False discovery rates for spatial signals

The problem of multiple testing for the presence of signal in spatial data can involve numerous locations. Traditionally, each location is tested separately for signal presence, but then the findings are reported in terms of clusters of nearby locations. This is an indication that the units of interest for testing are clusters rather than individual locations. The investigator may know a priori these more natural units or an approximation to them. We suggest testing these cluster units rather than individual locations, thus increasing the signal-to-noise ratio within the unit tested as well as reducing the number of hypothesis tests conducted. Because the signal may be absent from part of each cluster, we define a cluster as containing a signal if the signal is present somewhere within the cluster. We suggest controlling the false discovery rate (FDR) on clusters (i.e., the expected proportion of clusters rejected erroneously out of all clusters rejected) or its extension to general weights (WFDR). We introduce a powerful two-stage testing procedure and show that it controls the WFDR. Once the cluster discoveries have been made, we suggest "cleaning" locations in which the signal is absent. For this purpose, we develop a hierarchical testing procedure that first tests clusters, then tests locations within rejected clusters. We show formally that this procedure controls the desired location error rate asymptotically, and conjecture that this is also so for realistic settings by extensive simulations. We discuss an application to functional neuroimaging that motivated this research and demonstrate the advantages of the proposed methodology on an example.