Many fields of science nowadays gather data at a very fine resolution but do inference at a higher aggregated level. For example, in neuroimaging data are gathered at the level of 3 mm × 3 mm × 3 mm voxels, but the relevant biology happens at the level of cm-scale brain areas; in genetics, data are gathered at the level of single-DNA-base polymorphisms, but interesting questions happen at the level of genes or even gene groups; in spatial statistics, data may be gathered at street level but interesting questions are about neighbourhoods or regions. Often, there is not just one natural way to aggregate data to prepare for inference. Multiple alternative criteria could be used to drive the grouping. Aggregation to large regions may give low specificity; more limited aggregation may give low power.
This talk presents how Closed Testing can be used to analyze this type data at all resolutions simultaneously. The method allows the choice how and how much to aggregate to be chosen freely by the researcher, in a data-dependent way, while still strictly controlling the probability of false positive findings. This allows researchers to adapt the inference to the amount and the shape of the signal that is present in the data: the stronger the signal, the better it will be pinpointed by the closed testing procedure.
I will review the general idea and theory of closed testing and recent progress in method development in this area. Several example contexts illustrate the wide applicability of all-resolutions inference.