# Subdistribution/cause-specific survival functions

How is the subdistribution survival function, associated with the subdistribution hazard function, interpreted? Is it just the probability of failing from the specific cause after the current time? I know it was developed to allow to estimate covariate effects on the incidence, but I'm not sure if that also means the survival function associated with it is interpretable as a normal survival function.

And then, is the survival function associated with the cause-specific hazard not interpretable as the probability of failing from the specific cause after the current time?

• I suggest you check out the Fine-Gray section in this vignette: cloud.r-project.org/web/packages/survival/vignettes/compete.pdf I'm getting the impression that you can't treat the cumulative sub distribution hazard curves as survival curves since they might add up to more than 1. Commented Sep 3, 2022 at 6:24

Fine and Gray used the word "subdistribution" to represent the cumulative incidence function of one of two or more competing events. Their method allows separate modeling of each event type in a way that readily allows for evaluating covariate associations with the risk of an event type. That's unlike the combined treatment in cause-specific modeling, in which the hazard of one event type implicitly includes the hazards of all the others.

To do that, however, one needs to continue including individuals who have already experienced another type of event in the risk set for the event in question. They noted the difficulty in interpreting the hazard function $$\lambda_1(t)$$ for event type $$\varepsilon=1$$ on page 497:

Clearly, the risk set associated with the hazard $$\lambda_1$$ is unnatural, as in reality those individuals who have already failed from causes other than $$\varepsilon = 1$$ prior to time $$t$$ are not "at risk" at $$t$$.

That makes is difficult to interpret a "survival curve" derived from a cumulative subdistributional hazard. As Iyar Lin points out in a comment, the R competing risks vignette illustrates that this separate modeling of event types can allow the sum of the probabilities of the individual events to exceed 1, which doesn't happen with combined cause-specific modeling. Austin et al recently explored that limitation in detail. Despite this, they conclude:

the [Fine-Gray] model can still be useful for at least two reasons. First, as noted above, it allows one to determine which variables are associated with increased or decreased risk. Knowledge of this cannot be extracted from cause‐specific hazard models in the presence of competing risks. Second, when the focus is on a single event type, the subdistribution hazard model allows for estimation of subject‐specific estimates of absolute risk. While there may be bias in some of these estimates, these estimates are still preferable to those obtained using a single Cox proportional hazards model that ignores the presence of competing risks.

That's pretty much what Fine and Gray said originally (page 498), that their method

is intended to be a convenient empirical representation for the cumulative risk of a competing risk and should be evaluated on the extent to which it permits the analyst to assess the effect of covariates on the cumulative incidence function...

Jason P. Fine and Robert J. Gray. A Proportional Hazards Model for the Subdistribution of a Competing Risk Journal of the American Statistical Association 94: 496-509, 1999.

Peter C Austin, Ewout W Steyerberg, and Hein Putter. Fine-Gray subdistribution hazard models to simultaneously estimate the absolute risk of different event types: Cumulative total failure probability may exceed 1. Stat Med 40: 4200-4212, 2021.

• Thanks for the answer; besides the subdistribution model, I seem to remember reading something about the cause-specific hazard not even having a survival function that is estimable but maybe it actually said it wasn't interpretable. Do you know about this? So for the subdistribution hazard you can get the (biased) survival function by $exp{-\int_{0}^{x} h_k(t) dt}$ -- does trying this with the cause-specific hazard not get you anything that is useful? Commented Sep 5, 2022 at 21:51
• @user352188 a survival curve for one event type might be interpretable in a combined cause-specific model, but the associations of covariates with specific event types are then difficult to interpret. In such a model, a change in a covariate might increase the apparent risk of one event type primarily because it decreases the risk of a competing event, while having no direct influence on the first event type of interest. The Fine-Gray model more directly provides specific associations between event types and covariates, but with the drawbacks noted in the answer.
– EdM
Commented Sep 5, 2022 at 23:06