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

γ(t,x)=γ0(t)eβx

Partial likelihood is given by:

L(β)=j=1rexp(βxj)iϵRjwji(βxi)

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:

wij=Ĝ (tj)Ĝ (min(tj,ti))

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

The score (U) statistic has the form:

U(β)=j=1r{XjiϵRjwjixj(βxi)iϵRjwji(βxi)}

The model based estimate of the CIF is defined

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

and

Ĥ
can be calculated using a Breslow-type estimator:

Ĥ (t;X0β̂ )=tjt{eβx0iϵRjeβxi}

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