Regression Models with Ordinal Variables

Abstract

Most discussions of ordinal variables in the sociological literature debate the suitability of linear regression and structural equation methods when some variables are ordinal. Largely ignored in these discussions are methods for ordinal variables that are natural extensions of probit and logit models for dichotomous variables. If ordinal variables are discrete realizations of unmeasured continuous variables, these methods allow one to include ordinal dependent and independent variables into structural equation models in a way that (I) explicitly recognizes their ordinality, (2) avoids arbitrary assumptions about their scale, and (3) allows for analysis of continuous, dichotomous, and ordinal variables within a common statistical framework. These models rely on assumed probability distributions of the continuous variables that underly the observed ordinal variables, but these assumptions are testable. The models can be estimated using a number of commonly used statistical programs. As is illustrated by an empirical example, ordered probit and logit models, like their dichotomous counterparts, take account of the ceiling andfloor restrictions on models that include ordinal variables, whereas the linear regression model does not. Empirical social research has benefited during the past two decades from the application of structural equation models for statistical analysis and causal interpretation of multivariate relationships (e.g., Goldberger and Duncan, 1973; Bielby and Hauser, 1977). Structural equation methods have mainly been applied to problems in which variables are measured on a continuous scale, a reflection of the availability of the theories of multivariate analysis and general linear models for continuous variables. A recurring methodological issue has been how to treat variables measured on an ordinal scale when multiple regression and structural equation methods would otherwise be appropriate tools. Many articles have appeared in this journal (e.g., Bollen and Barb,