From: On hazard ratio estimators by proportional hazards models in matched-pair cohort studies
Type | Number of pairs | Observed data in the pair | Observed time | Y _{ k1} | Y _{ k0} | L _{ k }(β) |
---|---|---|---|---|---|---|
1 | n _{1} | Exposed gets event first, followed by unexposed event | X _{ k1} < X _{ k0} | 1 | 1 | \(\frac{{{\text{e}}^{\beta } }}{{1 + {\text{e}}^{\beta } }}\) |
2 | n _{2} | Unexposed gets event first, followed by exposed event | X _{ k1} > X _{ k0} | 1 | 1 | \(\frac{1}{{1 + {\text{e}}^{\beta } }}\) |
3 | n _{3} | Exposed gets event first, followed by unexposed censored | X _{ k1} < X _{ k0} | 1 | 0 | \(\frac{{{\text{e}}^{\beta } }}{{1 + {\text{e}}^{\beta } }}\) |
4 | n _{4} | Unexposed gets event first, followed by exposed censored | X _{ k1} > X _{ k0} | 0 | 1 | \(\frac{1}{{1 + {\text{e}}^{\beta } }}\) |
5 | n _{5} | Exposed is censored first, followed by unexposed event | X _{ k1} < X _{ k0} | 0 | 1 | 1 |
6 | n _{6} | Unexposed is censored first, followed by exposed event | X _{ k1} > X _{ k0} | 1 | 0 | 1 |
7 | n _{7} | Exposed is censored first, followed by unexposed censored | X _{ k1} < X _{ k0} | 0 | 0 | 1 |
8 | n _{8} | Unexposed is censored first, followed by exposed censored | X _{ k1} > X _{ k0} | 0 | 0 | 1 |
9 | n _{9} | Exposed and unexposed are censored simultaneously | X _{ k1} = X _{ k0} = τ | 0 | 0 | 1 |