Model-based clustering of multivariate ordinal data relying on a stochastic binary search algorithm

We design a probability distribution for ordinal data by modeling the process generating data, which is assumed to rely only on order comparisons between categories. Contrariwise, most competitors often either forget the order information or add a non-existent distance information. The data generating process is assumed, from optimality arguments, to be a stochastic binary search algorithm in a sorted table. The resulting distribution is natively governed by two meaningful parameters (position and precision) and has very appealing properties: decrease around the mode, shape tuning from uniformity to a Dirac, identifiability. Moreover, it is easily estimated by an EM algorithm since the path in the stochastic binary search algorithm can be considered as missing values. Using then the classical latent class assumption, the previous univariate ordinal model is straightforwardly extended to model-based clustering for multivariate ordinal data. Parameters of this mixture model are estimated by an AECM algorithm. Both simulated and real data sets illustrate the great potential of this model by its ability to parsimoniously identify particularly relevant clusters which were unsuspected by some traditional competitors.