1

Suppose I am modeling survival with the hazard rate specified using logistic regression, and the model is adequate for the data. Now add censoring, and the model formulation becomes a bit more cumbersome and the implementation a bit more involved. That holds if we assume censoring to be independent of the survival process, but I guess gets even worse if we relax that assumption.

On the other hand, could we perhaps treat censoring as a competing risk and go for a multinomial logit model? Both interpretation and implementation would be pleasantly easy again. Moreover, I think we would not have to assume the survival and censoring processes to be independent.

Is it an OK approach, or am I missing something (perhaps some implicit assumptions that can be problematic)?

Richard Hardy
  • 67,272

1 Answers1

2

What's typically of interest in survival analysis is the process underlying the distribution of events in time. The tools of survival analysis are designed to deal with missing (censored) event times in a way that provides reliable estimates of the event process itself.

What you describe is a model of whether you observe an event or a censoring time. By itself it wouldn't directly describe the event process if censoring is informative. The review by Leung et al. provides an introduction the problems introduced by making incorrect assumptions about censoring.

Modeling the distribution of censoring times, however, can be a first step toward dealing with potentially informative censoring. As Tutz and Schmid explain in Chapter 4, inversely weighting cases with respect to their probabilities of having been censored over time (inverse probability of censoring weighting, IPCW) can "'reconstruct' the characteristics of the unknown [due to censoring] full data sample by using the weights" (page 89).

Although Tutz and Schmid present IPCW in the context of estimating prediction errors, in some circumstances it can correct for informative censoring. Robins and Finkelstein illustrate this in "Correcting for Noncompliance and Dependent Censoring in an AIDS Clinical Trial with Inverse Probability of Censoring Weighted (IPCW) Log-Rank Tests," Biometrics 56: 779-788 (2000).

Hernán and Robins devote much of their book "Causal Inference: What If" to the problems introduced by censoring. Chapters 8 and 12 explain the problems for analysis that doesn't explicitly involve time (e.g., continuous or binary outcomes when some individuals are lost to follow up); Chapters 21 and 22 cover situations where censoring is a function of time, as in survival analysis.

EdM
  • 92,183
  • 10
  • 92
  • 267
  • Thank you! Still trying to get some more intuition. The tools Tutz and Schmid describe in their book seem to assume independence between censoring and the event of interest (p. 4: We will most often assume that the survival time is independent of the censoring time). So I thought multinomial logic provides a more flexible and realistic alternative to these methods. Now I understand that having a wrong model for censoring may mess up what we learn about the event of interest. At the same time, I think the problems caused by assuming independent censoring may be larger than the ones... – Richard Hardy Oct 15 '22 at 13:40
  • ...I see on the surface of the multinomial logit. And the notion that censoring is a competing risk appears very intuitive to me. Also, while modeling censoring times and then using them for IPCW makes sense, is a corresponding adjustment not happening automatically in the multinomial logit model? – Richard Hardy Oct 15 '22 at 13:45
  • @RichardHardy I suppose you could consider censoring as a competing risk in terms of observation times of events versus censoring. But the multinomial model doesn't deal with the fundamental problem of informative censoring: if the fact that there is right-censoring at some time $t$ for an individual is associated with the time that the event of interest would occur at some point in the future. A multinomial discrete-time model by itself only estimates the probability of censoring at time $t$; it doesn't further incorporate that information into the model of event times per se. – EdM Oct 15 '22 at 14:11
  • Hmm... The multinomial logit model does not specify the event time $T$ as a function of the censoring time $C$, nor the other way around, and that is a drawback. On the other hand, at least the model does not implicitly or explicitly prohibit the event time $T$ from being a function of the censoring time $C$ or vice versa, or does it? I mean, if $T$ is a function of $C$, does that automatically mean a multinomial logit is the wrong model? (I am a bit lost in this unfamiliar territory.) Also, what would be the simplest model that models $T$ as a function of $C$? – Richard Hardy Oct 15 '22 at 15:19
  • I thought more about this and have to agree with you. – Richard Hardy Oct 16 '22 at 16:18
  • Since you have helped me repeatedly this year, I wonder if you could take a look at some of my recent questions. There is a bounty on one of them expiring in 2 days. – Richard Hardy Nov 04 '22 at 08:45
  • @RichardHardy I'm well beyond looking for bounties on this site, but I am interested in your questions. I'm currently getting ready for a trip so I can't attend to them immediately. I'll be bringing an electronic copy of Tutz and Schmid along and hope to address some of your questions in the next week or two. – EdM Nov 04 '22 at 16:14
  • I appreciate that! – Richard Hardy Nov 04 '22 at 16:16