Resolving the tail instability in weighted log-rank statistics for clustered survival data

In this note, we consider weighted log-rank statistics applied to clustered survival data with variable cluster sizes and arbitrary treatment assignments within clusters. Specifically, we verify that the contribution over the time interval for which the risk set proportion is arbitrarily small (the so-called “tail instability”) is asymptotically negligible. These results were claimed but not proven by Gangnon and Kosorok [2004. Sample-size formula for clustered survival data using weighted log-rank statistics. Biometrika 91, 263-275.] who developed sample size formulas in this context. The main difficulty is that standard martingale methods cannot be used on account of the dependencies within clusters, and new methods are required.