  Research on cluster analysis for categorical data continues to develop, with
new clustering algorithms being proposed. However, in this context, the
determination of the number of clusters is rarely addressed. In this paper, we
propose a new approach in which clustering of categorical data and the
estimation of the number of clusters is carried out simultaneously. Assuming
that the data originate from a finite mixture of multinomial distributions, we
develop a method to select the number of mixture components based on a minimum
message length (MML) criterion and implement a new expectation-maximization
(EM) algorithm to estimate all the model parameters. The proposed EM-MML
approach, rather than selecting one among a set of pre-estimated candidate
models (which requires running EM several times), seamlessly integrates
estimation and model selection in a single algorithm. The performance of the
proposed approach is compared with other well-known criteria (such as the
Bayesian information criterion-BIC), resorting to synthetic data and to two
real applications from the European Social Survey. The EM-MML computation time
is a clear advantage of the proposed method. Also, the real data solutions are
much more parsimonious than the solutions provided by competing methods, which
reduces the risk of model order overestimation and increases interpretability.
