 Efron approximation for tied event times

For $d$ events tied within a risk set, put $u_i=\exp ({\beta }^T{z}_i)$, $S={\sum }_{i\in R}{u}_i$ and $U_D={\sum }_{i\in D}{u}_i$. The Efron approximation to Cox partial likelihood removes the average tied-event hazard multiplier from the denominator at each successive event. Unlike the Breslow approximation for tied event times, it accounts approximately for depletion of the risk set within the group. Factors independent of $\beta$ can be dropped for estimation. This is an approximation for coarsened continuous event times, not a full interval-observation likelihood.