Exact Permutation Algorithm for Paired Observations: A General and Efficient Version

Abstract

For the better part of a century, methods have been illustrated for the enumeration of all possible permutations of cases from which an exact characterization of the likelihood of obtaining results as or more extreme as that observed may be determined without having to rely on parametric assumptions or schemes that may be only asymptotically correct. The challenge is the computational intensity associated with these methods, which is largely overcome with the wide availability of inexpensive, powerful computational resources. The algorithm presented here is given in two versions, one a general form that can be adapted to a wide variety of permutation tests and a specialized one that is efficient for the exact analog to the dependent-t test. The application is illustrated using Charles Darwin’s Zea mays data, which presents a modest task of accounting for 2¹⁵ = 32,768 permutations. The resultant algorithm improves on that of Odiase and Ogbonmwan and is presented in syntax that may be run in R, the open source statistical package."