Breslow approximation for tied event times ($\frac{\exp ({\sum }_{i\in D_j}{x}_i^T\beta )}{({\sum }_{i\in R_j}{e}^{x_i^T\beta }{)}^{d_j}}$)

When $d_j$ events occur at the same recorded time, the Breslow approximation to Cox partial likelihood uses the factor $\exp ({\sum }_{i\in D_j}{x}_i^T\beta )/({\sum }_{i\in R_j}{e}^{x_i^T\beta }{)}^{d_j}$, where $R_j$ is the risk set and $D_j$ the event set. It approximates the ordering of tied events rather than treating their within-tie order as known.