Comparison of threshold selection methods for microarray gene co-expression matrices.

Abstract

Network and clustering analyses of microarray co-expression correlation data often require application of a threshold to discard small correlations, thus reducing computational demands and decreasing the number of uninformative correlations. This study investigated threshold selection in the context of combinatorial network analysis of transcriptome data.
Six conceptually diverse methods - based on number of maximal cliques, correlation of control spots with expressed genes, top 1% of correlations, spectral graph clustering, Bonferroni correction of p-values, and statistical power - were used to estimate a correlation threshold for three time-series microarray datasets. The validity of thresholds was tested by comparison to thresholds derived from Gene Ontology information. Stability and reliability of the best methods were evaluated with block bootstrapping.Two threshold methods, number of maximal cliques and spectral graph, used information in the correlation matrix structure and performed well in terms of stability. Comparison to Gene Ontology found thresholds from number of maximal cliques extracted from a co-expression matrix were the most biologically valid. Approaches to improve both methods were suggested.
Threshold selection approaches based on network structure of gene relationships gave thresholds with greater relevance to curated biological relationships than approaches based on statistical pair-wise relationships.

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BMC Research Notes
Open Access
Short Report
Comparison of threshold selection methods for microarray gene
co-expression matrices
Bhavesh R Borate1, Elissa J Chesler3, Michael A Langston2,
Arnold M Saxton*4 and Brynn H Voy3
Address: 1Genome Science and Technology Program, University of Tennessee, Knoxville, Tennessee, USA, 2Department of Electrical Engineering
and Computer Science, University of Tennessee, Knoxville, Tennessee, USA, 3Oak Ridge National Laboratory, Systems Genetics Group, Biosciences
Division, Oak Ridge, Tennessee, USA and 4Department of Animal Science, University of Tennessee, Knoxville, Tennessee, USA
Email: Bhavesh R Borate - boratebr@mail.nih.gov; Elissa J Chesler - elissa.chesler@jax.org; Michael A Langston - langston@eecs.utk.edu;
Arnold M Saxton* - asaxton@utk.edu; Brynn H Voy - bhvoy@utk.edu
* Corresponding author
Abstract
Background: Network and clustering analyses of microarray co-expression correlation data often
require application of a threshold to discard small correlations, thus reducing computational
demands and decreasing the number of uninformative correlations. This study investigated
threshold selection in the context of combinatorial network analysis of transcriptome data.
Findings: Six conceptually diverse methods - based on number of maximal cliques, correlation of
control spots with expressed genes, top 1% of correlations, spectral graph clustering, Bonferroni
correction of p-values, and statistical power - were used to estimate a correlation threshold for
three time-series microarray datasets. The validity of thresholds was tested by comparison to
thresholds derived from Gene Ontology information. Stability and reliability of the best methods
were evaluated with block bootstrapping.
Two threshold methods, number of maximal cliques and spectral graph, used information in the
correlation matrix structure and performed well in terms of stability. Comparison to Gene
Ontology found thresholds from number of maximal cliques extracted from a co-expression matrix
were the most biologically valid. Approaches to improve both methods were suggested.
Conclusion: Threshold selection approaches based on network structure of gene relationships
gave thresholds with greater relevance to curated biological relationships than approaches based
on statistical pair-wise relationships.
Introduction
To extract gene networks from microarray data, correla-
tions are often used as a measure of gene co-expression. A
typical microarray with 20,000 gene probes will produce
200 million correlations. Correlations below a threshold
value, closer to zero, will be less meaningful. Hard and
soft threshold approaches have been applied to biological
data. Hard thresholds discard gene pairs with correlation
below the threshold, while soft thresholds use the correla-
tion value to weight gene network relationships. Zhang
and Horvath [1] concluded that soft thresholds based on
aggregate, modular relationships between genes gave
Published: 2 December 2009
BMC Research Notes 2009, 2:240doi:10.1186/1756-0500-2-240
Received: 27 August 2009
Accepted: 2 December 2009
This article is available from: http://www.biomedcentral.com/1756-0500/2/240
© 2009 Saxton et al; licensee BioMed Central Ltd.
This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0),
