 Solution

The Log-rank test compares observed failures in each group with the numbers expected under equal hazards, conditioning at every event time on the risk set and total failures. Its score sums $O_{1j}-Y_{1j}{d}_j/Y_j$ and is standardized by its null hypergeometric variance. It is most powerful for approximately proportional hazards. Strongly crossing survival curves can produce large positive and negative contributions that cancel, so very different distributions may yield a weak log-rank statistic.