# Type of censoring in discrete time survival

*I have a prospective longitudinal study. In this study, the patients come to the hospital every three months for check-ups. T0 ( one week before surgery), T3 (Three months after surgery), T6 (6 months after surgery), T9,T12,,,,,,,T24(24 months after surgery). So there are 9 time points.

We are interested in a predictive Cox regression including a binary time dependent covariate and a continuous independent variable. Cases who dropped out before event, completed follow up (T24) event-free, or showed specific level of the time dependent covariate at the end of the study are censor.

*It seems that in the counting process method, there are only right censors. And in order to use the discrete time survival, the following conditions must be satisfied (please correct me if I am not right)

1. the same time interval for all patients
2. A limited number of time points
3. Interval censors*

1. I am not sure about the type of censoring in my study. Can we use the discrete time survival method if there are both right censors and interval censors in the study?

2. If the study allows patients to rejoin the trial after missing one interval point (all obs are missing for one time period), I was wondering how we should evaluate them. Can we still use discrete time survival analysis when there is missing info. For example, the patient did not show up for his T6 appointment (six months after surgery), but he returned to study at T9.

The event variable is progressing disease (YES-NO), and the main covariate (time dependent variable-variable of interest) is mental abilities(Yes-No). We are investigating if changing the state of mental abilities can predict rising disease.

• I started to edit my answer to address your comments on it, but I don't completely understand what you mean by "Cases who... showed specific level of the time dependent covariate at the end of the study are censored." That seems a little strange. It would help (for this and your other questions) if you could say more about the nature of the time-varying covariate(s) and the event. I infer that the event isn't fatal, and is reported or found at a follow up visit. When/how is a time-varying covariate measured, relative to the visits and the reporting of the interval containing the event?
– EdM
Commented Jun 7 at 15:05
• @Thank you so much for your comment. The event variable is progressing disease (YES-NO), and the main covariate (time dependent variable-variable of interest) is cognitive function (Yes-No). We are investigating if changing the state of cognitive functions can predict rising disease. If the study allows patients to rejoin the trial after missing one interval point (quit the study, return to study), I was wondering how we should evaluate them. For example, the patient did not show up for his T6 appointment (six months after surgery), but he returned to study at T9. Commented Jun 7 at 17:47
• @ EdM, Another question: when we say in discrete time intervals, the intervals must be the same for all subjects, what exactly does it mean? In my scenario, T0 (one day before surgery) would be a different day for each participant. For example, one person's surgery date is in June, whilst another's is in July. So T3 indicates three months after surgery. However, for the first ID, the T3 date would be in September, whereas for the second ID, it would be November. Still, can we presume the time intervals are the same? Commented Jun 7 at 17:48

Yes, you can consider a discrete-time model here. Interval censoring means that you know an event happened within some time interval, but you don’t know exactly when. That’s how you treat event times in discrete-time models.

Right censoring is when you only have a lower limit to an event time, because of dropout or the administrative end of follow up. Individuals with right-censored times are removed from analysis after you no longer have information about their event times. In a discrete-time model you need to think about the interval at which you cease including an individual with a right-censored time in the analysis. Those who complete the full study follow up without an event should be included in all time intervals. Those who drop out part-way should be removed after the last interval during which you know that you had information.

In response to edited question and comments

Don't forget the importance of having a clear definition of the time = 0 reference time for the survival analysis. Statements of time values or time intervals are then all with respect to that reference time. If the reference time is the date of surgery, and the times intervals between visits are the same for all individuals, then the time intervals relative to the reference are the same for purposes of discrete-time survival analysis even if the calendar dates differ.

If you have individuals who miss some follow-up visits and then come back with an event, you have wider spreads of interval censoring for those individuals. You might be better off using models that can handle arbitrary interval censoring, like those provided by the icenReg package, instead of discrete-time analysis.

Now that you've provided more information about your study, it's clear that it requires a lot of care in thinking about and setting up the data beyond these issues of interval censoring of event times. One big question is the time interval in which you should first include a change in cognitive status that was identified at a follow-up visit. You don't know when, during the interval since the prior visit, that change in status occurred. So do you mark cognitive function during the time interval prior to identifying the change in status as YES or NO?

Also, I question the binary treatment of cognitive function as a predictor. There are graded measures of cognitive function, like the Montreal scale; an arbitrary cutoff of a graded score is not typically a wise choice. Furthermore, a graded score would be more amenable to the type of joint modeling of covariates and events that might handle this situation better.

I'm glad to have been able to provide help up to this point to clarify the principles involved, but you should get some experienced local statistical advice before you proceed with designing and performing this study. The issues are too complicated and specific to the field of study to provide detailed help on a site like this.

• Thank you for your comment. I'm glad to see it. So if a patient has shown up at T6 but has not experienced the event but has not come up to the hospital at T9 to do the test then we need to keep all of his records up to T6 in the analysis - please correct me if I am wrong. Commented Jun 6 at 13:02
• what if a person who could not come to the hospital at T6 and missed the appointment could come back to the hospital at T9 to continue the study. what kind of design analysis could we have for this case? Commented Jun 6 at 13:06
• @Stat2024 your first comment is correct. The situation in the second comment might be a bit trickier with a time-varying covariate. I’ll come back to that when I have more time. Can there be more than 1 event per individual?
– EdM
Commented Jun 6 at 14:58
• @ EdM, thank you so much for your time, Yes, there is just one event Commented Jun 6 at 17:27
• @ EdM, I will accept your comment but I would be very thankful if you could later add some comments on my question regarding returning to the hospital at T9. Also do you have any advice regarding my next question? stats.stackexchange.com/questions/648604/… Commented Jun 6 at 17:37