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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more robust results, but data reduction by a hard thresh-
old is often essential for computational tractability of
graph algorithms.
We focus on relevance networks, created by applying a
hard threshold to the gene expression correlation matrix
[2], then extracting gene networks. The resulting networks
have been well documented in recent literature to yield
sets of co-expressed genes [3-5]. Relevance networks are
easily converted to graphs, with genes as vertices, only
connected by an edge if their correlation is above the
threshold. A clique is a sub-graph in which all nodes are
connected to each other [6]. A disadvantage of using
cliques is the computational requirements, which grow
exponentially with number of genes. Thus hard threshold
selection is required when performing clique extraction
on microarray data.
Current approaches to threshold selection are typically
statistically based, and do not fully reflect the connectivity
of the data [7]. Methods based on statistical arguments
may not necessarily yield biologically significant relation-
ships [3,8].
Some studies used an arbitrary threshold correlation such
as 0.80 [9]. Moriyama et al. [10] obtained random corre-
lation distributions for gene pairs by permuting their
expression values and defended their choice of threshold
based on statistical significance. Lee et al. [11] used the
top 1% of correlations (absolute value) to build a co-
expression network. Voy et al. [3] used distribution of cor-
relations of genes with buffer spots on the arrays to select
a threshold correlation value of 0.875.
However, using connectivity of the data to derive thresh-
olds has been suggested. Langston et al. [12] recom-
mended use of ontological
significance and various graph structural attributes to
arrive at a correlation threshold. Palla et al. [13] found
that a threshold based on clique size was effective at sepa-
rating networks.
distance, statistical
Here two threshold selection methods based on correla-
tion graph structure are compared with common statisti-
cally based methods. The graph based methods used
spectral properties [14] or number of cliques to select a
threshold. Objectives were to compare the various hard
threshold methods for validity (retention of biological
information), stability, and reliability.
Methods
Datasets
Three yeast S. cerevisiae time-series datasets were chosen
for this study: 31 arrays for Anoxia state [15], 21 arrays for
Reoxygenation state [15] and 18 arrays from yeast cultures
synchronized using Alpha-factor arrest [16]. Data are
available on Gene Expression Omnibus under GSE2246,
GSE2267 and GSE22. Extensive GO annotation for S. cer-
evisiae genes influenced the selection. Exploratory data
analyses within each dataset using PCA, box plots and
pair-wise correlations between arrays found no outlier
arrays. Quantile plots showed data were normally distrib-
uted, and distribution of correlations among gene expres-
sion profiles had the expected bell-shaped curve, so all
data were used.
Software
Software written by Langston and colleagues (University
of Tennessee) was used, including Datagen version 1.4a
for computing correlations, maximal clique enumeration
code version 2.0.1 [17], spectral analysis code [14], and
GO Pairwise Similarity analysis code version 1.0. Matrix
calculations for spectral graph analysis were carried out in
MATLAB 7.0. P-values were calculated in SAS version 9.1
(Cary; NC). Statistical power was calculated using PASS
statistical software http://www.ncss.com/pass.html.
Threshold Estimation
Six conceptually different approaches were evaluated:
1) Numbers of maximal cliques were calculated at each
potential correlation threshold, starting at r = 0.99. The
threshold was lowered, in steps of 0.01, and number of
maximal cliques increased due to greater connections
among genes. When clique number increased two times
(Maximal Clique-2) or three times (Maximal Clique-3)
the previous value, that correlation was chosen as the
threshold.
2) For each potential threshold correlation value, spectral
graph theory [18] was used to decompose the resulting
graph into eigenvalues and eigenvectors, which were used
to enumerate spectral clusters [19]. As the potential
threshold was incrementally lowered in steps of 0.01, a
peak in the number of clusters occurs, and the threshold
is chosen to maximize cluster number. Details are in [14].
3) Correlations of control spots with all other genes on
the array were calculated, creating a null distribution. The
99th percentile correlation value (absolute value) of this
distribution gave the threshold.
4) The top 1% of all correlations (absolute value) among
genes was used to estimate a threshold [11]. Correlations
were ranked, and the correlation at the 99th percentile
was the threshold estimate. Note that the control spot
method uses a different subset of correlations (only with
control spots), whereas this method uses all correlations
among genes.

