Functional magnetic resonance imaging (fMRI) is an advanced technique in obtaining images of brains that are undertaking tasks. It can be very useful in studies of many diseases that are related to malfunction of the brain structures. Whether or not volume elements (voxels) within a region or regions of the brain of a subject are responding to the designed stimulus challenges could be used to distinguish between brain activations of unaffected patients and those with brain abnormalities. Appropriate statistical methodologies to analyze fMRI data play an essential role in investigating the status of brain voxels.

Using fMRI techniques, signals are recorded sequentially at hundreds of thousands voxel locations of the brain. These fMRI data are very complex with a low signal-to-noise ratios and both spatial and temporal correlations. One of the most popular methods for analyzing fMRI data uses a general linear model (GLM) for univariate time series of fMRI signals at each voxel location of the brain. The spatial correlations are not explicitly modeled and are only indirectly used to adjust the thresholds of the individual test statistics for each voxel.

This dissertation offers a strong alternative, which does incorporate both the temporal correlations and the spatial correlations in a spatiotemporal model with spatially varying parameters. The available scientific information about the fMRI data collection process helps to specify the components in the model. Spatial correlation structures of the model parameters are quantified by spatial semivariograms. A Bayesian hierarchical modeling approach is used to take into account the spatial dependency among the parameters in the model fitting. The results show that this Bayesian hierarchical model is able to provide strong evidence of the presence of activated voxels.