Learning Latent and Hierarchical Structures in Cognitive Diagnosis Models

TL;DR

This paper introduces a new method for jointly learning the latent attributes and hierarchical structures in Cognitive Diagnosis Models directly from data, reducing reliance on subjective pre-specification.

Contribution

It proposes a penalized likelihood approach with an EM algorithm to simultaneously estimate the number of attributes and their hierarchical relationships, with proven statistical consistency.

Findings

The method accurately recovers latent structures in simulations.

It effectively estimates the number of attributes in real data.

Demonstrates improved structure learning over existing methods.

Abstract

Cognitive Diagnosis Models (CDMs) are a special family of discrete latent variable models that are widely used in modern educational, psychological, social and biological sciences. A key component of CDMs is a binary $Q$-matrix characterizing the dependence structure between the items and the latent attributes. Additionally, researchers also assume in many applications certain hierarchical structures among the latent attributes to characterize their dependence. In most CDM applications, the attribute-attribute hierarchical structures, the item-attribute $Q$-matrix, the item-level diagnostic model, as well as the number of latent attributes, need to be fully or partially pre-specified, which however may be subjective and misspecified as noted by many recent studies. This paper considers the problem of jointly learning these latent and hierarchical structures in CDMs from observed data with minimal model assumptions. Specifically, a penalized likelihood approach is proposed to select the number of attributes and estimate the latent and hierarchical structures simultaneously. An efficient expectation-maximization (EM) algorithm and a latent structure recovery algorithm are developed, and statistical consistency theory is also established under mild conditions. The good performance of the proposed method is illustrated by simulation studies and a real data application in educational assessment.