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I am looking at the time required by judges to reach decisions. Each judge assesses a number of applicants and can either approve or not approve the application. The case is finalized when the judge renders his report, which may be some time after the hearing. A number of cases were still open at the end of the study period.

I want to estimate the average time required for cases to move through the system. In addition, I would like to see if cases that are refused take longer than cases that are approved. (Judges seem to spend longer writing up the reports of those they eventually fail to approve, or seek extra documentation).

Obviously, I don't know if the cases that were still open when the study ended would have been approved or not, so the covariate (approve/don't approve) is censored along with the data.

Is there anything I can do about this?

Placidia
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  • Is each judge treating just one applicant? Do we have a problem with the 'non-informative censoring' assumption in survival? Did all applicants start the process at the same time? – Michael M May 07 '14 at 06:09
  • Each judge treats many applicants in the study period (around 30 each). Some of the cases are finalized (accept/reject) - others are still open. – Placidia May 07 '14 at 12:22

2 Answers2

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@jsk has the key in their comment to @Alexis' answer. The appropriate type of survival analysis to use in this case is Competing Risks. You have three possible outcomes: a) accepted, b) rejected, and c) right-censored.

The key is that accepted/rejected is not a single covariate but rather are two competing risks. This is pretty easy in most statistical software. For example, in R's survival package, you simply code the event as a factor with levels censored, accepted, and rejected. (censored must be the first level, other levels are assumed to be competing risks.)

Wayne
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  • Thanks for answering this. The analysis that prompted my question has been overtaken by events, but I have just been handed a new data set with similar demands. – Placidia Mar 01 '17 at 19:24
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If I understand you, this is pretty standard survival analysis/event history analysis right-censoring stuff; Kaplan-Meyer, discrete-time hazard models etc. all estimate "whether and when" an event occurs while accounting for right-censoring of event occurrence (i.e your case case approval) by incorporating the shrinkage of the sample at risk of event over time due to both event occurrence and due to censoring.

The Wikipedia article gives a decent intro. And you might check out Singer, J. D. and Willett, J. B. (2003). Applied longitudinal data analysis: Modeling change and event occurrence. Oxford University Press, New York, NY, which goes into detail on discrete-time event history models, and has a decent enough section on Cox proportional-hazards models.

Alexis
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    Are you sure that this is appropriate for a standard survival analysis? It seems there are two mutually exclusive events that can occur. Perhaps a competing risk model would be more appropriate? – jsk May 07 '14 at 05:14
  • Oh. Am I misunderstanding? Ah yes. . . I think you may be right. Although, I wonder if it could be framed as a two-stage model: event history for whether/when a "decision" is made, and second stage for modeling what accept/reject? – Alexis May 07 '14 at 05:39
  • Not sure that would work. Sooner or later, every case is settled one way or another. If there is no relationship between time to judgment and decision, then censoring is non-informative. But if rejects take longer, say, a disproportionate number of the open cases will be for rejects (although we don't know which). I wonder what would happen if one attempted to impute the outcome of the open cases. – Placidia May 07 '14 at 12:27
  • Placida, that is really interesting: could you elaborate on this aspect of time dependence into you question? Also what about death, removal and retirement: surely not every case is ultimate decided by the original judge? – Alexis May 07 '14 at 12:56
  • I suppose a case might be transferred to another judge, but that's rare. The cases are typically decided within a few weeks. If rejectable cases take longer, then more rejectables will be censored, since the study will have ended before they are decided. The problem is that I can observed the decision status of cases closed before the study ends, but I can't observe that status for censored cases. Using the outcome of the case as a covariate seems dodgy to me, but untuitively, the data should give information on that issue. – Placidia May 07 '14 at 20:22