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The Scientific World Journal
Volume 2014 (2014), Article ID 698296, 23 pages
Research Article

Testing Spatial Symmetry Using Contingency Tables Based on Nearest Neighbor Relations

Department of Mathematics, Koç University, Sarıyer, 34450 Istanbul, Turkey

Received 23 August 2013; Accepted 22 October 2013; Published 19 January 2014

Academic Editors: A. Barra, S. Casado, and J. Pacheco

Copyright © 2014 Elvan Ceyhan. 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.


We consider two types of spatial symmetry, namely, symmetry in the mixed or shared nearest neighbor (NN) structures. We use Pielou’s and Dixon’s symmetry tests which are defined using contingency tables based on the NN relationships between the data points. We generalize these tests to multiple classes and demonstrate that both the asymptotic and exact versions of Pielou’s first type of symmetry test are extremely conservative in rejecting symmetry in the mixed NN structure and hence should be avoided or only the Monte Carlo randomized version should be used. Under RL, we derive the asymptotic distribution for Dixon’s symmetry test and also observe that the usual independence test seems to be appropriate for Pielou’s second type of test. Moreover, we apply variants of Fisher’s exact test on the shared NN contingency table for Pielou’s second test and determine the most appropriate version for our setting. We also consider pairwise and one-versus-rest type tests in post hoc analysis after a significant overall symmetry test. We investigate the asymptotic properties of the tests, prove their consistency under appropriate null hypotheses, and investigate finite sample performance of them by extensive Monte Carlo simulations. The methods are illustrated on a real-life ecological data set.