Weighted correlation network analysis

Weighted correlation network analysis, also known as weighted gene co-expression network analysis (WGCNA), is a widely used data mining method especially for studying biological networks based on pairwise correlations between variables. While it can be applied to most high-dimensional data sets, it has been most widely used in genomic applications. It allows one to define modules (clusters), intramodular hubs, and network nodes with regard to module membership, to study the relationships between co-expression modules, and to compare the network topology of different networks (differential network analysis). WGCNA can be used as data reduction technique (related to oblique factor analysis ), as clustering method (fuzzy clustering), as feature selection method (e.g. as gene screening method), as framework for integrating complementary (genomic) data (based on weighted correlations between quantitative variables), and as data exploratory technique.^[1] Although WGCNA incorporates traditional data exploratory techniques, its intuitive network language and analysis framework transcend any standard analysis technique. Since it uses network methodology and is well suited for integrating complementary genomic data sets, it can be interpreted as systems biologic or systems genetic data analysis method. By selecting intramodular hubs in consensus modules, WGCNA also gives rise to network based meta analysis techniques ^[2]

History

The WGCNA method was developed by Steve Horvath, a professor of human genetics at the David Geffen School of Medicine at UCLA and of biostatistics at the UCLA Fielding School of Public Health and his colleagues at UCLA, and (former) lab members (in particular Peter Langfelder, Bin Zhang, Jun Dong). Much of the work arose from collaborations with applied researchers. In particular, weighted correlation networks were developed in joint discussions with cancer researchers Paul Mischel, Stanley F. Nelson, and neuroscientists Daniel H. Geschwind, Michael C. Oldham (according to the acknowledgement section in ^[1]). There is a vast literature on dependency networks, scale free networks and coexpression networks.

Comparison between weighted and unweighted correlation networks

A weighted correlation network can be interpreted as special case of a weighted network, dependency network or correlation network. Weighted correlation network analysis can be attractive for the following reasons:

• The network construction (based on soft thresholding the correlation coefficient) preserves the continuous nature of the underlying correlation information. For example, weighted correlation networks that are constructed on the basis of correlations between numeric variables do not require the choice of a hard threshold. Dichotomizing information and (hard)-thresholding may lead to information loss.^[3]
• The network construction is highly robust results with respect to different choices of the soft threshold.^[3] In contrast, results based on unweighted networks, constructed by thresholding a pairwise association measure, often strongly depend on the threshold.
• Weighted correlation networks facilitate a geometric interpretation based on the angular interpretation of the correlation, chapter 6 in.^[4]
• Resulting network statistics can be used to enhance standard data-mining methods such as cluster analysis since (dis)-similarity measures can often be transformed into weighted networks.,^[5] chapter 6 in ^[4]
• WGCNA provides powerful module preservation statistics which can be used to quantify whether can be found in another condition. Also module preservation statistics allow one to study differences between the modular structure of networks.^[6]
• Weighted networks and correlation networks can often be approximated by "factorizable" networks.^[4][7] Such approximations are often difficult to achieve for sparse, unweighted networks. Therefore, weighted (correlation) networks allow for a parsimonious parametrization (in terms of modules and module membership) (chapters 2, 6 in ^[1]) and ^[8]

Method

First, one defines a gene co-expression similarity measure which is used to define the network. We denote the gene co-expression similarity measure of a pair of genes i and j by $s_{ij}$. Many co-expression studies use the absolute value of the correlation as an unsigned co-expression similarity measure,

$s^{unsigned}_{ij}=|cor(x_i,x_j)|$

where gene expression profiles $x_{i}$ and $x_{j}$ consist of the expression of genes i and j across multiple samples. However, using the absolute value of the correlation may obfuscate biologically relevant information, since no distinction is made between gene repression and activation. In contrast, in signed networks the similarity between genes reflects the sign of the correlation of their expression profiles. To define a signed co-expression measure between gene expression profiles $x_{i}$ and $x_{j}$ , one can use a simple transformation of the correlation:

$s^{signed}_{ij}=0.5+0.5 cor(x_i,x_j)$

As the unsigned measure $s^{unsigned}_{ij}$ , the signed similarity $s^{signed}_{ij}$ takes on a value between 0 and 1. Note that the unsigned similarity between two oppositely expressed genes ($cor(x_i,x_j) = -1$) equals 1 while it equals 0 for the signed similarity. Similarly, while the unsigned co-expression measure of two genes with zero correlation remains zero, the signed similarity equals 0.5.

