关键词: 潜变量建模, 属性层级结构, 假设检验, 贝叶斯结构学习, 参数估计

Abstract: Attribute hierarchical structure (AHS) characterizes the prerequisite relationships among attributes. It plays an important role in diagnostic test design, instructional design and assessment, learning path planning, and personalized learning recommendation.
Latent variable models serve as the foundation for parameterized approaches to AHS analysis, as they determine the objectives, analytical perspectives, and implementation strategies of these methods. From the perspective of latent variable modeling, this study systematically reviews methods for estimating and testing AHS according to three methodological perspectives: hypothesis testing, Bayesian structure learning and parameter estimation.
Specifically, two major frameworks have been developed for testing AHS. The first evaluates AHS by comparing the model fit of nested hierarchical structures, including the likelihood ratio test (LRT), parametric bootstrap LRT, and non-parametric bootstrap LRT. The second tests AHS by examining whether the structural parameters corresponding to all impossible attribute mastery patterns under a prespecified AHS are significantly equal to zero, primarily using the Wald test. The empirical distribution of the LRT statistic often deviates substantially from its asymptotic distribution. Although parametric and non-parametric bootstrap LRTs improve statistical inference, they are computationally intensive in large-scale applications. In contrast, the Wald test is computationally simpler, provides better control of Type I error rates, achieves higher statistical power, and performs better than the LRT under small-sample conditions.
Three major strategies have been proposed for estimating AHS. The first is based on structural parameter testing, including the Z test and the iterative Z test. The second relies on Bayesian network structure learning algorithms, such as the K2 algorithm (Cooper & Herskovits, 1992), Hill Climbing (HC) (Scutari & Denis, 2014), and Max-Min Hill Climbing (MMHC) (Tsamardinos et al., 2006). The third is based on parameter estimation and includes methods based on penalized marginal maximum likelihood estimation (MMLE), such as Regularized Latent Class Modeling (RLCM) (Wang & Lu, 2021), Latent Variable Selection (LVS) (Wang & Lu, 2021), and the Penalized Likelihood Approach (PLA) (Ma et al., 2022), as well as Bayesian joint estimation methods, including the General Bayesian Estimation Method (GBEM) (Chen & Wang, 2023) and the Novel Bayesian Estimation Method (NBEM) (Wang et al., 2026). Taken together, these methods each have their own strengths and limitations. These methods also differ in terms of analytical strategies, evaluation criteria, and classification accuracy across different types of AHS, reflecting their distinctive characteristics.
It is worth noting that recent studies have further extended AHS analysis to the joint estimation or refinement of the Q-matrix and AHS (Lee & Gu, 2024; Ma et al., 2022; Wang et al., 2026; Wang & Sun, 2025). In particular, the Latent Class Bayesian Network (LCBN) proposed by Lee and Gu models structural parameters under AHS constraints, substantially reducing the number of structural parameters and providing a promising solution for high-dimensional attribute settings. In addition, the Attribute Correlation Intensity Matrix (ACIM) proposed by Yan (2022), as a representative non-parametric approach, performs well without imposing stringent sample-size requirements. Collectively, these studies represent important directions for the future development of AHS analysis.
Overall, research on AHS has evolved along several dimensions. In terms of research objectives, it has progressed from AHS validation to AHS estimation, joint estimation of the number of attributes and AHS, and joint estimation of the Q-matrix and AHS. In terms of research content, the focus has shifted from describing the external forms of AHS to characterizing probabilistic relationships within its internal structure. Methodologically, the underlying modeling framework has evolved from single latent variable modeling frameworks based on Latent Class Models (LCMs), Diagnostic Classification Models (DCMs), or Bayesian Network Models (BNMs) toward integrated modeling frameworks. In practice, appropriate methods should be selected based on prior information, including sample size, attribute specification, the number of attributes, the Q-matrix, the measurement model and AHS.
Future research should continue to optimize key methodological components, enrich experimental conditions, expand application scenarios, and develop innovative research perspectives. For example, promising directions include extending AHS analysis to polytomous items, polytomous attributes, multiple-strategy responses, longitudinal assessment, and computerized adaptive testing; integrating the strengths of existing methods to develop more effective analytical approaches; exploring new strategies for AHS analysis from the perspective of latent variable modeling; and expanding modeling frameworks through machine learning algorithms. These all represent important avenues for advancing methodological research on AHS analysis.

Key words: latent variable modeling, attribute hierarchical structure, hypothesis testing, Bayesian structure learning, parameter estimation