Module preservation statistics
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Module preservation statistics. Steve Horvath University of California, Los Angeles. Module preservation is often an essential step in a network analysis. Construct a network Rationale: make use of interaction patterns between genes. Identify modules

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Module preservation statistics

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Module preservation statistics

Steve Horvath

University of California, Los Angeles

Module preservation is often an essential step in a network analysis

Construct a network

Rationale: make use of interaction patterns between genes

Identify modules

Rationale: module (pathway) based analysis

Relate modules to external information

Array Information: Clinical data, SNPs, proteomics

Gene Information: gene ontology, EASE, IPA

Rationale: find biologically interesting modules

  • Study Module Preservation across different data

  • Rationale:

  • Same data: to check robustness of module definition

  • Different data: to find interesting modules

Find the key drivers of interesting modules

Rationale: experimental validation, therapeutics, biomarkers

Motivational example: Studying the preservation of human brain co-expression modules in chimpanzee brain expression data. Modules defined as clusters(branches of a cluster tree)Data from Oldam et al 2006

Preservation of modules between human and chimpanzee brain networks

Standard cross-tabulation based statistics have severe disadvantages


  • only applicable for modules defined via a clustering procedure

  • ill suited for making the strong statement that a module is not preserved

    We argue that network based approaches are superior when it comes to studying module preservation

Is my network module preserved and reproducible?Langfelder et al PloS Comp Biol. 7(1): e1001057.

Broad definition of a module

  • Abstract definition of module=subset of nodes in a network.

  • Thus, a module forms a sub-network in a larger network

  • Example: module (set of genes or proteins) defined using external knowledge: KEGG pathway, GO ontology category

  • Example: modules defined as clusters resulting from clustering the nodes in a network

  • Module preservation statistics can be used to evaluate whether a given module defined in one data set (reference network) can also be found in another data set (test network)

Networkof cholesterol biosynthesis genes


female liver network (reference)

Looks most similar to male liver network


  • How to measure relationships between different networks (e.g. how similar is the female liver network to the male network)?

  • Answer: network concepts aka statistics

Connectivity (aka degree)

  • Node connectivity = row sum of the adjacency matrix

    • For unweighted networks=number of direct neighbors

    • For weighted networks= sum of connection strengths to other nodes


  • Density= mean adjacency

  • Highly related to mean connectivity

Network concepts to measure relationships between networks

Numerous network concepts can be used to measure the preservation of network connectivity patterns between a reference network and a test network

  • E.g. Density in the test set

  • cor.k=cor(kref,ktest)

  • cor(Aref,Atest)

Module preservation

in different types of networks

  • One can study module preservation in general networks specified by an adjacency matrix, e.g. protein-protein interaction networks.

  • However, particularly powerful statistics are available for correlation networks

    • weighted correlation networks are particularly useful for detecting subtle changes in connectivity patterns. But the methods are also applicable to unweighted networks (i.e. graphs)

Network-based module preservation statistics

  • Input: module assignment in reference data.

  • Adjacency matrices in reference Aref and test data Atest

  • Network preservation statistics assess preservation of

    • 1. network density: Does the module remain densely connected in the test network?

    • 2. connectivity: Is hub gene status preserved between reference and test networks?

    • 3. separability of modules: Does the module remain distinct in the test data?

Several connectivity preservation statistics

For general networks, i.e. input adjacency matrices

  • cor.kIM=cor(kIMref,kIMtest)

    • correlation of intramodular connectivity across module nodes

  • cor.ADJ=cor(Aref,Atest)

    • correlation of adjacency across module nodes

      For correlation networks, i.e. input sets are variable measurements

  • cor.Cor=cor(corref,cortest)

  • cor.kME=cor(kMEref,kMEtest)

    One can derive relationships among these statistics in case of weighted correlation network

  • Choosing thresholds for preservation

    statistics based on permutation test

    For correlation networks, we study 4 density and 4 connectivity preservation statistics that take on values <= 1

    Challenge: Thresholds could depend on many factors (number of genes, number of samples, biology, expression platform, etc.)

    Solution: Permutation test. Repeatedly permute the gene labels in the test network to estimate the mean and standard deviation under the null hypothesis of no preservation.

    Next we calculate a Z statistic

    Permutation test for estimating Z scores

    Gene modules in Adipose

    • For each preservation measure we report the observed value and the permutation Z score to measure significance.

