Item response models for count data

Abstract

Item response theory (IRT) represents a statistical framework within which responses to psychological tests can be modelled. A psychological test consists of a set of items (e.g., tasks to solve or statements to rate) to which a person taking the test responds. IRT assumes that responses are influenced by respondents' latent traits (e.g., personality traits or cognitive abilities) as well as by items' characteristics (e.g., difficulty). IRT models exist for a variety of different response types; the focus of this thesis lies on count responses. These can for example be generated by cognitive tests measuring idea fluency (counts: number of ideas), as process data during test taking (counts: number of clicks), or by reading proficiency assessments (counts: number of errors). Previously comparatively understudied, the field of count item response theory (CIRT) has witnessed a steady increase in interest in recent years. As a result, a number of new CIRT models have been proposed that address limitations of previously existing CIRT models, broadening the empirical applicability of CIRT.

An important concern regarding modelling of counts is their dispersion: The most common distribution for counts, the Poisson distribution, assumes its mean equals its variance (so called equidispersion). By relying on the Poisson distribution, prominent CIRT models assume such equidispersion for responses (conditional on the latent trait(s)). Research has found this assumption empirically violated for some tests. A recently introduced unidimensional CIRT model using the Conway-Maxwell-Poisson (CMP) distribution instead, accommodates over- and underdispersed conditional responses as well. Nonetheless, the model maintains some of the restricting assumptions of previous models. Thus, even with new model proposals, CIRT still offers less modelling flexibility than IRT for other response types (such as binary responses).

The present cumulative thesis aims to address three such gaps in the CIRT landscape. In the first article, I propose a unidimensional CIRT model with a conditional CMP response distribution which extends a previously proposed model through the inclusion of another item parameter (i.e., a discrimination parameter). As such a model has previously not been computable with existing estimation methods, I derive a maximum likelihood estimation procedure to this end, using the Expectation-Maximization (EM) algorithm. In the second article, we propose two extensions of this model which allow the inclusion of item- and person-specific covariates, respectively. Therewith, we allow to investigate explanations for differences between items and participants, respectively. Again, we provide corresponding estimation methods. In the third article, we generalize the unidimensional CIRT model proposed in the first article to a multidimensional count item response model framework, with a focus on exploratory models. We provide a respective estimation procedure, of which we additionally develop a lasso-penalized variant. The articles in this thesis are accompanied by the development of an R package that implements the proposed models and estimation methods.