When the Mean is Misleading: A Guide to Ordered Regression for Meaningfully Modeling Ordinal Outcomes

Objectives: To demonstrate the utility of ordered regression models for analyzing ordinal outcomes frequently encountered in criminology, comparing their performance in representing data patterns and estimating effect magnitudes against conventional linear models commonly used in practice. Methods: The study employs simulation to illustrate how linear and ordered probit models represent five different ordinal data distributions. Subsequently, it reanalyzes national survey data (N=1,150) on racial differences in fear of police from Pickett et al. (2022). Bayesian linear models (item-specific, multilevel, normed-scale) are compared to Bayesian cumulative probit models using model diagnostics (posterior predictive checks; LOO-CV) and visualizations (mosaic plots; predicted probabilities). Results: Simulations confirm linear models poorly represent non-normal ordinal distributions, while ordered probit models accurately recover observed patterns. The reanalysis reveals linear approaches substantially underestimate racial disparities in fear of police compared to ordered probit models, which align closely with observed data and show superior predictive fit (ΔELPD > 1900). For instance, ordered probit models estimate nearly one-third (~33%) of Black participants report being “very afraid” of being killed by police; comparable linear model estimates were much lower (~11%). Conclusions: Appropriate analysis of ordinal data using ordered regression models allows for more accurate quantification of effect magnitudes and distributional patterns than conventional linear models, moving beyond potentially misleading average effects or vague directional inferences. This enhances analytical precision, providing a stronger foundation for theory and policy. Limitations include the focus on cumulative probit models and a single empirical example. Further research should explore alternative ordinal model specifications and applications.