False discovery rate revisited: FDR and topological inference using Gaussian random fields

In this note, we revisit earlier work on false discovery rate (FDR) and evaluate it in relation to topological inference in statistical parametric mapping. We note that controlling the false discovery rate of voxels is not equivalent to controlling the false discovery rate of activations. This is a problem that is unique to inference on images, in which the underlying signal is continuous (i.e., signal which does not have a compact support). In brief, inference based on conventional voxel-wise FDR procedures is not appropriate for inferences on the topological features of a statistical parametric map (SPM), such as peaks or regions of activation. We describe the nature of the problem, illustrate it with some examples and suggest a simple solution based on controlling the false discovery rate of connected excursion sets within an SPM, characterised by their volume.