A Test for Symmetry in Contingency Tables

Abstract Suppose that a sample of N individuals is classified according to two arguments, A and B, in an m by m contingency table and suppose that the categories of classification A are associated in some unique way with the categories of classification B. For example, in the study in which this problem arose the categories were related to the position of plates of stickleback (Gasterosteus) and the two arguments corresponded to right and left sides; it was logical to associate identical patterns on opposite sides. Denote by Pij the probability that a randomly chosen individual belongs to the ith. row and jth column of the table. As usual, a dot indicates summation over the replaced index.