# Discrete time survival analysis or Cox regression?

In an RCT, I want to find out whether the treatment (treatment vs control) has an effect on the uptake of aftercare (yes/no + time). I have five measurement points, which are not equidistant (i.e., baseline, 3 weeks, 6 weeks, 12 weeks and 24 weeks after randomization). There are some questions regarding the difference between Cox regression and discrete-time survival analysis here, but I still wonder whether discrete-time survival analysis would be better in my case. As I read, there are two advantages of discrete-time over Cox regression: 1) handling of ties and 2) no proportionality of hazards. Regarding 1): the handling of ties should be no problem in interval-censored Cox regression. Regarding 2): as I normally would assume PH, but the time periods of my five measurement points are not equal in length, discrete-time survival analysis would be favored. Is that true?

EDIT: I am only interested in the first uptake of aftercare (or, conversely, no uptake at all until end of measurement period), so I don't care about whether aftercare was continued or not after uptake at other measurement timepoints, so the outcome can't change over time)

• For the outcome, "uptake of aftercare," can someone flip back and forth between yes and no over time? Please edit the question to explain in more detail how "uptake of aftercare" is evaluated and whether that measure can change over time.
– EdM
Commented May 21, 2023 at 10:49

With only 5 time points and at most one event per individual, a discrete-time model would be the most natural. Interval-censored Cox regression is possible, but that's probably better reserved for situations where the time intervals differ among individuals. When the time intervals are the same for everyone, using binomial regression with a complementary log-log link provides a time-grouped proportional hazards model that's what you would get from an interval-censored Cox model. See this page. Other links in the binomial regression are possible, but wouldn't have the same proportional hazards interpretation.

The different lengths of the time periods don't really matter, unless you explicitly model time as other than categorical in the binomial discrete-time survival model. A Cox model per se doesn't directly evaluate event times at all. It only uses the order of events in time. The survival curves you can generate from a Cox model simply re-express the ordered events in terms of the times at which they occurred.

• Dear EdM, thanks a lot for your answers.I read the linked site and the book you recommended there. Two further questions would be: 1) The choice of the model highly depends on whether time intervals are the same for everyone or not. In my case, each individual has the 5 time points mentioned, however, missing data are frequently present (up to 30% missing). Does this have any implication on the choice of the model (or that using MI data is more favorable?)?. 2) my true underlying time variable is continuous (i.e., days), however it's only measured at 5 time points. Are there any consequences? Commented Jun 19, 2023 at 10:45
• @Sebastian the discrete-time analysis is OK with respect to both issues. With continuous time sampled at a few time points, a complementary log-log link in a binomial regression is equivalent to a Cox model on the data ground by time. See this page. Depending on the nature of the missing data you might need to use multiple imputation, but binomial regression is still appropriate for the multiple imputed data sets.
– EdM
Commented Jun 19, 2023 at 13:48
• Thanks! This really helps alot! I'm not sure where to adress this question but maybe you could help further: In continuous outcome, I understand the concept of right censoring. In discrete time survival analysis, however, I do not understand what the correct way of right censoring is. E.g., if someone misses time point 4, but has data for 1, 2, 3 and 5, is the correct right censoring measurement point 5? This is important as I wanted to compute the interval-censored cox model using icenReg in R, which forces me to choose a timepoint of right censoring Commented Jun 21, 2023 at 17:12
• @Sebastian if you have only 5 time points that are the same for all individuals (even if they aren't equidistant in time), a binomial model with a complementary log-log link should be equivalent to what you would get with an icenReg Cox model. Handling the missing data might be tricky. In your example, if someone might have taken up aftercare in a way that could have shown up at time point 4 but not be seen at time point 5, then either use multiple imputation or set the right censoring to time point 3.
– EdM
Commented Jun 21, 2023 at 17:55
• in the meanwhile I got a logistic regression with clog-log link and an interval-censored model using the icenReg package. However, the results differ quite significantly. They only differ slightly between a usual cox-model (no interval censoring) and the logistic regression with clog-log link. Is there any possibility to share the code with you to dig in why the two models, interval-censored cox and logistic regression with clog-log link, differ even if the theoretical background would be the same? Commented Jul 22, 2023 at 15:51