  Categorical responses arise naturally within various scientific disciplines.
In many circumstances, there is no predetermined order for the response
categories, and the response has to be modeled as nominal. In this study, we
regard the order of response categories as part of the statistical model, and
show that the true order, when it exists, can be selected using
likelihood-based model selection criteria. For predictive purposes, a
statistical model with a chosen order may outperform models based on nominal
responses, even if a true order does not exist. For multinomial logistic
models, widely used for categorical responses, we show the existence of
theoretically equivalent orders that cannot be differentiated based on
likelihood criteria, and determine the connections between their maximum
likelihood estimators. We use simulation studies and a real-data analysis to
confirm the need and benefits of choosing the most appropriate order for
categorical responses.
