= Cox partial likelihood
{c}
{title2=$L_p(\beta)$}
In a <Cox proportional-hazards model>, $h_i(t)=h_0(t)\exp\{\beta^Tz_i(t)\}$. 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> is
$$
L_p(\beta)=\prod_j
\frac{\exp\{\beta^Tz_{i_j}(t_j)\}}
{\sum_{i\in R_j}\exp\{\beta^Tz_i(t_j)\}}.
$$
Each 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.<A package for survival analysis in R by Terry Therneau>[Sections 1.4 and 5.1, pp. 90–91]{ref}
Back to article page
