What are cure models?
Cure models are a special type of survival analysis model where it is assumed that there is a proportion of “Immune”, or “Cured “subjects, called generically as “Long-Term Survivors”. I.e.: those cured subjects will possibly not experience the event under investigation, and thus the survival curve will eventually reach a plateau (Maller and Zhou, 1996).
Although very popular topics within statistical literature, cure models are not as widely known in the clinical literature.
Class of Cure models
There are 2 major classes of cure models, mixture and non-mixture models (Megan Othus et al, 2012). Mixture cure models, as the name suggests, explicitly model survival as a mixture of 2 types of patients: those who are cured and those who are not cured. Typically, the probability a patient is cured is modeled with logistic regression. The second component of the model is a survival model for patients who are not cured. There are many options for this, but 2 common models are the Weibull and the Cox models.
In words, a mixture cure model can be written as follows:
Probability Alive at time t = (Probability Cured) + [(Probability not cured) * (Probability Alive at time t if not Cured)]
Non-mixture survival models can be written as follows:
Probability Alive at time t = (Probability Cured) to the power (1-sx(t))
Where 1−S×(t) is an exponent of the probability of being cured and S×(t) is a survival function.
Cure models provide systematic methodology for analyzing survival data which contains cured subjects. Clinical and epidemiological as well as financial and marketing researches are subject to such data, for instance:
- Cancer data,
- Default/loss of follow up in TB or HIV treatment,
- Default and fraud detection in banking,
In oncology, cure models can be used to investigate the heterogeneity between cancer patients who are long-term survivors and those who are not. A straightforward way to identify whether a particular dataset might have a subset of long-term survivors is to look at the survival curve. If the survival curve has a plateau at the end of the study, a cure model may be an appropriate and useful way to analyze the data.
As standard survival models, such as the Cox model, do not assume heterogeneity in data, it appears obvious to apply cure models as an alternative in order:
- To better describe survival data distributions and thereby
- To better estimate the Hazard Rate as well as the mixture effect of the events being analyzed.
The Expectation Maximization (EM) algorithm through SAS NLMIXED Procedure can be used to compute the target estimates: the Risk Ratio (RR) of the non-cured fraction and the Odd Ratio (OR) of Cured fraction.
As a Conclusion
Cure models can be useful tools to analyze and describe various survival and time to event data when they contain immune or cured subjects.
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