Group Comparison of Eigenvalues and Eigenvectors of Diffusion Tensors

Diffusion tensor imaging (DTI) data differ from most medical images in that values at each voxel are not scalars, but 3×3 symmetric positive definite matrices called diffusion tensors (DTs). The anatomic characteristics of the tissue at each voxel are reflected by the DT eigenvalues and eigenvectors. In this article we consider the problem of testing whether the means of two groups of DT images are equal at each voxel in terms of the DT’s eigenvalues, eigenvectors, or both. Because eigendecompositions are highly nonlinear, existing likelihood ratio statistics (LRTs) for testing differences in eigenvalues or eigenvectors of means of Gaussian symmetric matrices assume an orthogonally invariant covariance structure between the matrix entries. While retaining the form of the LRTs, we derive new approximations to their true distributions when the covariance between the DT entries is arbitrary and possibly different between the two groups. The approximate distributions are those of similar LRT statistics computed on the tangent space to the parameter manifold at the true value of the parameter, but plugging in an estimate for the point of application of the tangent space. The resulting distributions, which are weighted sums of chi-squared distributions, are further approximated by scaled chi-squared distributions by matching the first two moments. For validity of the Gaussian model, the positive definite constraints on the DT are removed via a matrix log transformation, although this is not crucial asymptotically. Voxelwise application of the test statistics leads to a multiple-testing problem, which is solved by false discovery rate inference. The foregoing methods are illustrated in a DTI group comparison of boys versus girls.