Spatial Confidence Regions for Excursion Sets with False Discovery Rate Control

TL;DR

The paper tackles spatial inference in neuroimaging by constructing confidence regions for excursion sets at a threshold $c$ with false discovery rate control, addressing the null-hypothesis fallacy common in large-sample brain analyses. It introduces one-sided, directionally controlled tests to form separate upper and lower confidence regions and extends to a joint framework that integrates both directions via a Benjamini–Hochberg procedure, with an optional two-stage adaptive procedure to boost power. The methodology is validated through extensive simulations that demonstrate robust FDR control and improved power, and is applied to Human Connectome Project fMRI data where the FDR-controlled regions yield tighter, more interpretable localization than prior methods like SSS. The work provides practical gains for accurate brain mapping and offers flexible options between separate and joint error control, with potential extensions to other effect measures and dependency structures.

Abstract

Identifying areas where the signal is prominent is an important task in image analysis, with particular applications in brain mapping. In this work, we develop confidence regions for spatial excursion sets above and below a given level. We achieve this by treating the confidence procedure as a testing problem at the given level, allowing control of the False Discovery Rate (FDR). Methods are developed to control the FDR, separately for positive and negative excursions, as well as jointly over both. Furthermore, power is increased by incorporating a two-stage adaptive procedure. Simulation results with various signals show that our confidence regions successfully control the FDR under the nominal level. We showcase our methods with an application to functional magnetic resonance imaging (fMRI) data from the Human Connectome Project illustrating the improvement in statistical power over existing approaches.

Figures (18)

• Figure 1: An illustration of confidence regions, developed in this work, applied to task-based fMRI: red area denotes upper confidence region $\hat{\mathcal{A}_c^+}$, blue area denotes lower confidence region $\hat{\mathcal{A}_c^-}$, and the yellow area denotes the point estimate for the excursion set $\hat{\mathcal{A}_c}$. The rows differ in threshold $c$ in the % blood-oxygen-level-dependent (BOLD) change, and the columns differ in confidence region construction methods.
• Figure 2: Upper confidence region schematic: (a) Example signal and the projection of the excursion set above $c$ ($\mathcal{A}_c$). (b) A hypothetical upper confidence region $\hat{\mathcal{A}}_c^+$ superimposed on the ground truth $\mathcal{A}_c$. (c) The division of (b) into regions corresponding to those specified in Table \ref{['table-upper_division']} as a visual representation of false positives (green) and false non-positives (orange).
• Figure 3: Lower confidence region schematic: (a) Example signal and the projection of the excursion set at and above $c$ ($\mathcal{\underline{A}}_c$). (b) A hypothetical lower confidence region $\hat{\mathcal{A}}_c^-$ superimposed on the ground truth $\mathcal{\underline{A}}_c$. (c) The division of (b) into regions corresponding to those specified in Table \ref{['table-lower_division']} as a visual representation of false positives (green) and false non-positives (orange).
• Figure 4: Upper and lower confidence regions from joint control method schematic: (a) Example signal and the projection of the excursion set above $c$ ($\mathcal{A}_c$) and at and above $c$ ($\mathcal{\bar{A}}_c$). (b) Hypothetical upper lower confidence regions $\hat{\mathcal{A}}_c^+$ and $\hat{\mathcal{A}}_c^-$ superimposed on the ground truth $\mathcal{A}_c$ and $\mathcal{\bar{A}}_c$ respectively. (c) The division of (b) into regions corresponding to those specified in Table \ref{['table-joint_division']} as a visual representation of false positives (green) and false non-positives (orange).
• Figure 5: Synthetic images used for confidence region illustration. The first row shows ramp, step, and circle signals. The second row shows the same synthetic images with added smoothed Gaussian noise. The pixels in the noise field follows $N(0, 1.5^2)$, constituting an uncorrelated Gaussian noise field. The noise field is smoothed using FWHM 8 pixels.