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I am fitting a subdistributional hazards model (Fine, Grey 1999) for multiple causes of death in a mortality incidence model following diagnosis of cancer. The researchers are interested in a nomogram, basically a type of risk prediction that allows for multinomial failures. I am struggling to connect cumulative incidence (a pivotal concept to competing risk models analogous to survival in a Cox model).

Are the coefficients output by a subdistributional hazards model still interpreted as a log-hazards ratio as with a Cox model? I observe that taking 1-Survival in a traditional survival model yields an incidence curve when only one outcome is available. I note that the output from this type of model outputs results which do not require the survival curve to approach 1. The multiple cumulative incidence curves "appear" proportional in the same way that S-shaped survival curves appear proportional when stratified by Cox model coefficients (e.g. proportional hazards).

If such an interpretation is acceptable, then of what is the hazard exactly? It is easy to think of a hazard when it refers to either death or censoring. But allowing for multiple outcomes confuses me somewhat.

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After reading the article from Fine and Grey, it seems that they define the subdistributional hazard, $\lambda_1(t; Z)$, as the following:

$$\lambda_1(t; Z) = \lim_{\Delta t \rightarrow 0} Pr\left( t \le T \le t + \Delta t ; \epsilon = 1 | T \ge t \cup (T \le t \cap \epsilon \ne 1) \right)$$

Where T is the observed failure time, Z is a covariate vector, $\epsilon$ is a cause of death indicator taking levels $1, \ldots, K$ exclusively.

The intepretation of this subdistributional hazard function is the instantaneous risk of death from cause 1 given you are either still alive, or you've already died of something else. In effect, it averages across these two possibilities in such a way that a high risk of dying previously from other causes lowers your hazard for that specific failure. This is reflected in the "risk sets" by including previous failures somewhat erroneously in the denominator.

The interpretation of the model coefficients, then, is subdistributional hazard ratios which should approximate relative incidence rates.

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  • $\begingroup$ Yes, the risk set is somehow "wrong", in the sense that also people that are not actually at risk are included. The Fine and Grey approach allows to determine the effect on global survival of each risk factor, but for instantaneous individual risk the cause-specific hazard is better. I'd say that, from the viewpoint of each single risk factor, the former approach treats individuals having events from other causes as people never having the event, and the latter one just consider them as censored cases. $\endgroup$ – Federico Tedeschi Jun 27 '17 at 13:48
  • $\begingroup$ Also relevant: onlinelibrary.wiley.com/doi/10.1002/sim.7501/… $\endgroup$ – AdamO Sep 25 '17 at 22:49

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