Cox partial likelihood

The Cox partial likelihood conditions each event on its risk set, eliminating the baseline hazard.

At an untied event time, the Cox partial likelihood weights the current risk set by $w_{jk}=e^{{\beta }^T{z}_k}/{\sum }_{\ell \in R_j}{e}^{{\beta }^T{z}_{\ell }}$. Differentiation gives the displayed score function and the negative Hessian as a sum of weighted covariances. The score is the sum of Schoenfeld functions, so Schoenfeld residuals sum to zero at a finite interior unpenalized fit. At arbitrary trial coefficients or boundary fits, a zero sum is not a general identity.

In a discrete-time model with independent event indicators whose odds are $a{u}_i$, condition on exactly $d$ events in the risk set $R$. The common factor $a^d$ cancels and the conditional probability of the event set $D$ is the displayed ratio. The denominator is an elementary symmetric polynomial and can be evaluated by recursion rather than enumerating every subset. This conditional model is distinct from summing a continuous-time rank likelihood over all unobserved within-tie orders.

For $d$ events tied within a risk set, put $u_i=\exp ({\beta }^T{z}_i)$, $S={\sum }_{i\in R}{u}_i$ and $U_D={\sum }_{i\in D}{u}_i$. The Efron approximation to Cox partial likelihood removes the average tied-event hazard multiplier from the denominator at each successive event. Unlike the Breslow approximation for tied event times, it accounts approximately for depletion of the risk set within the group. Factors independent of $\beta$ can be dropped for estimation. This is an approximation for coarsened continuous event times, not a full interval-observation likelihood.

When $d_j$ events occur at the same recorded time, the Breslow approximation to Cox partial likelihood uses the factor $\exp ({\sum }_{i\in D_j}{x}_i^T\beta )/({\sum }_{i\in R_j}{e}^{x_i^T\beta }{)}^{d_j}$, where $R_j$ is the risk set and $D_j$ the event set. It approximates the ordering of tied events rather than treating their within-tie order as known.

Assume independent individual event times with positive time-constant hazard multipliers and a common baseline hazard whose cumulative hazard tends to infinity. Transforming each event time by that cumulative hazard then gives independent exponential event times with corresponding rates, so all event orders exist. Their complete-order probabilities select each next label proportionally to its remaining multiplier. Marginalizing over the position of a deleted label leaves the order law of the other exponential times, hence the same sequential formula without that label. This proves the deletion identity used when unobserved ranks are summed out. It does not supply a likelihood function for censoring times; ignoring censoring in observed survival analysis still needs independent censoring.