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.
Highlights
Cognitive Diagnosis Models (CDMs) are a popular family of discrete latent variable models that have been widely used in social and biological sciences
In educational assessments, the latent attributes are assumed to be mastery or deficiency of target skills (Junker and Sijtsma, 2001; de la Torre, 2011); in psychiatric diagnosis, they are modeled as presence or absence of some underlying mental disorders (Templin and Henson, 2006; de la Torre et al, 2018); and in epidemiological and medical measurement studies, the latent attributes are interpreted as existence or nonexistence of some disease pathogens (Wu et al, 2016b,a; O’Brien et al, 2019)
We focus on the problem of hypothesis testing for latent hierarchical structures
Summary
Cognitive Diagnosis Models (CDMs) are a popular family of discrete latent variable models that have been widely used in social and biological sciences. Testing the sparsity structure of the proportion parameter vector in CDMs is closely related to the problem of testing the number of components in finite mixture models and latent class models (Nylund et al, 2007; Chen, 2017). We do not recommend using the nonstandard limiting distribution to conduct the hypothesis testing in practice Instead, based on these findings, we propose to use resampling-based methods to test hierarchical structures.
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