Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 34 / 5

Let $R(t^{\ast })$ be the risk set immediately before $t^{\ast }$, conditional on the preceding history. Only its members can experience the next event. In a short interval $[t^{\ast },t^{\ast }+dt)$, the conditional probability that subject $j$ fails is $h_0(t^{\ast })e^{\beta z_j}dt+o(dt)$, whereas the probability of one event from the risk set is $h_0(t^{\ast }){\sum }_{i\in R(t^{\ast })}{e}^{\beta z_i}dt+o(dt)$. Dividing and taking the small-interval limit givesIt is zero outside the risk set. The common baseline hazard cancels. This is the conditional event-label probability underlying the Cox partial likelihood. Conditioning at an exact continuous event time is understood by this limiting conditional-intensity argument, rather than as division by the zero unconditional probability of an event at a fixed time. We use the standard simple event process with no simultaneous events and independent censoring.

Average the covariate over its possible event subjects using the conditional probabilities from part (a). Define the risk-weighted sumswith $z_i^0=1$. ThenThis is the mean covariate of the next event subject, conditional on the current risk set and an event. It is generally not the unweighted average over the original sample. For $\beta =0$ it reduces to the unweighted current risk-set mean.

The difference $s(t^{\ast },\beta )$ is the observed event subject's covariate minus its conditional expected value under the working coefficient. It is the Schoenfeld function; at a fitted coefficient it becomes a Schoenfeld residual. If the conditional event-label distribution is generated by coefficient $\beta$, part (b) givesHence its unconditional expectation is zero whenever it is integrable. Centering is at the true coefficient: data generated under ${\beta }_0$ do not generally make $s(t^{\ast },\beta )$ mean zero at an arbitrary trial value $\beta \ne {\beta }_0$.

The same quantity is the score function of one conditional event factor, sinceThis connects its conditional centering to the usual zero-mean likelihood-score identity.

For the distinct observed event times $a_k$, let $i_k=\pi (a_k)$ and let $R_k$ be the risk set immediately before that event. The Cox partial likelihood isThe different risk sets already incorporate each event and any intervening censoring. Differentiating this log-likelihood givesTherefore a finite interior maximum-likelihood estimate from the Cox partial likelihood satisfies the first-order conditionMoreoverEach summand is the conditional event-covariate variance, so the partial likelihood is log-concave. If some informative risk set has unequal covariates, the curvature is strictly negative and an existing finite root is the unique maximizer.

The finite-interior qualification is necessary. With two at-risk subjects having covariates zero and one, suppose the subject with covariate one is the sole observed event and the other is subsequently censored. Then $L_p(\beta )=e^{\beta }/(1+e^{\beta })$ increases strictly and has its supremum only as $\beta \to +\infty$; the score $1/(1+e^{\beta })$ has no finite zero. If every event risk set has identical covariates, the coefficient is instead unidentifiable and the score is identically zero. The displayed fitted-score equation applies to the regular case in which the proportional-hazards estimate is finite.