 Paired log-rank reduction to a sign test

With one A and one B observation per stratum and no ties, the stratified log-rank statistic receives a nonzero contribution only when the earlier observation is a failure while both remain at risk. It is $+1/2$ if A fails first and $-1/2$ if B fails first. After the first observation, only one member remains and every further observed-minus-expected contribution is zero. Under an exchangeable null with independent censoring, conditional on being informative the two signs have equal probability, with mean zero and variance $1/4$. Thus standardization is the normal approximation to a sign test on informative pairs. Failure-time distances do not affect this statistic.