# Cox model - unsure of time unit of analysis

I am running a survival analysis (Cox model) on time to event in cancer patients. The start of followup is end of treatment. Tests for the event (recurrence) are performed every 6 months from cancer diagnosis, and the exact date (day,month,year) of these tests are known for each patient. I am unsure about the unit of analysis and the computing the time to event variable required for the Cox model. Because we know the start date of followup and date of tests, we can calculate the time to event variable in units of days. However, I don't think this is the correct approach as surely this would imply that a test was performed everyday over followup, and not every 6 months as we have. If an event is detected on a testing date, all we know is that the event occurred somewhere between the previous (event free) and current (event detected) test dates. Considering this, would it make sense, in this context, to compute the time to event variable first in days, but then bin this into intervals, each of length 6 months? The unit of analysis will then be a half year, so interpretation of hazards will be e.g. X events per person at risk per half year?

I am thinking to do an ordinary Cox model on time to detection and recognise this is detection and not recurrence per se. (See comment from @EdM). We have date and cause of death. I’m unsure how to incorporate this information. For example, for patients who don’t get the event (recurrence) nor die, the end of follow-up is their last test date. For patients who do not have event by last test date, but then die afterwards, should end of follow-up be death date? I’m presuming not, if cause of death is cancer, since recurrence by definition must have occurred before death. But for non cancer related death, should end of follow-up be date of death, or last test date? We don’t know for sure if recurrence might have occurred between last test date (which was negative for the event) and date of death.

Cox survival regressions don't directly model time at all. They just use the rank-ordering in time of event times, so you can use any time scale that makes sense. You can get predictions of survival over time from those models, but those predictions will just use whatever time scale you chose.

That said, an issue raised in the answer from Roger V. should be considered. Technically, in this situation, what you have is interval-censored event times: you have lower and upper limits for the time to recurrence, but not the exact time. Often, in practice in cancer studies, that is ignored so that you model the time to detection of recurrence rather than the time to recurrence itself.

There are ways to model interval-censored values in continuous time. The R icenReg package can handle that. If you have defined 6-month intervals shared by all participants, then you might consider a discrete-time survival model, implemented via binomial regressions with a complementary log-log link. That gives you a "grouped" proportional hazards model. See this page, for example.

In response to edited question

This report explains major types of survival outcomes and the difficulties in evaluating and applying them, in the context of The Cancer Genome Atlas. What you are evaluating seems most like the "progression-free interval" (PFI) those authors used:

PFI is the period from the date of diagnosis until the date of the first occurrence of a new tumor event (NTE), which includes progression of the disease, locoregional recurrence, distant metastasis, new primary tumor, or death with tumor. Patients who were alive without these event types, or died without tumor were censored. (Emphasis added.)

If death is from cancer and there was no prior evidence of recurrence, then for PFI you would use the date of death as an event time. If death is from another cause, then you censor the PFI at time of death.

Depending on details of your study design, there might be some additional issues to consider, to minimize the risk of informative censoring. In particular, was there something associated with your ability to have information after the last test date that might somehow be related to progression? See this review for a good introduction to problems arising from censoring in general, and this commentary more specifically related to cancer research.

• Thanks @EdM. Even considering the rank ordering you mentioned, I'm still uncertain about using days. For example, we might calculate the time to event for one patient as 40 days and another as 50 days. Or we could bin both into the same value i.e. 1 half year. Surely these 2 approaches will lead to different results in the Cox model? Commented May 22 at 14:29
• @user167591 yes, binning event times will lead to different results. A Cox model works best with continuous time, however, and typically needs to implement assumptions if there are tied event times. If you have exact times of detecting recurrence I think it would be best to use those and just recognize that you are modeling time to observing recurrence. If you want to model time to develop recurrence then use the interval censoring approaches I suggest.
– EdM
Commented May 22 at 16:50
• Thanks EdM! May I please ask how you would deal with censoring? For non events we could follow up until last test date. In a few cases there are deaths before first test date. Commented May 24 at 10:58
• @user167591 deaths in discrete-time models are recorded as being within a specific time interval; early deaths would go into the first time interval. Cases with censored event times in those models are just omitted from the calculations for time intervals after the censoring time. If you are working in continuous time, it's might be possible to get dates of death from public records. If you can't, then you can use continuous-time interval-censoring approaches as implemented in the icenReg package linked in the answer, or recognize that you are modeling time until you learn of death.
– EdM
Commented May 24 at 18:30
• please see my updates on the original question Commented May 25 at 11:30

I suppose that that tests are performed approximately every six months (rather than exactly every 180 days), so using months or even 6-month units as a measure of time would be appropriate.

I don't think there is any problem with using days - but this is excessive in terms of precision and may be misleading.

As far as the interpretation of the results is concerned, one should indeed keep in mind that the event didn't occur on exactly this day, but somewhere within a 6-th month interval before it.

For this I would prefer to use a discrete time survival model, with the unit of time being each visit. Now if you want to use days then there is probably going to be a problem with tied survival times. This means using the most effective method of dealing with tied survival times which is usually called Exact.