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Cited by 476Bioinformatics and Genomic NetworksGene expression and cancer classificationGene Regulatory Network Analysis

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Comparison of co-expression measures: mutual information, correlation, and model based indices

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Song, Lin, Langfelder, Peter, Horvath, Steve·2012·BMC Bioinformatics
DOI 10.1186/1471-2105-13-328PMC PMC3586947
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Abstract

Background: Co-expression measures are often used to define networks among genes. Mutual information (MI) is often used as a generalized correlation measure. It is not clear how much MI adds beyond standard (robust) correlation measures or regression model based association measures. Further, it is important to assess what transformations of these and other co-expression measures lead to biologically meaningful modules (clusters of genes). Results: We provide a comprehensive comparison between mutual information and several correlation measures in 8 empirical data sets and in simulations. We also study different approaches for transforming an adjacency matrix, e.g. using the topological overlap measure. Overall, we confirm close relationships between MI and correlation in all data sets which reflects the fact that most gene pairs satisfy linear or monotonic relationships. We discuss rare situations when the two measures disagree. We also compare correlation and MI based approaches when it comes to defining co-expression network modules. We show that a robust measure of correlation (the biweight midcorrelation transformed via the topological overlap transformation) leads to modules that are superior to MI based modules and maximal information coefficient (MIC) based modules in terms of gene ontology enrichment. We present a function that relates correlation to mutual information which can be used to approximate the mutual information from the corresponding correlation coefficient. We propose the use of polynomial or spline regression models as an alternative to MI for capturing non-linear relationships between quantitative variables. Conclusion: The biweight midcorrelation outperforms MI in terms of elucidating gene pairwise relationships. Coupled with the topological overlap matrix transformation, it often leads to more significantly enriched co-expression modules. Spline and polynomial networks form attractive alternatives to MI in case of non-linear relationships. Our results indicate that MI networks can safely be replaced by correlation networks when it comes to measuring co-expression relationships in stationary data.

