Structural equation models for functional data

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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.