Cause-Specific Hazard Function:

The cause-specific hazard function denotes the instantaneous rate of occurrence of the kth event in subjects who are currently event free (ie, in subjects who have not yet experienced any of the different types of events). It is defined as:

h^{CS}_k(t) = \underset{\Delta t \rightarrow 0}{lim} \frac{Pr \left[ t \leq T \leq t+\Delta t , D = k  | T \geq t \right]}{\Delta t}

Sub-Distribution Hazard Rate Function:

h^{SD}_k(t) = \underset{\Delta t \rightarrow 0}{lim} \frac{Pr \left[ t \leq T \leq t+\Delta t , D = k  | T > t \cup (T< t \cap K \neq k) \right]}{\Delta t}

In settings in which competing risks are present, two different hazard regression models are available: modeling the cause-specific hazard and modeling the subdistribution hazard function. Both models account for competing risks, but do so by modeling the effect of covariates on different hazard functions. Consequently, each model has its unique interpretation. We refer to these two models as cause-specific hazard models and subdistribution hazard models. The second model has also been described as a CIF regression model. The latter name makes explicit the link between the subdistribution hazard and the effect on the incidence of an event. That is, one may directly predict the cumulative incidence for an event of interest using the usual relationship between the hazard and the incidence function under the proportional hazards model. Thus, the subdistribution hazard model allows one to estimate the effect of covariates on the cumulative incidence function for the event of interest