Many people are familiar with survival models. Survival models measure the probability of survival to a given time period. The “problem” addressed by these models is that some people are “censored”, in other words, the do not die in the sample time period. Although longer survival is good in practice, for statisticians it is problematic because one does not observe how long a person lives for.
Standard survival models can be written formally as:
- S(t)=S*(t)R(t)
where S(t) is the probability of surviving to time t, S*(t) is the baseline expected survival (e.g., survival without having a disease) and R(t) e.g., the effect of having a disease on survival probabilities.
The corresponding hazard functions are:
- h(t)=h*(t) + λ(t)
where h(t) is the all-cause hazard (mortality) rate, h*(t) is the expected hazard (mortality) rate, and λ(t) is the excess hazard (mortality) rate associated with the disease of interest.
However, what if there is some probability each period that you can be cured of a disease? A cure fraction model takes a standard survival model and adapts it for this case. As described in Lambert (2007):
Cure models are a special type of survival analysis model where it is assumed that there are a proportion of subjects who will never experience the event and thus the survival curve will eventually reach a plateau. In population based cancer studies, cure is said to occur when the mortality (hazard) rate in the diseased group of individuals returns to the same level as that expected in the general population.
Assume that in any time period, the likelihood of being cured is π. Then in this case, one can model survival as:
- S(t)=S*(t){π+(1-π)Su(t)}
where Su(t)is the survival function for the uncured individuals. The corresponding hazard function is:
The mixture cure fraction model assumes that at diagnosis there is a group of individuals who experience no excess mortality compared to the general population.
Paul C. Lambert derives additional fractional cure survival models in his 2007 Stata Journal article.
- Paul C. Lambert. Modeling of the cure fraction in survival studies. The Stata Journal (2007), 7, Num 3, 351-375.