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A covariate is an observed explanatory variable used to model the conditional distribution of a response. A vector of covariates can enter a linear predictor, a treatment-effect adjustment or a hazard multiplier. Conditioning on a covariate does not itself establish that its effect is causal.

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.

In a proportional hazards model $h_i(t)=h_0(t){\varphi }_i$, the hazard multiplier ${\varphi }_i>0$ is the relative hazard against the reference multiplier one. In a Cox proportional-hazards model it is $e^{x_i^T\beta }$. Ratios of two time-constant multipliers give constant hazard ratios; time-dependent multipliers require a corresponding extension.

A proportional hazards model assigns individual $i$ the hazard function $h_i(t)=h_0(t){\varphi }_i$, where $h_0$ is the baseline hazard for multiplier one and ${\varphi }_i>0$ is a time-constant hazard multiplier. In a Cox proportional-hazards model, ${\varphi }_i=e^{x_i^T\beta }$ for a covariate vector $x_i$ and coefficient vector $\beta$; ratios $h_i(t)/h_k(t)={\varphi }_i/{\varphi }_k$ are constant whenever the baseline is nonzero.

Assume independent individuals, independent censoring conditional on the covariates, and no tied events. Immediately before an event at $t$, condition on its observed risk set $R(t)$ and one event in a small time interval. The individual event probability is $h_0(t){\varphi }_{i}dt+o(dt)$, while the total is $h_0(t){\sum }_{k\in R(t)}{\varphi }_{k}dt+o(dt)$. Their limiting ratio is ${\varphi }_i/{\sum }_{k\in R(t)}{\varphi }_k$. Multiplication over observed events gives the Cox partial likelihoodThe unspecified baseline hazard cancels. This is the usual successive conditional event contribution defining the partial likelihood, not the full event-time likelihood conditioned simultaneously on all event times.

For the four individuals the successive event risk sets are $\{a,b,c,d\}$, $\{c,d\}$ and $\{d\}$. Individual $b$ leaves at its right censoring time and contributes no event numerator. ConsequentlyThe last event contributes one and carries no further relative-risk information.

Because $t^{\ast }>t_b>t_a$, the three possible complete event orders are $t_a<t^{\ast }<t_c<t_d$, $t_a<t_c<t^{\ast }<t_d$, and $t_a<t_c<t_d<t^{\ast }$. Write $K={\varphi }_a/({\varphi }_a+{\varphi }_b+{\varphi }_c+{\varphi }_d)$ and $V={\varphi }_b+{\varphi }_c+{\varphi }_d$. Their partial likelihoods, with the final singleton factor omitted, are respectivelyAdding first $L_2$ and $L_3$ gives $K{\varphi }_c/V$, and thereforeThusThis is Cox rank-likelihood deletion consistency: summing out the unobserved position of $b$ leaves the relative order information in the observed events. It is an algebraic marginalization of complete-order probabilities under time-constant hazard multipliers. It does not include the probability density function of the actual censoring time, establish the distribution of $t^{\ast }$ given all observed times, or justify informative censoring. Ignoring the censoring mechanism in the survival analysis still requires independent censoring given the modeled covariates.

The ordinary Nelson–Aalen estimator adds $d_j/r_j$ at event time $t_j$. Under a fitted proportional hazards model, replace the number at risk by the sum of fitted hazard multipliers:This is the Breslow estimator of the integrated baseline hazard, using fitted regression coefficients obtained by maximizing the Cox partial likelihood. The denominator arises because the total event intensity is $h_0(t){\sum }_{i\in R(t)}{\varphi }_i$; with every multiplier equal to one it reduces to the Nelson–Aalen estimator.

The estimator is right-continuous, so ${\hat{H}}_0(t_a)$ already includes the event at $t_a$. Subtracting it leaves only the events at $t_c$ and $t_d$, whose risk sets are $\{c,d\}$ and $\{d\}$. HenceThere is no increment at the censoring time $t_b$. If the lower endpoint had instead been $t_a-$, an additional $1/({\hat{\varphi }}_a+{\hat{\varphi }}_b+{\hat{\varphi }}_c+{\hat{\varphi }}_d)$ would be present. The final singleton event can increase the baseline hazard estimate even though its partial likelihood contribution is one.