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Constructive Q-Matrix Identifiability via Novel Tensor Unfolding.

Yuqi Gu1

  • 1Department of Statistics, https://ror.org/00hj8s172Columbia University, USA.

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|January 6, 2026
PubMed
Summary
This summary is machine-generated.

This study introduces a new tensor-unfolding method to identify the Q-matrix in cognitive diagnostic models (CDMs). This approach precisely recovers the Q-matrix and attribute numbers, applicable to various CDMs and response types.

Keywords:
Algebraic statisticsCognitive diagnostic modelConstructive proofIdentifiabilityQ-matrixTensor unfolding

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Area of Science:

  • Psychometrics
  • Cognitive Science
  • Data Science

Background:

  • Cognitive Diagnostic Models (CDMs) are crucial for understanding student knowledge structures.
  • The Q-matrix, a key component of CDMs, defines the relationship between skills and items.
  • Identifiability of the Q-matrix is essential for valid model interpretation and application.

Purpose of the Study:

  • To establish a novel identifiability theory for the Q-matrix in CDMs.
  • To develop a constructive proof strategy using tensor unfolding.
  • To extend identifiability analysis beyond existing limitations in CDMs.

Main Methods:

  • Representing joint response distributions as J-way tensors.
  • Employing tensor unfolding techniques to decompose the J-way tensor into matrices.
  • Analyzing the rank properties of these matrices to identify the Q-matrix.

Main Results:

  • A constructive, population-level procedure to exactly recover the Q-matrix is established.
  • The method successfully identifies both the Q-matrix and the number of latent attributes.
  • The theory accommodates linear and nonlinear CDMs, main or saturated effects, and polytomous responses.

Conclusions:

  • The novel tensor-unfolding approach provides a unified and strengthened identifiability guarantee for diverse CDMs.
  • This work offers rigorous theoretical foundations for Q-matrix identifiability.
  • It paves the way for practical Q-matrix estimation using tensor unfolding techniques.