Ordinal models can be seen as being composed from simpler, in particular binary models. This view on ordinal models allows to derive a taxonomy of models that includes basic ordinal regression models, models with more complex parameterizations, the class of hierarchically structured models, and the more recently developed finite mixture models. The structured overview that is given covers existing models and shows how models can be extended to account for further effects of explanatory variables. Particular attention is given to the modeling of additional heterogeneity as, for example, dispersion effects. The modeling is embedded into the framework of response styles and the exact meaning of heterogeneity terms in ordinal models is investigated. It is shown that the meaning of terms is crucially determined by the type of model that is used. Moreover, it is demonstrated how models with a complex category‐specific effect structure can be simplified to obtain simpler models that fit sufficiently well. The fitting of models is illustrated by use of a real data set, and a short overview of existing software is given.This article is categorized under: Statistical Models > Fitting Models Data: Types and Structure > Categorical Data Statistical Models > Generalized Linear Models

A new diagnostic tool for the identification of differential item functioning (DIF) is proposed. Classical approaches to DIF allow to consider only few subpopulations like ethnic groups when investigating if the solution of items depends on the membership to a subpopulation. We propose an explicit model for differential item functioning that includes a set of variables, containing metric as well as categorical components, as potential candidates for inducing DIF. The ability to include a set of covariates entails that the model contains a large number of parameters. Regularized estimators, in particular penalized maximum likelihood estimators, are used to solve the estimation problem and to identify the items that induce DIF. It is shown that the method is able to detect items with DIF. Simulations and two applications demonstrate the applicability of the method.

In the last two decades, regularization techniques, in particular penalty-based methods, have become very popular in statistical modelling. Driven by technological developments, most approaches have been designed for high-dimensional problems with metric variables, whereas categorical data has largely been neglected. In recent years, however, it has become clear that regularization is also very promising when modelling categorical data. A specific trait of categorical data is that many parameters are typically needed to model the underlying structure. This results in complex estimation problems that call for structured penalties which are tailored to the categorical nature of the data. This article gives a systematic overview of penalty-based methods for categorical data developed so far and highlights some issues where further research is needed. We deal with categorical predictors as well as models for categorical response variables. The primary interest of this article is to give insight into basic properties of and differences between methods that are important with respect to statistical modelling in practice, without going into technical details or extensive discussion of asymptotic properties.