $S^*(t)$ is the survivor function of the non-cured (susceptible) group. Covariates typically enter $\pi$ through a binomial/logistic regression.
Promotion-time model (Chen-Ibrahim-Sinha 1999): a biologically-motivated alternative in which a latent number of "carcinogenic" cells $N \sim \text{Poisson}(\theta)$ each have i.i.d. promotion times; the event time is the minimum, and a subject is cured iff $N=0$ . This gives $S_{\text{pop}}(t) = \exp(-\theta F(t))$ with cure fraction $e^{-\theta}$ .
The two formulations are mathematically equivalent (any mixture model can be rewritten in promotion-time form and vice versa), but the promotion-time version has a proportional-hazards structure and better Bayesian behavior.

How It Works

Promotion-time construction

Let $F(t) = 1 - S(t)$ be the CDF of a promotion time. Given $N=k$ latent cells with i.i.d. promotion times, the survivor function conditional on $N=k$ is $S(t)^k$ . Marginalizing over $N \sim \text{Poisson}(\theta)$ :

$S_{\text{pop}}(t) = \sum_{k=0}^{\infty} S(t)^k \frac{\theta^k e^{-\theta}}{k!} = \exp\{-\theta(1 - S(t))\} = \exp\{-\theta F(t)\}.$

The population density and hazard are

$f_{\text{pop}}(t) = \theta f(t)\,e^{-\theta F(t)}, \qquad h_{\text{pop}}(t) = \frac{f_{\text{pop}}(t)}{S_{\text{pop}}(t)} = \theta f(t),$

so with $\theta_i = \exp(x_i'\beta)$ the model is proportional hazards in the cure parameter — a property the mixture model cannot attain when covariates enter $\pi$ .

Bayesian estimation

Treating the latent counts $N_i$ as auxiliary data gives a tractable complete-data likelihood and an efficient data-augmentation / Gibbs sampling scheme. A key advantage of the promotion-time model is that it yields proper posteriors under improper noninformative priors (including a uniform prior on $\beta$ ), whereas the mixture model generally requires proper priors. Informative priors can be elicited from historical studies via a power prior.

Why It Matters

Cure-rate models are the standard tool when survival data exhibit a long-run plateau (a nonzero fraction never fails), common in cancer trials. The promotion-time formulation made fully Bayesian, covariate-dependent cure modeling practical: proportional hazards, closed latent-variable Gibbs updates, valid noninformative-prior inference, and natural historical-data priors.

Open Questions

Identifiability of the cure fraction is fragile when follow-up is short (insufficient plateau); it relies on the tail of $S(t)$ .
Choice of the promotion-time distribution $F$ (Weibull, piecewise-exponential, nonparametric) and its interaction with the cure fraction.
Extending to correlated / frailty and multivariate survival settings.