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5) A p-value for every correlation was computed, testing if
the correlation was zero (Fisher's z-transformation).
Threshold estimate was the correlation value correspond-
ing to the critical Bonferroni p-value, 0.05/number of cor-
relations. This threshold will remove any correlations that
are statistically equal to zero.
6) Statistical power calculations were used to find the cor-
relation value that gave an 80% chance of rejecting the
null hypothesis, Ho: correlation = 0. Type I error rate in
these calculations was Bonferroni-adjusted to correct for
multiple testing.
Further details on computing these threshold estimation
methods are in the Additional file 1.
Performance Evaluation
Performance of the threshold estimation methods was
evaluated by comparison to a biologically based Gene
Ontology threshold. GO
gene_ontology_edit.obo.2008-05-01.gz. The biological
meaning for each correlation bin (in 0.01 increments)
was the average of functional similarity scores for all gene
pairs within that correlation bin. Functional similarity for
a pair of genes was defined as log(n/N)/log(2/N), where n
is the number of genes in the lowest GO category that con-
tained both genes, and N is the total number of genes
annotated for the organism. The formula normalizes
Functional similarity to a 0 to 1 range, and a value of 1
means the GO category contained only the two genes
being considered (perfect similarity). GO threshold esti-
mate was defined as the correlation at which change in
average functional similarity exceeded median change
plus half its standard deviation, thus identifying where
biological information begins to accumulate.
data used was
To study stability of the methods, 10,000 block bootstrap
samples were created by sampling arrays with replace-
ment from each block. Blocks were defined to be 2 or 3
adjacent time periods, such that each block contained 3 or
4 arrays. Block bootstrapping was necessary to preserve as
much as possible the time-course dependency structure of
the experiments [20]. For each of the 10,000 samples, a
threshold estimate was calculated by each method, and
the distribution of these thresholds was used to compare
threshold methods for stability.
Results
Functional similarity scores for the three datasets are dis-
played in Figure 1. Changes in scores across correlation
values were similar for all datasets, and the lack of GO
term relationship for negative correlations is striking.
Because of this, the GO threshold was defined by the
curve for positive correlations. Biological relationship
begins to increase sharply above a correlation value of
0.80, and this produced the GO thresholds in Table 1.
Estimated thresholds obtained by each method are listed
in Table 1 for the three datasets. If estimated threshold is
higher than the biological threshold, false negatives will
occur, because data reduction by the higher threshold will
remove real relationships. Conversely, using a threshold
below the biological threshold will create false positives,
and relationships that are not real would be included in
the network. In discovery-based settings, false positives
are more acceptable, as they can be removed with further
validation. Thus methods that estimate a lower threshold
are preferred. Maximal Clique-2 and Spectral Clustering
performed better than the other methods, based on
summed absolute deviations from GO threshold (Table
1). Maximal Clique-2 was further from the GO threshold,
but might be preferred since it never exceeded that thresh-
old.
The estimated threshold derived for selected methods for
each dataset is compared to bootstrap distributions in
Table 2. The best methods from above, Maximal Clique-2
and Spectral Clustering, and two other methods for com-
parative purposes were chosen for this analysis. The boot-
strap mean was never less than the estimated threshold,
and occasionally was two standard deviations above. This
upward bias in correlation is expected, as each time period
had a limited number of arrays, making it likely that the
identical array would be resampled. However, Maximal
Clique and Spectral Clustering methods showed more
resistance to this bias. The bootstrap standard deviation
measures ability of the methods to produce similar
Change in GO functional similarity score across correlation values
Figure 1
Change in GO functional similarity score across cor-
relation values. Lines represent Anoxia dataset (solid line),
Reoxygenation dataset (dashed line) and Alpha dataset (dot-
ted line).

