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Letand let $C$ be the common partial likelihood contribution from the first $n-3$ observations. Since $t_{n-2}>x_{n-2}$, the three possible complete-data tail orderings and their partial likelihoods areTheir sum isWhen individual $n-2$ is right-censored at $x_{n-2}$, that individual leaves the risk set before the event at $x_{n-1}$, so the directly calculated Cox partial likelihood is also $C,a/(a+b)$. Summing over the unobserved compatible event orderings therefore reproduces the censored-data partial likelihood.

A semiparametric proportional hazards model specifieswith finite-dimensional parameter $\beta$ but an unspecified baseline hazard $h_0$. A partial likelihood uses a component of the data likelihood that depends on $\beta$ while eliminating the nuisance function. In the Cox proportional-hazards model, conditioning on which member of each risk set experiences the event produces the Cox partial likelihood.