This paper discusses the detection of brain activations in functional neuroimaging, particularly in fMRI and PET. It introduces a hierarchy of statistical inference levels—voxel, cluster, and set levels—to assess the sensitivity and power of different methods for detecting activations. The key idea is that set-level inference, which considers the number of clusters, is more powerful than cluster-level inference, which in turn is more powerful than voxel-level inference. However, set-level inference has less localizing power compared to voxel-level inference. Set-level inference involves determining the probability of observing a certain number of clusters with a specific size above a threshold. This approach allows for statistical inference about distributed activations, particularly in fMRI. The paper also discusses the importance of spatial extent and signal characteristics in determining the most powerful thresholds for each level of inference. For fMRI, where signals are large relative to resolution, the optimal extent threshold should be greater than the expected number of voxels per cluster. For PET, where signals are small, the extent threshold should be smaller. The paper emphasizes that set-level inferences are more powerful for distributed signals and can provide a more comprehensive characterization of activation profiles. However, they require careful interpretation to avoid overestimating regional specificity. The paper also addresses the interpretation of set-level inferences, noting that they can be used in an omnibus sense when the number of observed clusters significantly exceeds the expected number by chance. This distinction is crucial for determining the validity of distributed activation inferences. The study concludes that set-level inferences are particularly useful in fMRI and can provide a more accurate and comprehensive understanding of brain activations.This paper discusses the detection of brain activations in functional neuroimaging, particularly in fMRI and PET. It introduces a hierarchy of statistical inference levels—voxel, cluster, and set levels—to assess the sensitivity and power of different methods for detecting activations. The key idea is that set-level inference, which considers the number of clusters, is more powerful than cluster-level inference, which in turn is more powerful than voxel-level inference. However, set-level inference has less localizing power compared to voxel-level inference. Set-level inference involves determining the probability of observing a certain number of clusters with a specific size above a threshold. This approach allows for statistical inference about distributed activations, particularly in fMRI. The paper also discusses the importance of spatial extent and signal characteristics in determining the most powerful thresholds for each level of inference. For fMRI, where signals are large relative to resolution, the optimal extent threshold should be greater than the expected number of voxels per cluster. For PET, where signals are small, the extent threshold should be smaller. The paper emphasizes that set-level inferences are more powerful for distributed signals and can provide a more comprehensive characterization of activation profiles. However, they require careful interpretation to avoid overestimating regional specificity. The paper also addresses the interpretation of set-level inferences, noting that they can be used in an omnibus sense when the number of observed clusters significantly exceeds the expected number by chance. This distinction is crucial for determining the validity of distributed activation inferences. The study concludes that set-level inferences are particularly useful in fMRI and can provide a more accurate and comprehensive understanding of brain activations.