Event-related potential (ERP) studies are a subset of experimental frameworks within the field of electroencephalography (EEG) that focus on ERPs, the electrical potential outputted by a subject's brain when presented with an implicit task in the form of stimuli. Data comprise an ERP repetition observed for each stimulus across electrodes on the scalp, producing a complex data structure consisting of a functional, longitudinal and spatial dimension. In typical ERP studies, the dimension of data is reduced into a single measure for each subject by cross-sectionally averaging ERP across longitudinal and spatial repetitions. Features are then extracted from the averaged ERP and analyzed using simple statistical methods, ignoring additional information that may be found in the collapsed dimensions. In this dissertation, three types of methodology are proposed for preserving and analyzing the lost dimensions of ERP data. The first method, moving average processed ERP (MAP-ERP), is a two-step approach comprised of a meta-preprocessing step to preserve longitudinal information and a weighted mixed effects regression framework to allow modeling of the resulting meta-preprocessed data. The proposed robust functional clustering (RFC) algorithm identifies substructures in features of the longitudinal ERP processes while accounting for subject-level covariance heterogeneity induced by meta-preprocessing. Finally, the proposed multidimensional functional principal components analysis (MD-FPCA) utilizes a two-stage procedure to summarize important characteristics across all three dimensions of the ERP data structure into an interpretable, low-dimensional form. Proposed methods are applied to a study on neural correlates of visual implicit learning in young children with autism spectrum disorder (ASD). Applications of the proposed methods reveal meaningful trends and substructures in the implicit learning processes of ASD children when compared to typically developing controls. Results indicate proposed methodology effectively preserves important information contained within the multiple dimensions of ERP.

Mobile technology has become increasingly popular in the past decade through the combination of device portability and advances in network and internet technology. Smart phones, in particular, are at the forefront of these technologies, enabling users to remotely track and further involve themselves in the management of personal health through participatory mobile health applications. Thus far, the majority of participatory applications currently provide users with data visualizations displaying information germane to the user's medical condition, however, there is still a need for in-depth exploratory and inference-based data analysis using advanced statistical methods to maximize the discernment of potential implications carried in these data streams. This paper provides an overview of the structure of participatory data using an asthma management application as an initial platform and discusses several directions for statistical analysis motivated by three usage cases: individuals using the application, creators of the application, and the scientific community. Methods include functional and semi-parametric data analysis, mixed modeling, and clustering methods to model variables representing asthma wellness as a function of subject-specific, population level, and latent spatiotemporal factors. Societal implications are also discussed.