A novel sparse partial correlation method for simultaneous estimation of functional networks in group comparisons
Xiaoyun Liang1, David Vaughan2,3, Alan Connelly1,4, and Fernando Calamante1,4

1Imaging Division, Florey Institute of Neuroscience and Mental Health, Melbourne, Australia, 2Epilepsy Division, Florey Institute of Neuroscience and Mental Health, Melbourne, Australia, 3Department of Neurology, Austin Health, Melbourne, Australia, 4Department of Medicine, University of Melbourne, Melbourne, Australia

### Synopsis

We propose a novel approach, Graphical-LAsso with Stability-Selection (GM-GLASS), by employing sparse group penalties for simultaneously estimating networks from healthy control and patient groups. Simulations demonstrate that both GM-GLASS and JGMSS outperform Fisher Z-transform. Our in vivo results further show that GM-GLASS yields highest contrast of network metrics between groups, demonstrating the superiority of GM-GLASS in detecting significance group differences over JGMSS and Fisher Z-transform. Overall, by controlling confounding variations between subjects, and therefore enhancing the statistical power, our simulated and in vivo results demonstrate that GM-GLASS provides a robust approach for conducting group comparison studies.

### Purpose

Functional networks at group-level can be used to gain insight into complex brain function, including in group comparison studies. Joint graphical model with stability selection (JGMSS) was shown to greatly control confounding variations when computing group-level connectomic networks [1]. However, JGMSS deals with one single group at a time. Recently, a fused graphical-lasso model (FGM) was proposed to robustly estimate group-level graphs from multiple groups simultaneously [2,3]. We aim to simultaneously estimate individual- and group-level networks from 2 groups using FGM. We propose a novel approach, Group-fused-Multiple-Graphical-LAsso with Stability-Selection (GM-GLASS), by employing sparse group penalties for simultaneously estimating connectivity matrices from within 2 contrast groups, e.g. normal control (NC) and patient (PT) group, while controlling for confounding variations.

### Methods

GM-GLASS can be described as follows (see Fig. 1 for flowchart): $2K$ datasets, $X^{(1)},...X^{(K)},X^{(K+1)},...,X^{2K}$, for $2$ groups (first $K$ from group $1$, last $K$from group $2$), $X^{(k)}$: $n_k\times p$ matrix, $n_k$: number of time-points, $p$: number of regions (nodes). The empirical covariance-matrix for $X^{(k)}$ is: $S^{(k)}=1/n_{k}(X^{(k)})^{T}X^{(k)}$, and the maximum-likelihood estimate of $(S^{(k)})^{-1}$ is [2,3]: $max_\left\{Θ\right\}(\sum_kn_{k}(logdetΘ^{(k)}-tr(S^{(k)}Θ^{(k)}))-P(\left\{Θ\right\}))$, $k=1,...,2K$, with the constraint that all $Θ^{(k)}$ are positive-definite; the fused regularization $P(\left\{Θ\right\})$ [2,3] is reformulated as: $P(\left\{Θ\right\})=\lambda_1\sum_k\sum_{i\neq j}|Θ_{ij}^{(k)}|+\lambda_2\sum_{k_1}\sum_{i\neq j}|Θ_{ij}^{(k_1)}-Θ_{ij}^{(k_1+K)}|$ with $\lambda_1,\lambda_2$: nonnegative regularization-parameters, and $k=1,...,2K$, and $k_1=1,...,K$. With stability selection [4], the data are subsampled many times and all positive-correlations that occur in a large fraction of the resulting selection sets are selected. The combination patterns (i.e. the order of subjects) in the second part of $P$ are shuffled for each resampling to avoid bias. For a given $(\lambda_1,\lambda_2)$, $S^{\lambda_1,\lambda_2}=\left\{(m,n):1\leq(m,n)\leq M;Θ(m,n)<0\right\}$. For a cut-off $P_{thr}\in(0,1)$ and a set of regularization parameters, the set of stable positive-correlations are as follows: $\widehat{S}_{stable}=\left\{(m,n): max_{\forall(\lambda_1,\lambda_2)}\widehat{P}_{m,n}^{\lambda_1,\lambda_2}\geq P_{thr}\right\}$.

