The functional magnetic resonance imaging (fMRI) technique is widely used in the medical field because it allows the in vivo investigations of human cognition, emotions, and behaviors at the neural level. One primary objective is to study brain activation, which can be achieved through a conventional two-stage approach. We consider the individualized voxel-specific modeling in the first stage and group-level inference in the second stage. Existing methods, in general, rely on pre-determined parameters or domain knowledge, which may not properly incorporate the unique features from different studies or cohorts, and thus also leads to some gaps in the inference for activated regions. This dissertation focuses on Bayesian approaches to fill the gaps in statistical inference at all levels, as well as accounting for the various information carried out by the data.

Cluster-wise statistical inference is the most widely used technique for fMRI data analyses. It consists of two steps: i) primary thresholding that excludes less significant voxels by a pre-specified cut-off (e.g., p<0.001); and ii) cluster-wise thresholding that is often obtained by counting the number of intra-cluster voxels which surpass a voxel-level statistical significance threshold.

The selection of the primary threshold is critical because it determines both statistical power and false discovery rate. However, in most existing statistical packages, the primary threshold is selected based on prior knowledge (e.g., p<0.001) without considering the information in the data. Thus, in the first project, we propose a data-driven approach to algorithmically select the optimal primary threshold based on an empirical Bayes framework. We evaluate the proposed model using extensive simulation studies and real fMRI data. In the simulation, we show that our method can effectively increase statistical power while controlling the false discovery rate. We then investigate the brain response to the dose effect of chlorpromazine in patients with schizophrenia by analyzing fMRI scans and generating consistent results.

In Chapter 3, we focus on controlling the FWER by conducting cluster-level inference. The cluster-extent measure can be sub-optimal regarding the power and false positive error rate because the supra-threshold voxel count neglects the voxel-wise significance levels and ignores the dependence between voxels. Based on the information that a cluster carries, we provide a new Integrated Cluster-wise significance Measure (ICM) for cluster-level significance determination in cluster-wise fMRI analysis by integrating cluster extent, voxel-level significance (e.g., p-values), and activation dependence between within-cluster voxels. We develop a computationally efficient strategy for ICM based on probabilistic approximation theories. Consequently, the computational load for ICM-based cluster-wise inference (e.g., permutation tests) is affordable. We validate the proposed method via extensive simulations and then apply it to two fMRI data sets. The results demonstrate that ICM can improve power with well-controlled FWER.

The above chapters focus on the cluster-extent thresholding method, while the Bayesian hierarchical model can also efficiently handle high-dimensional neuroimaging data. Existing methods provide voxel-specific and pre-determined regional (region of interest (ROI)) inference. However, the activation clusters may be across multiple ROIs or vary from studies and study cohorts. To provide the inference and build the bridge between voxels, unknown activation clusters, targeted regions, and the whole brain, we propose the Dirichlet Process Mixture model with Spatial Constraint (DPMSC) in Chapter 4. The spatial constraint is based on the Euclidean distance between two voxels in the brain space. With such a constraint added at each iteration in Markov Chain Monte Carlo (MCMC), our DPMSC can efficiently remove the single voxel or small noise clusters, as well as provide a whole contiguous cluster that belongs to the same component in the mixture model. Finally, we provide a real data example and simulation studies based on various dataset features.