Log-Linear and Log-Multiplicative Association Models for Categorical Data

The chapter reviews uni- and multidimensional association models (AMs) that provide a parsimonious modelling of interactions between categorical variables. AMs go beyond the standard log-linear modeling framework to model non-saturated models that exist between a saturated and an independence model, including also log-nonlinear models. AMs are presented here both graphically and algebraically to aid interpretation and help in understanding the features and structures of the associations. The chapter discusses the connection of AMs with graphical models and item response theory (IRT) models developed in Psychometrics that open new insights in the understanding and modeling of associations of categorical variables. Estimation methods within the maximum likelihood estimation framework including composite likelihood methods, goodness-of-fit tests and measures, as well as R packages available, are discussed. The models are fitted to data from massively open online courses (MOOCs) using the R package logmulti and on the data from the depression, anxiety and stress scale (DASS) using the R package pleLMA.