Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 207 / 5

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.