Weighted log-rank tests for censored data under Wei's urn design: saddlepoint approximation and confidence intervals

The randomization designs in clinical trials provide probabilistic basis for the statistical inference of the permutation tests. One of the widely used designs to avoid the problems of imbalance and selection bias for one of the treatments is Wei's urn design. In this article, the saddlepoint approximation is suggested to approximate the p-values of the weighted log-rank class of two-sample tests under Wei's urn design. To show the accuracy of the proposed method and to clarify its procedure, two sets of real data are analyzed, and a simulation study is conducted using different sample sizes and three different life time distributions. Through the illustrative examples and simulation study, a comparison is made between the proposed method and the traditional method, which is the normal approximation method. All of these procedures confirmed that the proposed method is more accurate and efficient than the normal approximation method in approximating the exact p-value of the considered class of tests. As a result, the nominal $$ confidence intervals for the treatment effect are determined.