On a Generalization of Local Independence in Item Response Theory Based on Knowledge Space Theory

  • Stefano NoventaEmail author
  • Andrea Spoto
  • Jürgen Heller
  • Augustin Kelava


Knowledge space theory (KST) structures are introduced within item response theory (IRT) as a possible way to model local dependence between items. The aim of this paper is threefold: firstly, to generalize the usual characterization of local independence without introducing new parameters; secondly, to merge the information provided by the IRT and KST perspectives; and thirdly, to contribute to the literature that bridges continuous and discrete theories of assessment. In detail, connections are established between the KST simple learning model (SLM) and the IRT General Graded Response Model, and between the KST Basic Local Independence Model and IRT models in general. As a consequence, local independence is generalized to account for the existence of prerequisite relations between the items, IRT models become a subset of KST models, IRT likelihood functions can be generalized to broader families, and the issues of local dependence and dimensionality are partially disentangled. Models are discussed for both dichotomous and polytomous items and conclusions are drawn on their interpretation. Considerations on possible consequences in terms of model identifiability and estimation procedures are also provided.


knowledge space theory item response theory Rasch models local independence BLIM Graded Response Model 



n Parameters normal ogive model, for the 4PNO see Eq. (3)


n Parameters logistic model, for the 4PL see Eqs. (1) and (2)


Basic Local Independence Model, see Eq. (15)


Category response function


Continuation ratio models


Graded Response Model, see Eq. (5)


Item characteristic curve


Item response theory


Knowledge space theory


Local dependence


Logistic knowledge structure, see Definition 1


Partial credit model


Probabilistic knowledge structure, see Eq. (14)


Simple learning model, see Eq. (19)


State characteristic curve (or subtest characteristic curve).



We wish to thank the two anonymous reviewers of the journal for their insight into the work and their helpful comments and suggestions. In particular, we would like to thank the second reviewer for pointing out the connection with Holland (1981).


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© The Psychometric Society 2018

Authors and Affiliations

  1. 1.Methods CenterUniversität TübingenTübingenGermany
  2. 2.Department of General PsychologyUniversity of PadovaPaduaItaly
  3. 3.Faculty of PsychologyUniversität TübingenTübingenGermany

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