Some Log-Linear and Log-Nonlinear Models for Ordinal Scales With Midpoints, With an Application to Public Opinion Data

Michael Sobel *

Ordinal response scales with a middle category are widely used in public opinion studies, psychology, medicine, computed tomography and other fields. The usual models in the statistical literature for ordinal response variables treat the case where the scale has a natural middle category no differently from the case where the scale does not have a middle category. This paper proposes new models for the analysis of ordinal response scales with middle categories, applying these to data collected in 1993-1994 on American opinion toward the balance between environmental quality and economic prosperity. Some of the models should also be useful when the scale does not have a natural middle category. The models are easily used to address issues of concern in empirical work—for example, stochastic ordering among covariate classes and asymmetry about the middle category. Log-linear models are considered in Section 2. The relationship between the normal distribution and a quadratic log-linear model with known scores, discussed in this section, is the basis for Section 3, which considers a log-nonlinear model with unknown scores estimated from the data. Section 4 shows how generalized log-linear and generalized log-nonlinear models can be used to simultaneously study whether the response is below, at, or above the midpoint, and the conditional distribution of responses above (below) the midpoint. These models are also useful when the response scale is viewed as nested and/or the response process is sequential.

* University of Arizona