How does ordinal regression compare to quantile regression?

Question

I am familiar with ordinal regression and quantile regression at a high level, but would like a deeper understanding of the two beginning on how they differ. Can someone compare and contrast the two, perhaps with a reference(s) to an article that compares them? TIA

Answer 1

Chapter 15 of the second edition of Frank Harrell's Regression Modeling Strategies textbook and chapter 11 of the most recent version of his class notes provide succinct comparisons. Either ordinal (e.g., via the orm() function of the rms package) or quantile regression methods can model continuous outcomes, so a comparison does make sense.

Quoting from pages 360 and 361 of the textbook:

Quantile regression... makes no distributional assumptions other than continuity of $Y$, while having all the usual right hand side assumptions... Using quantile regression, we directly model the median [or any other quantile] as a function of covariates so that only the $X\beta$ structure need be correct.

A different robust semiparametric regression approach than quantile regression is the cumulative probability ordinal model...While quantile regression has no restriction in the parameters when modeling one quantile versus another, ordinal cumulative probability models assume a connection between distributions of $Y$ for different $X$. Ordinal regression even makes one less assumption than quantile regression about the distribution of $Y$ for a specific $X$: the distribution need not be continuous. (Emphasis added, to highlight a major difference from quantile regression.)

A semi-parametric ordinal regression model "contains an intercept for every unique $Y$ value less one" (legend to Table 15.1 in the textbook) and assumes a link function that provides the further association with covariates $X$ (Equation 15.2):

$$ \text{Prob}[Y \ge y_i|X]= F (\alpha_{y_i} + X\beta).$$

Typical quantile regression requires no such assumptions. There's a variant of quantile regression that restricts predictor coefficients to be the same across multiple quantiles (see footnote b on page 361), however, so the distinction isn't necessarily that sharp.

I understand that quantile regressions are typically fit via linear programming, while ordinal regression uses maximum likelihood. That's also not such a strict distinction, however, as quantile regression can be formulated as maximizing the likelihood for Laplace-distributed errors.