International Journal of Mathematics and Mathematical Sciences
Volume 2010 (2010), Article ID 769780, 10 pages
Research Article

Invalid Permutation Tests

Department of Family & Community Medicine, University of Arizona, Tucson, AZ 85724, USA

Received 25 December 2009; Accepted 15 March 2010

Academic Editor: Andrei Volodin

Copyright © 2010 Mikel Aickin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Permutation tests are often presented in a rather casual manner, in both introductory and advanced statistics textbooks. The appeal of the cleverness of the procedure seems to replace the need for a rigorous argument that it produces valid hypothesis tests. The consequence of this educational failing has been a widespread belief in a “permutation principle”, which is supposed invariably to give tests that are valid by construction, under an absolute minimum of statistical assumptions. Several lines of argument are presented here to show that the permutation principle itself can be invalid, concentrating on the Fisher-Pitman permutation test for two means. A simple counterfactual example illustrates the general problem, and a slightly more elaborate counterfactual argument is used to explain why the main mathematical proof of the validity of permutation tests is mistaken. Two modifications of the permutation test are suggested to be valid in a very modest simulation. In instances where simulation software is readily available, investigating the validity of a specific permutation test can be done easily, requiring only a minimum understanding of statistical technicalities.