The Log‐Linear Cognitive Diagnostic Model ( LCDM ) as a Special Case of the General Diagnostic Model ( GDM )

Abstract

Diagnostic models combine multiple binary latent variables in an attempt to produce a latent structure that provides more information about test takers' performance than do unidimensional latent variable models. Recent developments in diagnostic modeling emphasize the possibility that multiple skills may interact in a conjunctive way within the item function, while individual skills still may retain separable additive effects. This extension of either the conjunctive deterministic‐input‐noisy‐and (DINA) model to the generalized version (G‐DINA) or the compensatory/additive general diagnostic model (GDM) to the log‐linear cognitive diagnostic model (LCDM) is aimed at integrating models with conjunctive skills and those that assume compensatory functioning of multiple skill variables. More recently, a result was proven mathematically that the fully conjunctive DINA model, which combines all required skills in a single binary function, may be recast as a compensatory special case of the GDM. This can be accomplished in more than one form such that the resulting transformed skill‐space definitions and design (Q) matrices are different from each other but mathematically equivalent to the DINA model, producing identical model‐based response probabilities. In this report, I extend this equivalency result to the LCDM and show that a mathematically equivalent, constrained GDM can be defined that yields identical parameter estimates based on a transformed set of compensatory skills.

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