Model-Based Clustering using Adjacent-Categories Logit Models via Finite Mixtures

Many researchers treat ordinal variables as continuous or nominal. Losing ordering information makes statistical analysis and inference weaker, which is not ideal when we try to get as much information as possible from our data. Clustering for ordinal data faces many challenges as well. Traditional clustering methods such as hierarchical clustering (Murtagh, 1983; Murtagh and Contreras, 2012), centroid-based clustering, and association analysis (Strehl et al., 1999) are not likelihood-based, and statistical inference is not available. In this dissertation, we apply clustering via finite mixtures to the adjacent-categories logit model (Agresti, 1999) for ordinal data, an extension of the likelihood-based clustering methods in Pledger and Arnold (2014); Fernandez et al. (2016).

Our data form a matrix where the rows are subjects, and the columns are a set of ordinal responses by those subjects to, say, the questions in a questionnaire. By implementing model-based clustering via a finite mixture model, the subjects (the rows of the matrix) and/or the questions (the columns of the matrix) are grouped into a finite number of clusters. The Expectation-Maximisation (EM) algorithm is utilised to estimate the model parameters. Several different stopping criteria for the EM algorithm are proposed and studied for our model. Additionally, asymptotic standard errors are provided by the standard information-based method. For model selection, six commonly used information criteria are evaluated through an empirical study. After information criteria are used to select the best model, the randomised quantile residual is proposed for model diagnostic and model checking. About the applications, we illustrate the models through two real-life data examples.