Saddlepoint approximations for linear rank tests with left-truncated, right-censored, and cross-sectional data under randomized block design

Left-truncated data arise when events are only recorded if they occur after a pre-specified time point, while right censoring occurs when the exact event time is not fully observed. Cross-sectional data refer to data collected at a single time point without follow-up. This paper proposes saddlepoint approximations (SPA) for the mid p values of four linear rank test statistics, namely TᵇLR , TᵇWLR , TᵇLRC , and TᵇWC , applied to these data types under a randomized block design (RBD). These test statistics are newly adapted to the RBD framework, and the Skovgaard SPA formula is applied to derive accurate approximations for their mid p values. The accuracy of the proposed SPA is compared against the standard normal approximation (NA) via extensive simulation studies under extreme value and logistic distributions, and illustrated through real data examples. Results demonstrate that the SPA consistently provides more accurate approximations to the mid p values compared to the NA across all simulation scenarios and real data examples.