Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 207 / 5 / b / i

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- Solution i

Solution

Let

R_{i} = {j : x_{j} \geq x_{i}}

be the risk set at an observed event time

x_{i}

. In the Cox proportional-hazards model, conditional on one member of this set failing then and on the preceding history, the probability that it is individual

i

is

\frac{e ^{β z_{i}} h _{0} ( x _{i} )}{\sum _{j \in R_{i}} e ^{β z_{j}} h _{0} ( x _{i} )} = \frac{e ^{β z_{i}}}{\sum _{j \in R_{i}} e ^{β z_{j}}} .

The baseline hazard cancels. Multiplying these conditional contributions gives the Cox partial likelihood

L_{p} (β) = i : v_{i} = 1 \prod \frac{e ^{β z_{i}}}{\sum _{j : x_{j} \geq x_{i}} e ^{β z_{j}}} .

The no-ties assumption avoids the need for a tied-event approximation. This construction assumes independent censoring conditional on the modelled covariates. The printed phrase calling

z

a scalar parameter is read as a scalar observed covariate

z_{i}

;

β

is the unknown regression parameter.

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