A novel sparse partial correlation method for simultaneous estimation of functional networks in group comparisons

Xiaoyun Liang^{1}, David Vaughan^{2,3}, Alan Connelly^{1,4}, and Fernando Calamante^{1,4}

*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.

[1] Liang X, Connelly A, Calamante F. A novel method for robust estimation of group fucntional connectivity based on a joint graphical models approach. Proc. ISMRM p.3059 (2014).

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[4] Meinshausen N, Buhlmann P. Stability selection. Journal of the Royal Statistical Society Series B-Statistical Methodology 72(4): 417-73 (2010).

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[11] Besson P, Dinkelacker V, Valabregue R, et al. Structural connectivity differences in left and right temporal lobe epilepsy. NeuroImage 100: 135-44 (2014).

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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