# Cumulative Incidence vs. Kaplan Meier to estimate probability of failure

To estimate the probability failure in medical sciences, it is not atypical to use 1-KM. However this does not account for competing risks, such as death by natural causes or causes unrelated with the disease, which preclude the event of interest. Thus 1-KM provides an inadequate measure, and cumulative incidence curves, such as the ones used in cmprsk (in R).

My question is that a lot of medical literature still reports KM curves or even 1-KM. Does this mean that the results reported in a lot of medical literature may be inaccurate (or more precisely over-estimated)? Or are there reasons why 1-KM would be preferred?

Furthermore, if there is a difference between 1-KM and the cumulative indices curve what other parts of your analysis are also effected (i.e. discrimination, calibration...)?

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Kaplan-Meier is sensible as long as the censoring is independent of the failure time. Is it a problem with the dependence ? –  Stéphane Laurent May 4 '12 at 18:08
Yes. For example in the case of cancer, a patient may experience a recurrence after treatment. If the patients dies from a cause that is not disease-specific (for example he gets hit by a bus, or dese of a heart attack) than the probability of an event occuring appears to be overestimated: uvm.edu/~rsingle/stat380/F04/papers/… –  user4673 May 4 '12 at 18:22
It is possible to fit a model with a dependence between the failure time and the censoring given by an Archimedean copula. Type "copula-graphic estimator" as a keyword for Google. If for some patients you possibly have the time to the recurrence and the time to end of the follow-up, then you might be interested in semi-competing risks. –  Stéphane Laurent May 4 '12 at 18:34

Let me state up front that I don't have answers for all of your questions. I'm not as strong on competing risks as simpler applications of survival analysis. So, I will just throw out a couple of pieces of information here that may be helpful. I suspect KM curves are more common because they are older and conceptually easier to understand (for both the researcher and the consumer of research). If the competing risks are truly independent, then I believe the KM estimates should be unbiased. That is, a plausible reason why people may prefer KM curves is that many people already understand them, and if those patients who had died due to other causes would have followed the same path as everyone else if they hadn't, the KM curves usefully illustrate what was learned from the study.

Regarding the question of whether there is over-estimation in the literature, one relevant fact, distinct from these issues, is that for practical purposes 'significance' is often required for publication. This guarantees that the literature is biased (specifically over-estimated), an issue known as the file drawer problem.

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You should look at the work of Jason Fine on competing risk modeling. Inplace of Kaplan-Meier there is the cumulative incidence function also analogous to the Hazard function in survival analysis is the cause specific hazard function. There is a competing risk model called Gray-Fine model that he uses. I heard him speak on this recently at a meeting in Connecticut at Yale. If you email me I can send you the lecture slides. Here is some information on references and a link to Bob Gray's website that I took off one of his slides.

INFERENCE FOR CIF: ONE SAMPLE, RIGHT CENSORING • Naive approach to estimation of Fk using Kaplan-Meier (KM) is invalid with dependent risks • Even if risks independent, KM estimates distribution of Tk , where death from other causes is impossible • Valid estimator (equivalent to MLE) obtained by substituting KM estimator of S and NA estimatorof k in Fk (Aalen, 1978; Gray, 1988; Pepe, 1991) • Special case of general counting process framework in ABGK, with technical issues handled via martingale results and alternative product integral representation of CIF • Available in R function “cuminc” on Bob Gray’s website, biowww.dfci.harvard.edu/ gray

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