**Project Heads**

Fabian Telschow

**Project Members**

N.N.

**Project Duration**

01.10.2021 − 30.09.2024

**Located at**

HU Berlin

We develop a mathematical frameworks for inference on level sets of random fields based on controlling FDR-type error criteria. This will allow for quantication of uncertainties in imaging data and its practical value will be demonstrated on large gold standard 3D fMRI imaging data sets.

**Background**

Mass-univariate and cluster testing strategies corrected by multiple testing procedures are widely used

statistical tools for localized inference on random fields (e.g., functional magnetic resonance imaging (fMRI) [1]. However, these methods suffer from at least two fundamental limitations: First, with sufficient sample sizes there is enough statistical power to reject the null hypothesis everywhere, making it difficult to localize effects of interest in large sample size applications.

Second, significant p-values in cluster-size inference only indicate that a cluster

is larger than chance and therefore giving neither a notion of confidence for repeated sampling of the actual size nor of the location of a detected signal.

In [2] these limitations have been addressed using Confidence Probability (CoPE) sets, which formalize localized inference for the excursion set of an underlying signal above a threshold c. These methods have been generalized in [3] and [4] to cover 3D fMRI imaging experiments and allowing to replace the signal by effect size measures. The latter enables better comparability of experiments across different MRI scanners. CoPE sets control an error criterion analogous to the family-wise error rate (FWER) in statistical testing. The FWER, however, is known to be too conservative in applications if the emphasis is on detection of true effects rather than guarding against any single false detection. Hence parts of the neuroimaging literature shifted to testing significance of topological features like clusters [5] which suffer from the localization and size limitations discussed earlier.

**Scope**

The false discovery rate (FDR) is another error criterion proposed to solve the low detection power of FWER methods in classical testing. It increases detection power by controlling the expected number of false discoveries instead of

the probability of having at least one false discovery. The main objectives of the project are therefore (i) developing a sound mathematical framework and theory of CoPE sets using an error cirterion similar to FDR and removing thereby the limitations of topological testing and the low detection power of current CoPE sets, (ii) provide user-friendly software in form of an MatLab toolbox for imaging data up to dimension 3 and (iii) demonstrate usability, validity and performance of the proposed methods in neuroimaging using for example the UK biobank.

**External Website**

**Selected Publications
**

[1] Thomas E Nichols. Multiple testing corrections, nonparametric methods, and random field theory. Neuroimage, 62(2):811–815, 2012.

[2] Max Sommerfeld, Stephan Sain, and Armin Schwartzman. Confidence regions for spatial excursion sets from repeated random field observations, with an application to climate. Journal of the American Statistical Association, 113(523):1327–1340, 2018.

[3] Alexander Bowring, Fabian JE Telschow, Armin Schwartzman, and Thomas E Nichols.

Spatial confidence sets for raw effect size images. NeuroImage, 203:116187, 2019.

[4] Alexander Bowring, Fabian JE Telschow, Armin Schwartzman, and Thomas E Nichols. Confidence sets for cohen’s d effect size images. NeuroImage, 226:117477, 2021.

[5] Justin R Chumbley and Karl J Friston. False discovery rate revisited: Fdr and topological inference using gaussian random fields. Neuroimage, 44(1):62–70, 2009.

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