Using a GLM to Decompose the Symmetry Model in Square Contingency Tables with Ordered Categories

In this paper, we are employing the generalized linear model (GLM) in the form lij=Xλ to decompose the symmetry model into the class of models discussed in Tomizawa (1992). In this formulation, the random component would be the observed counts fij with an underlying Poisson distribution. This approach utilizes the non-standard log-linear model and our focus in this paper therefore relates to models that are decompositions of the complete symmetry model. That is, models that are implied by the symmetry models. We develop factor and regression variables required for the implementation of these models in SAS PROC GENMOD and SPSS PROC GENLOG. We apply this methodology to analyse the three 4×4 contingency table, one of which is the Japanese Unaided distance vision data. Results obtained in this study are consistent with those from the numerous literature on the subject. We further extend our applications to the 6×6 Brazilian social mobility data. We found that both the quasi linear diagonal-parameters symmetry (QLDPS) and the quasi 2-ratios parameter symmetry (Q2RPS) models fit the Brazilian data very well. Parsimonious models being the QLDPS and the quasi-conditional symmetry (QCS) models. The SAS and SPSS programs for implementing the models discussed in this paper are presented in Appendices A, B and C.