Subdistribution

Subdistribution is a term used in survival analysis with competing risks. It refers to the subdistribution hazard, the instantaneous rate of occurrence of a specific type of failure in the presence of other competing failure types. The main object of interest is the cumulative incidence function (CIF) for that cause, which is the probability that the event of interest has occurred by time t.

In competing risks settings, there are several possible failure types J. The cause-specific hazard h_j(t) considers only individuals who are still event-free. The subdistribution hazard h_j^*(t) relates to the CIF for cause j by focusing on the rate at which the CIF accumulates, and it is defined in a way that keeps individuals who have already experienced a competing event in the risk set for modeling purposes. This contrasts with standard hazard concepts and leads to different regression formulations.

The most common modeling approach for subdistribution hazards is the Fine-Gray model. It specifies h_j^*(t | X) = h_j0^*(t) exp(β^T X), where X are covariates and β are regression coefficients. Coefficients in this model quantify the effect of covariates on the cumulative incidence of the cause of interest, with interpretations tied to the cumulative incidence rather than to a traditional survival probability alone.

Estimation and interpretation in practice use specialized methods and software. Packages implementing the Fine-Gray approach include the cmprsk package in R, which provides regression analysis for subdistribution hazards. Subdistribution modeling is widely used in medical research to assess cause-specific risks when competing events (such as death from other causes) may preclude the event of interest.

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