  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.
