Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 41 / 4 / d / Solution

Only sets 4, 5, 7 and 8 have an earlier failure with both members at risk. Sets 4 and 8 have B fail first and favor A, giving $6+3=9$ informative pairs. Sets 5 and 7 have A fail first and favor B, giving $18+9=27$. Both-censored pairs contribute zero; in sets 3 and 6 the earlier observation is censoring, so the later failure occurs with no remaining comparator and also contributes zero.

There are thus 36 informative independent strata. The stratified log-rank statistic, its null variance, and its standardized value areEquivalently the squared statistic is nine. The normal approximation gives a two-sided p-value $2[1-\Phi (3)]<0.0027$ using the stated bound. The exact conditional sign test instead gives $2{\sum }_{k=0}^9\left(\genfrac{}{}{0ex}{}{36}{k}\right)2^{-36}\approx 0.00393$; the slight difference is discreteness rather than a different direction of effect. There is strong evidence favoring Method B, which is associated with the later failure in three quarters of the informative pairs, under the test assumptions.