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Figure 1 Relating mutual information based adjacencies to the Pearson correlation and biweight midcorrelation in simulation. Each point corresponds to a pair of numeric vectors x and y with length m = 1000 . These pairs of vectors were simulated to exhibit different correlations. AUV1, AUV2, cor, bicor are abbreviations for AMI,UniversalVersion1, AMI,UniversalVersion2, Pearson correlation and biweight midcorrelation, respectively. (A) MI-based adjacency AMI,UniversalVersion2 versus absolute Pearson correlation. Spearman correlation of the two measures and the corresponding p-value are shown at the top, implying a strong monotonic relationship. The red line shows the predicted AMI,UniversalVersion2 according to Fcor−MI(Eq. 18). Note that the prediction function is highly accurate in simulation. (B) Observed AMI,UniversalVersion2 versus its predicted value. The straight line has slope 1 and intercept 0. (C) Observed Pearson correlation (x-axis) and the corresponding bicor values (y-axis). The straight line has slope 1 and intercept 0. These 2 measurements are practically indistinguishable when x and y are normally distributed. (D) AMI,UniversalVersion2 versus bicor. Spearman correlation and p-value of the 2 measurements are presented at the top, and predicted AMI,UniversalVersion2 are shown as the red line. (E) AMI,UniversalVersion2 versus AMI.SymmetricUncertainty. (F) AMI,UniversalVersion2 versus AMI,UniversalVersion1.
Figure 1 Relating mutual information based adjacencies to the Pearson correlation and biweight midcorrelation in simulation. Each point corresponds to a pair of numeric vectors x and y with length m = 1000 . These pairs of vectors were simulated to exhibit different correlations. AUV1, AUV2, cor, bicor are abbreviations for AMI,UniversalVersion1, AMI,UniversalVersion2, Pearson correlation and biweight midcorrelation, respectively. (A) MI-based adjacency AMI,UniversalVersion2 versus absolute Pearson correlation. Spearman correlation of the two measures and the corresponding p-value are shown at the top, implying a strong monotonic relationship. The red line shows the predicted AMI,UniversalVersion2 according to Fcor−MI(Eq. 18). Note that the prediction function is highly accurate in simulation. (B) Observed AMI,UniversalVersion2 versus its predicted value. The straight line has slope 1 and intercept 0. (C) Observed Pearson correlation (x-axis) and the corresponding bicor values (y-axis). The straight line has slope 1 and intercept 0. These 2 measurements are practically indistinguishable when x and y are normally distributed. (D) AMI,UniversalVersion2 versus bicor. Spearman correlation and p-value of the 2 measurements are presented at the top, and predicted AMI,UniversalVersion2 are shown as the red line. (E) AMI,UniversalVersion2 versus AMI.SymmetricUncertainty. (F) AMI,UniversalVersion2 versus AMI,UniversalVersion1.CC BY · PubMed Central
Figure 2 Comparison of correlation and mutual information based co-expression measures in 8 empirical data sets. Absolute value of bicor versus AMI,UniversalVersion2 for all probe pairs in each data set. The Spearman correlation and corresponding p-value between the two measures are shown at the top. The two measures show different levels of monotonic relationships in data sets. The red curve predicts AMI,UniversalVersion2 from bicor based on Eq. 18. The blue circle highlights the probe pair with the highest AMI,UniversalVersion2 z-score among those with insignificant bicor z-scores (less than 1.9 ); the red circle highlights the probe pair with the highest bicor z-score among those with insignificant AMI,UniversalVersion2 z-scores (less than 1.9 ).
Figure 2 Comparison of correlation and mutual information based co-expression measures in 8 empirical data sets. Absolute value of bicor versus AMI,UniversalVersion2 for all probe pairs in each data set. The Spearman correlation and corresponding p-value between the two measures are shown at the top. The two measures show different levels of monotonic relationships in data sets. The red curve predicts AMI,UniversalVersion2 from bicor based on Eq. 18. The blue circle highlights the probe pair with the highest AMI,UniversalVersion2 z-score among those with insignificant bicor z-scores (less than 1.9 ); the red circle highlights the probe pair with the highest bicor z-score among those with insignificant AMI,UniversalVersion2 z-scores (less than 1.9 ).CC BY · PubMed Central
Figure 3 Comparison of predicted and observed AMI,UniversalVersion2 in 8 empirical data sets. In all data sets, prediction from bicor based on Eq. 18 and observed AMI,UniversalVersion2 are highly correlated (the Pearson correlation and corresponding p-value shown at top). Line y=x is added. Blue and red circles have the same meaning as in Figure 2.
Figure 3 Comparison of predicted and observed AMI,UniversalVersion2 in 8 empirical data sets. In all data sets, prediction from bicor based on Eq. 18 and observed AMI,UniversalVersion2 are highly correlated (the Pearson correlation and corresponding p-value shown at top). Line y=x is added. Blue and red circles have the same meaning as in Figure 2.CC BY · PubMed Central
Figure 4 Gene expression of example probe pairs for which the correlation and mutual information based measures disagree. (A) Gene expression of probe pairs highlighted by blue circles in Figure 2. (B) Gene expression of probe pairs highlighted by red circles in Figure 2. The Pearson correlation, bicor, AMI,UniversalVersion2 values and z-scores of the latter two measures are shown at the top. Mutual information is susceptible to outliers, sometimes detects unusual patterns that are hard to explain, and often misses linear relations that are captured by bicor.
Figure 4 Gene expression of example probe pairs for which the correlation and mutual information based measures disagree. (A) Gene expression of probe pairs highlighted by blue circles in Figure 2. (B) Gene expression of probe pairs highlighted by red circles in Figure 2. The Pearson correlation, bicor, AMI,UniversalVersion2 values and z-scores of the latter two measures are shown at the top. Mutual information is susceptible to outliers, sometimes detects unusual patterns that are hard to explain, and often misses linear relations that are captured by bicor.CC BY · PubMed Central
Figure 5 Module identification based on various network inference methods in simulation with non-linear gene-gene relationships. The data set is composed of 200 genes across 200 samples. 3 true modules are designed. Two of them, labeled with colors turquoise and blue, contain linear and non-linear (quadratic) gene-gene relationships. For each adjacency, the clustering tree and module colors are shown. True simulated module assignment is shown by the first color band underneath each tree. On top of each panel is the Rand index between inferred and simulated module assignments.
Figure 5 Module identification based on various network inference methods in simulation with non-linear gene-gene relationships. The data set is composed of 200 genes across 200 samples. 3 true modules are designed. Two of them, labeled with colors turquoise and blue, contain linear and non-linear (quadratic) gene-gene relationships. For each adjacency, the clustering tree and module colors are shown. True simulated module assignment is shown by the first color band underneath each tree. On top of each panel is the Rand index between inferred and simulated module assignments.CC BY · PubMed Central
Figure 6 Gene ontology enrichment analysis comparing AMI,UniversalVersion2 with bicor based adjacencies in 8 empirical data sets. 5 best GO enrichment p-values from all modules identified using each adjacency are log transformed, pooled together and shown as barplots. Error bars stand for 95% confidence intervals. On top of each panel is a p-value based on multi-group comparison test. TOM outperforms the others in all 8 data sets.
Figure 6 Gene ontology enrichment analysis comparing AMI,UniversalVersion2 with bicor based adjacencies in 8 empirical data sets. 5 best GO enrichment p-values from all modules identified using each adjacency are log transformed, pooled together and shown as barplots. Error bars stand for 95% confidence intervals. On top of each panel is a p-value based on multi-group comparison test. TOM outperforms the others in all 8 data sets.CC BY · PubMed Central
Figure 7 Gene ontology enrichment analysis comparing TOM with MI based adjacencies in 8 empirical data sets. 5 best GO enrichment p-values from all modules identified using each adjacency are log transformed, pooled together and shown as barplots. Error bars stand for 95% confidence intervals. On top of each panel is a p-value based on multi-group comparison test. TOM outperforms the others in 5 data sets. ARACNE(ε = 0 ) wins in two data sets, making it the second best.
Figure 7 Gene ontology enrichment analysis comparing TOM with MI based adjacencies in 8 empirical data sets. 5 best GO enrichment p-values from all modules identified using each adjacency are log transformed, pooled together and shown as barplots. Error bars stand for 95% confidence intervals. On top of each panel is a p-value based on multi-group comparison test. TOM outperforms the others in 5 data sets. ARACNE(ε = 0 ) wins in two data sets, making it the second best.CC BY · PubMed Central
Figure 8 Fitting polynomial and spline regression models to measure non-linear relationships. (A-B) A pair of simulated data x,y(black dots) with the black curve illustrating the true expected value E(y) given x, where E(y|x) = cos(x2)2. The red curve shows the fit of a polynomial regression model with degree d = 4 . The blue curve shows the fit of a cubic spline regression model with 2 knots. Fitting indices of the two models are shown at the top. In simulation, polynomial and cubic spline regression models can properly discover non-linear relations. (C-D) Comparisons of regression models and mutual information based co-expression measures in the ND data set. Co-expression of probe pairs is measured with polynomial (d = 3)/cubic spline regressions (x-axis) and mutual information AMI,UniversalVersion2(y-axis). The Spearman correlation and p-value of the two measures are shown at the top. (E-F) Comparisons in the mouse muscle data set. AMI,UniversalVersion2 has a stronger correlation with regression models than with bicor, indicating that the first two measures can capture certain common non-linear patterns.
Figure 8 Fitting polynomial and spline regression models to measure non-linear relationships. (A-B) A pair of simulated data x,y(black dots) with the black curve illustrating the true expected value E(y) given x, where E(y|x) = cos(x2)2. The red curve shows the fit of a polynomial regression model with degree d = 4 . The blue curve shows the fit of a cubic spline regression model with 2 knots. Fitting indices of the two models are shown at the top. In simulation, polynomial and cubic spline regression models can properly discover non-linear relations. (C-D) Comparisons of regression models and mutual information based co-expression measures in the ND data set. Co-expression of probe pairs is measured with polynomial (d = 3)/cubic spline regressions (x-axis) and mutual information AMI,UniversalVersion2(y-axis). The Spearman correlation and p-value of the two measures are shown at the top. (E-F) Comparisons in the mouse muscle data set. AMI,UniversalVersion2 has a stronger correlation with regression models than with bicor, indicating that the first two measures can capture certain common non-linear patterns.CC BY · PubMed Central
Figure 9 Rand indices in simulations with various number of observations. Simulation sample size versus Rand indices between inferred and simulated module assignments from different network inference methods. increase as the simulation data set contains more samples. Non-linear measures, especially polynomial and spline regression models, outperform other measures as sample size increases.
Figure 9 Rand indices in simulations with various number of observations. Simulation sample size versus Rand indices between inferred and simulated module assignments from different network inference methods. increase as the simulation data set contains more samples. Non-linear measures, especially polynomial and spline regression models, outperform other measures as sample size increases.CC BY · PubMed Central

