Semiparametric Ordinal Regression Models for Continuous Y: Are Linear Models Obsolete?

Semiparametric ordinal regression models such as the proportional odds and proportional hazards models have long been used to analyze discrete or censored data. Just as the Cox model generalizes the log-rank test, the most commonly used ordinal model, the proportional odds model, generalizes the Wilcoxon/Kruskal-Wallis tests. Because semiparametric models do not assume a distribution for Y given X, they also allow for major discontinuities in the distribution of Y (e.g., clumping at zero). These models are more robust than parametric models, and free the analyst from having to decide on a transformation for Y. Ordinal models are frequently better choices than Poisson or negative binomial models when there is major zero-inflation.

A hindrance in analyzing continuous Y is that there may be thousands of unique Y values, necessitating inclusion of thousands of intercepts in the model. Fortunately, the portion of the information matrix corresponding to the intercept parameters is tri-band diagonal, so inverting the matrix is almost instant once the sparse nature of the matrix is capitalized upon. The new orm function in the R rms package uses this approach, just as SAS JMP did more than 20 years ago (Sall, 1991).

This talk includes a case study in development of a screening model to predict current HbA1c using NHANES data, and covers model diagnostics and some statistical and computing issues. The case study demonstrates how ordinal models allow the analyst to easily obtain an array of estimates from quantiles to exceedance probabilities to moments. Ordinal regression is not only a possible replacement for linear models but is more efficient at estimating quantiles than quantile regression in some cases, and get around quantile regression's assumption that Y is completely continuous.

The talk also includes a diversion about evils of dichotomization in medical diagnosis.

The full case study may be found at http://biostat.mc.vanderbilt.edu/CourseBios330 under Handouts.