Analysis of paired ordered categorical data in a factorial design

In clinical trials and behavioral sciences, there exist situations where paired responses are obtained from each subject on an ordinal scale. Existing methods for analyzing such data in a factorial design are reviewed and new methods are developed with a special emphasis on pre- and post-treatment responses. The distribution of a square table is decomposed into successively independent pairs of discordant vectors, and assuming a logistic model, a statistics is computed to measure the shift in the marginal distributions. The approach is similar to McCullagh's approach (1). Two other criteria are proposed, one based on a Lehmann alternative used for comparing two distribution functions and the other based on a proportional odds model. These criteria are applied to the marginal distributions of a square table. For each case, a statistic measuring lack of marginal homogeneity and its variance are computed for each independent square table of a factorial design. Given such statistics, one can estimate a set of linear contrasts and compute its dispersion matrix for making inference. A numerical example is given.