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BMC Bioinformatics. 2007; 8(Suppl 6): S6.

Published online 2007 September 27. doi: 10.1186/1471-2105-8-S6-S6

PMCID: PMC1995543

Harmen J Bussemaker: ude.aibmuloc@4002bjh; Lucas D Ward: ude.aibmuloc@5012wdl; Andre Boorsma: ln.avu.ecneics@amsroob

Otto Warburg International Summer School and Workshop on Networks and
Regulation

Peter F Arndt and Martin Vingron

Copyright ©2007 Bussemaker 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.

This article has been cited by other articles in PMC.

The genomewide pattern of changes in mRNA expression measured using DNA microarrays is typically a complex superposition of the response of multiple regulatory pathways to changes in the environment of the cells. The use of prior information, either about the function of the protein encoded by each gene, or about the physical interactions between regulatory factors and the sequences controlling its expression, has emerged as a powerful approach for dissecting complex transcriptional responses.

We review two different approaches for combining the noisy expression levels of multiple individual genes into robust pathway-level differential expression scores. The first is based on a comparison between the distribution of expression levels of genes within a predefined gene set and those of all other genes in the genome. The second starts from an estimate of the strength of genomewide regulatory network connectivities based on sequence information or direct measurements of protein-DNA interactions, and uses regression analysis to estimate the activity of gene regulatory pathways. The statistical methods used are explained in detail.

By avoiding the thresholding of individual genes, pathway-level analysis of differential expression based on prior information can be considerably more sensitive to subtle changes in gene expression than gene-level analysis. The methods are technically straightforward and yield results that are easily interpretable, both biologically and statistically.

Many of the popular methods for analyzing DNA microarray expression data, from clustering [1] to more sophisticated machine-learning approaches [2-5], require expression data over a large number of different conditions as input. However, it is common to only have expression data for a few different strains and/or conditions. In this case, what is of interest are the changes in mRNA abundance for each gene, usually represented as the logarithm of the fold-change between test and control. The traditional way of analyzing such data is to first identify significantly up- and down-regulated genes, and subsequently to characterize these sets in terms of enrichment for functional annotation [6] or upstream promoter elements [7-9]. However, by requiring statistically significant differential expression at the level of individual genes, a lot of information about differential expression will be lost that could have been detected using analysis methods working at the level of pathways.

To understand this, assume that we are comparing two conditions and that the measurement
error for the fold-change of individual genes is 20%. Now consider a specific pathway
consisting of 100 genes that are all upregulated by 10%. This level of differential
expression is well within the noise for individual genes, none of which will therefore
be classified as significantly induced. However, the error in the *average
*expression of 100 *randomly *chosen genes will be on the order of
20%/$\sqrt{100}$
= 2%. The 10% change in expression at the level of the whole pathway therefore
corresponds to five units of standard error and is highly statistically significant.

In recent years, two distinct classes of methods have been developed that use prior information about how genes can be viewed as belonging to different regulatory or functional pathways (Figure (Figure1).1). This information can be used to score differential expression at the pathway level rather than at the gene level. The first class of methods represents pathways as gene sets, to which individual genes either belong or do not belong. One well-known source of such gene sets is the Gene Ontology (GO) project [6], where the classification is based on the function of the proteins encoded by each gene. The second class of methods takes a more sophisticated approach by assigning a regulatory susceptibility to each gene, quantifying how strongly this gene is expected to respond to a change in the activity of a specific regulatory pathway. For example, the affinity of a gene's promoter sequence for a specific transcription factor (TF) could be predicted using consensus motifs or weight matrices [10] and be used to predict the response of that gene to changes in TF activity.

In this review, we describe how such pathway-level analyses can be implemented
mathematically. It is helpful to understand that, in general, information about genes
comes in two different types: *categorical *information of boolean type ("true"
or "false"), which tells us whether or not a gene belongs to a specific gene set; and
*quantitative *information, e.g., the mRNA expression log-ratio between two
conditions for a gene or the ChIP-chip [11] fold
enrichment for the gene's promoter region. Given any two distinct features
characterizing each gene, their genomewide statistical association can be scored using
an appropriate statistical test (Table (Table11).

Suppose that we want to know whether a specific set of genes of interest is
statistically enriched for genes with a specific annotation in Gene Ontology. In this
case, both features (namely, "does the gene belong to the set of genes of interest"
and "is the gene associated with GO term X") are categorical, and the appropriate
statistic is the *overlap *between both gene sets. Let the total number of
genes in set *A *be *a*, the total number of genes in set *B *be
*b*, and the total number of genes in the genome be *n*.
Furthermore, let the overlap *x *denote the number of genes shared between
*A *and *B*. If the two sets are chosen randomly and independently,
the average overlap will be:

$$x=\frac{ab}{n}$$

(1)

