Set-level threshold-free tests on the intrinsic volumes of SPMs

Gareth R Barnes,Karl Friston,Guillaume Flandin,Gerard R Ridgway,Mark Woolrich

doi:10.1016/j.neuroimage.2012.11.046

Abstract

Conventionally, set-level inference on statistical parametric maps (SPMs) is based on the topological features of an excursion set above some threshold—for example, the number of clusters or Euler characteristic. The expected Euler characteristic—under the null hypothesis—can be predicted from an intrinsic measure or volume of the SPM, such as the resel counts or the Lipschitz–Killing curvatures (LKC). We propose a new approach that performs a null hypothesis omnibus test on an SPM, by testing whether its intrinsic volume (described by LKC coefficients) is different from the volume of the underlying residual fields: intuitively, whether the number of peaks in the statistical field (testing for signal) and the residual fields (noise) are consistent or not. Crucially, this new test requires no arbitrary feature-defining threshold but is nevertheless sensitive to distributed or spatially extended patterns. We show the similarities between our approach and conventional topological inference—in terms of false positive rate control and sensitivity to treatment effects—in two and three dimensional simulations. The test consistently improves on classical approaches for moderate (> 20) degrees of freedom. We also demonstrate the application to real data and illustrate the comparison of the expected and observed Euler characteristics over the complete threshold range.

Highlights

Random field theory is used in neuroimaging to account for the intrinsic smoothness of images, when making inferences on the basis of statistical parametric maps (Kilner and Friston, 2010; Worsley et al, 1996, 2004)
We show that a simple multivariate test comparing the Lipschitz–Killing curvatures (LKC) of the residual fields and the Statistical parametric maps (SPMs) provides good control over false positive rates, and show that the method is comparable with techniques that rely on featuredefining thresholds
The resulting SPM will no longer be well described by the Gaussian Kinematic Formula—because it no longer conforms to a central statistical field under the null hypothesis

Summary

Introduction

Random field theory is used in neuroimaging to account for the intrinsic smoothness of images, when making inferences on the basis of statistical parametric maps (Kilner and Friston, 2010; Worsley et al, 1996, 2004). Random field theory allows one to make predictions about the Euler characteristic (EC) of the excursion set produced by thresholding a random field (intuitively, the number of blobs minus the number of holes of the excursion set). This prediction is useful, because—at high threshold—holes disappear and the expected EC approximates the number of maxima one would expect under the null hypothesis. In this work we introduce an approach that exploits random field theory without the need for a particular threshold

Methods

Results

Conclusion