Title: Representing Probabilistic Models of Knowledge Space Theory by
Multinomial Processing Tree Models

Authors: Daniel W. Heck, Stefano Noventa

Affiliation: Philipps-Universität Marburg; Universität Tübingen

Abstract:
Knowledge Space Theory (KST) aims at modelling the hierarchical relations
between items or skills in a learning process. Individuals are classified
according to their knowledge state, i.e. partially ordered latent classes
representing the collection of mastered items. Conditional probability
parameters are introduced to model transitions from these latent knowledge
states to observed response patterns. Since these models account for discrete
data by assuming a finite number of latent states, they can be represented by
Multinomial Processing Tree (MPT) models. Extending previous work on the link
between MPT and KST models for procedural assessments of knowledge, we prove
that standard probabilistic models of KST such as the Basic Local Independence
Model (BLIM) and the Simple Learning Model (SLM) can be represented as
specific instances of MPT models. Given this close link, MPT methods may be
applied to address theoretical and practical issues in KST, Cognitive
Diagnostic Models, and Item Response Theory (IRT). For instance,
model-selection methods recently implemented for MPT models (e.g., the Bayes
factor) might allow researchers to test and account for violations of local
independence, a fundamental assumption in Item Response Theory (IRT) and
psychological testing in general. By highlighting the MPT-KST link and its
implications for IRT, we hope to facilitate an exchange of theoretical
results, statistical methods, and software across these different domains of
mathematical psychology and psychometrics.