Competing Risks Model

Definition

Competing risks (multiple modes of failures) model is designed to accommodate the multiple causes to the same event.

Let $(T_i,K_i),i=1,2,\dots ,n$ be the survival time and failure mode (competing risk type) for each individual, where $K_i\in \{1,2,\dots ,k\}$

The mode-specific hazard function is defined as ${\lambda }_j(t)={\lim }_{\Delta t\to 0}\frac{P(T<t+\Delta t,K=j|T>t)}{\Delta t}$ where $j=1,2,\dots ,k$

The marginal hazard function and marginal cumulative hazard function are defined as $\lambda (t)=\sum _{j=1}^k{\lambda }_j(t)$ $\Lambda (t)=\sum _{j=1}^k{\Lambda }_j(t)$

Likelihood

We can observe $(Y_i,{\delta }_i,K_i),i=1,2,\dots ,n$, where $Y_i=\min (T_i,C_i)$, ${\delta }_i=I(T_i\le C_i)$ is censoring indicator, and $K_i$ is failure mode.

The likelihood function is defined as $L={\prod }_{i=1}^n{\prod }_{j=1}^k{f}_j(y_i{)}^{{\delta }_{ij}}S(y_i{)}^{1-{\delta }_i}$ where ${\delta }_{ij}=I(T_i\le C_i,K_i=j)$

Estimators

The Nelson-Aalen Estimator for competing risk model is defined as ${\hat{\Lambda }}_j(t)=\sum _{i:y_i\le t}\frac{{\delta }_{ij}}{n_i}$ where $j=1,2,\dots ,k$

The survival function is estimated by $\hat{S}(t)=\exp \left[-\sum _{j=1}^k{\hat{\Lambda }}_j(t)\right]$

The sub-distribution function is estimated by ${\hat{F}}_j(t)=\sum _{i:y_i\le t}\hat{S}(t_i)\frac{{\delta }_{ij}}{n_i}$ where $j=1,2,\dots ,k$