Residuals and diagnostics for multinomial regression models

In this paper, we extend the concept of a randomized quantile residual to multinomial regression models. Customary diagnostics for these models are limited because they involve difficult-to-interpret residuals and often focus on the fit of one category versus the rest. Our residuals account for associations between categories by using the squared Mahalanobis distances of the observed log-odds relative to their fitted sampling distributions. Aside from sampling variation, these residuals are exactly normal when the data come from the fitted model. This motivates our use of the residuals to detect model misspecification and overdispersion, in addition to an overall goodness-of-fit Kolmogorov-Smirnov test. We illustrate the use of the residuals and diagnostics in both simulation and real data studies.