Inverse generalised spin models of answers to questionnaires
Network psychometrics conceptualises psychological constructs as emergent properties of systems of interacting items. Energy-based probabilistic models have gained popularity as models of these interactions, but their psychometric application has so far been limited to binary responses, bilinear interactions, and approximated inference methods. To fill these gaps, we here infer and analyse three generalized-spin models of ordinal questionnaire data: the Ising, Blume-Capel (BC), and Blume-Emery-Griffiths (BEG) models. These are maximum-entropy models that accommodate ordinal responses on Likert-type scales with an arbitrary number of options, allowing for single-site anisotropy (BC, BEG) and bi-quadratic item interactions (BEG). We prove the concavity of the maximum likelihood estimation of their parameters, as well as the gauge invariance of the Ising and BC models. We introduce a stochastic gradient ascent algorithm for maximum likelihood inference, and apply this procedure to eleven psychometric and sociological questionnaire datasets. Evaluating the predictive ability of the inferred models reveals that the BEG model systematically outperforms Factor Analysis and the other spin models in capturing the distributions of factors and distances of subject answers to the mean, across all datasets. By leveraging a competitive interplay between the quadratic and bi-quadratic energy terms, the BEG model uniquely captures individual average-extremist response styles, alongside standard latent factor positioning. Moreover, only the spin models can account for the non-concavity and multi-modality of factor histograms in the most polarizing questionnaires. Finally, the analysis reveals other highly non-linear traits of ordinal data---such as the fat-tailed distribution of Mahalanobis distances to the mean---that escape satisfactory description by both factor and spin models.