6.3 Weighted log-rank tests

The log-rank test gives every event time equal weight. A weighted log-rank test uses \(\sum _jw_j(d_{1j} - E_{1j})\) with variance \(\sum _jw_j^2V_j\):

\(w_j = 1\): the log-rank test, best for proportional hazards;
\(w_j = n_j\): the Gehan–Wilcoxon test, which weights early times, when many are at risk;
\(w_j = \hat S(t_{(j)})\): the Peto–Peto test, also emphasising early differences but less dependent on the censoring pattern (R’s rho = 1);
\(w_j = \hat S(t_{(j)})^\rho (1 - \hat S(t_{(j)}))^\gamma \): the Fleming–Harrington family, which can emphasise early (\(\rho > 0\)) or late (\(\gamma > 0\)) differences.

The choice must be made before seeing the data, from what effect is expected; trying several and reporting the smallest \(p\)-value is not legitimate. When the curves cross, as for the two regimens in veteran (Figure 6), early and late differences of opposite sign cancel in the log-rank statistic, and no single test is satisfactory; the right response is to describe the curves, survival at fixed times, or a restricted mean, rather than to rely on one statistic.

log- rank χ2 = 0.008
dˆS1020304050600.0.0.1sttePay(t000000257astets)55nd:oa 0–rd.3P:8e 0ato.5t χ5902atda=9y00s,.da08y.17s,10a.0t73a60t360 — Figure 6: Kaplan–Meier curves for the two chemotherapy regimens of the veteran trial. The standard regimen is better for about the first five months and the test regimen afterwards. The early and late differences cancel in the log-rank test; the Peto–Peto test, weighting early deaths, gives a larger statistic but still no significant difference.

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