Yuqi Gu is a fifth-year PhD student in the Department of Statistics at the University of Michigan, advised by Gongjun Xu. Her research interests include statistical machine learning, latent variable models, and psychometrics. Yuqi is passionate about developing theoretically principled and computationally efficient methods to solve problems of scientific interests. She has been working to provide rigorous theory and methods that guide scientific practices in education and psychology, and she hopes to bridge psychometrics and modern statistical machine learning in her future research endeavors. Yuqi received a BS in Mathematics from Tsinghua University in 2015. She will join the Department of Statistics at Columbia University in 2021 as an assistant professor.
Uncovering Hidden Fine-Grained Scientific Information: Structured and Hierarchical Latent Attribute Models
In the era of data science, latent variable models have witnessed a tremendous surge of interest from a wide range of scientific applications and machine learning problems. Under this background, in modern psychological and biomedical research with diagnostic purposes, researchers often formulate the key task as inferring the fine-grained latent information under structural constraints. These structural constraints usually come from the domain experts’ prior knowledge or insight. The emerging family of structured latent attribute models (SLAMs) accommodates these modeling needs; SLAMs have received substantial attention in psychology, education, and epidemiology. These models bring exciting opportunities and unique challenges. In particular, with high-dimensional latent attributes and structural constraints encoded by a structural matrix, one needs to balance the gain in the model’s explanatory power and interpretability, against the difficulty of understanding and handling the complex model structure.
Specifically, a SLAM models how multivariate discrete observations depend on certain multivariate discrete latent attributes. This dependence takes a highly structured fashion and is summarized by a structural matrix of binary entries. A hierarchical latent attribute model (HLAM) further builds upon this and additionally models the attribute hierarchy: the hierarchical constraints on which configurations of the latent attributes are allowed. In the educational context, such hierarchy is crucial for practitioners to understand the psychological ordering of cognitive competencies. For example, this can reveal the information that mastering certain skill attributes serves as the prerequisite for mastering some others.
One challenge in modern applications of latent attribute models is the high dimensionality of the latent patterns that result from combinations of a large number of discrete attributes. The number of potential latent patterns can be much larger than the sample size. Another challenge is that the structural matrix and the attribute hierarchy often suffer from potential misspecification, or they are even completely unknown in some applications. A key question is then how to efficiently estimate both the structural matrix and the attribute hierarchy from noisy observations. More fundamentally, it is an important yet open theoretical question whether and when the latent structural matrix and the attribute hierarchy are identifiable and uniquely recoverable. Establishing identifiability without assuming any knowledge of the structural matrix and the attribute hierarchy is a technically very challenging task. Moreover, computationally, there is no existing method to simultaneously and efficiently estimate both the structural matrix and the attribute hierarchy.
This talk focuses on hierarchical latent attribute models from theoretical, methodological, and computational perspectives. Theoretically, I present sufficient and almost necessary conditions for identifying the attribute hierarchy, the structural matrix, and all the related model parameters in an HLAM. The derived identifiability conditions advance the theoretical knowledge and provide insights into real designs of diagnostic tests. Methodologically, I develop a statistically consistent method to select significant latent patterns in high dimensions. Computationally, I propose a scalable algorithm to simultaneously recover both the structural matrix and the attribute hierarchy. The application of the proposed methodology to the data from an international educational assessment uncovers meaningful knowledge structures of the student population.
1. Y. Gu and G. Xu, Identification and Estimation of Hierarchical Latent Attribute Models, arXiv:1906.07869, 2020+.
2. Y. Gu and G. Xu, Partial Identifiability of Restricted Latent Class Models, Annals of Statistics, to appear, 2020+.
3. Y. Gu and G. Xu, Sufficient and Necessary Conditions for the Identifiability of the Q-matrix, Statistica Sinica, to appear, 2020+.
4. Y. Gu and G. Xu, Learning Attribute Patterns in High-Dimensional Structured Latent Attribute Models, Journal of Machine Learning Research, 20(115):1−58, 2019.
5. Y. Gu and G. Xu, The Sufficient and Necessary Condition for the Identifiability and Estimability of the DINA Model, Psychometrika, 84(2), 468–483, 2019.