Structural equation models for functional data

心理学和各个领域的研究人员使用结构方程模型(SEM)来说明和测试观察变量和潜在理论结构(通常称为潜在变量)之间的复杂路径分析关系。由于技术的进步，研究人员越来越多地以曲线、曲面或图像的形式收集数据，这些数据随着时间、空间或其他连续性的变化而变化。心理学中收集的这种功能数据的几个例子包括运动控制、欺骗检测、音乐感知、凝视跟踪和功能神经成像的数据。与传统的多变量数据相比，函数数据可以通过高频重复测量来表征，该高频重复测量反映了一个平滑但往往复杂的函数，该函数被假定为生成函数。由于这些独特的特点，一直需要有效地分析功能数据并从这些数据中获得有洞察力的信息。特别是，心理学家对功能数据中复杂的相互依赖关系的规范和测试产生了兴趣。例如，认知神经心理学家使用功能磁共振成像来收集血氧水平依赖(BOLD)信号，这些信号反映了大脑空间元素中的神经活动，称为体素，在一些时间点(扫描)。然后，他们想要测试不同大脑区域在完成认知任务中的重要性，以及它们之间的方向关系。扫描电子显微镜是分析神经功能成像中所谓的有效连通性的自然选择。然而，粗体信号是时间(扫描)和空间(体素)的双变量函数。结构方程目前主要用于多变量数据的分析，因此不能很好地适用于函数数据的分析。因此，拟议研究计划的长期目标是开发用于功能数据分析的扫描电子显微镜。这将进一步推动结构方程的理论和实证创新，这是我多年来一直追求的一个总的研究目标。该计划有两个短期目标：(1)它为同时在两个连续体(例如，时间和空间)上变化的二元函数数据开发了一个通用的结构方程框架--功能结构方程(FSEM)，以及(2)它的目的是扩展功能结构方程以解决高级问题，并增强其通用性和灵活性。这些问题包括对簇级异质性、多级数据、高阶潜在变量和潜在调节变量的分析。该计划涉及FSEM及其扩展的理论发展和实证应用。理论发展包括模型和优化算法的数学推导，以及算法在计算机程序中的实现。经验应用包括通过对各种模拟和真实数据的应用来系统地调查所提出的技术的性能。该程序将在结构方程建模和函数数据分析这两个统计领域做出原创性的理论贡献，因为它将扩展结构方程模型处理函数数据的能力，并拓宽函数数据分析的范围，使其超越传统的回归和数据约简分析。此外，该程序将为研究人员提供一种宝贵的手段，用于检验函数数据和潜在变量之间的各种假设关系。它将有助于吸引和培训来自加拿大和海外的优秀学生，他们有兴趣为这两个统计领域的最新发展做出贡献，同时为与心理学和各个领域的研究人员合作创造大量机会。

Researchers in psychology and various fields have used structural equation modeling (SEM) for the specification and testing of complex path-analytic relationships between observed variables and underlying theoretical constructs, often called latent variables. Owing to advances in technology, researchers have increasingly collected data in the form of curves, surfaces, or images that vary over time, space, or other continua. A few examples of such functional data collected in psychology include data from motor control, deception detection, musical perception, gaze-tracking, and functional neuroimaging. As compared to conventional multivariate data, functional data can be characterized by high-frequency repeated measurements that reflect a smooth but often intricate function, which is assumed to generate them. Due to these distinctive characteristics, there has been a continuing need to analyze functional data effectively and gain insightful information from them. In particular, psychologists have grown an interest in the specification and testing of complex interdependencies in functional data. For example, cognitive neuropsychologists use functional magnetic resonance imaging to collect blood-oxygen level dependent (BOLD) signals that reflect neural activity in spatial elements of the brain, called voxels, over a number of time points (scans). They then want to test the importance of different brain regions in completing a cognitive task, as well as their directional relationships. SEM can be a natural choice for the analysis of such so-called effective connectivity in functional neuroimaging. However, BOLD signals are a bivariate function of time (scan) and space (voxel). SEM is currently geared for the analysis of multivariate data, so that it is not well-suited to the analysis of functional data. Thus, the long-term objective of the proposed research program is to develop SEM for the analysis of functional data. This will further theoretical and empirical innovation in SEM, which is a general research objective that I have pursued over the past years. The proposed program has two short-term objectives: (1) it develops a general SEM framework, Functional SEM (FSEM), for bivariate functional data that vary over two continua simultaneously (e.g., time and space) and (2) it aims to extend FSEM to address advanced issues and enhance its generality and flexibility. These issues include the analyses of cluster-level heterogeneity, multilevel data, higher-order latent variables, and latent moderator variables. The proposed program involves the theoretical development and empirical application of FSEM and its extensions. The theoretical development includes the mathematical derivation of models and optimization algorithms, as well as the implementation of the algorithms into computer programs. The empirical application involves systematic investigations into the performance of the proposed techniques through their application to various simulated and real data. The proposed program will make original, theoretical contributions to two statistical domains of structural equation modeling and functional data analysis, because it will expand the capacity of SEM to deal with functional data and broaden the scope of functional data analysis beyond conventional regression and data-reduction analyses. Moreover, the program will provide researchers with a valuable means for examining various hypothesized relationships between functional data and latent variables. It will contribute to attracting and training outstanding students from Canada and abroad, who are interested in contributing to the latest developments in the two statistical domains, while creating ample opportunities for collaboration with researchers in psychology and various fields.