Next, an adjacency matrix (network), $A=[a_{ij}]$, is used to quantify how strongly genes are connected to one another. $A$ is defined by thresholding the co-expression similarity matrix $S = [s_{ij}]$ . 'Hard' thresholding (dichotomizing) the similarity measure $S$ results in an unweighted gene co-expression network. Specifically an unweighted network adjacency is defined to be 1 if $s_{ij}>\tau$ and 0 otherwise. Because hard thresholding encodes gene connections in a binary fashion, it can be sensitive to the choice of the threshold and result in the loss of co-expression information.^[3] The continuous nature of the co-expression information can be preserved by employing soft thresholding, which results in a weighted network. Specifically, WGCNA uses the following power function assess their connection strength:

${\textstyle a_{ij} = (s_{ij})^\beta }$,

where the power $\beta$ is the soft thresholding parameter. The default values $\beta=6$ and $\beta=12$ are used for unsigned and signed networks, respectively. Alternatively, $\beta$ can be chosen using the scale-free topology criterion which amounts to choosing the smallest value of $\beta$ such that approximate scale free topology is reached.^[3]

Since $log (a_{ij}) = \beta log (s_{ij})$, the weighted network adjacency is linearly related to the co-expression similarity on a logarithmic scale. Note that a high power $\beta$ transforms high similarities into high adjacencies, while pushing low similarities towards 0. Since this soft-thresholding procedure applied to a pairwise correlation matrix leads to weighted adjacency matrix, the ensuing analysis is referred to as weighted gene co-expression network analysis.

A major step in the module centric analysis is to cluster genes into network modules using a network proximity measure. Roughly speaking, a pair of genes has a high proximity if it is closely interconnected. By convention, the maximal proximity between two genes is 1 and the minimum proximity is 0. Typically, WGCNA uses the define the topological overlap measure (TOM) as proximity.^[9][10] which can also be defined for weighted networks.^[3] The TOM combines the adjacency of two genes and the connection strengths these two genes share with other "third party" genes. The TOM is a highly robust measure of network interconnectedness (proximity). This proximity is used as input of average linkage hierarchical clustering. Modules are defined as branches of the resulting cluster tree using the dynamic branch cutting approach ^[11] Next the genes inside a given module are summarize with the module eigengene, which can be considered as the best summary of the standardized module expression data.^[4] The module eigengene of a given module is defined as the first principal component of the standardized expression profiles. To find modules that relate to a clinical trait of interest, module eigengenes are correlated with the clinical trait of interest, which gives rise to an eigengene significance measure. One can also construct co-expression networks between module eigengenes (eigengene networks), i.e. networks whose nodes are modules ^[12] To identify intramodular hub genes inside a given module, one can use two types of connectivity measures. The first, referred to as $kME_i=cor(x_i,ME)$, is defined based on correlating each gene with the respective module eigengene. The second, referred to as kIN, is defined as a sum of adjacencies with respect to the module genes. In practice, these two measures are equivalent.^[4] To test whether a module is preserved in another data set, one can use various network statistics, e.g. $Zsummary$.^[6]

Applications

WGCNA has been widely used for analyzing gene expression data (i.e. transcriptional data), e.g. to find intramodular hub genes.^[2][13]

It is often used as data reduction step in systems genetic applications where modules are represented by "module eigengenes" e.g.^[14][15] Module eigengenes can be used to correlate modules with clinical traits. Eigengene networks are coexpression networks between module eigengenes (i.e. networks whose nodes are modules) . WGCNA is widely used in neuroscientific applications, e.g.^[16][17] and for analyzing genomic data including microarray data (Kadarmideen et al. 2011), single cell RNA-Seq data (Kogelman et al. 2014) ^[18] DNA methylation data,^[19] miRNA data, peptide counts ^[20] and microbiota data (16S rRNA gene sequencing).^[21] Other applications include brain imaging data, e.g. functional MRI data ^[22]

R software package

The WGCNA R software package ^[23] provides functions for carrying out all aspects of weighted network analysis (module construction, hub gene selection, module preservation statistics, differential network analysis, network statistics). The WGCNA package is available from the Comprehensive R Archive Network (CRAN), the standard repository for R add-on packages.

