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.

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.

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.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.