# Time-to-event outcomes where some patients are 'cured'

When I first learnt about survival analysis, I remember being told about an assumption that everyone will eventually have the event, it’s just a matter of how long it takes (and whether you are able to follow them for that long) . Obviously this is true for one particular time-to-event outcome, it might be a reasonable assumption for others, but I’m wondering about a situation in which we know that it is clearly not the case.

The study is an RCT looking at whether a new treatment is protective against a particular post-operative complication. Although the complication would manifest early (21 days follow-up are planned, but they don’t expect any new events after ~5 days) and drop-outs are very unlikely, they would like to consider time-to-event rather than a simple binary outcome because — if it can’t be avoided altogether — it is better to have the event later rather than sooner, since the potential complications of the event are lessened. After 21 days, the wound would have completely healed and the patient is no longer ‘at risk’ of the event, no matter how long they are followed. This is expected to apply to the vast majority of patients (>75\%).

At the moment I’m just trying to plan the analysis, so I don’t have any data to check things like the proportional hazards assumption. But I was hoping to get some ideas and/or references on the best way to go about it.

My initial thoughts:

• It would be important to detect any difference between the survival curves, so a log-rank test might be a robust first option. But I would like to account for some pre-specified baseline covariates, some of which are continuous.
• Maybe a standard survival model is fine, in the sense that the \mathrm{Pr}(T < \infty) = 1 assumption that I’m worried about actually doesn’t matter?
• I’ve seen competing risks, and in particular the Fine & Gray proportional subdistribution hazards model, presented as a model in which one random variable determines the type of event that an individual has, and a second determines the time until that event. This almost has an appealing parallel with my scenario, but the ‘competing event’ here would be ‘event-free survival’, which doesn’t have a time tied to it.
• I am vaguely aware of survival models that account for the possibility of ‘cure’, which seems to be exactly what I’m after (although it might be better termed ‘immunity’ in my case, as we would also expect many event-free patients in the placebo arm). If that sounds right, could anyone point me in the direction of some literature that would be relevant to my simple scenario?

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A recent case study would be: https://onlinelibrary.wiley.com/doi/abs/10.1002/pst.1840

Further starting points are

https://onlinelibrary.wiley.com/doi/abs/10.1002/sim.4780110710

https://www.jstor.org/stable/2677127

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This isn’t exactly what you’re discussing, but the recent article by Michael Harhay and colleagues about “ventilator free days” as an outcome in critical care trials may be of some relevance, since your problem sounds quite similar:

https://www.atsjournals.org/doi/abs/10.1164/rccm.201810-2050CP

The article linked above describes a fairly similar issue - not exactly alike, but similar. Patients with ARDS are started on a ventilator; trials want to measure how quickly they can get the patient “off” the ventilator (so fewer days on ventilator are good); but, patients that died quickly are a bad outcome, so we don’t want that to count as a “good” thing if a patient is only on ventilator for 1 day because they died. After 28 days, most patients would be off the ventilator, so additional time beyond that is usually considered superfluous and additional “ventilator free days” after that time are not counted.

Harhay’s article gives some concrete recommendations for use of ventilator-free days with practical suggestions to account for its problems. I think it may be useful in the problem you’ve outlined here, though I’m still trying to think through it a bit…

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