Statistical Modeling of fMRI Data Sample Clauses

Statistical Modeling of fMRI Data. ‌ After obtaining the pre-processed fMRI BOLD time series, the statistical analysis proceeds with the intra-subject modeling, which assesses the single-subject data for the evidence against the null hypothesis of no task- or stimulus-related effects, and then generates the statistic output image for each subject. This intra-subject model can also be embedded in the group model to form a single model investigating all subjects simultaneously for longitudinal data analysis, but normally, it would be more intuitive and more efficient to separate this single statistical model into two parts: first level or intra-subject modeling, and group level or between-subject mod- xxxxx, and extract the information needed from the first level to yield the group data. For each voxel, the first-level or intra-subject modeling is to build and fit the general linear model (GLM) to a single subject’s fMRI BOLD time series with one scan per timepoint (Frackowiak et al., 2004). Assume that the sample size is n. The first- level GLM of subject i (i = 1, . . . , n) for a particular voxel can be described in a matrix form as follows: Yi = Xiβi + si, si ∼ N(0, Vi), where Xx is an si × 1 vector of BOLD time series, Xi is an si × pi design matrix with the use of the predicted hemodynamic response to each task or stimulus as a regressor, βi is the unknown time-invariant pi × 1 regression parameter vector (or effect sizes for all tasks or stimuli), si is the within-subject random error, and Vi is the within-subject covariance matrix (Xxxxxxxx et al., 2003; Frackowiak et al., 2004; Xxxxxx and Xxxxxxx, 2006). Here, si denotes the number of scans for subject i, which can vary between subjects, and pi specifies the number of explanatory variables related to the tasks or stimuli according to the experimental design. Denote the overall voxel-wise output statistic values for the first-level inference as a vector ϕ = (ϕ1, ϕ2, . . . , ϕn)T from these n subjects after ignoring the voxel index. Here ϕ consists of all the information of interest extracted from the first-level inference and its element ϕi (i = 1, . . . , n) can be true values of the parameter vector (ϕi = βi), estimates of the parameter vector (ϕi = βi), contrast (ϕi = c βi), or contrast estimate (ϕi = c βi), where the contrast coefficient vector c is pre-defined according to the experimental requirement. The matrix form of group-level or between-subject GLM for a specified voxel is ϕ = Gθ + η, η ∼ N(0, M), where G is the (Σn pi) ...