Regression with an ordered categorical response

A survey on Mseleni joint disease in South Africa involved the scoring of pelvic X-rays of women to measure osteoporosis. The scores were ordinal by construction and ranged from 0 to 12. It is standard practice to use ordinary regression techniques with an ordinal response that has that many categories. We give evidence for these data that the constraints on the response result in a misleading regression analysis. McCullagh's proportional-odds model is designed specifically for the regression analysis of ordinal data. We demonstrate the technique on these data, and show how it fills the gap between ordinary regression and logistic regression (for discrete data with two categories). In addition, we demonstrate non-parametric versions of these models that do not make any linearity assumptions about the regression function.