Understanding the Importance of "Cure Models" in Survival Analysis

Question

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?

Answer 1

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.

In response to comments

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.