The term latent variable model (LVM) refers to any statistical procedure that utilizes information contained in a set of observed variables to construct a set of underlying latent variables that drive the observed values and associations. Independent component analysis (ICA) is a LVM that separates recorded mixtures of signals into independent source signals, called independent components (ICs). ICA is popular tool for separating brain signals of interest from artifacts and noise in electroencephalogram (EEG) data. Due to challenges in the estimation of uncertainties in ICA, standard errors are not generally estimated alongside ICA estimates and thus ICs representing brain signals of interest cannot be distinguished through a statistical hypothesis testing framework. In Chapter 2 of this dissertation, we propose a bootstrapping algorithm for ICA that produces bootstrap samples that retain critical correlation structures in the data. These are used to compute uncertainties for ICA parameter estimates and to construct a hypothesis test to identify ICs representing brain activity, which we demonstrate in the context of EEG functional connectivity. In Chapter 3, we extend this bootstrapping approach to accommodate pre-ICA dimension reduction procedures, and we use the resulting method to compare popular strategies for pre-ICA dimension reduction in EEG research. In the final chapter, we turn our attention to another LVM, factor analysis, which utilizes the covariance structure of a set of correlated observed variables to model a smaller number of unmeasured underlying variables. A spatial factor analysis (SFA) model can be used to quantify the social vulnerability of communities based on a set of observed social variables. Current SFA methodology is ill-equipped to handle spatial misalignment in the observed variables. We propose a joint spatial factor analysis model that identifies a common set of latent variables underlying spatially misaligned observed variables and produces results at the level of the smallest spatial units, thereby minimizing loss of information. We apply this model to spatially misaligned data to construct an index of community social vulnerability for Louisiana, which we integrate with Louisiana flood data to identify communities at high risk during natural disasters, based on both social and geographic features.Doctor of Philosoph
