A Modification of McFadden's $R^2$ for Binary and Ordinal Response Models

TL;DR

This paper introduces a simple modification to McFadden's $R^2$ for categorical response models, adjusting for response categories and providing a more consistent goodness-of-fit measure for binary and ordinal models.

Contribution

The paper proposes a modified $R^2$ measure that accounts for the number of response categories and rescales values, improving goodness-of-fit assessment for categorical models.

Findings

Modified $R^2$ outperforms traditional measures in simulations.

The measure is invariant to the number of response categories.

Real data application confirms improved fit assessment.

Abstract

A lot of studies on the summary measures of predictive strength of categorical response models consider the likelihood ratio index (LRI), also known as the McFadden-$R^2$, a better option than many other measures. We propose a simple modification of the LRI that adjusts for the effect of the number of response categories on the measure and that also rescales its values, mimicking an underlying latent measure. The modified measure is applicable to both binary and ordinal response models fitted by maximum likelihood. Results from simulation studies and a real data example on the olfactory perception of boar taint show that the proposed measure outperforms most of the widely used goodness-of-fit measures for binary and ordinal models. The proposed $R^2$ interestingly proves quite invariant to an increasing number of response categories of an ordinal model.