Measure of Departure from Average Cumulative Symmetry for Square Contingency Tables with Ordered Categories

Problem statement: For square contingency tables with ordered categories, we are interested in considering a structure of weak symmetry when Bowker's symmetry model does not hold and in measuring the degree of departure from weak symmetry. Approach: The present study considered the average cumulative symmetry model that has a weaker restriction than the structure of symmetry. It also gave a measure to represent the degree of departure from average cumulative symmetry. When the conditional symmetry and the cumulative linear diagonals-parameter symmetry holds, the proposed measure can measure what degree of departure from the symmetry is. Results: The proposed model and the measure were applied and analyzed (1) for the data of 4×4 contingency table of unaided distance vision of 7477 women aged 30-39 employed in Royal Ordnance factories in Britain from 1943-1946 and (2) the data of 4×4 contingency table of the 59 matched pairs using from dose levels of conjugated oestrogen. Conclusion: The proposed model is useful when the symmetry model does not hold and the proposed measure is useful for comparing the degree of departure from the weak symmetry model in several tables. Especially the proposed measure is useful to measure the degree of departure from symmetry when the conditional symmetry (the cumulative diagonals-parameter symmetry) model holds.