Table 1.

Assumptions, Uses, and Advantages of 3 Different Regression Approaches to Modeling Competing Risks

	Approach
	Cause-specific Proportional Hazards Model	Subdistribution Proportional Hazards Model	Parametric Mixture Model
Model assumption	Assumes proportionality of the cause-specific hazard, as this model is exactly the same as conducting a regular proportional hazards model in which individuals with the competing event are censored at that time point.	Assumes that the subdistribution hazards are proportionala	Assumes that the investigator has correctly specified both the distribution for the event of interest and the competing event
Are these assumptions reasonable?	As with all proportional hazards models, the analyst should evaluate whether the proportionality assumption is met. In practice, this assumption is often violated.	The assumption that the subdistribution is proportional should be assessed.	Correctly specifying the distribution of events is difficult for any parametric model.
	Nevertheless, the investigator should acknowledge that there was some indication of nonproportionality and report the csRH as this is the weighted average over follow-up.	This can be done by assessing the residuals that are returned by the “crr” function in R against the unique failure times (16). This is analogous to examining the Schoenfeld residuals from a regular proportional hazards model.	The correct specification of the distribution may be made more tenable by utilizing a flexible distribution that can accommodate various shapes of the hazard function.
	Alternatively, violation of the proportional hazards assumption may be mitigated by including an interaction between variables and time to allow the csRH to vary over time.	Alternatively, proportionality may be assessed by evaluating the log(−log) transformation of the nonparametric cumulative incidence function estimators (2, 10, 14, 15) stratified by exposure variable. The step function curves should be separated by a constant difference.	A generalized gamma distribution is an example of a flexible distribution that can accommodate increasing, decreasing, arc-shaped- (increasing then decreasing), and bathtub-shaped (decreasing then increasing) hazard functions.
		Nevertheless, the investigator should acknowledge that there was some indication of nonproportionality and report the sdRH as this is the weighted average over follow-up.
		When the proportionality assumption is violated, this may be mitigated by including an interaction between variables and time.
What is the model useful for?
Measuring the association?	Yes, the csRH is a measure of association. It implies that, among any individuals who survive all events up to some unspecified time t, those with the exposure have a cause-specific hazard rate of csRH × the cause-specific hazard rate of those who do not have the exposure.	Yes, the sdRH is a measure of association. However, it is a measure of association that is due to both the association of the exposure at the event of interest and the possibly differential impact of competing events on the risk set for exposed and unexposed individuals.	Yes, both the csRH and sdRH are estimable.
Evaluating the risk of the event?	No, the csRH by itself cannot be used to predict whether the event will be observed. Whether the event will be observed is a function of both the csRH associated with the event of interest and the csRH associated with the competing event.	Yes, because the sdRH intrinsically accounts for the competing event by modifying the risk set at time t; a sdRH >1 indicates that those with exposure will be seen to have a quicker time to event in the study population. Similarly, a sdRH <1 indicates a longer time to event for those exposed.	Yes, in addition to the sdRH, the sdCIF is directly estimable.
	csRH >1 does not necessarily imply that the sdCIFexposed > the sdCIFunexposed and vice versa.	sdRH >1 does imply that the sdCIFexposed > the sdCIFunexposed and vice versa.
What is the model's advantage?	It measures the association of an exposure on the corresponding event in which the competing event contributes only by passively removing individuals from the risk set.	It measures the association of an exposure to the corresponding event in which the competing event actively contributes to the risk set.	The model can obtain the csRH, the sdRH, and the subdistribution cumulative incidence, as well as the cause-specific hazard and subdistribution hazards all as functions of time.
	The model does not have to correctly specify the unspecified baseline cause-specific hazard function.	The model does not have to correctly specify the unspecified baseline subdistribution hazard function.	The model does not require the assumption of proportional hazards over time.
			When correctly specified, parametric models generally tend to have more power than semi- or nonparametric models.

Abbreviations: _csRH, cause-specific relative hazard; _sdCIF, subdistribution of the cumulative incidence function; _sdRH, subdistribution relative hazard.

^aThe subdistribution proportional hazards model assumes that the transformation of the subdistribution cumulative incidence functions as log(−log) transformation results in a constant difference between curves (16).