Bioinformatics Advance Access originally published online on December 20, 2006
Bioinformatics 2007 23(4):401-407; doi:10.1093/bioinformatics/btl633
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Enrichment or depletion of a GO category within a class of genes: which test?
Équipe de Statistique Appliquée 10 rue Vauquelin, 75005 Paris, France
1 Laboratoire de Neurobiologie et Diversité Cellulaire, École Supérieure de Physique et de Chimie Industrielles (ESPCI) 10 rue Vauquelin, 75005 Paris, France
*To whom correspondence should be addressed.
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ABSTRACT |
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 TOP ABSTRACT 1 INTRODUCTION2 PROBLEM STATEMENT3 CANDIDATE FORMULATIONS4 TESTS AND P-VALUES5 SUMMARY AND DISCUSSION6 NUMERICAL ILLUSTRATIONS7 CONCLUSIONREFERENCES |
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Motivation: A number of available program packages determine the significant enrichments and/or depletions of GO categories among a class of genes of interest. Whereas a correct formulation of the problem leads to a single exact null distribution, these GO tools use a large variety of statistical tests whose denominations often do not clarify the underlying P-value computations.
Summary: We review the different formulations of the problem and the tests they lead to: the binomial, 
2, equality of two probabilities, Fisher's exact and hypergeometric tests. We clarify the relationships existing between these tests, in particular the equivalence between the hypergeometric test and Fisher's exact test. We recall that the other tests are valid only for large samples, the test of equality of two probabilities and the 2-test being equivalent. We discuss the appropriateness of one- and two-sided P-values, as well as some discreteness and conservatism issues.
Contact: isabelle.rivals{at}espci.fr
Supplementary information: Supplementary data are available at Bioinformatics online.
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1 INTRODUCTION |
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TOPABSTRACT1 INTRODUCTION2 PROBLEM STATEMENT3 CANDIDATE FORMULATIONS4 TESTS AND P-VALUES5 SUMMARY AND DISCUSSION6 NUMERICAL ILLUSTRATIONS7 CONCLUSIONREFERENCES |
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A common problem in functional genomic studies is to detect significant enrichments and/or depletions of Gene Ontology (GO) categories within a class of genes of interest, typically the class of significantly differentially expressed (DE) genes. Many GO processing tools perform this task using various statistical tests refered to as: the binomial test, the
2-test, the equality of two probabilities test, Fisher's exact test and the hypergeometric test (see Table 1). The authors of some packages claim the advantages of the test(s) they propose, often seemingly contradicting each other. For example, Zeeberg et al. (2003) favor Fisher's exact test: ‘Unlike the Z-statistics with the hypergeometric distribution, and tests based on it, Fisher's exact test is appropriate even for categories containing a small number of genes’, whereas for Martin et al. (2004) the hypergeometric test is most appropriate: ‘On the average, the hypergeometric distribution seems to be both the most adapted model and the most powerful test’. Moreover, even though the most recent review papers use a number of criteria to exhaustively compare the different existing tools (Khatri and Draghici, 2005), they do not discuss in detail the identity and approximation relationships existing between the different tests. This is precisely the aim of the present paper.
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2 PROBLEM STATEMENT |
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TOPABSTRACT1 INTRODUCTION2 PROBLEM STATEMENT3 CANDIDATE FORMULATIONS4 TESTS AND P-VALUES5 SUMMARY AND DISCUSSION6 NUMERICAL ILLUSTRATIONS7 CONCLUSIONREFERENCES |
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We consider a total population of genes, e.g. the genes expressed in a microarray experiment, and we are interested in the property of a gene to belong to a specific GO category. The aim is to establish whether the class of the DE genes presents an enrichment and/or a depletion of the GO category of interest with respect to the total gene population.
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3 CANDIDATE FORMULATIONS |
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TOPABSTRACT1 INTRODUCTION2 PROBLEM STATEMENT3 CANDIDATE FORMULATIONS4 TESTS AND P-VALUES5 SUMMARY AND DISCUSSION6 NUMERICAL ILLUSTRATIONS7 CONCLUSIONREFERENCES |
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Let H0 denote the null hypothesis that the property for a gene to belong to the GO category of interest and that to be DE are independent, or equivalently that the DE genes are picked at random from the total gene population. We consider successively the hypergeometric, the comparison of two probabilities, and the 2 x 2 contingency table formulations of the above problem, and introduce the exact or approximate null distributions they lead to.
Notations (see Table 2): the total number of genes is denoted by n, the total number of genes belonging to the GO category of interest by n+1, the number of DE genes by n1+: n, n+1 and n1+ are hence fixed by the experiment. The number of DE genes belonging to the GO category is denoted by n11.
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3.1 Hypergeometric formulation
The hypergeometric formulation is directly derived from the problem statement.
3.1.1 Exact null distribution
If H0 is true, the random variable N11 whose realization1 is the observed value n11, has a hypergeometric distribution with parameters n, n1+, and n+1, which we denote by N11 
Hyper(n, n1+, n+1), with:
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| (1) |
Hyper(n, n+1, n1+).
