Simplifying the calculation of the P-value for Barnard's test and its derivatives

Unconditional non-asymptotic methods for comparing two independent binomial proportions have the drawback that they take a rather long time to compute. This problem is especially acute in the most powerful version of the method (Barnard, 1947). Thus, despite being the version which originated the method, it has hardly ever been used. This paper presents various properties which allow the computation time to be drastically reduced, thus enabling one to use not only the more traditional and simple versions given by McDonald et al. (1977) and Garside and Mack (1967), but also the more complex original version of Barnard (1947).