Ordinal Regression Models for Continuous Scales

Abstract

Ordinal regression analysis is a convenient tool for analyzing ordinal response variables in the presence of covariates. In this paper we extend this methodology to the case of continuous self-rating scales such as the Visual Analog Scale (VAS) used in pain assessment, or the Linear Analog Self-Assessment (LASA) scales in quality of life studies. These scales measure subjects' perception of an intangible quantity, and cannot be handled as ratio variables because of their inherent nonlinearity. We express the likelihood in terms of a function connecting the scale with an underlying continuous latent variable and approximate this function either parametrically or non-parametrically. Then a general semi-parametric regression framework for continuous scales is developed. Two data sets have been analyzed to compare our method to the standard discrete ordinal regression model, and the parametric to the non-parametric versions of the model. The first data set uses VAS data from a study on the efficacy of low-level laser therapy in the treatment of chronic neck pain; the second comes from a study on chemotherapy treatments in advanced breast cancer and looks at the impact of different drugs on patients' quality of life. The continuous formulation of the ordinal regression model has the advantage of no loss of precision due to categorization of the scores and no arbitrary choice of the number and boundaries of categories. The semi-parametric form of the model makes it a flexible method for analysis of continuous ordinal scales.