Abstract
The purpose of this publication is to propose a permutation test to detect the departure from symmetry in multidimensional contingency tables. The proposal is a multivariate extension of McNemar’s test. McNemar’s test could be applied to 2 × 2 contingency tables. The proposal may be also treated as a modification of Cochran’s Q test which is used for testing dependency for multivariate binary data. The form of the test statistics that allows us to detect the departure from counts symmetry in multidimensional contingency tables is presented in the article. The permutation method of observations was used to estimate the empirical distribution of the test statistics. The considerations were supplemented with examples of the use of a multivariate test for simulated and real data. The application of the proposed test allows us to detect the asymmetrical distribution of counts in multivariate contingency tables.
Highlights
McNemar’s test was proposed in 1947 (McNemar, 1947)
The use of the Cochran Q test leads to the detection of existing differences in the percentage of re‐ sponses for individual variables, and the proposed test, like the Bowker test, lets us detect asymmetry of counts in multivariate contingency tables
If nij is the count of i‐th row and j‐th column in the contingency table, the test statistic could be written in the following form (Bowker, 1948):
Summary
McNemar’s test was proposed in 1947 (McNemar, 1947) It is a statistical test used for paired nominal data. This test is applied to 2 × 2 contingency tables with a binomial outcome, with matched pairs of subjects, to determine whether there is marginal homogeneity. Bowker (1948) presented a generalisation of McNemar’s test for k (k > 2) variables. The proposed test, like the Cochran Q test, leads to testing the null hypothesis on the independ‐ ence of k (k > 2) binary variables. The use of the Cochran Q test leads to the detection of existing differences in the percentage of re‐ sponses for individual variables, and the proposed test, like the Bowker test, lets us detect asymmetry of counts in multivariate contingency tables
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