Classical and Bayesian Inference in fMRI - Advance Image Processing in Magnetic Resonance Imaging - page 540

are not independent by virtue of continuity in the original data. Provided the data
are sufficiently smooth, the GRF correction is less severe (i.e., is more sensitive)
than a Bonferroni correction for the number of voxels. As noted in the preceding
text, the GRF theory deals with the multiple comparisons problem in the context
of continuous, spatially extended statistical fields, in a way that is analogous to
the Bonferroni procedure for families of discrete statistical tests. There are many
ways to appreciate the difference between GRF and Bonferroni corrections.
Perhaps the most intuitive is to consider the fundamental difference between an
SPM and a collection of discrete T values. When declaring a connected volume
or region of the SPM to be significant, we refer collectively to all the voxels that
comprise that volume. The false positive rate is expressed in terms of connected
(excursion) sets of voxels above some threshold, under the null hypothesis of no
activation. This is not the expected number of false positive voxels. One false
positive region may contain hundreds of voxels, if the SPM is very smooth. A
Bonferroni correction would control the expected number of false positive voxels,
whereas GRF theory controls the expected number of false positive regions.
Because a false positive region can contain many voxels, the corrected threshold
under a GRF correction is much lower, rendering it much more sensitive. In fact
the number of voxels in a region is somewhat irrelevant because the correction
is a function of smoothness. The GRF correction discounts voxel size by express-
ing the search volume in terms of smoothness or resolution elements (resels).
This intuitive perspective is expressed formally in terms of differential topology
using the Euler characteristic (23). At high thresholds the Euler characteristic
corresponds to the number of regions exceeding the threshold.
There are only two assumptions underlying the use of the GRF correction:
(1) The error fields (but not necessarily the data) are a reasonable lattice approx-
imation to an underlying random field with a multivariate Gaussian distribution
and (2) these fields are continuous, with a differentiable and invertible autocor-
relation function. A common misconception is that the autocorrelation function
has to be Gaussian. It does not. The only way in which these assumptions can
be violated is if the data are not smoothed (with or without subsampling to
preserve resolution), violating the reasonable lattice assumption or the statistical
model is misspecified so that the errors are not normally distributed. Early for-
mulations of the GRF correction were based on the assumption that the spatial
correlation structure was wide-sense stationary. This assumption can now be
relaxed due to a revision of the way in which the smoothness estimator enters
the correction procedure (28). In other words, the corrections retain their validity,
even if the smoothness varies from voxel to voxel.

17.5 POSTERIOR PROBABILITY MAPPING

Despite its success, SPM has a number of fundamental limitations. In SPM, the
p value, ascribed to a particular effect, does not reflect the likelihood that the effect
is present but simply the probability of getting the observed data in the effect's
absence. If sufficiently small, this p value can be used to reject the null hypothesis