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

Table of contents
- Solution b

Solution

Because

t^{*} > t_{b} > t_{a}

, the three possible complete event orders are

t_{a} < t^{*} < t_{c} < t_{d}

,

t_{a} < t_{c} < t^{*} < t_{d}

, and

t_{a} < t_{c} < t_{d} < t^{*}

. Write

K = ϕ_{a} / (ϕ_{a} + ϕ_{b} + ϕ_{c} + ϕ_{d})

and

V = ϕ_{b} + ϕ_{c} + ϕ_{d}

. Their partial likelihoods, with the final singleton factor omitted, are respectively

L_{1} = K \frac{ϕ _{b}}{V} \frac{ϕ _{c}}{ϕ _{c} + ϕ _{d}}, L_{2} = K \frac{ϕ _{c}}{V} \frac{ϕ _{b}}{ϕ _{b} + ϕ _{d}}, L_{3} = K \frac{ϕ _{c}}{V} \frac{ϕ _{d}}{ϕ _{b} + ϕ _{d}} .

Adding first

L_{2}

and

L_{3}

gives

K ϕ_{c} / V

, and therefore

L_{1} + L_{2} + L_{3} = K \frac{ϕ _{c}}{V} (\frac{ϕ _{b}}{ϕ _{c} + ϕ _{d}} + 1) = K \frac{ϕ _{c}}{ϕ _{c} + ϕ _{d}} .

Thus

L_{1} + L_{2} + L_{3} = L_{a, b, c, d} .

This 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^{*}

given all observed times, or justify informative censoring. Ignoring the censoring mechanism in the survival analysis still requires independent censoring given the modeled covariates.

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