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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]

Terry Therneau is a statistician working on survival analysis and statistical software. His Mayo Clinic profile identifies continuous-time multi-state models for time-to-event data as a focus, connecting statistical methods with the analysis of transitions between observed states.^[ref]

His A package for survival analysis in R by Terry Therneau documents tools for Cox proportional-hazards models and the handling of censoring. A censored observation still conveys information: a subject observed without an event through time $t$ contributes a survival probability through $t$, rather than being discarded. The software's treatment of risk sets and tied event times makes the statistical model's assumptions part of the computation.^[ref]