Residuals and Diagnostics for Ordinal Regression Models: A Surrogate Approach

Area of Science:

• Statistics
• Biostatistics
• Econometrics

Background:

• Ordinal outcomes are prevalent in research, yet diagnosing regression model assumptions is challenging.
• Existing methods lack effective tools for validating ordinal regression models.
• Ordinal variables represent ordered categories, not true numerical values, complicating residual analysis.

Purpose of the Study:

• To propose a novel surrogate approach for defining residuals for ordinal outcomes.
• To evaluate the theoretical and graphical properties of the proposed residuals.
• To assess the utility of the residuals in detecting model misspecifications.

Main Methods:

• Defined a continuous surrogate variable (S) for the ordinal outcome (Y).
• Derived residuals based on the surrogate variable within cumulative link regression models.
• Investigated theoretical properties and conducted numerical studies for diagnostic power.

Main Results:

• The surrogate residual exhibits properties similar to residuals for continuous outcomes.
• Demonstrated power in detecting misspecifications in mean structures, link functions, heteroscedasticity, proportionality, and mixed populations.
• Enabled development of goodness-of-fit measures.

Conclusions:

• The proposed surrogate residual offers a powerful tool for ordinal regression model diagnostics.
• It provides deeper insights into model misspecifications compared to previous methods.
• This approach aids in identifying specific areas for model improvement beyond simple hypothesis testing.