Time intervals without events in event history models with competing risks (e.g., Cox, piecewise constant, logit)

Question

I want to evaluate factors influencing the time to degree for college students in their master's degree with panel data. The dependent variable is time to degree in months (reference: student is still studying). However, I must respect the time to college dropout as a competing event. The data is in person-month (long) format with time-varying and time-constant independent variables. I have observed a cohort that started in the winter semester of 2017/18 for six semesters. The standard period of study is four semesters. There are no graduations within the first 12 months (first two semesters) because students usually graduate in the third, fourth, or fifth semester. However, students can drop out at any time.

How can I handle such data in an estimation? I have estimated a piecewise constant model in Stata, and the first 12 months are automatically discarded - because there are no graduations. The Cox model yields similar results. Is this problematic?

What would be alternatives? I have considered defining a long first interval and then using the monthly data (first two semesters; months 13, 14, ...). I have also thought about starting process time with the third semester (13th month) - however, I have many students who have responded to the questionnaire in the first or second semester - and indicated graduation later on but have not responded to the questionnaire in the third semester. I do not want to discard these observations.

What would you recommend me to do?

Answer 1

A lack of events simply means that there is no evidence for a hazard of an event during the time period of interest. In a continuous-time Cox model, there is 0 hazard between event times. One might consider that an even more extreme example of what you describe. Furthermore, a Cox model doesn't directly include time in the calculation, only the ordering of events in time. Any survival curves over time are estimated from the model results, based on the observed event times.

A continuous-time model like a Cox model isn't appropriate here. I don't use Stata so I'm not sure how it implements the "piecewise constant" model. I suspect that it is also at its base a continuous-time model.

What you want is a true discrete-time model. That's a set of binomial models (for individual event types) or multinomial models (for both event types together), evaluated for each time period (semester) along with the covariate values in place during that period. Depending on the nature of the questionnaires, when questionnaire data are missing for an individual in a semester you could either carry forward prior responses or use multiple imputation to build a model that includes all individuals while accounting for the potential errors in the estimated questionnaire responses.