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

At each event time within a stratum, let $r_A,r_B$ be the group risk sets and let $d$ be the total number of observed failures. Under the null of equal hazard functions, the expected A count is $d{r}_A/(r_A+r_B)$. The log-rank statistic adds observed minus expected A counts over event times; the stratified log-rank statistic first makes these comparisons within each patient stratum, then adds across patients. Let $V$ be the sum of the corresponding null conditional variances. The standardized statistic $U/\sqrt{V}$ has an approximate standard normal distribution under the null, and its square has an approximate one-degree-of-freedom chi-squared distribution. Compare it with the appropriate quantile for a two-sided test, under the usual independent censoring and exchangeability assumptions.

No: with one pair per stratum, the statistic uses failure order but discards the size of the time difference. The paired log-rank reduction to a sign test assigns the same contribution when A fails just before B as when A fails much earlier. This deliberately avoids a model for the failure-time distances, but it does not extract every kind of evidence in the recorded times. An accelerated failure time model, for example, could use log-time differences in uncensored pairs under additional distributional assumptions.

Use the A-group observed-minus-expected convention for the stratified log-rank statistic. At the first failure both teeth are at risk, so A has one observed event and null expectation $1/2$, giving $+1/2$. At the later B failure, A is no longer at risk; the A contribution is zero. This patient's contribution is $+1/2$, favoring B. The opposite group convention reverses the sign but gives the same test.

A nonzero contribution occurs when a failure, rather than a censoring, is the earlier observation while both members remain at risk. By the paired log-rank reduction to a sign test, the two possible contributions are $+1/2$ for an earlier A failure and $-1/2$ for an earlier B failure. Under the exchangeable null and the usual independent censoring assumptions they are equally likely, conditional on being informative. HenceOnce the first member leaves the risk set, the second member's later observation adds no group-comparison information within this stratum.

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.