 Cox partial likelihood

In a Cox proportional-hazards model, $h_i(t)=h_0(t)\exp \{{\beta }^T{z}_i(t)\}$. At distinct event times $t_j$, let $i_j$ identify the event subject and let $R_j$ be the risk set immediately before $t_j$. With covariates evaluated just before the event, the untied Cox partial likelihood isEach ratio is the conditional probability that the next single failure belongs to the observed subject, given its event time and current risk set. Substituting the hazards into this ratio cancels their common baseline hazard $h_0(t_j)$. With three subjects whose scalar covariates are $0,0,1$, the probability that the third subject fails next is $e^{\beta }/(2+e^{\beta })$.

A risk set includes subjects who have entered observation and remain event-free and uncensored at that instant; delayed entry and time-dependent covariates must be represented accordingly. The usual inference requires appropriate independent censoring conditional on the modeled information. Tied recorded event times require a specified treatment, such as the Breslow approximation for tied event times, the Efron approximation for tied event times, or an exact tied-set conditional likelihood for its discrete-time model. These need not give the same objective as an arbitrarily ordered sequence of failures.^[ref]