    • Each Z score provides answer to “Is the module significantly better than a random sample of genes?”

    • Summarize the individual Z scores into a composite measure called Z.summary

    • Zsummary < 2 indicates no preservation, 2<Zsummary<10 weak to moderate evidence of preservation, Zsummary>10 strong evidence

    Overview of Preservation statistics

    Module preservation statistics are often closely related

    Message: it makes sense to aggregate the statistics

    into “composite preservation statistics”

    Clustering module preservation statistics based on correlations across modules

    Red=density statistics

    Blue: connectivity statistics

    Green: separability statistics

    Cross-tabulation based statistics

    Composite statistic in correlation networks based on Z statistics

    Analogously define composite statistic: medianRank

    Gene modules in Adipose

    • Based on the ranks of the observed preservation statistics

    • Does not require a permutation test

    • Very fast calculation

    • Typically, it shows no dependence on the module size

    Summary preservation

    • Network based preservation statistics measure different aspects of module preservation

      • Density-, connectivity-, separability preservation

    • Two types of composite statistics: Zsummary and medianRank.

    • Composite statistic Zsummary based on a permutation test

      • Advantages: thresholds can be defined, R function also calculates corresponding permutation test p-values

      • Example: Zsummary<2 indicates that the module is *not* preserved

      • Disadvantages: i) Zsummary is computationally intensive since it is based on a permutation test, ii) often depends on module size

    • Composite statistic medianRank

      • Advantages: i) fast computation (no need for permutations), ii) no dependence on module size.

      • Disadvantage: only applicable for ranking modules (i.e. relative preservation)

    Application:Modules defined as KEGG pathways.Comparison of human brain (reference) versus chimp brain (test) gene expression data.Connectivity patterns (adjacency matrix) is defined as signed weighted co-expression network.

    Preservation of KEGG pathwaysmeasured using the composite preservation statistics Zsummary and medianRank

    • Humans versus chimp brain co-expression modules

    Apoptosis module is least preserved

    according to both composite preservation statistics

    Visually inspect connectivity patterns of the apoptosis module in humans and chimpanzees

    Weighted gene co-expression module.

    Red lines=positive correlations,

    Green lines=negative cor

    Note that the connectivity patterns look very different.

    Preservation statistics are ideally suited to measure differences in connectivity preservation

    Literature validation:Neuron apoptosis is known to differ between humans and chimpanzees

    • It has been hypothesized that natural selection for increased cognitive ability in humans led to a reduced level of neuron apoptosis in the human brain:

      • Arora et al (2009) Did natural selection for increased cognitive ability in humans lead to an elevated risk of cancer? Med Hypotheses 73: 453–456.

    • Chimpanzee tumors are extremely rare and biologically different from human cancers

    • A scan for positively selected genes in the genomes of humans and chimpanzees found that a large number of genes involved in apoptosis show strong evidence for positive selection (Nielsen et al 2005 PloS Biol).

    Application: Studying the preservation of a female mouse liver module in different tissue/gender combinations.Module: genes of cholesterol biosynthesis pathway Network: signed weighted co-expression networkReference set: female mouse liverTest sets: other tissue/gender combinationsData provided by Jake Lusis

    Networkof cholesterol biosynthesis genes


    female liver network (reference)

    Looks most similar to male liver network

    Note that Zsummary

    is highest

    in the male liver network

    Jeremy Miller, et al Dan Geschwind (2010)Divergence of human and mouse brain transcriptome highlights Alzheimer disease pathways.PNAS 2010

    Slide from Jeremy Miller

    Why compare human and mouse brain transcription?

    • 1) Module membership (kME) in conserved modules may be used to identify reliable markers for cell types and cellular components.