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threshold estimates from randomized arrays. Again the
network-based methods showed the lowest standard devi-
ations, and highest stability. All methods showed poorest
performance with the Alpha dataset, possibly due to its
unreplicated design. This makes it less likely that all time
levels would be represented in the bootstrap samples,
whereas the other datasets had glucose and galactose bio-
logical replicates.
Discussion
The two network-based methods, Maximal Clique-2 and
Spectral Clustering, performed very well in terms of boot-
strap stability and biological validity. Though Maximal
Clique-2 method gave thresholds close to the biological
threshold, and always below, the method had slightly
higher bootstrap standard deviations. The robustness of
the Maximal Clique-2 algorithm could be enhanced by
exclusion of smaller cliques in the graph, for example
cliques of size 3. Spectral Clustering thresholds were on
average closer to biological thresholds, but too often
exceeded it. However, if all thresholds for Spectral Cluster-
ing were lowered by 0.05, it would have been clearly the
best method. Further fine-tuning of the parameters in the
algorithm (size of sliding window, different tolerance lev-
Table 1: Estimated threshold for each method by dataset, with methods sorted by the sum of absolute deviations from the GO
functional similarity threshold.
MethodAnoxiaReoxygenation AlphaAbsolute deviations from GO threshold
GO Functional Similarity0.97 0.92 0.85
Spectral Clustering 0.93
0.97a
0.89
0.04+0.05+0.04 = 0.13
Maximal Clique-20.900.910.740.07+0.01+0.11 = 0.19
Power 0.88
0.940.96
0.09+0.02+0.11 = 0.22
Bonferroni adjustment0.85
0.93 0.95
0.12+0.01+0.10 = 0.23
Control-Spot0.930.83 0.700.04+0.09+0.15 = 0.28
Maximal Clique-3 0.870.890.60 0.10+0.03+0.25 = 0.38
Top 1 Percent 0.81 0.81 0.720.16+0.11+0.13 = 0.40
aThresholds above the GO functional similarity threshold are in bold.
Table 2: Summary of bootstrap results compares the estimated threshold with the bootstrap distribution for the four selected
methods.
MethodDatasetEstimated ThresholdBootstrap MeanDifferencea
Bootstrap Standard Deviation
Maximal Clique-2
Anoxia
Reoxy
Alpha
0.90
0.91
0.74
0.91
0.93
0.78
-0.01
-0.02
-0.04
0.015
0.009
0.057
Spectral Clustering
Anoxia
Reoxy
Alpha
0.93
0.97
0.89
0.95
0.97
**0.95
-0.02
0.00
-0.06
0.012
0.011
0.017
Top 1 Percent
Anoxia
Reoxy
Alpha
0.81
0.81
0.72
0.83
0.84
**0.79
-0.02
-0.03
-0.07
0.011
0.016
0.027
Control Spot
Anoxia
Reoxy
Alpha
0.93
0.83
0.70
0.95
**0.90
**0.82
-0.02
-0.07
-0.12
0.015
0.034
0.043
a Estimated threshold minus bootstrap mean.
** Estimated threshold is more than 2 std. deviations from bootstrap mean.

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els for cluster formation) may improve the method's
validity. In a recent paper, Almendral and Díaz-Guilera
[21] documented the sensitivity of the non-zero eigen-
value to network changes. All methods had subjective set-
tings, and further work on many more species and
experiments would be needed to establish best choices.
The results from this study complement the work of
Zhang and Horvath [1] which concluded that thresholds
based on the scale-free topology - the formation of hubs
and densely-connected sub-graphs - produced more
robust results. The statistically-based methods studied
here are directly dependent on the correlation distribution
and thus were unable to capture biological relationships.
Although the Control-Spot method is based on logical
reasoning, the high correlation of control spots with other
genes on the arrays weakened the method's validity. The
Top 1% Correlations method is arbitrary, and failed to
capture biological relationships. Statistical considerations
used for the Power and Bonferroni methods were also not
able to identify biological relationships, reflecting the
well-known discrepancy between biological and statistical
significance. Experiments that are small will produce
thresholds that are too high, while large experiments will
give excessively low thresholds, even though the biologi-
cal relationships are the same.
The GO similarity measure of biological validity we have
used, however, is by no means perfect and is just one way
of quantifying biological information. Khatri and Dragh-
ici [22] have listed limitations of GO in detail. We also
found low GO scores at high negative correlations as com-
pared to the high GO score associated with high positive
correlations for all three datasets. The drop in GO score at
high negative correlations could be due to several reasons,
for example experimental and analytical limitations to
detect biologically negative correlations among genes,
and limited gene annotations [11]. As the quantification
of biological information in data gets more precise, the
selection of thresholds should become easier. In fact, note
that a method like the GO threshold used here would be
a logical choice if GO information were complete and
accurate.
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
BRB wrote code for the analyses, summarized results, and
drafted the paper. All authors were involved in study
design, and read and approved the final manuscript.
Additional material
Acknowledgements
This research has been supported in part by the National Institutes of
Health under grants P01DA015027-01, R01HD052472-02, R01MH074460-
01, U01AA013512 and U01AA013641-04 and by the UT-ORNL Science
Alliance. Dr. E.J. Chesler was supported by NIAAA Integrative Neuro-
science Initiative on Alcoholism under grants U01AA13499 and
U24AA13513. This research used resources of the National Energy
Research Scientific Computing Center, which is supported by the Office of
Science of the U.S. Department of Energy under Contract No. DE-AC02-
05CH11231. Additional support was provided by the University of Tennes-
see Genome Science and Technology program. John Eblen, Andy Perkins,
Gary Rogers and Yun Zhang helped with basic issues of algorithm synthesis.
Drs. Bing Zhang and Roumyana Yordanova provided valuable comments on
certain aspects of this study.
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Additional file 1
Methodology for Threshold Estimation. Details on the six threshold esti-
mation methods are presented in a computationally oriented manner.
Click here for file
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0500-2-240-S1.PDF]

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