Latent Variable Modeling and Statistical Learning

Latent variable models play an important role in psychological and educational measurement, which attempt to uncover the underlying structure of responses to test items. This thesis focuses on the development of statistical learning methods based on latent variable models, with applications to psychological and educational assessments. In that connection, the following problems are considered. The first problem arises from a key assumption in latent variable modeling, namely the local independence assumption, which states that given an individual's latent variable (vector), his/her responses to items are independent. This assumption is likely violated in practice, as many other factors, such as the item wording and question order, may exert additional influence on the item responses. Any exploratory analysis that relies on this assumption may result in choosing too many nuisance latent factors that can neither be stably estimated nor reasonably interpreted. To address this issue, a family of models is proposed that relax the local independence assumption by combining the latent factor modeling and graphical modeling. Under this framework, the latent variables capture the across-the-board dependence among the item responses, while a second graphical structure characterizes the local dependence. In addition, the number of latent factors and the sparse graphical structure are both unknown and learned from data, based on a statistically solid and computationally efficient method. The second problem is to learn the relationship between items and latent variables, a structure that is central to multidimensional measurement. In psychological and educational assessments, this relationship is typically specified by experts when items are written and is incorporated into the model without further verification after data collection. Such a non-empirical approach may lead to model misspecification and substantial lack of model fit, resulting in erroneous interpretation of assessment results. Motivated by this, I consider to learn the item - latent variable relationship based on data. It is formulated as a latent variable selection problem, for which theoretical analysis and a computationally efficient algorithm are provided.