Modeling the hazar of the cummulative incidence function (CIF) (Fine and Gray, 1999):

$$\gamma(t,x)\,=\,\gamma_{0}(t)e^{\beta x}$$

Partial likelihood is given by:

$$L(\beta)\,=\,\prod_{j=1}^r\,\frac{exp(\beta x_{j})}{\sum_{i\epsilon R_{j}}w_{ji}(\beta x_{i})}$$

The risk set is formed of those who did not experience an event by time t and of those who experienced a competing risk event by time t. Those who expererienced other types of events remain in the risk set all the time. However, individuals experiencing a competing risk envent do not participate fully in the partial likelihood.

Where the weights are defined:

$$w_{ij}\,=\,\frac{\hat{G}(t_{j})}{\hat{G}(min(t_{j},t_{i}))}$$

$\hat{G}$ is the Kaplan-Meier estimate of the survivor function of the censoring distribution.

The score (U) statistic has the form:

$$U(\beta)\,=\,\sum_{j=1}^r\,\left\{X_{j}\,-\,\frac{\sum_{i\epsilon R_{j}}w_{ji}x_{j}(\beta x_{i})}{\sum_{i\epsilon R_{j}}w_{ji}(\beta x_{i})}\right\}$$

The model based estimate of the CIF is defined

$$F(t) = 1 - exp(-H(t)),$$

and

$$\hat{H}$$ can be calculated using a Breslow-type estimator:

$$\hat{H}\,(\text{t;} X_{0}\,\hat{\beta}) \,=\,\sum_{t_{j}\leq t}\left\{\frac{e^{\beta x_{0}}}{\sum_{i\epsilon R_{j}}\,e^{\beta x_{i}}}\right\}$$

Go back