Neuroimaging produces data that are continuous in one or more dimensions.
This calls for an inference framework that can handle data that approximate
functions of space, for example, anatomical images, time--frequency maps and
distributed source reconstructions of electromagnetic recordings over time.
Statistical parametric mapping (SPM) is the standard framework for whole-brain
inference in neuroimaging: SPM uses random field theory to furnish p-values
that are adjusted to control family-wise error or false discovery rates, when
making topological inferences over large volumes of space. Random field theory
regards data as realizations of a continuous process in one or more dimensions.
This contrasts with classical approaches like the Bonferroni correction, which
consider images as collections of discrete samples with no continuity
properties (i.e., the probabilistic behavior at one point in the image does not
depend on other points). Here, we illustrate how random field theory can be
applied to data that vary as a function of time, space or frequency. We
emphasize how topological inference of this sort is invariant to the geometry
of the manifolds on which data are sampled. This is particularly useful in
electromagnetic studies that often deal with very smooth data on scalp or
cortical meshes. This application illustrates the versatility and simplicity of
random field theory and the seminal contributions of Keith Worsley
(1951--2009), a key architect of topological inference.Comment: Published in at http://dx.doi.org/10.1214/10-AOAS337 the Annals of
Applied Statistics (http://www.imstat.org/aoas/) by the Institute of
Mathematical Statistics (http://www.imstat.org