This makes sense: if a fraction *b/n *of all genes belongs to set *B
*then the expected fraction of genes in set *A *that also belongs to set
*B *equals *x/a*. In the case of over-representation, when *x
*> <*x*>, the P-value that quantifies how likely it is to get at least
the same number of overlapping genes by chance, is given by

$${P}_{\text{over}}(x)={\displaystyle \sum _{{x}^{\prime}=x}^{\mathrm{min}(a,b)}H({x}^{\prime}|a,b,n)}$$

(2)

where *H *is the hypergeometric distribution given by

$$H(x|a,b,n)=\frac{\left(\begin{array}{c}a\\ x\end{array}\right)\left(\begin{array}{c}n-a\\ b-x\end{array}\right)}{\left(\begin{array}{c}n\\ b\end{array}\right)}$$

(3)

and

$$\left(\begin{array}{c}n\\ k\end{array}\right)=\frac{n!}{k!(n-k)!}$$

(4)

It is also possible to have significant under-representation (*x *<
<*x*>). In that case, the P-value is given by

$${P}_{\text{under}}(x)={\displaystyle \sum _{{x}^{\prime}=0}^{x}H({x}^{\prime}|a,b,n)}$$

(5)

This use of the cumulative hypergeometric distribution is also known as "Fisher's exact test." The test is by nature non-parametric because both input features are non-parametric. Under specific conditions the hypergeometric distribution may be approximated by the binomial or chi-square distribution. Several implementations of this approach are reviewed by Khatri and Draghici [12]. Since typically a large number of gene sets are scored in parallel, the p-values must be corrected for multiple testing. Grossman et al. [13] recently addressed technical complications arising from the strong overlap between the hierarchically organized Gene Ontology categories.

An early example of the use of predefined gene sets to analyze differential
expression at the pathway level can be found in Lascaris et al. [14]. The authors used a z-score to represent the
difference between the average expression in a gene set *S *consisting of
*n *genes and the genomewide mean *μ*:

$$z=\frac{{\overline{x}}_{S}-\mu}{{\sigma}_{\overline{x}}}$$

(6)

Here ${\sigma}_{\overline{x}}$
= *σ*/$\sqrt{n}$
is the standard error of the mean, *σ *being the standard deviation of
the genomewide distribution of log-ratios. The same metric is used by the "parametric
analysis of gene expression" (PAGE) method of Kim and Volsky [15]. For larger gene sets, however, the standard t-test for
the difference between means yields more accurate results [16]. The t-test, in general, scores the statistical
association between a categorical and quantitative feature. The categorical feature
is used to partition the set of all genes, *G*, into two complementary subsets
*S *and S'. $\overline{S}$The
*t *statistic measures the difference between the means of the two subsets
in units of its standard error:

$t=\frac{{\overline{x}}_{S}-{\overline{x}}_{{S}^{\prime}}}{\sigma \sqrt{\frac{1}{\left|S\right|}+\frac{1}{\left|{S}^{\prime}\right|}}}$

Here $\overline{x}$_{S
}and ${\overline{x}}_{{S}^{\prime}}$
are the mean expression value of genes in set *S *and *S'*,
respe$\overline{x}$ctively, and the standard error of the
difference is given by

$$s=\sqrt{\frac{(|S|-1){\sigma}_{S}^{2}+(|{S}^{\prime}|-1){\sigma}_{{S}^{\prime}}^{2}}{\left|S\right|+\left|{S}^{\prime}\right|-2}}$$

(8)

with *σ*_{S }and *σ*_{S'
}the standard deviation of the expression values of the genes within set
*S *and *S'*, respectively. Using a t-distribution with *n *-
2 degrees of freedom, each t-value can be converted to a p-value, which should again
be corrected for multiple testing.

Figure Figure22 shows a side-by-side comparison of Fisher's exact test and the t-test for a specific combination of GO category and genomewide differential expression profile. Fisher's exact test can only be applied once a set of "genes of interest" has been defined. We thresholded the fold-induction of individual genes to define this gene set, and computed GO category enrichment P-values at different thresholds (solid line/symbols). The smallest, most significant, P-value is obtained at an individual-gene threshold significantly below 2-fold induction, satisfied by over 500 genes. In general, the optimal threshold will depend on both the GO category and the expression data. By contrast, the two-sample t-test uses the expression value for all genes; no threshold for individual genes is required, an important practical advantage. While the optimal P-value from Fisher's exact test is slightly smaller than that of the two-sample t-test (dashed line), this seeming advantage disappears as soon as multiple-testing correction associated with the required threshold optimization is taken into account. Note that at the commonly used threshold of 2-fold induction, the two-sample t-test performs dramatically better.

Other statistical tests have also been used to detect differential expression of gene sets based on the distribution of expression values. The original version of the "gene set enrichment analysis" (GSEA) method [17] used the Kolmogorov-Smirnov (KS) statistic to test whether the distribution of expression levels in a specific gene set was different from that of all genes; this approach was later found to require a modification to work reliably [18]. The Wilcoxon-Mann-Whitney test, a non-parametric equivalent of the t-test that uses expression values only to rank the genes, has also been applied to this problem [19].

