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Suppose there is a university that has data on different students. The university is trying to build a statistical model that will help them understand which "types of students" (e.g. males, low income, etc.) are at higher risk of dropping out of the university - based on the results of this model, the university would try to help these "types of students" prior to them dropping out in the future.

The university has 10 years of historical data (e.g. 2008 - 2018). Let's say that students can either "drop out" of the university, "successfully graduate" from the university, be "expelled" from the university for bad behavior, or "currently enrolled in the university" (e.g. as of 2018, the student was still currently enrolled in the university). The university has data (e.g. age, gender, employment, placement exam score - recorded once a year on the first day of school, September 1st) on students up until a year before one of these events happen. The data looks something like this (let's imagine that on September 1st, the student are required to take a placement exam and receive a score):

  student_id gender age date_measurements_taken employement Placement_Exam Year_Expelled Year_Drop_Out Year_Graduated
1          1   male  19             Sept-1-2010         yes      73.39544            NA          2013             NA
2          1   male  20             Sept-1-2011         yes      54.40925            NA          2013             NA
3          1   male  21             Sept-1-2012          no      43.70675            NA          2013             NA
4          2 female  24             Sept-1-2009         yes      84.69812          2011            NA             NA
5          2 female  25             Sept-1-2010         yes      92.73473          2011            NA             NA
6          3   male  19             Sept-1-2010          no      65.44966            NA            NA           2014
7          3   male  20             Sept-1-2011          no      77.61102            NA            NA           2014
8          3   male  21             Sept-1-2012          no      69.92497            NA            NA           2014
9          3   male  22             Sept-1-2013          no      84.74314            NA            NA           2014

My initial idea was to fit a Discrete Time Survival Model to this data (e.g. c-log-log link) or to just treat this problem as continuous time and use a classical Cox Proportional Hazards - this way, I could find out the effect of different covariates on the hazard, and if these effects were statistically significant.

The problem I have is deciding how to define the "Event" and "Censoring" - two issues come to mind:

  • Issue 1: In this example, the university only records data on the first of September and does not record the exact date at which the student drops out, nor does the university record measurements at the the exact time at which the student drops out. As an example, the student may not be employed on September 1st - but starts working in November, can't handle the stress and ends dropping out by March.

  • Issue 2: It seems to me that in the traditional sense, "Drop Out" (e.g. prematurely dropping out of a medical study) is generally considered to be "Censoring". However in my case, I am interested in identifying "the last known September 1st in which the student was present at the university" (so that the university has a few months to take some action)

With this being said, I thought of the following approach. In this problem, I would define:

  • Event: "Drop Out" (occurring at "Year_Drop_Out")
  • Censoring: "Graduation" (occurring at "Year_Graduated"), "Expelled" (occurring at "Year_Expelled") , "Currently Enrolled At University"

Thus, it would appear as though I have interchanged "Censor" and "Event" definitions in this problem - can someone please tell me if the modelling framework that I have proposed in this question is logical and corresponds to a Survival Model?

Notes:

  • I would obviously restructure my data according to the necessary software package/library specifications prior to starting analysis
  • In the comments, a "Competing Risks Survival Model" was suggested for this problem. While in theory I think this is a suitable approach - suppose the university is only interested in studying "students who drop out" vs. "students who do not drop out". Given this, is the approach I have outlined suitable?
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    $\begingroup$ These seem like competing "risks" requiring more than 1 type of event. Have you looked into the methods explained by the R competing risks vignette? $\endgroup$
    – EdM
    Dec 4, 2022 at 16:15
  • $\begingroup$ @ EdM: thank you for your reply! I looked at this ... in my case, I am not particularly interested in knowing about graduation or being expelled - just dropping out. In this case, does it make sense to "clump" graduation, currently enrolled and getting expelled into a single "state" (censored) and dropping out as the event? $\endgroup$
    – stats_noob
    Dec 4, 2022 at 17:13
  • $\begingroup$ Conceptually, does it make sense - "creating a survival model to estimate the start of the last year the student will be in prior to dropping out"? $\endgroup$
    – stats_noob
    Dec 4, 2022 at 17:18
  • $\begingroup$ I think this is a common problem - when measurement on an individual patient are only available at some time point before the actual event occurs? Is this a major violation of survival analysis assumptions? $\endgroup$
    – stats_noob
    Dec 4, 2022 at 17:21

1 Answer 1

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Most useful here might be a multi-state model. In particular, someone who "drops out" might re-enroll and then be once again "at risk" of the other fates. The institution could be interested in evaluating factors associated with re-enrollment and eventual success.

That said, you can define the "event" as anything that makes sense. If all you care about is the first drop-out event and not subsequent re-enrollment, you could model this as a competing risks problem.

Even with competing risks, however, in a proportional hazards setting you get the correct hazard ratios for the event of interest if you censor at the times of other events that aren't of interest (in addition to right-censoring at last observation times of those still at risk for all event types). See the R survival package's competing risks vignette, for example. When you do that censoring you don't get estimates of the probability of being in the other states over time, which are often of interest.

You speak of a "Discrete Time Survival Model" and a "Cox Proportional Hazards" model as being synonymous. Although you are correct that a Cox model is evaluated at discrete event times, it's considered to be working at samples of an underlying continuous time. What's conventionally considered a discrete-time survival model typically considers only a restricted number of time points, as what you might have with panel data.

Your hypothetical data, with events apparently only known to have occurred within an academic term or academic year, is what's considered interval censored: you have both lower and upper limits to event times. The discrete-time model most closely related to a Cox model in this case is a binomial regression with a complementary log-log link, on properly formatted data. See this page.

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