多层有序分类数据的隶属度分析
Grade of membership analysis for multi-layer ordinal categorical data
AI总结：
本文提出多层GoM模型，用于分析多层有序分类数据，通过GoM-DSoG方法估计混合隶属度，并验证了其在确定潜在类别数量上的有效性。
摘要：
考虑一组参与相同心理测试的个体（受试者），在不同时间点进行多次测试，其中每个问题的选择具有隐含的顺序性。观察到的响应可以记录在多个响应矩阵中，称为多层有序分类数据，其中层指的是时间点。假设每个受试者都有一个共同的混合隶属度，跨所有层共享，使其能够隶属于多个潜在类别，具有不同的权重。等级隶属度（GoM）分析的目标是从数据中估计这些混合隶属度。当测试仅进行一次时，数据变为传统的单层有序分类数据。GoM模型是描述具有潜在混合隶属度结构的单层分类数据的流行选择。然而，GoM无法处理多层有序分类数据。在本工作中，我们提出了一种新的模型，多层GoM，将GoM扩展到多层有序分类数据。为了估计共同的混合隶属度，我们提出了一种新的方法，GoM-DSoG，基于去偏的Gram矩阵求和。我们建立了在多层GoM模型下GoM-DSoG的每个性别收敛率。我们的理论结果表明，较少的无回答、更多的受试者、更多的项目和更多的层对GoM分析是有益的。我们还提出了一种选择潜在类别的方法。广泛的实验研究验证了理论发现，并显示GoM-DSoG在与其竞争者相比时的优越性，以及我们方法在确定潜在类别的准确性。
英文摘要：
Consider a group of individuals (subjects) participating in the same psychological tests with numerous questions (items) at different times, where the choices of each item have an implicit ordering. The observed responses can be recorded in multiple response matrices over time, named multi-layer ordinal categorical data, where layers refer to time points. Assuming that each subject has a common mixed membership shared across all layers, enabling it to be affiliated with multiple latent classes with varying weights, the objective of the grade of membership (GoM) analysis is to estimate these mixed memberships from the data. When the test is conducted only once, the data becomes traditional single-layer ordinal categorical data. The GoM model is a popular choice for describing single-layer categorical data with a latent mixed membership structure. However, GoM cannot handle multi-layer ordinal categorical data. In this work, we propose a new model, multi-layer GoM, which extends GoM to multi-layer ordinal categorical data. To estimate the common mixed memberships, we propose a new approach, GoM-DSoG, based on a debiased sum of Gram matrices. We establish GoM-DSoG's per-subject convergence rate under the multi-layer GoM model. Our theoretical results suggest that fewer no-responses, more subjects, more items, and more layers are beneficial for GoM analysis. We also propose an approach to select the number of latent classes. Extensive experimental studies verify the theoretical findings and show GoM-DSoG's superiority over its competitors, as well as the accuracy of our method in determining the number of latent classes.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
