skip to main content skip to footer

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

von Davier, Matthias
Publication Year:
Report Number:
ETS Research Report
Document Type:
Page Count:
Subject/Key Words:
Diagnostic Classification Models, General Diagnostic Model (GDM), Log-Linear Cognitive Diagnostic Model (LCDM), Model Equivalency, Parameter Redundancy, Identifiability, Test-Taker Performance


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.

Read More