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I have a question on how to account theoretically for the risk of competing event in a specific setting.

Suppose we have a cohort of patients at high risk of both infection-related mortality and non-infection related mortality.

We randomize these patients to receive treatment A, which has an effect in reducing the risk of infection-related mortality, but not on the non-infection related one. Follow-up length is long, let's say 10 year.

When we draw survival curves for infection-related mortality, we observe an initial reduction in the risk in patients randomised to A; however, this effect dilutes after 4 years, and the curves approaches after that time.

My guess is that patients receiving A are protected for infection-related mortality, but starts to develop non-infection mortality, get censored, and this influence the survival curves (as well as the incidence rate of events). Indeed, if we observe patients long enough, we will end up having all patients either died of infection-related mortality or censored (i.e., died for other reasons).

Is this an actual problem when looking at cause-specific risk of death, in the context of competing events? How can I account for such bias? Would a Cox-regression analysis be influenced by this potential bias if we have a long enough follow-up?

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My guess is that patients receiving A are protected for infection-related mortality, but starts to develop non-infection mortality, get censored, and this influence the survival curves (as well as the incidence rate of events). Indeed, if we observe patients long enough, we will end up having all patients either died of infection-related mortality or censored (i.e., died for other reasons).

That's why it's not a good idea to censor one type of event time at the occurrence of a different type of event. As the R vignette on competing risks says in Setion 2.2:

A common mistake with competing risks is to use the Kaplan-Meier separately on each event type while treating other event types as censored...We thus have an unreliable estimate of an uninteresting quantity.

You need to use a true competing-event analysis, as explained in detail in the vignette. It outlines both multi-state rate models (Section 3) and the Fine-Gray subdistributional hazard model (Section 4).

As the vignette points out in Section 3, for a Cox model you will get the same regression coefficient estimate for an event type whether you use a multi-state rate model or censor when other event types occur. The latter censoring approach, however, does not provide the correct probabilities of being in each state at any given time. The Fine-Gray model allows for individuals to remain at risk for one type of event even after experiencing a terminal event of a different type. The assumptions needed for a Fine-Gray model might, however, be harder to meet in practice; see Section 4 of the vignette.

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