Nonstationary cluster-size inference with random field and permutation methods
Introduction
Whether functional or structural, detecting changes in brain images is a central problem in neuroimaging. Cluster-size tests, pioneered by Poline and Mazoyer (1993) and Friston et al. (1994), have been widely used in such investigations because of increased sensitivity to spatially extended signals, compared to voxel-intensity tests Friston et al., 1996, Poline et al., 1997. Different implementations of cluster-size tests have been developed, including simulation-based tests Forman et al., 1995, Ledberg et al., 1998, Poline and Mazoyer, 1993, Roland et al., 1993, random field theory-based (RFT) tests Cao and Worsley, 2001, Worsley et al., 1996, and permutation tests Holmes et al., 1996, Nichols and Holmes, 2002.
One of the assumptions usually required in a cluster-size test is stationarity or uniform smoothness. This assumption is crucial because when it is violated, the sensitivity and the specificity of the test can depend on local smoothness of images (Worsley et al., 1999). In smooth regions, clusters tend to be large even in the absence of true signals, thus resulting in increased false positives. On the other hand, in rough regions, clusters tend to be small, and even a true positive cluster may be too small to be detected, resulting in reduced power. Because of such bias, Ashburner and Friston (2000) discourage use of cluster-size tests in voxel-based morphometry (VBM) data that are known to exhibit nonstationary. Even in a typical BOLD fMRI data set, the stationarity assumption is questionable (see Fig. 1), yet this assumption is not routinely assessed, and in our experience, rarely true.
To address this problem associated with nonstationarity, Worsley et al. (1999) suggested adjusting cluster sizes according to the local smoothness at each voxel. With the RFT framework by Cao (1999), this approach has been implemented (Worsley, 2002). However, this test is subject to various random field assumptions: Images have to be a lattice approximation of a smooth random field, the cluster defining threshold needs to be sufficiently high, and the cluster-size distribution is considered to approximately follow a known parametric form. Though nonstationarity is accounted for, the test is still restricted by such stringent assumptions.
Therefore, as an alternative to the nonstationary RFT test, we propose a nonstationary permutation test. Using Worsley et al.'s roughness adjusted cluster sizes, we adjust sensitivity of the test according to image smoothness. Since the permutation cluster-size test requires few assumptions and it is robust and exact (Hayasaka and Nichols, 2003), we avoid stringent restrictions of the RFT test. We validate the nonstationary permutation test by noise image simulations, both stationary and nonstationary, and compare its performance to the RFT counterpart. We further validate both nonstationary tests by applying them to a PET data set and examine their sensitivity relative to stationary cluster-size tests.
This paper is structured as following: In Methods and materials, we describe briefly both nonstationary RFT and permutation tests. Furthermore, we describe the validation of both tests in simulations and in data analysis. In Results, findings from the simulations and the data analysis are presented. In Discussion, we examine the findings from the simulation and draw conclusions.
Section snippets
Model
We assume that voxel intensities of a brain image can be expressed as a linear modelwhere v = (x,y,z) ∈ 3 is an index for voxels, Y(v) = {Y1(v), Y2(v),…,Yn(v)}′ is a vector of observed voxel intensities at v from n scans, X is a known design matrix of size n × p, β(v) is a p-dimensional vector of unknown parameters, σ(v) is an unknown standard deviation at v, and ε(v) = {ε1(v), ε2(v), …, εn(v)}′ is a vector of unknown random errors with unit variance. Images are denoted by
Simulation-based validation
The results from the stationary simulation are shown in Fig. 4. The RFT test is found to be anticonservative for the highest threshold (0.0001) for any smoothness or df. For lower thresholds, the test is found to be conservative for low smoothness, and as smoothness increases, the test becomes less conservative. However, this increase in the rejection rates surpasses the designed significant level 0.05, and the test becomes anticonservative for high smoothness. This trend is particularly
Nonstationary permutation test
The nonstationary permutation test shows excellent control of FWER for all smoothness, df, and cluster defining thresholds considered. Note that it shows even more accurate control of FWER than the stationary permutation test. At very low smoothness, the stationary permutation test controls FWER conservatively because of discreteness in the cluster-size distribution: Most of clusters have the same size, 1, 2, or 3 voxels, and there is no critical cluster size that can exactly control FWER
Acknowledgements
This Human Brain Project/Neuroinformatics research is funded by the National Institute of Mental Health, National Institute on Aging, and the National Institute of Biomedical Imaging and Bioengineering. This research was also supported in part by a VA/DoD collaborative program grant to IL. The authors would like to thank Dr. Stephan F Taylor and Dr. James Becker for providing us the data sets for the illustration in Fig. 1.
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