Accurate approximations to the distribution of a statistic testing symmetry in contingency tables

Abstract

This manuscript examines this task of approximating significance levels for a test of symmetry in square contingency tables. The null sampling distribution of this test statistic is the same as that of the sum of squared independent centered binomial random variables, weighted by their separate sample size; each of these variables may be taken to have success probability half. This manuscript applies an existing asymptotic correction to the standard chi-squared approximation to the distribution of the quadratic form of a random vector confined to a multivariate lattice, when the quadratic form is formed from the inverse variance matrix of the random vector. This manuscript also investigates non-asymptotic corrections to approximations to this distribution, when the separate binomial sample sizes are small.