# Understanding the Importance of "Cure Models" in Survival Analysis

I am trying to better understand the importance of "Cure Models".

Reading some introductory materials on this topic, its mentioned that traditional Survival Models rely on the assumption that all observations being modelled are expected to eventually experience the "event". Cure Models are apparently intended to be used in instances where this is not the case and model the dataset using a mixture-model approach where the sub-population that experiences the event is modelled alongside the sub-population that does not experience the event.

Based on this, I had the following questions:

• As such, how can we intuitively understand that regular Survival Models require all observations to eventually experience the event? Is there some mathematical formula which demonstrates this fact?

• How strong is this assumption? Is there some mathematical formula which can be used to illustrate what happens when this condition is not met?

• And in general, how does modelling both sub-populations using a mixture model approach (as is done in Cure Models) help deal with this situation? Informally, using a mixture model approach makes sense - but I was wondering if there is some mathematics which can show why Cure Models are useful in this context.

Thanks!

Note:

• Consider this situation: Suppose a significant percent of your observations are "Right Censored" (i.e. were not recorded to experience the event during the medical study) - but in theory, they could have experienced the event. In such a case, are Cure Models needed - or could we proceed with standard Survival Models?
• I think the assumption you have about Survival models that "all observations are expected to experience the event" is not correct. Observations might or might not experience he event, even if you follow up your observations for an infinite amount of time. Of course, if your event is death and your observations are people living in a city, eventually everyone must experience the event. However, imagine your event is defined as 'first asthma attack", then many subjects will never experience it. Commented Jun 27, 2023 at 9:12
• @jmarkov : thank you so much for pointing this out! Commented Jun 27, 2023 at 16:32

The survival function over time, $$S(t)$$, is the complement of the cumulative distribution of event probabilities over time, $$F(t)$$: $$S(t)=1-F(t)$$.

If the survival function never reaches a value of $$0$$, $$S(\infty)>0$$, then the corresponding event-probability distribution is that of an improper random variable in that $$F(\infty)-F(-\infty)<1$$. That's the fundamental mathematical issue at play.

A standard parametric survival model, like a Webull or lognormal model, is based on the assumption that the underlying event-probability distribution is proper. If it isn't, then you have to model both the probability of no event and the probability of events over time given that there is an event. That's the basis of a cure model as outlined on this page, which provides a link to a recent review.

In a Cox proportional hazards model, however, you can get away without using a cure model if the instantaneous relative hazards are the same as the ultimate relative hazards of an event. In that case, you estimate $$S(t)$$ from the cumulative hazard function, via $$S(t)=\exp(-H(t))$$. You never take the estimate of $$H(t)$$ out to $$t \to \infty$$, as it's limited to the time period of observation until the last event. If the proportional hazards assumption holds over that time period, you get useful estimates of regression coefficients and associated hazard ratios without modeling the "cure" fraction separately.

A standard parametric survival model requires a proper distribution and is fit via maximum likelihood. In a cure model, the full likelihood involves both the likelihood of being in the "cured" group and the likelihood of having an event if an individual isn't in that group. That last part is based on a proper distribution of event times within the group of "non-cured" individuals. The "outcome" random variable is then a composite of the "cured" and the "time-to-event-if-not" outcomes, providing overall a proper random variable with probability 1 of some outcome eventually.

If you don't account for "cure" in a parametric model where it's important, your estimated survival curves at late times will approach 0 even if the data, due to the "cure," can't.

If only a few individuals have an event it might not be possible to evaluate whether a "cure" state is involved. Distinguishing right-censoring from "cure" typically requires applying some understanding of the underlying subject matter and evaluating survival out to very long times.

The semi-parametric nature of the Cox model is essentially what I noted above. You are not directly modeling the baseline hazard, so if the baseline hazard is 0 beyond a certain time then all might be OK--if the proportional hazards assumption holds. My explanation in terms of the cumulative hazard (which is estimated outside of the Cox modeling process per se) is just another way of stating that.

• @ EdM: Thank you for your detailed analysis! This makes a lot of sense! Commented Jun 27, 2023 at 16:21
• 1) Is this assumption about the "improper random variable" a "paper tiger"? By that I mean - while now it makes sense to me that the basic definition of a Survival Function would in fact require all observations to eventually experience the event - suppose we decide to proceed anyways with a Non-Cure Model .... how much of a problem is this likely to result in? What kind of problems could this result in (e.g. "invalid inference") - can these problems be detected in the outputs? Commented Jun 27, 2023 at 16:27
• 2) I assume this problem might not be as bad when only few observations experience the event compared to when more observation experience the event? Commented Jun 27, 2023 at 16:27
• 3) The idea of Cure Models is to address situations when observations are "believed" to never have the possibility to experience the event - but consider this situation: Suppose a significant percent of your observations are "Right Censored" (i.e. were not recorded to experience the event during the medical study) - but in theory, they could have experienced the event. In such a case, are Cure Models needed - or could we proceed with standard Survival Models? Commented Jun 27, 2023 at 16:30
• 4) If you have time, could you please go over the last point again on why Cure Models might not be needed when you decide to use a Cox-Ph model? At first I thought the reason was because of the Semi-Parametric nature of the Survival Function - but now it seems like the reason is because you are indirectly deriving the Survival Function from the Hazard Function (and the Hazard Function is not subject to this limitation)? I would have thought that improper Random Variable issue would still arise in this case? Commented Jun 27, 2023 at 16:35