Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 34 / 5 / a

Let $R(t^{\ast })$ be the risk set immediately before $t^{\ast }$, conditional on the preceding history. Only its members can experience the next event. In a short interval $[t^{\ast },t^{\ast }+dt)$, the conditional probability that subject $j$ fails is $h_0(t^{\ast })e^{\beta z_j}dt+o(dt)$, whereas the probability of one event from the risk set is $h_0(t^{\ast }){\sum }_{i\in R(t^{\ast })}{e}^{\beta z_i}dt+o(dt)$. Dividing and taking the small-interval limit givesIt is zero outside the risk set. The common baseline hazard cancels. This is the conditional event-label probability underlying the Cox partial likelihood. Conditioning at an exact continuous event time is understood by this limiting conditional-intensity argument, rather than as division by the zero unconditional probability of an event at a fixed time. We use the standard simple event process with no simultaneous events and independent censoring.