A Note on the Relationship of the Shannon Entropy Procedure and the Jensen–Shannon Divergence in Cognitive Diagnostic Computerized Adaptive Testing

Abstract

The purpose of this study is to investigate the relationship between the Shannon entropy procedure and the Jensen–Shannon divergence (JSD) that are used as item selection criteria in cognitive diagnostic computerized adaptive testing (CD-CAT). Because the JSD itself is defined by the Shannon entropy, we apply the well-known relationship between the JSD and Shannon entropy to establish a relationship between the item selection criteria that are based on these two measures. To understand the relationship between these two item selection criteria better, an alternative way is also provided. Theoretical derivations and empirical examples have shown that the Shannon entropy procedure and the JSD in CD-CAT have a linear relation under cognitive diagnostic models. Consistent with our theoretical conclusions, simulation results have shown that two item selection criteria behaved quite similarly in terms of attribute-level and pattern recovery rates under all conditions and they selected the same set of items for each examinee from an item bank with item parameters drawn from a uniform distribution U(0.1, 0.3) under post hoc simulations. We provide some suggestions for future studies and a discussion of relationship between the modified posterior-weighted Kullback–Leibler index and the G-DINA (generalized deterministic inputs, noisy “and” gate) discrimination index.