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

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- Solution ii

Solution

Taking logarithms of the Cox partial likelihood gives

S (β) = \sum_{i = 1}^{n} v_{i} [β z_{i} - lo g (\sum_{j : x_{j} \geq x_{i}} e^{β z_{j}})] .

Differentiate each term, using the chain rule for the logarithm:

S^{'} (β) = i = 1 \sum n v_{i} [z_{i} - \frac{\sum _{j : x_{j} \geq x_{i}} z _{j} e ^{β z_{j}}}{\sum _{j : x_{j} \geq x_{i}} e ^{β z_{j}}}] .

This score function is the sum over failures of the observed covariate minus its current risk-weighted mean. Define that mean as

\overline{z}_{β} (x_{i})

; observations with

v_{i} = 0

have no direct event term, although they occur in the relevant risk sets.

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