A Novel Test Statistic allowing a General Linear Contrast Vector for Local Canonical Correlation Analysis in fMRI

Introduction Local canonical correlation analysis (CCA) is a multivariate method that takes into account the spatial correlation of neighboring voxels rather than treating each voxel time series individually. Mathematically, CCA is a generalization of the general linear model (GLM), and is defined by maximizing the correlation of a linear combination of voxel time series in a local region and a linear combination of set of temporal basis functions (regressors). The unknown coefficients of these linear combinations are determined by CCA. In this research, we derive a novel statistic using the vector of the temporal regressors in CCA that project the original space of basis functions onto a vector space maximizing the correlation with the observed voxel time series and allows for an arbitrary temporal contrast vector. This statistic makes CCA feasible for multiple-regressor designs. A non-parametric approach [1] is adapted to estimate the family-wise error rate of the newly developed CCA statistic. Theory Considering a group of K local neighboring voxels, the multivariate multiple-regression model can be written as: Y=XB+E (1) where X is fixed (i.e. the n×p design matrix), Y=(y1,..., yK) is the matrix containing K neighboring voxels, B=(β1,..., βK) is the parameter matrix to be estimated, and E=(ε1,..., εK) is the error matrix. The least-squares solution of the model (1) is B*=(X’X)X’Y (2) We can see that the estimator in Eq. (2) is just the matrix form of the GLM leading to equivalent solutions. However, the hypothesis tests in the multivariate case are different on including the interactions among voxels, such as Wilks’ Λ [2]. In order to increase detection power of weak activations, local spatial smoothing is usually applied to decrease the noise variance. Let α be the vector containing the spatial smoothing coefficients. Multiplication of both sides of Eq. (2) with α gives Yα=Xβ+ε (3) where β≡Bα and ε≡Eα. In conventional fixed Gaussian smoothing, both Y and α are fixed and treated as known, and only β has to be estimated, i.e. y≡Yα=Xβ+ε (*). In the formalism of CCA, both α and β have to be determined simultaneously to achieve maximum canonical correlation. The vector α can be treated as a locally adaptive spatial (smoothing) kernel (with positive constraint). Based on CCA with the unit variance requirement of canonical variables, the solution of (3) can be derived as 1 1 1 c xx xy c c r r − = = β S S α βα % % % % (4)