Analyzing ordinal categorical responses using multilevel models with an application related to dancing

Abstract

Ordinal categorical outcomes frequently arise in applied research, particularly in settings where responses are recorded using rating scores rather than continuous measurements. In many practical applications, the data may be nested in higher level structures, with observations clustered within higher-level units, leading to dependence that cannot be adequately handled by standard regression models. This paper analyses ordinal categorical responses within a multilevel modelling framework, with particular emphasis on likelihood-based estimation methods for generalized mixed-effects models and their practical application to dance competition data. The study first develops the theoretical foundation of two-level models for ordinal responses assuming a multinomial distribution and a logit link function. Particular attention is given to random intercept and random coefficient structures, the interpretation of between-cluster variation, and the role of intra-class correlation in assessing dependence within hierarchical data. The paper then discusses the estimation and inferential techniques used for multilevel ordinal models and explores procedure to overcome difficulties that arise when the marginal likelihood involves integrals over random effects that do not have a closed-form solution. The study examines various numerical integration methods used in marginal likelihood estimation, with emphasis on Gaussian quadrature and Gauss-Hermite quadrature. The construction of quadrature rules and their role in approximating intractable integrals are discussed in detail, together with the use of modified Newton-Raphson procedures for maximising the approximated likelihood. In addition, Bayesian ideas are introduced in the context of predicting random effects, where empirical Bayes estimates (posterior means) are used to obtain cluster-specific predictions within the fitted models. These modelling methods are applied to a dance competition dataset. Since the dancing performance scores awarded by judges had a left skewed non-normal distribution, it was decided to categorise these scores to five ordinal response categories and analysed using multilevel logit models. The hierarchical structure is represented by individual performers (level-1 units) nested within dance types (level-2 units) allowing two-level models to be fitted. The models are implemented in Stata using the GLLAMM software, which provides flexible likelihood-based estimation for multilevel models with ordinal responses using numerical integration. The analysis demonstrates how multilevel models for ordinal responses can be used to account for clustering, estimate random-effects variability, compare alternative model structures, and interpret the effects of explanatory variables in a practically meaningful way. Overall, the paper provides a methodological and applied examination of multilevel modelling for ordinal categorical data. It shows that when ordinal responses are analysed within a hierarchical framework, likelihood-based estimation supported by quadrature methods offers a rigorous and practical approach for modelling complex dependence structures and obtaining interpretable statistical inferences.