Subdistribution hazard ratios: incidence of different events

Question

As I understand it, the subdistribution hazard model gives hazard ratios which tell you the direction of the effect of a covariate on the event occurrence, as defined in the paper below. If the SD hazard ratio for age is 1.2 for event type 1, an increase of a year means the probability of event type 1 (at any particular time) increases.

• "the covariates in [the subdistribution hazard] model can be interpreted as having an effect on the cumulative incidence function or on the probability of events occurring over time."

Now, what I'm massively confused about is that the paper below uses an example where there are 2 event types (cardiovascular and noncardiovascular).

They say the following:

• "we conclude that a 10‐year increase in age is associated with an increase in the incidence of cardiovascular death"

AND

• "similarly, we conclude that a 10‐year increase in age is associated with an increase in the incidence of noncardiovascular death".

How is it possible, with only 2 event types, that the same variable increases the probability of BOTH event types? If one probability goes up, the other has to come down, right? The only thing getting in the way of event 1 occurring is if event 2 occurs first. Of course there's a probability that neither occur at any given moment, but one event occurring would prevent the other from occurring, so how can the same variable have a SD hazard ratio > 1?

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Austin PC, Fine JP. Practical recommendations for reporting Fine-Gray model analyses for competing risk data. Stat Med. 2017 10.1002/sim.7501

Answer 1

Think about it this way first: as one gets older, the risk of death increases. That's true whether the death is from a cardiovascular event or from another cause. That's true whether you are thinking about this in the context of a cause-specific model or a subdistribution model for competing events.

There's no problem with a covariate like age at study entry affecting both events in the same direction, with either type of model. It's often helpful to think of a 2-competing-risk model as a 3-state model. In that interpretation, the "competing risk" that's decreasing with age is the "risk" of staying in the event-free state. The question of interest might be which event type is more associated with age at study entry.

Interpretation gets tricky in subdistribution models, as the coefficients have a somewhat strange meaning to those who aren't familiar with them. Quoting from the paper you cite:

The exponentiated regression coefficient from a Fine-Gray subdistribution hazard model denotes the magnitude of the relative change in the subdistribution hazard function associated with a 1-unit change in the given covariate. Therefore, one is reporting the relative change in the instantaneous rate of the occurrence of the event in those subjects who are event-free or who have experienced a competing event. (Emphasis added.)

So the subdistribution hazards, at a fundamental level, aren't really for "competing events" in the way you might usually think about them. A subdistribution hazard represent the apparent hazard for a hypothetical population that includes both those who are event-free and those who have already experienced the other event!

I'd recommend also reading the treatment of competing risks for the R survival package. It turns out that interpretation of hazard ratios in subdistribution models can perhaps be more directly related to the ultimate risk of each type of event than those in cause-specific models, but they can lead to things like the summed probability of all event types exceeding 1.