References

1. Horvath S (2011). Weighted Network Analysis: Applications in Genomics and Systems Biology. Springer Book. 1st Edition., 2011, XXII, 414 p Hardcover ISBN 978-1-4419-8818-8 website
2. Langfelder P, Mischel PS, Horvath S (2013) When Is Hub Gene Selection Better than Standard Meta-Analysis? PLoS ONE 8(4): e61505. doi:10.1371/journal.pone.0061505 PMCID: PMC3629234 PloS One
3. Zhang B, Horvath S (2005) A General Framework for Weighted Gene Co-Expression Network Analysis", Statistical Applications in Genetics and Molecular Biology: Vol. 4: No. 1, Article 17 PMID 16646834
4. Horvath S, Dong J (2008) Geometric Interpretation of Gene Coexpression Network Analysis. PLoS Comput Biol 4(8): e1000117 PMID 18704157 PMCID: PMC2446438 Plos Comp Biol
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15. Plaisier CL, Horvath S, Huertas-Vazquez A, Cruz-Bautista I, Herrera MF, Tusie-Luna T, Aguilar-Salinas C, Pajukanta P (2009) A systems genetics approach implicates USF1, FADS3 and other causal candidate genes for familial combined hyperlipidemia. PloS Genetics;5(9):e1000642
16. Voineagu I, Wang X, Johnston P, Lowe JK, Tian Y, Horvath S, Mill J, Cantor R, Blencowe BJ, Geschwind DH (2011) Transcriptomic analysis of autistic brain reveals convergent molecular pathology. Nature. PMID 21614001
17. Hawrylycz MJ, Lein ES, Guillozet-Bongaarts AL, Shen EH, Ng L, Miller JA, van de Lagemaat LN, Smith KA, Ebbert A, Riley ZL, Abajian C, Beckmann CF, Bernard A, Bertagnolli D, Boe AF, Cartagena PM, Chakravarty MM, Chapin M, Chong J, Dalley RA, Daly BD, Dang C, Datta S, et al, Koch C, Grant SG, Jones AR (2012) An anatomically comprehensive atlas of the adult human brain transcriptome. Nature. 2012 Sep 20;489(7416):391-9. doi: 10.1038 nature11405. PMID 22996553 Nature
18. Xue Z, Huang K, Cai C, Cai L, Jiang CY, Feng Y, Liu Z, Zeng Q, Cheng L, Sun YE, Liu JY, Horvath S, Fan G. (2013) Genetic programs in human and mouse early embryos revealed by single-cell RNA?sequencing. Nature. 2013 Jul 28. doi: 10.1038/nature12364 PMID 23892778 Nature
19. Horvath S, Zhang Y, Langfelder P, Kahn RS, Boks MP, van Eijk K, van den Berg LH, Ophoff RA. Aging effects on DNA methylation modules in human brain and blood tissue. Genome Biol. 2012 Oct 3;13(10):R97. PMID 23034122 Genome Biology
20. Shirasaki DI, Greiner ER, Al-Ramahi I, Gray M, Boontheung P, Geschwind DH, Botas J, Coppola G, Horvath S, Loo JA, Yang XW. (2012) Network organization of the huntingtin proteomic interactome in Mammalian brain. Neuron. 2012 Jul 12;75(1):41-57. PMID 22794259 Neuron
21. Tong M, Li X, Wegener Parfrey L, Roth B, Ippoliti A, Wei B, Borneman J, McGovern DP, Frank DN, Li E, Horvath S, Knight R, Braun J (2013) A modular organization of the human intestinal mucosal microbiota and its association with inflammatory bowel disease. PLoS One. 2013 Nov 19;8(11):e80702. doi: 10.1371/journal.pone.0080702. PMID 24260458 PMC
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23. Langfelder P, Horvath S (2008) WGCNA: an R package for weighted correlation network analysis. BMC Bioinformatics 2008, 9:559 PMID 19114008 PMCID: PMC2631488 BMC Bioinformatics