A Conway-Maxwell-Multinomial Distribution for Flexible Modeling of Clustered Categorical Data

TL;DR

This paper introduces the Conway-Maxwell-multinomial distribution, a flexible model for clustered categorical data that captures both positive and negative associations, with efficient estimation methods and practical applications.

Contribution

It proposes the CMM distribution, extending multinomial models to handle various association levels, and provides methods for estimation and real data demonstrations.

Findings

CMM can model both positive and negative associations.

Efficient maximum likelihood estimation method developed.

Demonstrated flexibility through real data examples.

Abstract

Categorical data are often observed as counts resulting from a fixed number of trials in which each trial consists of making one selection from a prespecified set of categories. The multinomial distribution serves as a standard model for such clustered data but assumes that trials are independent and identically distributed. Extensions such as Dirichlet-multinomial and random-clumped multinomial can express positive association, where trials are more likely to result in a common category due to membership in a common cluster. This work considers a Conway-Maxwell-multinomial (CMM) distribution for modeling clustered categorical data exhibiting positively or negatively associated trials. The CMM distribution features a dispersion parameter which allows it to adapt to a range of association levels and includes several recognizable distributions as special cases. We explore properties of CMM, illustrate its flexible characteristics, identify a method to efficiently compute maximum likelihood (ML) estimates, present simulations of small sample properties under ML estimation, and demonstrate the model via several data analysis examples.