# Learning Latent and Hierarchical Structures in Cognitive Diagnosis Models

arXiv: https://arxiv.org/abs/2104.02143 · HTML: https://mlsanity.com/paper/2104.02143/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 61 references that are in the map.
- First posted on arXiv 5 Apr 2021.

## Builds on (most cited first, 5 of 5)

- [Identifiability of restricted latent class models with binary responses](https://mlsanity.com/paper/1603.04140/) (2016)
- [Nested Partially-Latent Class Models for Dependent Binary Data; Estimating Disease Etiology](https://mlsanity.com/paper/1510.08862/) (2015)
- [The Sufficient and Necessary Condition for the Identifiability and Estimability of the DINA Model](https://mlsanity.com/paper/1711.03174/) (2017)
- [Learning Attribute Patterns in High-Dimensional Structured Latent Attribute Models](https://mlsanity.com/paper/1904.04378/) (2019)
- [Learning Large Q-matrix by Restricted Boltzmann Machines](https://mlsanity.com/paper/2006.15424/) (2020)

## Built on by (most cited first, 2 of 2)

- [End-to-end Graph Learning Approach for Cognitive Diagnosis of Student Tutorial](https://mlsanity.com/paper/2411.00845/) (2024)
- [On Theoretical Identifiability of Discrete Latent Causal Graphical Models](https://mlsanity.com/paper/2505.18410/) (2025)

## Abstract (arXiv)

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.