# Ordinal
regression analysis for continuous scales

Ordinal regression analysis is a convenient tool for analyzing
ordinal response variables in the presence of covariates. 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. Subjects are
typically given a linear scale of 100 mm and asked to put a mark where
they perceive themselves. These scales measure subjects’ perception of
an intangible quantity, and cannot be handled as ratio variables because
of their inherent nonlinearity. Instead we treat them as ordinal
variables, measured on a continuous scale. We express the likelihood in
terms of a function (the “g function”) connecting the scale with an
underlying continuous latent variable. In the current version the g
function is expressed with monotone increasing I-splines (Ramsey 1988).
The link function is the inverse of the CDF of the assumed underlying
distribution of the latent variable. Currently the logit link, which
corresponds to a standard logistic distribution, is implemented. (This
implies a proportional odds model.) The likelihood is maximized using
the MI algorithm (Ma, 2010). Fixed- and mixed-effects models are
implemented in the function *ocm*.

### References

Manuguerra M, Heller GZ (2010). Ordinal Regression Models for
Continuous Scales, *The International Journal of Biostatistics*:
6(1), Article 14.

Heller, GZ, Manuguerra M, Chow R (2016). How to analyze the
Visual Analogue Scale: Myths, truths and clinical relevance,
*Scandinavian Journal of Pain*, Volume 13, 67 - 75

Ma, J. (2010). Positively Constrained Multiplicative Iterative
Algorithm for Maximum Penalized Likelihood Tomographic Reconstruction,
*Nuclear Science* 57 (1): 181-92.

Ramsay, J. O. (1988). Monotone regression splines in action.
*Statistical science*, 425-441.