3.1.2 Approximate null distribution
For a large sample, N11 has approximately a binomial distribution with parameters n1+ and n+1/n: N11 Bi(n1+, n+1/n). Note that if n1+ n+1/n is also large, the binomial approximation can further be approximated by a Gaussian distribution.
3.2 Comparison of two probabilities formulation
In a second formulation, we consider two samples, that of the DE genes of size n1+, among which n11 genes belonging to the GO category of interest, and that of the not DE genes of size n2+, among which n21 genes belonging to the GO category. The proportions of genes belonging to the GO category in the two samples are thus f1 = n11/n1+ (DE genes) and f2 = n21/n2+ (not DE genes). Let p1 and p2 denote the probabilities to belong to the GO category in the two samples; then N11 Bi(n1+, p1) and N21 Bi(n2+, p2). In this formulation, the null hypothesis H0 is the equality of the two probabilities p1 = p2 = p, i.e. there is neither enrichment nor depletion in the sample of DE genes with respect to that of the not DE genes.
3.2.1 Approximate null distribution
The case of large samples arises frequently. Then, the binomial distributions can be approximated with Gaussian distributions. Under H0, n1+ and n2+ being large, the probability p can be correctly estimated with f = (n11 + n21)/(n1+ + n2+) = n+1/n, leading to the approximately normally distributed variable:
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| (2) |
3.2.2 Exact null distribution
Without approximating the binomial distribution, and taking into account that N11 + N21 = n+1, we naturally obtain N11 Hyper(n, n1+, n+1) (see (Fisher, 1935; Lehman, 1986) for the complete computation with the binomial distribution conditionally on N11 + N21 = n+1). Hence, the exact distribution of N11 under H0 is as before the hypergeometric distribution.
3.3 Contingency table formulation
A third formulation is based on Table 2 seen as a 2 x 2 contingency table. Let again H0 denote the hypothesis that the property to belong to the GO category of interest and that to be DE are independent.
3.3.1 Approximate null distribution
The case of a large sample is frequently considered where, if H0 is true, the following variable is asymptotically 2 distributed with one degree of freedom (Mood et al., 1974):
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3.3.2 Exact null distribution
Whatever the sample size, Fisher's formula gives the probability of the observed configuration of the contingency table under H0 (Fisher, 1935; Mood et al., 1974; Agresti, 2002):
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Hyper(n, n1+, n+1):
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3.4 Summary
Under H0, i.e. assuming the independence of the property to belong to the GO category of interest and of the property to be DE, or equivalently assuming p1 = p2 where p1 is the probability of the DE genes to belong to the GO category, and p2 the probability of the not DE genes to belong to the GO category, the exact distribution of N11 is the hypergeometric distribution N11 Hyper(n, n1+, n+1) which, if n is large, can be approximated with the binomial distribution Bi(n1+, n+1/n). If the two samples are large, it is also possible to exhibit an approximately normal variable Z or its square D2 = Z2, the latter being hence approximately 2 distributed with one degree of freedom.
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4 TESTS AND P-VALUES |
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TOPABSTRACT1 INTRODUCTION2 PROBLEM STATEMENT3 CANDIDATE FORMULATIONS4 TESTS AND P-VALUES5 SUMMARY AND DISCUSSION6 NUMERICAL ILLUSTRATIONS7 CONCLUSIONREFERENCES |
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Generally, when performing the test of a null hypothesis H0 against some alternative hypothesis Ha, one disposes of a realization x of a random variable X with known distribution under H0, the null distribution. One chooses a priori a probability
of type I error (the error to reject H0 whereas it is true) that must not be exceeded, also called significance level, the decision to reject H0 being taken when x falls in the critical region. In this context, the P-value is the minimum significance level for which H0 would be rejected, or equivalently, it is the probability, under H0, of the minimal critical region containing x.
The choice of a critical region in order to maximize the power of the test, and hence the choice of the corresponding P-value, depends on the alternative hypothesis Ha, which may be ‘enrichment’ (p1 > p2, one-sided test, critical region right), ‘depletion’ (p1 < p2 ‘one-sided’ test, critical region left), or ‘enrichment or depletion’ (p1 
p2, two-sided test, critical region left and right). Enrichment, depletion and enrichment or depletion are later denoted by E, D, and E/D, respectively.
4.1 One-sided tests
The one-sided P-value is defined as:
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’. Due to the discreteness, the actual significance level (or size of the test) is generally smaller than the nominal (desired) significance level , which results in a loss of power.
To minimize this loss, a good remedy is the use of mid-P-values (Agresti and Min, 2001; Agresti, 2002). The one-sided mid-P-value, which we denote by 
one, is defined as:
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| (8) |
Another remedy is randomization, with which any desired significance level can be achieved. However in practice, randomization having nothing to do with the data does not make much sense (Lehmann, 1986; Agresti, 2002).
