= Solution
In a <proportional hazards model>, individual $i$ has
$$
h_i(t)=h_0(t)\exp(x_i^T\beta).
$$
The 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
$$
\widehat H_0(t)=\sum_{t_j\leq t}
\frac{d_j}{\sum_{i\in R_j}\exp(x_i^T\widehat\beta)}.
$$
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.
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