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I am studying survival analysis and reading Tutz & Schmid "Modeling Discrete Time-to-Event Data" (2016). The authors often consider the censoring time $C$ to be independent of the time to event of interest (e.g. failure) $T$. At the same time they occasionally suggest each of $T$ and $C$ could be modelled using e.g. a logistic discrete hazard model, potentially using the same covariates $\mathbf{X}$. I do not see that this would enforce the independence of the model-implied $C$ and $T$.

What are some survival models and/or estimation methods that enforce the assumed independence between $C$ and $T$?

Richard Hardy
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  • The question is kind of broad, but this is because I am new to the survival literature. Sorry if it happens to be a duplicate. – Richard Hardy Aug 10 '22 at 14:37
  • They are speaking of assessing systematic patterns of follow-up time for different types (at baseline) of subjects. By examining $C$ vs. $X$ they are avoiding a discussion of $C$ vs. $T$ which is an unverifiable assumption as you don't observed failure time for censored observations. – Frank Harrell Aug 10 '22 at 14:38
  • @FrankHarrell, while the assumption is unverifiable, it would be nice if it were used consistently in all steps of a problem (or not used at all). I find it uncomfortable to make an assumption in step 1 but disregard it in step 2. It is my impression this is what is happening, or am I mistaken? – Richard Hardy Aug 10 '22 at 14:42
  • That's not the way I'd interpret their work. Looking at $C$ vs. $X$ is trying to address a different question. It's not that $C$ vs. $T$ is any less important. – Frank Harrell Aug 10 '22 at 15:17
  • @FrankHarrell, yes, I realize that $C$ vs. $\mathbf{X}$ is a different question, but it is not my concern. My concern is that $C$ is assumed to be independent of $T$ when deriving some results such as the likelihood of $T$ (and also the likelihood of $C$). Yet this is not enforced in modelling $T$ and $C$ later on. If we do not enforce that, how can we trust the likelihoods of $T$ and $C$ taken separately? – Richard Hardy Aug 10 '22 at 15:32
  • What do you mean by "enforce"? The independence of censoring and impending risk of failure is a fundamental assumption for everything we do. – Frank Harrell Aug 10 '22 at 15:57
  • @FrankHarrell, e.g. in linear regression models estimated by OLS, $X$ is assumed to be uncorrelated with $\varepsilon$. This is enforced by the OLS estimator so that $\hat\varepsilon \perp X$. This is roughly what I mean by enforce an assumption. – Richard Hardy Aug 11 '22 at 05:33
  • I think we're arguing about semantics (which doesn't make it less important). Assumptions that are not currently stated are still "enforced" (minor point). – Frank Harrell Aug 11 '22 at 11:44
  • Related: "Survival: treat censoring as a competing risk and use multinomial logit". – Richard Hardy Oct 11 '22 at 08:41

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