Hypothesis Testing for Hierarchical Structures in Cognitive Diagnosis Models

Ma, Chenchen; Xu, Gongjun

doi:10.6339/21-JDS1024

Journal of Data Science

Hypothesis Testing for Hierarchical Structures in Cognitive Diagnosis Models

Volume 20, Issue 3 (2022): Special Issue: Data Science Meets Social Sciences, pp. 279–302

Chenchen Ma Gongjun Xu

https://doi.org/10.6339/21-JDS1024

Pub. online: 14 October 2021 Type: Statistical Data Science

Open Access

Received
2 June 2021

Accepted
11 September 2021

Published
14 October 2021

Abstract

Cognitive Diagnosis Models (CDMs) are a special family of discrete latent variable models widely used in educational, psychological and social sciences. In many applications of CDMs, certain hierarchical structures among the latent attributes are assumed by researchers to characterize their dependence structure. Specifically, a directed acyclic graph is used to specify hierarchical constraints on the allowable configurations of the discrete latent attributes. In this paper, we consider the important yet unaddressed problem of testing the existence of latent hierarchical structures in CDMs. We first introduce the concept of testability of hierarchical structures in CDMs and present sufficient conditions. Then we study the asymptotic behaviors of the likelihood ratio test (LRT) statistic, which is widely used for testing nested models. Due to the irregularity of the problem, the asymptotic distribution of LRT becomes nonstandard and tends to provide unsatisfactory finite sample performance under practical conditions. We provide statistical insights on such failures, and propose to use parametric bootstrap to perform the testing. We also demonstrate the effectiveness and superiority of parametric bootstrap for testing the latent hierarchies over non-parametric bootstrap and the naïve Chi-squared test through comprehensive simulations and an educational assessment dataset.

Supplementary material

Supplementary Material

More comprehensive simulation results are presented in the supplementary material. Specifically, bootstrap results for DINA and GDINA models under both null hypothesis and alternative hypothesis with different sample sizes and noise levels are plotted there. We also include the codes for simulations and real data analysis.

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2022 The Author(s). Published by the School of Statistics and the Center for Applied Statistics, Renmin University of China.

Open access article under the CC BY license.

Keywords

bootstrapping latent hierarchical structure likelihood ratio test

Funding

This research is partially supported by National Science Foundation CAREER SES-1846747 and Institute of Education Sciences R305D200015.

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