The assignment of genes to gene sets is categorical: Either the gene belongs to the set, or it does not. However, gene sets are often a proxy for regulatory pathways. This is most obvious in the case of the gene sets based on ChIP-chip data [11], which were used by Boorsma et al. [16] to analyze differential mRNA expression using the t-test. The strict delineation of "targets" of a given TF based on thresholding of the ChIP-chip signals is an oversimplification. In reality, the degree to which the transcription rate for a given gene responds to a change in the activity of the TF depends in a continuous fashion on the binding affinity between the TF and the promoter DNA (as well as interactions with co-factors, chromatin, etc.). Thus, if an estimate of this affinity is used as a predictor for changes in transcription rate (and therefore expression), a single parameter that quantifies the global change in TF activity may explain a wide range of transcriptional responses across the genome. This intuition can be formalized in the form of a linear regression model:

(9)

where *C *is an intercept and *F *a slope estimating the change in TF
activity. The dependent ("response") variable *A*_{g }is the
mRNA expression log-ratio of gene *g *between conditions. The independent
("predictor") variable *N*_{g }represents the regulatory
network connectivity between the TF and the promoter region of gene *g*. For
given *A*_{g }and *N*_{g}, the
deviance *D *between the measured and predicted expression values

$$D={\displaystyle \sum _{g}{({A}_{g}-C-F{N}_{g})}^{2}}$$

(10)

is minimized. The solution is given by

$$F=\frac{AN-AN{N}^{2}-{N2}^{}=\frac{\delta A\delta N\delta {N}^{2}}{}}{}$$

(11)

and

(12)

where <*X*> = (1/*G*) ∑_{g
}*X*_{g }denotes an average over all genes
and *δX*_{g } *X*_{g }-
<*X*> denotes the deviation from the genomic mean, so that
<*δX*^{2}> equals the variance of *X*. Because we
are dealing with *univariate *regression (a single independent variable), the
Pearson correlation coefficient between *A *and *N*,

$$r=\frac{\delta A\delta N\sqrt{\delta {A}^{2}\delta {N}^{2}}}{}$$

(13)

can be directly related to the slope *F *by the following equation:

$$F=r\sqrt{\frac{\delta {A}^{2}\delta {N}^{2}}{}}$$

(14)

It can furthermore be shown that, in the univariate case, *R*^{2},
defined as the fraction of the variance in expression that can be explained by the
linear model, is given by the square of Pearson correlation:

$${R}^{2}=\frac{\mathrm{var}(C+FN)}{\mathrm{var}(A)}=\frac{{\delta A\delta N2}^{\delta {A}^{2}\delta {N}^{2}={r}^{2}}}{}$$

(15)

A transformation of *r *due to R.A. Fisher

$$t=\frac{1}{2}\mathrm{ln}\left(\frac{1+r}{1-r}\right)$$

(20)

yields a statistic *t *that is distributed according a t-distribution with
*n *- 3 degrees of freedom, and can thus be easily converted to a p-value.
Again, multiple testing will need to be accounted for whenever the association with
multiple features is scored in parallel.

There are many ways in which the regulatory network connectivities
*N*_{g }can be chosen. The first application of
regression analysis to microarray data, by Bussemaker et al. [20], used integer motif counts in promoter regions. Continuous
sequence scores based on position-specific scoring matrices (PSSMs) [21,22] and position-specific
affinity matrices (PSAMs) [23,24] have also been used. The values for *R*^{2
}obtained with such sequence-based predictors are typically in the range of
1–5%. Another possible choice for *N *are ChIP-chip enrichment
(log-)ratios [25,26].
As these values are relatively noisy experimental measurements, the values for
*R*^{2 }observed in this case are usually smaller (< 1%).

In this work, rather than providing a comprehensive review of all relevant literature, we have outlined two conceptually different approaches for scoring differential expression at the pathway level. These methods use prior information about how different genes relate to each other to reduce the dimensionality of the problem. This obviates the need to first obtain gene clusters or modules from expression data over multiple conditions, and thereby makes it possible to analyze each differential expression profile by itself in a condition-specific fashion.

HJB drafted the paper, which was edited and proofread by all authors. LDW and AB prepared Figure Figure11 and and2,2, respectively.

We thank members of the Bussemaker Lab for valuable discussions, and Barrett Foat for a critical reading of the manuscript. This work was supported by grants HG003008, CA121852, and GM074105 from the National Institutes of Health and grant APB.5504 from the Netherlands Foundation for Technical Research (STW).

This article has been published as part of *BMC Bioinformatics* Volume 8
Supplement 6, 2007: Otto Warburg International Summer School and Workshop on Networks
and Regulation. The full contents of the supplement are available online at
http://www.biomedcentral.com/1471-2105/8?issue=S6

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