The Cox proportional-hazards model iswhere is an unspecified baseline hazard and indicates treatment. With no tied events, the Cox partial likelihood isEquality of the two hazard functions is , testable with a likelihood-ratio, score, or Wald test from this likelihood.
Use each matched pair as a separate stratum with its own baseline hazard and fit one common treatment coefficient . The Stratified Cox model partial likelihood multiplies the within-pair risk-set contributions, eliminating all pair-specific baselines. Test by the corresponding conditional score, likelihood-ratio, or Wald test.
A pair is informative only when its first observed exit is an event while both members remain at risk; later events after one member has left contribute a factor one. The uninformative pairs are therefore pair 5, with tied events at , and pairs 10, 11, and 13, whose earlier observations are censorings: , , and . At both are in the risk set immediately before time 140, so pair 12 is informative.
Among the nine informative pairs, the active patient fails first once and the placebo patient fails first eight times. The conditional partial likelihood is proportional tosoUnder , the number of active-first failures is . The exact two-sided sign test hasso the data reject at the 5% level and favor lower hazard under active treatment.
Matching controls strong prognostic differences between patients and can greatly improve precision by making treatment comparisons within homogeneous pairs. It also protects against confounding when treatment assignment is not perfectly randomized globally.
Its disadvantage is loss of information: pairs with censoring before a comparison or tied outcomes contribute little or nothing, and the analysis cannot estimate the effects of pair-level matching variables. Poor matches and the logistics of paired enrollment can also reduce efficiency relative to a well-adjusted unpaired analysis.
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