Saddlepoint p-values for the class of bivariate two-sample tests under generalized randomized block design

Bivariate data arise in several research fields, such as clinical trials and reliability studies. In clinical trials, patients are distributed into treatment groups using randomization designs, which prevent selection bias and incidental bias. Generalized randomized block design is among clinical trials' most famous and widely used randomization designs. This article investigates the use of the saddlepoint method to approximate the underlying permutation distributions of bivariate two-sample tests under a generalized randomized block design. Additionally, the saddlepoint method is utilized to approximate the tail probability of these tests. Through comprehensive simulation studies, the accuracy of this approximation method is thoroughly evaluated, revealing a significant improvement in precision compared to the asymptotic normal approximation.