Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 207 / 5

In a proportional hazards model, individual $i$ hasThe exponential term is the hazard multiplier relative to the baseline hazard $h_0$. With $h_0$ unspecified, the Cox partial likelihood multiplies, over event times, the failing subject's multiplier divided by the sum of multipliers in the current risk set. After estimating $\beta$, the Breslow estimator is

A Stratified Cox model uses a separate baseline hazard $h_{0s}$ for each stratum but a common $\beta$. Its partial likelihood is the product of within-stratum partial likelihoods, and a separate integrated baseline hazard is estimated in each stratum.

In a matched pair, the only informative comparison occurs while both members remain at risk. If the exposed member fails first, the pair contributes $e^{\beta }/(1+e^{\beta })$; if the unexposed member fails first, it contributes $1/(1+e^{\beta })$. A censoring as the first recorded time gives no informative failure comparison, and any later one-person risk set contributes one. With only two observations per stratum, each stratum supplies almost no information about its arbitrary baseline hazard, so a useful common integrated baseline-hazard estimate is generally unavailable.

Among the 12 pairs with control time first, the eight in which that first time is an event contribute $(1+\beta {)}^{-1}$ each. Among the eight pairs with treated time first, the four in which that first time is an event contribute $\beta /(1+\beta )$ each. Later events in one-eye risk sets contribute one. HenceDifferentiating the log likelihood gives $4/\beta -12/(1+\beta )=0$, and therefore

Without randomization, treatment side can be associated with prognosis. Always treating the left eye confounds treatment with systematic left-right differences; choosing the worse eye creates severe confounding by indication, baseline imbalance, and possible regression toward the mean. The within-patient comparison then no longer identifies a treatment effect without stronger adjustment assumptions.