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Abstract:
Introduction:
The multiple comparison problem arises in the statistical analysis of fMRI data because independent statistical tests are performed at each voxel of an image. In order to reduce the number of tests, statistical inference is often performed at the cluster level where clusters are identified by thresholding the uncorrected map of z-values, and assuming that small clusters tend to be spurious while larger clusters are more reliable. The multiple comparison problem then still exists but is alleviated because the number of tests is reduced from many thousands to a few dozens. A common procedure is to control the 'false discovery rate' (BH-Fdr) (Benjamini and Hochberg, 1995) at the cluster level. Here we argue that BH-Fdr as implemented in major software systems (e.g. Chumbley et al 2009) is too conservative because it rests on assumptions which are unrealistic in this context. We propose a revised algorithm to solve this problem.
Methods:
BH-Fdr assumes that the null distribution is continuous and uniform in [0,1]. However, as we found using simulations, the distribution of cluster sizes is not uniform with small clusters being much more frequent than large ones. Thus, the corresponding null distribution is neither uniform nor continuous (fig 1,2 and Stelzer et al. 2013). To respect the specific nature of this distribution, we propose to use an empirical null density based on simulations. We employ this new null density within the framework of 'local fdr' (Efron 2007). Local fdr assumes a two-class model with a mixture density f(x) = p0 f0(x) + p1 f1(x) where f0 f1 are the null and non-null densities, and p0,p1 their priors. BH-Fdr uses the same model with f0, f1 replaced by their cumulative distributions where f0 is assumed to be uniform. In the context of fMRI, local fdr was previously proposed by Schwartzman et al (2009) for statistical inference at the voxel level. Here, we derive the empirical null by recording the sizes of randomly generated clusters and define fdr(x) = f0(x) / f(x). We call this new algorithm ``clusterFDR''.
Results:
We analyzed an fMRI experiment featuring an auditory paradigm described in Mueller et al (2011). In this experiment, 20 subjects (7 females) were presented with pieces of music versus scrambled music. Scanning was done at a 3T MedSpec 30/100 scanner (Bruker, Ettlingen, Germany) using a standard EPI sequence. For details see Mueller et al (2011). Using standard GLM-based data analysis, it was found that real music showed a stronger activation in left and right auditory cortices than scrambled music. Here, we also computed the reverse contrast (scrambled > real music) and found two clusters when BH-Fdr correction with an initial cluster threshold of z > 2.33 was used (figure 3). Using SPM8, spatial smoothness was found to be 6.3 voxels. We then tested clusterFDR on these data. To derive an empirical null distribution, we generated 200 images simulating zmaps with the same spatial smoothness, and obtained a histogram of cluster sizes via thresholding at z=2.33 and a null density function of corresponding p-values (figures 1,2). With our new clusterFDR, we could replicate the first finding. But in addition, we found four more clusters in the reverse contrast that had previously been overlooked (figure 3).
Conclusions:
As noted by Lieberman and Cunningham (2009), statistical procedures for analyzing fMRI data traditionally have been geared towards a rather strict exclusion of false positives. As a consequence, relevant aspects of the data may have been overlooked in the past (Gonzales et al 2012). Our new clusterFDR may help to remedy this problem. But let us note that statistical methodology can only be used to weed out truly random effects. It does not guard against false positives due to confounding effects.
