Extending the Basic Local Independence Model to Polytomous Data

A probabilistic framework for the polytomous extension of knowledge space theory (KST) is proposed. It consists in a probabilistic model, called polytomous local independence model, that is developed as a generalization of the basic local independence model. The algorithms for computing “maximum likelihood” (ML) and “minimum discrepancy” (MD) estimates of the model parameters have been derived and tested in a simulation study. Results show that the algorithms differ in their capability of recovering the true parameter values. The ML algorithm correctly recovers the true values, regardless of the manipulated variables. This is not totally true for the MD algorithm. Finally, the model has been applied to a real polytomous data set collected in the area of psychological assessment. Results show that it can be successfully applied in practice, paving the way to a number of applications of KST outside the area of knowledge and learning assessment.

Naïve Tests of Basic Local Independence Model’s Invariance

The basic local independence model (BLIM) is a probabilistic model for knowledge structures, characterized by the property that lucky guess and careless error parameters of the items are independent of the knowledge states of the subjects. When fitting the BLIM to empirical data, a good fit can be obtained even when the invariance assumption is violated. Therefore, statistical tests are needed for detecting violations of this specific assumption. This work provides an extension to theoretical results obtained by de Chiusole, Stefanutti, Anselmi, and Robusto (2013), showing that statistical tests based on the partitioning of the empirical data set into two (or more) groups are not adequate for testing the BLIM’s invariance assumption. A simulation study confirms the theoretical results.