# Survival Analysis model choice

I am trying to investigate the statistical difference between two groups in my population and exploring suitable survival models for analysis. Let me first outline the data I am working with to provide better context. There are two things that are important to explain about the data I work with.

The data contains information about participants' survival times until a particular event occurs, which is the standard survival data. However, it also includes participants for whom the event does not occur (I possess an indicator that confirms the non-occurrence of the event). I do not have information on the exact time when the event did not happen, only the indicator that it did not occur. In order to examine the statistical difference between the two populations, both of these things are important (i.e. the time-until-event, and also the event not happening).

With this in mind, I have been reading a lot the past few days about the possibilities. I have come across three potential approaches that seem relevant:

• Censoring: Considering the event not happening as censoring and employing standard survival models, such as the Cox model.
• Cure survival models: Utilizing cure survival models, which treats cases where the event does not occur as "cured."
• Competing risk models, where both the occurrence and non-occurrence of the event are treated as separate events.

I would like some help to put me in the right direction. What is the best model in order to examine the statistical difference between the two populations? If you could direct me to relevant literature that addresses a similar setup, it would be greatly appreciated.

• The statement "I do not have information on the exact time when the event did not happen, only the indicator that it did not occur" is a bit unclear to me. Does that mean that you have no information whatsoever on the last observation time for individuals without an event? Was there perhaps some defined period of time over which all event-free individuals were observed?
– EdM
Commented Jul 2, 2023 at 15:00
• There is no defined period in which all event-free individuals are observed. I know the start time for all observations. For the observations to whom the event occurs, I know the time at which the event occurs. Of the observations to whom the event does not occur, I know only that the event does not occur to them (i.e. not the time at which it becomes known that the event does not occur). I could try to estimate their end time using another variable, but this may not be completely accurate. Would the model choice be different if we knew the timestamp at which the event did not occur?
– John
Commented Jul 2, 2023 at 15:42

If you had the last follow-up times for the cases that didn't have the event, this would be a simple survival model with right-censored observations. For such individuals you would use that last follow-up time as the time outcome, and code those cases as right censored. Typically, one uses a variable with value 0 for a right-censored case and 1 for an event (although, as I recall, MATLAB might expect the reverse coding).

Otherwise you have a big problem. For someone without a recorded event, is it possible that she had the event right after time = 0 and you just didn't know it? In that case you certainly don't want to use a "cure model" to handle such individuals. On the other hand, if all those without recorded events truly were cured then a "cure model" might be quite appropriate.

A competing-risk model wouldn't work, as you don't have a time for the "competing risk" of right censoring.

It sounds like you might have some way of estimating the event times for those individuals. Even though each estimate might not be accurate, you could consider multiple imputation to take that inaccuracy into account. You produce multiple data sets, with the missing event times in each imputed from a data-based probabilistic distribution. You build your model on each of the imputed data sets, then combine the results in a way that takes the uncertainty in the imputations into account.

See Stef van Buuren's Flexible Imputation of Missing Data and many related pages on this site. That approach is based on an assumption of "missing at random" in a technical sense, in that the probability of a missing value is determined only by data that are available to model, not by other reasons for missingness. Evaluating that assumption requires careful application of your understanding of the subject matter, as there is no way to check it directly.

• Thanks for this elaborate answer. If I were to estimate the ‘end time’ for the observations for which I know for certain the event did not occur, and use a cure model, what would be the best way to assess the significance difference between the two groups that I want to compare? What I essentially want to show is that the two groups have a different hazard curve, but also that the ‘non-event’ occurrence is different for both populations.
– John
Commented Jul 2, 2023 at 17:37
• @Rik a cure model is essentially a combined model of "cure" probability along with a model of time-to-event probability for those who aren't cured. You can get separate coefficients with respect to group membership for each of those submodels. The R survival task view has links to some implementations.
– EdM
Commented Jul 2, 2023 at 18:09