  Cluster-level inference procedures are widely used for brain mapping. These
methods compare the size of clusters obtained by thresholding brain maps to an
upper bound under the global null hypothesis, computed using Random Field
Theory or permutations. However, the guarantees obtained by this type of
inference - i.e. at least one voxel is truly activated in the cluster - are not
informative with regards to the strength of the signal therein. There is thus a
need for methods to assess the amount of signal within clusters; yet such
methods have to take into account that clusters are defined based on the data,
which creates circularity in the inference scheme. This has motivated the use
of post hoc estimates that allow statistically valid estimation of the
proportion of activated voxels in clusters. In the context of fMRI data, the
All-Resolutions Inference framework introduced in [25] provides post hoc
estimates of the proportion of activated voxels. However, this method relies on
parametric threshold families, which results in conservative inference. In this
paper, we leverage randomization methods to adapt to data characteristics and
obtain tighter false discovery control. We obtain Notip, for Non-parametric
True Discovery Proportion control: a powerful, non-parametric method that
yields statistically valid guarantees on the proportion of activated voxels in
data-derived clusters. Numerical experiments demonstrate substantial gains in
number of detections compared with state-of-the-art methods on 36 fMRI
datasets. The conditions under which the proposed method brings benefits are
also discussed.