    • 2) Studying differences in network organization could provide a basis for better understanding diseases enriched in human populations, such as Alzheimer’s Disease

    Slide from J. Miller

    Co-expression modules based on multiple human and mouse gene expression data

    Human Brain Modules

    Mouse Brain Modules

    Human modules

    M7h and M9h

    were enriched

    with AD genes

    These modules

    could not be found

    In mouse brains

    Preservation of human network modules in mouse brains


    Human specific modules M9h and M7h are related to AD

    • Module preservation analysis identified two highly human-specific module, M9h and M7h

      • No clear functional annotation

    • Guilt by association approaches show these modules are related to neurodegenerative dementias

      • M9H showed significant overlap with an Alzheimer’s disease module that was identified using independent data sets run on different brain regions, on different platforms, and in different labs

      • M7h contained two intramodular hub genes related to AD and frontotemporal dementia (FTD) in humans: GSK3β and tau

    • These two modules provide key targets for furthering our understanding of neurodegenerative dementias

    Genetic Programs in Human and Mouse Early Embryos Revealed by Single-Cell RNA-Sequencing

    ZhigangXue, Kevin Huang, XiaofeiYe,

    et al

    Guoping Fan


    • Mammalian preimplantation development is a complex process involving dramatic changes in the transcriptional architecture.

    • Through single-cell RNA-sequencing (RNA-seq), we report here a comprehensive analysis of transcriptome dynamics from oocyte to morula in both human and mouse embryos.

    PCA of RNA seq data reveals known trajectory

    WGCNA analysis

    Module eigengenes vs stages

    Module preservation analysis

    Implementation and R software tutorials, WGCNA R library

    • General information on weighted correlation networks

    • Google search

      • “WGCNA”

      • “weighted gene co-expression network”

    • R function modulePreservation is part of WGCNA package

    • Tutorials: preservation between human and chimp brains

    Network Methods for Describing Sample Relationships in Genomic Datasets: Application to Huntington's Disease

    • Michael C Oldham et al BMC Syst Biol. 2012 PMID: 22691535

    Rich but complex HD data

    • Affymetrix microarray data from “the HD study”

      • Hodges et al: Regional and cellular gene expression changes in human Huntington’s disease brain. Hum Mol Genet 2006, 15(6):965-977

    • Brain samples of patients with HD (n = 44 individuals) and unaffected controls (n = 36 individuals, matched for age and sex)

    • caudate nucleus (CN), cerebellum (CB), primary motor cortex (Brodmann’s area 4; BA4), and prefrontal cortex (Brodmann’s area 9; BA9)

    • across five grades using Vonsattel’s neuropath criteria

    • Further, age, sex, the country where the experiment was performed (samples were processed in the United States and New Zealand) and the microarray hybridization batch

    Defining sample adjacency

    Why define this sample network adjacency measure?

    • Our proposed sample adjacency measure (based on β = 2) also has several other advantages.

      • it preserves the sign of the correlation

      • while any other power β could be used, the choice of β = 2 results in an adjacency measure that is close to the correlation when the correlation is large (e.g. larger than 0.6, which is often the case among samples in microarray data).

    • The adjacency measure allows one to define network concepts.


    • Gene connectivity = row sum of the adjacency matrix

      • For unweighted networks=number of direct neighbors

      • For weighted networks= sum of connection strengths to other nodes

      • Scaled connectivity:

    Clustering Coefficient

    Measures the cliquishness of a particular node

    « A node is cliquish if its neighbors know each other »

    This generalizes directly to weighted

    networks (Zhang and Horvath 2005)

    Clustering Coef of the black node = 0

    Clustering Coef = 1

    C(k) curve is a plot of scaled clustering coefficient versus scaled connectivity

    Sample network concepts reveal the profound effect of Huntington’s disease in caudate nucleus.

    Summary sample network

    • Z.k is a very useful measure for finding array outliers.

    • The correlation cor(K,C) between the connectivity and the clustering coefficient (two important network concepts) is a sensitive indicator of homogeneity among biological samples.

      • It can distinguish biologically meaningful relationships among subgroups of samples.

      • Advantage: This measure can highlight differences that cannot be found using differential expression

      • Disadvantage: It requires some work to figure out which genes lead to this effect.

      • Here: effect is concentrated in specific modules of genes

    • Sample network approach is implemented in an R function and tutorial


    Current and former lab members:

    • Peter Langfelder first author on many related articles

    • Jason Aten, Chaochao (Ricky) Cai, Jun Dong, Tova Fuller, Ai Li, Wen Lin, Michael Mason, Jeremy Miller, Mike Oldham, Chris Plaisier, Anja Presson, Lin Song, Kellen Winden, Yafeng Zhang, Andy Yip, Bin Zhang

    • Colleagues/Collaborators

    • Neuroscience: Dan Geschwind, Giovanni Coppola, Jeremy Miller, Mike Oldham, Roel Ophoff

    • Mouse: Jake Lusis, Tom Drake

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