Efficient Blockwise Permutation Tests Preserving Exchangeability

Abstract

In this paper, we present a new blockwise permutation test approach based on the moments of the test statistic. The method is of importance to neuroimaging studies. In order to preserve the exchangeability condition required in permutation tests, we divide the entire set of data into certain exchangeability blocks. In addition, computationally efficient moments-based permutation tests are performed by approximating the permutation distribution of the test statistic with the Pearson distribution series. This involves the calculation of the first four moments of the permutation distribution within each block and then over the entire set of data. The accuracy and efficiency of the proposed method are demonstrated through simulated experiment on the magnetic resonance imaging (MRI) brain data, specifically the multi-site voxel-based morphometry analysis from structural MRI (sMRI).

Keywords: Efficient nonparametric test; Pearson distribution series; moments; structural MRI; voxel-based morphometry.

Figure 1

Multi-site sMRI experiment: data generation process with ground truth. (a) and (b): inhomogeneity fields from site one and site two, respectively. (c): difference of bias fields in (a) and (b), with positive differences shown as light regions and negative differences as dark regions. (d): generated ground truth anatomical differences between two subject groups. (e): difference of bias fields in the regions having main effect (i.e. the red regions in (d)). (f) and (g): enlarged version of slice no. 10 in (d) and (e), respectively (i.e. the last slice of the second row in (d) and (e)).

Figure 2

Multi-site sMRI experiment: data analysis results comparing the effects of blockwise and multiple comparison (MP) correction. (a) and (b): results from our moments-based blockwise permutation tests without multiple comparison (MP) correction (a) and with false discovery rate (FDR) control at significance level α = 0.05 (b). (c) and (d): enlarged version of slide no. 10 in (a) and (b), respectively (i.e. the last slice of the second row in (a) and (b)). (e) and (f): results from our moments-based regular permutation tests without MP correction (e) and with FDR control at α = 0.05 (f). (g) and (h): enlarged version of slide no. 10 in (e) and (f), respectively (i.e. the last slice of the second row in (e) and (f)). Note that the regular permutation tests (see (f) and (h)) can only detect the regions with small or negligible bias field difference (gray level close to background in Figure 1e or Figure 1g.