COMPUTATIONAL STATISTICS & DATA ANALYSIS ELSEVIER Computational Statistics & Data Analysis 20 (1995) 643-656 Refined approximations to permutation tests for multivariate inference Frrdrrique Kazi-Aoual a'*, Simon Hitier b, Robert Sabatier c, Jean-Dominique Lebreton 2 a Unitd de Biom~trie, INRA/ENSA.M/UM II, 9 Place Viala, 34060 Montpellier cedex 1, France b Centre d'Ecologie Fonctionnelle et Evolutive, CNRS, BP 5051, 34 033 Montpellier cedex 1, France cLaboratoire de Physique Mol~culaire et Structurale, Facultb de Pharmacie, Universit~ Montpellier L 34 060 Montpellier cedex 1, France Received 1 April 1994; Revised 1 September 1994 Abstract Various authors have proposed approximations to permutation tests of independence between two data tables. We develop approximations based on explicit expressions of the first three moments of three different test statistics under the permutation distribution. The rejection level is then determined by using a Pearson-type III distribution matching the values of the first three moments. We present three examples in which the relative merits of the test statistics are examined and the results of the approximation procedure are compared with explicit permutation tests. Keywords: Permutation tests; Exact permutation tests; Monte-Carlo methods; Multivariate infer- ence; Randomization tests 1. Introduction Multivariate inference procedures, such as the one-way multivariate analysis of variance (MANOVA1; e.g., Krishnaiah, 1980), have been developed under the assumption of Gaussian distributions (e.g., Giri, 1977). This framework is fre- quently restrictive in biological applications. To avoid it, many authors have used explicit permutation tests, made possible by the availability of fast computers (see reviews by Edington, 1987; Manly, 1991). In this approach, to test for the indepen- dence of a table Y and a table X, one gets the observed distribution of a test statistic * Corresponding author. 0167-9473/95/$9.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0167-9473(94)00064-6