<p>Left-truncated data arise when events of interest are only recorded if they occur after a pre-specified time point. Events occurring before this point are excluded from the dataset, resulting in left truncation. Right censoring occurs when the exact timing of the event of interest is not fully observed, but the event is known to occur after a certain point. Cross-sectional data without follow-up refer to data collected at a single time point, without tracking individuals or events over a period. Linear rank tests are widely used for two-sample comparisons in such types of data. This paper compares the accuracy of saddlepoint and normal approximation methods in approximating the mid <i>p</i>-value of linear rank tests applied to left-truncated, right-censored, and cross-sectional data under a random allocation design. The comparison is demonstrated using real data examples and multiple simulation studies.</p>