The problem of multiple testing for the presence of signal in spatial data can involve a large number of 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 interests 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 hypotheses tests conducted. Since the signal may be absent from part of each cluster, we define a cluster as containing signal if the signal is present in at least one subset of the cluster. We introduce powerful adaptive procedures for controlling the false discovery rate (FDR) on clusters, i.e. the proportion of clusters rejected erroneously out of all clusters rejected, or the size weighted FDR on clusters, which captures the size of erroneously rejected clusters out of the total size of clusters rejected. Moreover, we prove that the adaptive weighted procedure controls the weighted FDR. 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 controls the expected proportion of locations in which false rejections occur under the fixed alternative model and show that it controls the desired error rates under less idealistic assumptions by extensive simulations. We discuss an application to functional neuroimaging which motivated this research and demonstrate the advantages of the proposed methodology on an example.
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