The log-rank test is the most widely used nonparametric method for testing treatment differences in survival between two treatment groups due to its efficiency under the proportional hazards model. Most previous work on the log-rank test has assumed that the samples from the two treatment groups are independent. This assumption is not always true. In multi-center clinical trials, survival times of patients in the same medical center may be correlated due to factors specific to each center. For such data, we can construct both stratified and unstratified log-rank tests. These two tests turn out to have very different powers for correlated samples. An appropriate linear combination of these two tests may give a more powerful test than either of the individual test. Under a bivariate frailty model, we obtain closed-form asymptotic local alternative distributions and the correlation coefficient between these two tests. Based on these results we construct an optimal linear combination of the two test statistics to maximize the local power. Simulation studies with Hougaard's model confirm our construction. We also study the robustness of the combined test by simulations.
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