This paper is about detecting activations in statistical parametric maps and considers the relative sensitivity of a nested hierarchy of tests that we have framed in terms of the level of inference (voxel level, cluster level, and set level). These tests are based on the probability of obtaining c, or more, clusters with k, or more, voxels, above a threshold u. This probability has a reasonably simple form and is derived using distributional approximations from the theory of Gaussian fields. The most important contribution of this work is the notion of set-level inference. Set-level inference refers to the statistical inference that the number of clusters comprising an observed activation profile is highly unlikely to have occurred by chance. This inference pertains to the set of activations reaching criteria and represents a new way of assigning P values to distributed effects. Cluster-level inferences are a special case of set-level inferences, which obtain when the number of clusters c = 1. Similarly voxel-level inferences are special cases of cluster-level inferences that result when the cluster can be very small (i.e., k = 0). Using a theoretical power analysis of distributed activations, we observed that set-level inferences are generally more powerful than cluster-level inferences and that cluster-level inferences are generally more powerful than voxel-level inferences. The price paid for this increased sensitivity is reduced localizing power: Voxel-level tests permit individual voxels to be identified as significant, whereas cluster-and set-level inferences only allow clusters or sets of clusters to be so identified. For all levels of inference the spatial size of the underlying signal f (relative to resolution) determines the most powerful thresholds to adopt. For set-level inferences if f is large (e.g., fMRI) then the optimum extent threshold should be greater than the expected number of voxels for each cluster. If f is small (e.g., PET) the extent threshold should be small. We envisage that set-level inferences will find a role in making statistical inferences about distributed activations, particularly in fMRI.
