Non-stationary FWHM and its effect on statistical inference of fMRI data

Keith Worsley
Department of Mathematics and Statistics and Montreal Neurological Institute, McGill University, Canada

Abstract
We propose an improved method for estimating the local FWHM or residual smoothness that corrects for two sources of bias encountered with fMRI or VBM data (but not usually PET data):

1. Low FWHM relative to voxel size, resulting in discontinuous spatial sampling.
2. Low residual degrees of freedom (df) from a mixed effects analyses for combining a small number of runs, sessions or subjects.

We then show how to account for this in statistical inference for T or F statistic images.

Estimating the local FWHM

To overcome the bias of the simple estimator of FWHM and resel density (=1/FWHM^3) based on numerical spatial derivatives of normalized linear model residuals (Kiebel et al., 1999) we propose:

1. A continuity correction based on a Taylor series expansion of the bias in terms of the variance matrix of the numerical derivatives. The correction is fast and accurate to within 1/10 of the voxel size for FWHM > voxel size.
2. A correction factor for both the simple estimator and its continuity correction that depends on the df, chosen so that the estimated FWHM is unbiased.

Statistical inference

Here the key quantity is the resels of the search region or cluster, defined as the sum over voxels of (resel density)*(voxel volume) (Worsley et al., 1999). Since they are small, cluster resels are the most random, so we propose a simple modification to their P-values for T or F statistics (Cao, 1999) that assumes that the estimated resel density is locally flat, an upper bound if this is not so. For mixed effects analyses, this correction depends on the df both before and after spatial regularization (Worsley et al. 2000).

Results

The above methods were implemented in MATLAB (http://www.math.mcgill.ca/keith/fmristat) and applied to an fMRI experiment in which a painful heat stimulus alternated with a warm (neutral) stimulus, interspersed by baselines (no stimulus). The voxel size was 2.3x2.3x7 mm and the data was smoothed in-slice by 6mm during motion correction. There were 4 runs of 120 frames (112 df per run).

The figure shows the estimated local FWHM (a) over scans (112*4=448 df), and (b) over runs (3 df) based on a mixed effects analysis of the 4 hot–warm pain effects. Note first the reassuring 6mm FWHM outside the brain, more evident in (c), which validates the correction factors. Note the increased FWHM (~10mm) in cortical areas, suggesting that the BOLD responses of neighboring cortical voxels are synchronized. The FWHM over runs is slightly higher than the FWHM over scans (d), suggesting that the pain response is slightly more synchronised than the scans.

There were 10 significant (P<0.05) clusters in the pain T statistic using the local FWHM, whereas there were 16 using the smaller 6mm uniform FWHM over the entire brain. This suggests that using the FWHM applied by motion correction results in too many false positive clusters.

References

Cao (1999). Advances in Applied Probability, 31:579-595
Kiebel et al., (1999) NeuroImage, 10:756-766
Worsley et al., (1999) Human Brain Mapping, 8:98-101
Worsley et al., (2000) NeuroImage, 11:S648
Figure 1: Estimated local FWHM for the fMRI data