Weighted log-rank tests for left-truncated data under Wei's urn design: saddlepoint p-values and confidence intervals

Left-truncated data are constructed from the time of an initial event and the time of the event of interest. That is to say, each individual represented by such data is exposed to an event prior to the event of interest, and both times are recorded. Weighted log-rank testing is commonly used for such data. In this paper, a saddlepoint approximation method is provided for computing p-values of the permutation distribution of tests from the weighted log-rank testing in the presence of left-truncated data and Wei's urn design. A simulation study is used to assess the efficiency of the saddlepoint approximation. The accuracy of the saddlepoint approximation in comparison to the normal approximation enables us to compute accurate confidence intervals for the treatment effect.