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Metadata

Authors
Song, L., Langfelder, P., & Horvath, S.
Affiliations
University of California, Los Angeles (US)
Corresponding author
Horvath, Steve
ORCID
Venue
BMC Bioinformatics
Publisher
Springer Science and Business Media LLC
Published
2012-12
Volume
13
Issue
1
Pages
328-328
ISSN
1471-2105 (ISSN-L)
DOI
10.1186/1471-2105-13-328
PMC
PMC3586947
OpenAlex
W2133254377
Cited by
460
References
78
Citations by year
2026: 9 · 2025: 27 · 2024: 34 · 2023: 35 · 2022: 48 · 2021: 49 · 2020: 51 · 2019: 47 · 2018: 36 · 2017: 33 · 2016: 38 · 2015: 25 · 2014: 18 · 2013: 8 · 2012: 2
FWCI
6.82
Type
journal article
Language
en
Open access
Licence
cc-by
Locations
  • BMC Bioinformatics (publishedVersion, cc-by)
  • Europe PMC (PubMed Central) (publishedVersion, cc-by)
  • PubMed Central (publishedVersion, cc-by)
  • PubMed (publishedVersion)
  • DOAJ (DOAJ: Directory of Open Access Journals) (submittedVersion)
  • eScholarship (California Digital Library) (submittedVersion)
Funders
  • National Institute on Drug Abuse (R01DA030913)
  • National Cancer Institute (P50CA092131)
  • National Institute of Neurological Disorders and Stroke (R01NS058980)
  • Clinical and Translational Science Institute, University of California, Los Angeles
Topics
  • Life Sciences › Biochemistry, Genetics and Molecular Biology › Molecular Biology › Bioinformatics and Genomic Networks
  • Life Sciences › Biochemistry, Genetics and Molecular Biology › Molecular Biology › Gene expression and cancer classification
  • Life Sciences › Biochemistry, Genetics and Molecular Biology › Molecular Biology › Gene Regulatory Network Analysis
Keywords
mutual information, gene co-expression network, correlation measures, gene modules, maximal information coefficient, spline regression, polynomial regression, gene ontology enrichment
MeSH terms
Algorithms, Multigene Family, Humans, Models, Genetic, Statistics as Topic, Gene Expression Profiling, Gene Regulatory Networks

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