Transformation invariant control of voxel-wise false discovery rate.

Multiple testing for statistical maps remains a critical and challenging problem in brain mapping. Since the false discovery rate (FDR) criterion was introduced to the neuroimaging community a decade ago, many variations have been proposed, mainly to enhance detection power. However, a fundamental geometrical property known as transformation invariance has not been adequately addressed, especially for the voxel-wise FDR. Correction of multiple testing applied after spatial transformation is not necessarily equivalent to transformation applied after correction in the original space. Without the invariance property, assigning different testing spaces will yield different results. We find that normalized residuals of linear models with Gaussian noises are uniformly distributed on a unit high-dimensional sphere, independent of t-statistics and F-statistics. By defining volumetric measure in the hyperspherical space mapped by normalized residuals, instead of the image's Euclidean space, we can achieve invariant control of the FDR under diffeomorphic transformation. This hyperspherical measure also reflects intrinsic "volume of randomness" in signals. Experiments with synthetic, semi-synthetic and real images demonstrate that our method significantly reduces FDR inconsistency introduced by the choice of testing spaces.