Toward a unified perspective on assessment models, Part III: Polytomous and continuous latent variables

In the past years, several theories for assessment have been developed within the overlapping fields of Psychometrics and Mathematical Psychology. The most notable are Item Response Theory (IRT), Cognitive Diagnostic Assessment (CDA), and Knowledge Structure Theory (KST). In spite of their common goals, these theories are largely independent and focus on different aspects. In Part I of this work, a general framework was introduced that, by means of two primitives (structure and process) and two operations (factorization and reparametrization), allows to derive the models of these theories. A two-processes approach provided a taxonomy encompassing IRT, CDA, and KST models. In Part II, the framework was used to derive KST and CDA models based on dichotomous latent variables. In this third contribution, IRT, CDA, and KST models based on polytomous and continuous latent variables are derived. Implications for IRT models are discussed. A special focus is on the implicit assumptions required to move from the discrete case to the continuous one.