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Suppose one had performed a competing risk analysis and the 10 year cumulative incidence for the event of interest is 60%. Am I correct in thinking that we CANNOT conclude that the event-free survival is 40% since, in competing risk situations, the survival and cumulative incidence are no longer inversely related? If that's right, am I right that we can't really say anything about event-free survival, but rather we should just discuss the cumulative incidence estimates?

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That is correct. For concreteness suppose that you are interested in the risk of cancer and there is a competing risk due to fatal heart attack. Event-free survival probability at time $t$ is the probability of not having cancer or a fatal heart attack before time $t$. It can be estimated by computing the Kaplan-Meier estimate for the time until the first of the two events.

On the other hand the cumulative incidence of cancer when fatal heart attack is taken as a competing event is the probability of developing cancer that preceeds heart attack death. Since death will mask some cancer cases, the cumulative incidence curve does not have to reach 1.0 as $t \rightarrow \infty$.

It is often more natural to model multiple event types using multistate transition models. Therneau, Crowson, and Atkinson have a wonderful article about competing risk and multistate models here. For example you could model the probability of going from a well state to a cancer state or to a fatal heart attack state, the probability of going from a cancer state to a fatal heart attack state, ... i.e., any transition probabilities you want as a function of time. From these you can come mean times in various states.

The article referenced above also explains why the commonly used Fine & Gray competing risk cumulative incidence model makes assumptions that are unlikely to hold.

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  • $\begingroup$ Thank you! In your first paragraph you seem to suggest event free survival is 1-KM estimate. Isn't it that the event free survival is just the KM estimate (not 1-KM, which gives cumulative incidence if there is just one type of event)? $\endgroup$
    – user167591
    Commented Jul 28, 2023 at 15:39
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    $\begingroup$ Thanks - fixing this now. Note that thanks is expressed by upvoting anwers, comments, questions. $\endgroup$ Commented Jul 28, 2023 at 17:44

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