Cluster-based statistics for brain connectivity in correlation with behavioral measures

Abstract

Graph theoretical approaches have successfully revealed abnormality in brain connectivity, in particular, for contrasting patients from healthy controls. Besides the group comparison analysis, a correlational study is also challenging. In studies with patients, for example, finding brain connections that indeed deepen specific symptoms is interesting. The correlational study is also beneficial since it does not require controls, which are often difficult to find, especially for old-age patients with cognitive impairment where controls could also have cognitive deficits due to normal ageing. However, one of the major difficulties in such correlational studies is too conservative multiple comparison correction. In this paper, we propose a novel method for identifying brain connections that are correlated with a specific cognitive behavior by employing cluster-based statistics, which is less conservative than other methods, such as Bonferroni correction, false discovery rate procedure, and extreme statistics. Our method is based on the insight that multiple brain connections, rather than a single connection, are responsible for abnormal behaviors. Given brain connectivity data, we first compute a partial correlation coefficient between every edge and the behavioral measure. Then we group together neighboring connections with strong correlation into clusters and calculate their maximum sizes. This procedure is repeated for randomly permuted assignments of behavioral measures. Significance levels of the identified sub-networks are estimated from the null distribution of the cluster sizes. This method is independent of network construction methods: either structural or functional network can be used in association with any behavioral measures. We further demonstrated the efficacy of our method using patients with subcortical vascular cognitive impairment. We identified sub-networks that are correlated with the disease severity by exploiting diffusion tensor imaging techniques. The identified sub-networks were consistent with the previous clinical findings having valid significance level, while other methods did not assert any significant findings.

Figure 1. Overview of the proposed method.

The method consists of two parts: correlation coefficient computation and multiple comparison correction with cluster-based statistics. In the former part, a partial correlation coefficient is calculated for each connection of the brain network with the behavioral measures. In this step, several compounding variables are taken as covariates in order to count their effects on the correlation coefficients. In the later part, we perform cluster-based correction for the multiple comparison of the correlation coefficients by adopting the supra-threshold cluster size test to our problem setting. In this approach, clusters are constructed by grouping together neighboring supra-threshold connections, and the p-values are estimated through permutation testing, forming a null distribution of the maximum cluster extent. The output of this step is a set of sub-networks consisting of neighboring connections that are significantly correlated with the behavioral measures.

Figure 2. Cluster-based statistics of correlation coefficients for multiple comparison correction.

The first step generates N permutation vectors by randomly reordering behavioral scores (the upper-left corner). Suppose there are n subjects in the given group. Then, the ith permutation vector PV _i, i = 1,2,…,N, has n elements, the ordering of n subjects’ behavioral scores. Note that the last permutation vector PV _N is constructed using the original ordering of the behavioral scores as usual in any permutation testing. We compute a partial correlation coefficient between an edge and the behavioral scores for every permutation vector. We repeat this procedure for every edge, resulting , i = 1,2,…,N and k = 1,2,…,m, where m is the total number of edges (the upper middle). In the second step, we extract sets of network edges of which correlation coefficient is beyond the initial threshold to form supra-threshold clusters. Denoted by the resulting cluster is corresponding to the jth cluster of the ith permutation vector PV _i, i = 1,2,…,N and j = 1,2,…c_i, where c_i is the number of identified clusters for PV _i. For a positive initial threshold, edges whose correlations were larger than it will form clusters, while for a negative threshold edges whose correlation is smaller than it will do. We employ the maximum cluster extent for the null permutation distribution by counting the number of edges in the largest connected sub-network of each permutation. represents the number of edges in , and represents the maximum cluster extent for PV _i (the upper-right corner)_. This representative statistic forms a null permutation distribution, which is shown as the histogram (the bottom). Finally, we estimate the significance level over the null distribution by computing the proportion of the number of entries whose maximal cluster extents are larger than the size of each identified sub-network, , (black entries in the histogram) to the number of entries, N.

Figure 3. The identified sub-network correlated with the disease severity: subcortical vascular mild cognitive impairment.

The figure shows in the lateral view of the left hemisphere (A), the transverse view of both hemispheres (B), and the lateral view of the right hemisphere (C). The identified connection was shown as an orange line, whose thickness represents the magnitude of its correlation coefficient between its edge weight and CDR-SOB. The identified node was shown with a colored sphere, whose color represents the lobe to which it belongs: frontal (cyan), limbic (blue), central (magenta), temporal (green), parietal and occipital (red).

Figure 4. The identified sub-network correlated with the disease severity: subcortical vascular dementia.

The figure shows in the lateral view of the left hemisphere (A), the transverse view of both hemispheres (B), and the lateral view of the right hemisphere (C). The identified connection was shown as an orange line, whose thickness represents the magnitude of its correlation coefficient between its edge weight and CDR-SOB. The identified node was shown with a colored sphere, whose color represents the lobe to which it belongs: frontal (cyan), limbic (blue), central (magenta), temporal (green), parietal and occipital (red).

Figure 5. Comparison to the other multiple comparison correction methods.

To compare with Bonferroni correction and FDR procedure, we drew histogram of p-values in log-scale whose correlation coefficients are negative, showing the thresholding p-value of Bonferroni correction with α = 0.10, (thin solid vertical line), and the maximum of uncorrected p-values of network connections in the proposed cluster-based correction, p _max (thick solid vertical line), for patients with svMCI (A) and SVaD (B). We note that the thresholding p-values of the FDR procedure with q = 0.05 and 0.1 cannot be shown in log-scale, because they both are exactly zero, leading no significant findings. To compare with extreme statistics, we drew the histogram of raw correlation coefficients (Spearman, partial correlation adjusting age and gender), showing 10% threshold of the extreme statistics, (thin solid vertical line), along with the initial threshold, (thick solid vertical line), in patients with svMCI (C) and SVaD (D), where dotted vertical line indicates the zero correlation coefficient.

Figure 6. Edge weights over the disease severity score for two representative edges of the identified sub-networks in patients with svMCI and SVaD: the edge between the medial surface of left superior frontal gyrus (SFGmed.L) and the left median cingulate and paracingulate gyri (DCG.L) in patients with svMCI (A) and SVaD (B); and the edge between the orbital part of left inferior frontal gyrus (ORBinf.L) and left insula (INS.L) in patients with svMCI (C) and SVaD (D).

A circle represents each subject, dotted horizontal lines represent zero edge weights, and solid lines represent linearly regressed lines. The noted correlation coefficient and p-values were calculated using Pearson correlation without any covariates. The former edge (A&B) belonged to the identified sub-network in patients with svMCI, and thus had a strong correlation coefficient in patients with svMCI; however, in patients with SVaD, about half of the subjects had zero or very low weights (20 over 45), resulting in a low correlation coefficient. The latter edge (C&D) belonged to the identified sub-network in patients with SVaD, and thus had a strong correlation coefficient in patients with SVaD; however, in patients with svMCI it is not the case.