Symmetry in square contingency tables: tests of hypotheses and confidence interval construction

Bowker's test, a generalization of McNemar's test, performs well under the hypothesis of symmetry, but the estimator of variance used in the test is biased when the table is asymmetric and this calls into question the test's performance in non-null situations. We seek an alternative to Bowker's test in search of methods for simultaneous inference that are valid when the hypothesis of symmetry is false. We apply multivariate normal theory to develop chi-square tests and simultaneous confidence intervals for inferences concerning symmetry in k X k contingency tables. We propose a modified Wald statistic as a competitor to Bowker's test. We also proffer quadratic estimators of confidence intervals. In large samples, the recommended test statistic rejects the null hypothesis at the stated level of significance when the null hypothesis is true and always rejects with greater power than Bowker's test. The proffered interval estimators provide good simultaneous coverage of the pairwise differences between the population proportions at the stated confidence level.