TESTS OF MARGINAL SYMMETRY AND QUASI-SYMMETRY IN 2-DIMENSIONAL AND 3-DIMENSIONAL CONTINGENCY-TABLES

When the two criteria of classificaion in a square contingency table are commensurable, it is of interest to investigate whether there are symmetric patterns in the data. In case the pattern is not completely symmetric, one likes to check whether, at least, the two sets of marginal totals have the same distribution (marginal symmetry) or the measures of association are symmetric (quasi-symmetry). This paper considers appropriate specifications of the hypotheses of different patterns of symmetry, and develops large-sample Wald statistics as efficient chi-square test criteria. This development is then extended to three-dimensional cubic contingency tables arising from three commensurable classifications. Both the two and three-dimensional cases are illustrated with numerical examples. It is pointed out that explicit test criteria are available in each and no iteration routine is needed to compute such Wald statistics, while iteration routines are needed to compute alternative equally efficient large-sample criteria such as those based on likelihood ratios. Another advantage in using Wald statistics is that either the marginal symmetry or the quasi-symmetry hypothesis can be tested on its own without making any extraneous assumption regarding the other.