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I am familiar with the literature on nonparametric estimation of the cumulative incidence function under a competing risk model. I'm looking for the more basic solution under exponential distributions of failure time for the various competing risks. I want to be able to estimate the hazard rate for each cause, and the cumulative incidence functions. I assume the hazard rate maximum likelihood estimates under constant hazard rate assumptions are of the form number of events by cause i divided by person-years of exposure but I'm not sure about the exact definition of that denominator.

I also have a secondary question. The definition of the cumulative incidence function under the competing risk models conditions on the cause of the event being a specific cause. I've never wrapped my head around whether this conditioning is presumptuous. Why would we ever know the cause of failure in advance if there is more than one cause?

Frank Harrell
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    I don't understand why you talk about "conditioning" in the case of Cumulative Incidence Function. The CIF for risk $j$ (being $J$ the variable indicating which risk will occur first) is given by $P[T \leq t, J=j]$. Thus, $lim_t\rightarrow \infty CIF(j) = P[J=j]$: there is no conditioning on the specific cause (if we knew in advance the cause is $j$, we wouldn't have to calculate its probability, and CIF would asymptotically tend to $1$). – Federico Tedeschi Jun 27 '17 at 13:36
  • I think that makes sense - thanks. The CIF would not tend to 1.0 if you don't condition on the event *occurring*. I was just referring to event $j$ being the one if an event did occur. – Frank Harrell Jun 27 '17 at 15:54
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    What happens if the event that occurs first is another one is that this person becomes "immortal" for risk $j$: s/he will remain in the risk set all time, without having any possibility to experience event $j$. This inclusion of people not at risk is recognized by Fine and Gray themselves. In my understanding, however, with their model things work well if we are just interested (for each time $t$ between $0$ and $T$) in the probability of each event (considering them as mutually exclusive) and (as $1-\sum_{j=1}^{J}P_{t}[j]$) the overall survival probability. – Federico Tedeschi Jun 28 '17 at 11:30
  • That's helpful, and points out why I always have difficulty with conceptualizing competing risks and methods for handling them. I find Terry Therneau's discussion (especially about transition models) helpful too: https://cran.r-project.org/web/packages/survival/vignettes/compete.pdf – Frank Harrell Jun 28 '17 at 11:39
  • Thank you for the link: I've printed down the paper and started to read it. You know: what I'm looking for is a "third way": "neither censoring nor immortality", i.e. neither "regular" cause-specific hazard (censoring the other outcome) nor the Fine-Gray approach (the CIF we talked about). What I'd like to answer is: "what would happen in an hypothetical world where the other events didn't exist?" I see that Thernau, however, finds this question rather ridiculous: "This author views the result in much the same light as discussions of survival after the zombie apocalypse". – Federico Tedeschi Jun 28 '17 at 12:50
  • My view is different: I think "statistics" rhymes with "counterfactual". I found that, starting from this paper: https://www.jstor.org/stable/pdf/2336666.pdf?refreqid=excelsior%3A2662291831971b76ae2c76815a0cb586 conditions for identification in the case of covariates (without further assumptions, like independence between events, as in censoring) have been found, but I don't know whether this "third way" has been somehow implemented in statistical software programmes. – Federico Tedeschi Jun 28 '17 at 12:56

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