Due to the increasing availability of highly structured data, data analysis methods that are computationally efficient are necessary to study the dependencies between functional responses. The methods developed in this dissertation are motivated by a set of electroencephalography (EEG) studies in children with autism spectrum disorder (ASD).
The first chapter of this body of work focuses on finding a set of components across dif- ferent conditions within multiple experimental tasks using principle ERP reduction (pERP- RED). These underlying components are referred to as principle ERPs (pERPs) and are found using principal component analysis and independent component analysis. The second chapter then provides a low-dimensional decomposition of the highly structured EEG data through multilevel hybrid principal component analysis (M-HPCA) by decomposing the total variation into between- and within-subjects variation using ideas from ANOVA and separating the variation further using vector and functional principal component analyses. Finally, the third chapter models associations between scalar outcomes, such as a diagno- sis, with region-referenced longitudinal functional predictors, such as EEG data collected across multiple visits, by introducing region-referenced longitudinal functional generalized linear models (RRLF-GLM). Utilizing a tensor product of discrete and continuous bases, the regression coefficient is able to capture the relationship that changes to the EEG response have on the outcome. Each of these three proposed methods can be used to understand the differences in brain responses between typically developing children and children with ASD.