Tests of symmetry with one-sided alternatives in three-way contingency tables

For a two-dimensional contingency table of probabilities, the concept of symmetry around the main diagonal is well defined. Statistical hypothesis test of symmetry versus positive bias have also been explored. For tables of higher (three or more) dimensions, however, different concepts of symmetry are available. In this study, we consider statistical inference procedures of symmetry in partial tables versus various biases in three-dimensional tables. We find the maximum likelihood estimates of the cell probabilities and the asymptotic distribution of the likelihood ratio test statistic in each case. Simulation studies are used to investigate the sizes and powers of the tests. The methodologies developed are applied on real data sets.