Statistical nonParametric Mapping: A quick primer
SnPM uses a permutation test to determine the signifiances of voxels and clusters of supra-threshold voxels. A basic understanding of the permutation test is essential for knowledgeable use of the SnPM toolbox. Fortunately, the basic concept is simple and intuitive: For example, consider a simple two sample problem, with one observation being made on each member of the two groups. If there is really no difference between the two groups, we would be fairly surprised if most of the group one observations were larger than the group two observations. Permutation tests provide a formal mechanism for quantifying this "surprise", leading to significance tests.
SPM - assumptions & low df problems...
The parametric approach embodied in SPM uses the General Linear Model to produce a statistic at every voxel, and assesses the resulting statistic image for significant regions using distributional approximations for a continuous random fields with the same marginal distribution and smoothness. This explicitly assumes that the data are derived from strictly stationary homogeneous discrete Gaussian random fields. Further, it is implicitly assumed that the statistic image is sufficiently smooth that it's properties may be approximated by a continuous random field. However t (& F) statistic images with low (denominator) degrees of freedom are very noisy (due to the unreliability of voxel variance estimates from low numbers of observations). These noisy t & F-statistic images have such low estimated smoothness that a continuous random field with the same smoothness would have features at sub-voxel resolution. The result is that these features affect the distribution of the maximal statistic of the continuous field, against which the voxel values of the statistic image are compared for significance, resulting in a conservative test (i.e. significance is under-estimated).
SnPM - a primer...
In contrast to the multitude of assumptions and approximations of the parametric approach, the non-parametric approach exploits the design of the experiment (randomisation test), or makes simple intuitive distributional assumptions, such as symmetry (permutation test). The thinking is simple: If there is really no experimental effect, then the experimental labels, whether A's and B's (for an activation study) or a set of real numbers (for a covariate study), are arbitrary. Any reallocation of the labels to the scans would lead to an equally plausible statistic image. So, considering the statistic images associated with all possible re-labellings of the data, we can derive the distribution of statistic images possible for this data. We can then test our hypothesis of no experimental effect by comparing the statistic for the actual labelling of the experiment with this distribution. Effectively we quantify our surprise at the observed findings on the basis of what we would expect were there no experimental effect! If, out of N possible relabellings the actual labelling gives the rth most extreme statistic, then the p-value is r/N. The details are worked out in the references.
SPM makes many strong assumptions about the nature of your data, while SnPM makes a few weak assumptions: SPM assumes that your data, at each voxel, are normally distributed and, across voxels, are derived from continuous random fields with a stationary covariance structure. SnPM assumes that, under the null hypothesis of no experimental effect, you can switch the experimental labels of your data and always expect the same (null) result.
The freedom of SnPM from parametric distributional assumptions allows us to pool variance estimates over neighbouring voxels, under the mild assertion that the true variance image is smooth, giving additional degrees of freedom. The Pseudo-t statistics computed with such a smoothed variance image don't exhibit the noise of low df t-statistic images, This variance smoothing circumvents the low df noisy statistic image problem discussed above, giving the non-parametric approach greater power than it's parametric counterpart.
Understanding the assumptions of SPM requires an understanding of normality, independence and the basics of Gaussian random field theory. Understanding the assumptions of SnPM requires an understanding of exchangability. See the PET example page for a brief introduction to exchangability and the literature for more detailed discussion.
Deptment of Statistics
University of Warwick
Handbook of fMRI Data Analysis by Russ Poldrack, Thomas Nichols and Jeanette Mumford