Links Between Binary and Multi-Category Logit Item Response Models and Quasi-Symmetric Loglinear Models

Caussinus’s loglinear model of quasi symmetry has interesting connections with models for within-subject effects with repeated categorical measurement. For binary responses, Tjur (1982) showed that estimates of main effect parameters in the quasi-symmetry model are also conditional maximum likelihood estimates of item parameters for a fixed effects treatment of subject terms in the Rasch item response model. He showed they are also nonparametric estimates of item parameters for a random effects treatment of subject terms in the Rasch model. I describe some generalizations of the quasi-symmetry model that have similar connections with generalizations of the Rasch model. These include a link between an ordinal quasi-symmetry model and an adjacent-categories logit model with random effects, and a link between a multivariate quasi-symmetry model and a logit random effects model for repeated measurement of a multivariate vector of binary responses.