Innovations In Clinical Neuroscience

NOV-DEC 2017

A peer-reviewed, evidence-based journal for clinicians in the field of neuroscience

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62 ICNS INNOVATIONS IN CLINICAL NEUROSCIENCE November-December 2017 • Volume 14 • Number 11–12 O R I G I N A L R E S E A R C H raters were held to this reliability standard. 29 A full description of the rater certification and training process for the PANSS is available in Swartz et al. 29 Network analysis. Network analysis was used to examine the structural/topological properties of the PANSS symptoms network at baseline and 18 months follow-up of Phase I and for treatment-resistant and -responsive patients. First, we constructed a symptom network for each patient group at the baseline and 18-month follow-up time-points by calculating the association between PANSS symptoms using the absolute value of the partial correlations among symptoms. Partial correlation estimates the level of association between two random variables while removing the effect of other variables. It was calculated based on the residual scores of the regression analysis as follows: where X and Y represent variables (i.e. PANSS items), Z represents the set of controlling variables, e represents the residual scores obtained from the regression analysis, a and b represent the regression analysis parameters, r represents the correlation coefficient value, and are the predicted values of X and Y estimated using regression analysis, and PC XY.Z represents the partial correlation value between variables X and Y while controlling for the effect of the variables in Z. After calculating the pairwise partial correlation between all symptoms, we obtained the connectivity matrix, which is a mathematical representation of a network, where rows i and columns j are the nodes of the network, and each cell w i j denotes the connectivity between nodes (i.e., PANSS items) i and j. Since our networks are weighted, each cell has a value between 0 and 1, representing the strength of connection between two nodes. Thus, the nodes of our network represent PANSS symptom items, and the edges represent the absolute value of the partial correlation between two symptoms. After constructing symptom networks using the absolute value of partial correlations, we calculated network properties to capture the interaction between different PANSS items in the treatment- responsive and treatment-resistant groups before and after treatment. Network variables analyzed in this study included density, average shortest path length (characteristic path length), average clustering coefficient, modularity, closeness centrality, and degree centrality, which are defined as follows: Network density is defined as the sum of the link weights divided by the number of all possible links. The higher density of a network suggests that nodes of the network are tightly connected to each other. The network density is calculated as follows: 16 where N represents the total number of nodes of the network, i and j represent nodes in the network, and w i j represents the weight of a link between i and j. The average shortest path length is another useful network measure that aims at quantifying the efficiency of information transfer in networks. The average shortest path length is defined as the mean of shortest path lengths between all possible pairs of nodes in the network. A smaller average shortest path length indicates more efficient information transfer in the network. Average shortest path length is calculated as follows: 16 where d i j is the shortest path from i to j, and N is the number of nodes in the network. In weighted networks, the distance between two nodes is defined as the inverse of corresponding link weight. 16 Therefore, the shortest path of weighted networks is based on the inverse of link weights. The clustering coefficient is a network measure to quantify the extent to which nodes tend to cluster together. 15 In other words, the clustering coefficient shows how well the neighborhood of one particular node is connected. The clustering coefficient ranges from 0 for the nonconnected neighborhood to 1 for the fully connected neighborhood (also known as a clique). The clustering coefficient of a node in a weighted graph is calculated as follows: where u, v,and i represent nodes in the network; k i represents weighted degree of node i; and W represents link weights in the network. The average clustering coefficient provides the overall clustering in the network, which is calculated as follows: 17 where N is the number of nodes in the network; and CC i is the clustering coefficient of node i. Many real-world networks tend to divide naturally into modules (also called clusters, communities, or groups). The strength of division of a network into modules is called modularity. 5 Community detection is a problem of finding maximal modularity. Higher modularity of a network is an indication of dense connections within modules and sparse connections between nodes from different modules. Modularity for weighted networks is calculated as follows: 18 where w i j is the link weight between nodes i and j; k i and k j are degrees of nodes i and j; c i and c j are the communities that nodes to which i and j belong; and m is equal to . The δ function δ(c i ,c j ) equals to one if c i = c j and 0 if c i ≠ c j . Finding the communities in a network can be formulated as an optimization problem. In this study, we used a greedy optimization method known as the Louvain method to find the optimal community structures. 18 The degree centrality of a node in the weighted undirected network is also called a "strength," which simply is the sum of the edge weights connected to a specific node. This simple measure provides useful information for identification of important nodes in the network and provides first clues about the structure of the network. The nodes with high degrees of centrality are known as hubs

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