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A Distinguished Speaker Series Seminar by Frank Harrell, Ph.D.

Ordinal regression is a kind of semiparametric regression model that is being used with increasing frequency. The most popular ordinal regression models are the proportional odds model (Walker and Duncan, 1967) and the proportional hazards model (Cox, 1972). Ordinal regression is very much preferred for a discrete ordered response variable (Y) or when there are floor or ceiling effects in a response scale.

Less well-known is that ordinal regression has excellent performance for continuous Y, which makes it a direct competitor to ordinary linear regression. In that setting, a major advantage is that the regression coefficients from fitting an ordinal model do not change if Y is monotonically transformed (logged, etc.). As a result, a major burden in determining how to transform Y is lifted from the analyst.

Other advantages include robustness to Y-outliers and automatically handling detection limits for Y. Ordinal regression allows the analyst to estimate effect ratios, probabilities of chosen Y levels, the mean of Y (when Y is interval-scaled), and quantiles of Y (when Y is continuous).

Just as the log-rank test is a special case of the Cox model, the Wilcoxon and Kruskal-Wallis nonparametric rank tests are special cases of the proportional odds (PO) model. Hence the two rank tests make the PO assumption just as the log-rank test makes the proportional hazards assumption. This will be explained in detail, along with how to convert between an odds ratio and a Wilcoxon statistic.

We will also discuss how to assess the impact of making the PO assumption, and we’ll show that the PO assumption is not needed to successfully use the PO model to infer which treatment group fares better. However, PO is needed to get accurate predicted probabilities of specific outcome categories. We will also show that even when PO is violated, analyzing Y as an unordered categorical variable can be worse.

This seminar will explore detailed ordinal regression case studies using R, and will introduce the Bayesian Wilcoxon test through a Bayesian PO model. It covers the necessary theory behind ordinal models but is more concerned with practical application.