Communications in Statistics—Theory and Methods, 41: 3162–3178, 2012 Copyright © Taylor & Francis Group, LLC ISSN: 0361-0926 print/1532-415X online DOI: 10.1080/03610926.2011.579380 Construction and Nonparametric Testing of Orthogonal Arrays through Algebraic Strata and Inequivalent Permutation Matrices ROSA ARBORETTI GIANCRISTOFARO 1 , ROBERTO FONTANA 2 , AND SUSANNA RAGAZZI 1 1 Department of Land, Environment, Agriculture and Forestry, University of Padova, Legnaro, Italy 2 Department of Mathematics, Politecnico di Torino, Torino, Italy In this article we use a new methodology, based on algebraic strata, to generate the class of all the orthogonal arrays of given size and strength. From this class we extract all the non isomorphic orthogonal arrays. Then, using all these non isomorphic orthogonal arrays, we suggest a method based on the inequivalent matrices permutations testing procedures Basso et al. (2004) in order to obtain separate permutation tests for the effects in unreplicated mixed level fractional factorial designs. In order to validate the proposed method we perform a Monte Carlo simulation study and find out that the permutation tests appear to be a valid solution for testing effects, in particular when the usual normality assumptions cannot be justified. Keywords Hilbert basis; Non isomorphic orthogonal arrays; Permutation tests; Polynomial counting functions; Unreplicated designs. Mathematics Subject Classification Primary 62K15; Secondary 62G10. 1. Introduction Analysis of unreplicated factorial and fractional factorial designs has been an extensively researched subject. This article develops separate permutation tests for the effects in three-level multifactorial designs, based on the concepts of non isomorphic orthogonal arrays and inequivalent matrices permutations (IMP) testing procedures. To avoid misunderstanding it should be pointed out that our approach can be adopted not only for unreplicated designs but also when some replications are available. We show our methodology on orthogonal arrays with 2 or 3 symbols Received February 19, 2011; Accepted April 5, 2011 Address correspondence to Susanna Ragazzi, Department of Land, Environment, Agriculture and Forestry, University of Padova, Viale dell’Università 16, Legnaro (PD) 35020, Italy; E-mail: susanna.ragazzi@unipd.it 3162