Graphical regression models for polytomous variables

When modeling the relationship between two nominal categorical variables, it is often desirable to include covariates to understand how individuals differ in their response behavior. Typically, however, not all the relevant covariates are available, with the result that the measured variables cannot fully account for the associations between the nominal variables. Under the assumption that the observed and unobserved variables follow a homogeneous conditional Gaussian distribution, this paper proposesRC(M) regression models to decompose the residual associations between the polytomous variables. Based on Goodman's (1979, 1985)RC(M) association model, a distinctive feature ofRC(M) regression models is that they facilitate the joint estimation of effects due to manifest and omitted (continuous) variables without requiring numerical integration. TheRC(M) regression models are illustrated using data from the High School and Beyond study (Tatsuoka & Lohnes, 1988). This article was accepted for publication, when Willem J. Heiser was the Editor ofPsychometrika. This research was supported by grants from the National Science Foundation (#SBR96-17510 and #SBR94-09531) and the Bureau of Educational Research at the University of Illinois. We thank Jee-Seon Kim for comments and computational assistance.