Weighted log-rank tests for left-truncated data: Saddlepoint p-values and confidence intervals

Left truncation is a feature of many clinical trials and survival analysis in which lifetimes are observed conditionally on being greater than random truncation times. The subjects of such studies are exposed to an event prior to the event of interest and both times are recorded for each subject. In other words, left-truncated data arise when we only observe these subjects whose failure event times are limited below by truncation event times. For such type of data, the most commonly used class of two-sample tests is the weighted log-rank class. The aim of this article is to use saddlepoint approximation techniques to calculate p-values from the null permutation distributions of tests from the weighted log-rank class. The saddlepoint p-values are almost closer to the exact mid-p-values. The weighted log-rank tests are inverted to determine nominal 95% confidence intervals for the treatment effect in the presence of left-truncated data. Such analytical inversions lead to permutation confidence intervals which are easily computed and virtually identical with the exact intervals that would usually require huge amounts of simulations.