Simulation: Simulations were conducted according to [6]. A structural-connectivity matrix [5] was employed to simulate data for NCs (see [6]); by randomly deleting (adding) $25$ existing (new) edges, patient data were then simulated, with $p=78$ and $n=200$. Ten datasets were generated (i.e. ‘subjects’) for each group. To perform stability selection, $n/2$ observations were randomly subsampled without overlapping for each dataset, repeating $100$ times to estimate probabilities over the regularization region, with the expected number of falsely-selected connections $V$ bounded by $E(V)=q^2/(2P_{thr}*r-r)$ [4], $r$: number of variables, $q$: average number of selected connections for a given range of $(\lambda_1,\lambda_2)$. The per-comparison-error-rate $(PCER=E(V)/r)$ is employed to control falsely-selected connections [4]. In-vivo data: Resting-state BOLD-fMRI data were acquired from $13$ mesial temporal lobe epilepsy (TLE) patients with right hippocampal-sclerosis, as well as $13$ age-matched healthy controls, on a 3T Siemens scanner: TR/TE=$3000/30$ms, $3$mm isotropic voxels, $44$ slices, matrix-size: $72×72$, $200$ time-points. Image analysis: images were preprocessed in a standard way [7]. With individual-level mean time-series within AAL template [8] as inputs, GM-GLASS was employed for estimating individual- and group-level functional networks. Since fMRI signal is not independent and not exchangeable, the time-series were divided into $20$ consecutive blocks ($200/20=10$). For comparisons, JGMSS [1] and Pearson-cross-correlation followed by Fisher-Z-transformation (Fisher Z-transform) [9] were also employed. For Fisher Z-transform, individual-level networks were averaged across the group, followed by thresholding the group-level networks with multiple-comparison correction.

### Results

Fig. 2 shows estimated networks from simulated normal and patient groups, demonstrating the capability of GM-GLASS in detecting network differences. The receiver-operating curves (ROCs) in Fig. 3 demonstrate that GM-GLASS and JGMSS estimate true connections from groups more reliably than Fisher Z-transform. Based on the ground truth, simulations have demonstrated that GM-GLASS and JGMSS achieve higher accuracy ($75$%) than Fisher Z-transform ($50$%) (Fig. 4). In-vivo results show that GM-GLASS has detected $3$ significantly different network metrics between groups, instead of $1$ for JGMSS and $0$ for Fisher Z-transform (Fig.5), further suggesting that GM-GLASS might have higher sensitivity in detecting group network differences.

### Discussion

Simulations demonstrate that GM-GLASS and JGMSS significantly outperform Fisher Z-transform (Fig. 3), and they are capable of detecting minor network differences (Fig. 2); only marginally better performance of GM-GLASS than JGMSS is achieved, likely due to high SNR of simulated data. However, for in vivo data, networks estimated using GM-GLASS are characterized by increased contrast of network metrics between groups compared to the other $2$ methods. More specifically, decreased local-efficiency and global-efficiency in right TLE are consistent with [10] and decreased network density is in line with [11]; these highlight the capability of GM-GLASS in detecting significant differences between groups that otherwise might not be detected using JGMSS or Fisher Z-transform. Overall, by controlling confounding variations between subjects, and therefore enhancing the statistical power, our simulated and in vivo results demonstrate that GM-GLASS provides a robust approach for conducting group comparison studies.

### Acknowledgements

We are grateful to the National Health and Medical Research Council (NHMRC) of Australia, the Australian Research Council (ARC), and the Victorian Government's Operational Infrastructure Support Program for their support.

### References

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### Figures

Figure 1: The flowchart shows the implementation of GM-GLASS in which both sparse individual- and group-level brain functional networks (Sstable) are calculated. Note: the second part of regularization term P was reformulated to incorporate shuffled network differences between groups, achieving the goal of reducing both within- and between-group confounding variability.

Figure 2: Orange and red colors indicate common true edges and truly detected edge differences between groups; blue and yellow colors represent false edges and falsely detected differences between groups (i.e. expected true common edges), with accuracy and sensitivity for NC and PT groups, respectively: 0.9924, 0.9938, 0.9666 and 0.9787.

Figure 3: The receiver-operating curves (ROC) for NC (left) and PT (right). Estimation of area under the curve (AUC) indicates that GM-GLASS yields the best performance, with accuracy and sensitivity for NC and PT groups, respectively: JGMSS: 0.9911, 0.9928, 0.9605 and 0.9757; Fisher Z-transform: 0.9770, 0.9852, 0.8815 and 0.8298.

Figure 4: Statistical differences of 4 individual-level network metrics estimated from simulated data, i.e. local efficiency (LE), global efficiency (GE), network density and clustering coefficient (CC), between NC and PT groups. Accuracy: 75% for GM-GLASS and JGMSS; 50% for Fisher Z-transform (FisherZ). Note: *p<0.05, ***<0.001 and ns – non-significant.

Figure 5: Statistical difference for in vivo data. For CC, all methods have consistently detected no difference between NC and TLE groups. GM-GLASS and JGMSS, but not FisherZ, have detected significant difference for GE between groups. However, for LE and network density, only GM-GLASS has detected significant difference between groups.

Proc. Intl. Soc. Mag. Reson. Med. 24 (2016)
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