If the approximately normal variable Z is considered, we have:
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2 distributed variable D2 is used, a one-sided test cannot be performed, since both enrichment (large observed n11) and depletion (small observed n11) lead to a large value of D2, i.e. there is a single critical region.
4.2 Two-sided tests
In the case of a two-sided test i.e. Ha = E/D, and of a discrete null distribution, there are several popular definitions of the P-value, see (Agresti, 1992, 2002). A first approach defines the two-sided P-value as twice the one-sided P-value:
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| (11) |
n11), P(N11 n11)] plus an attainable probability in the other tail that is as close as possible to, but not greater than, that one-tailed probability (Agresti, 2002). These definitions lead to equal P-values in the case of symmetric distributions, i.e. when n1+ = n2+; else, they possibly lead to different P-values and corresponding test results, each of them having advantages and disadvantages, due to the discreteness and skewness of the hypergeometric distribution. The problem is also that these P-values do not correspond to any well-defined two-sided test. This issue is discussed for example in (Dunne et al., 1996), where a two-sided P-value based on an uniformly most powerful unbiased test is proposed. However, this P-value is obtained with an iterative procedure, which makes this approach inadequate for the screening of hundreds of different GO categories.
Thus, if a single simple and computationally light (see subsection 6.3) procedure were to be recommended, we would advice the doubling approach, against which there is no strong argument, and using the mid-P-value, in order to reduce the discreteness and conservatism effects:
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| (12) |
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| (13) |
If the approximately normal variable Z is considered (a continuous and symmetrically distributed variable), we have:
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| (14) |
2 distributed variable D2 is considered, the P-value is computed as:
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| (15) |
4.3 One versus two-sided tests?
Consider a dataset consisting of tissues in a pathological condition and of normal tissues, and a GO category whose genes are directly affected by the condition, i.e. the genes belonging to this GO category are DE (either over- or under-expressed). Such a GO category is likely to be over-represented among the DE genes, i.e. an enrichment is expected. Thus, detecting an enrichment is desirable. On the other hand, consider a GO category such that the normal expression of the corresponding genes is necessary for the condition to develop, i.e. the genes belonging to this GO category are not DE. Such a GO category is likely to be under-represented among the DE genes, i.e. a depletion is expected. Thus, detecting a depletion is also desirable, even if there is a risk to detect the depletion of a GO category corresponding to genes whose normal expression is necessary to the mere survival of the specie.
Thus, both enrichments and depletions of GO categories are potentially of interest. Hence, unless there is a specific reason not to consider enrichment or depletion, the adequate alternative hypothesis is Ha = E/D, i.e. two-sided tests are appropriate.
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5 SUMMARY AND DISCUSSION |
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TOPABSTRACT1 INTRODUCTION2 PROBLEM STATEMENT3 CANDIDATE FORMULATIONS4 TESTS AND P-VALUES5 SUMMARY AND DISCUSSION6 NUMERICAL ILLUSTRATIONS7 CONCLUSIONREFERENCES |
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To summarize, there is a single exact null distribution of N11, the hypergeometric distribution, but different exact tests (exact in the sense that they are based on the exact null distribution), one or two-sided, and with several definitions of the P-value in the latter case. These tests can equally be called hypergeometric or Fisher's exact tests2. Thus, it is not justified to claim, as Masseroli et al. (2004) do, that ‘the
2 and Fisher's exact tests have less power than the hypergeometric and binomial distribution tests’. GFINDER and GOToolBox propose the hypergeometric test and Fisher's exact test as two alternative options: GFINDER indeed provides the same results for the two options (one-sided tests), but strangely enough, GOToolBox gives different results, whereas they should be identical for the same choice of a P-value (incorrect results given by some GO tools are detailed in the Appendix, which is available as supplementary data). The available GO tools often do not explicitly state which P-value is computed. For example, BINGO calls the test it performs ‘hypergeometric test’ (Maere et al. 2005), without saying that it is two-sided with the minimum-likelihood approach. According to both references and websites, we could establish that FuncAssociate, GFINDER and THEA provide only one-sided tests in both directions, while FuncSpec, EASEonline, GO Term Finder (CPAN), Term Finder (SGD), GOTM, L2L, Ontology Traverser and STEM only one-sided enrichment tests and that BINGO, DAVID, eGOn, 2004 FatiGo, GeneMerge, GoMiner, GOstat, GoSurfer, NetAffx and Onto-Express provide two-sided tests, the P-values being computed according to the minimum-likelihood approach when a discrete distribution is used.
As discussed in section 4.3, two-sided tests are usually most appropriate. Be it with the doubling or the minimum-likelihood approach to the P-value, the discreteness and conservatism effects can be efficiently dealt with using mid-P-values, a possibility that is not offered by any of the GO tools of Table 1.
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6 NUMERICAL ILLUSTRATIONS |
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TOPABSTRACT1 INTRODUCTION2 PROBLEM STATEMENT3 CANDIDATE FORMULATIONS4 TESTS AND P-VALUES5 SUMMARY AND DISCUSSION6 NUMERICAL ILLUSTRATIONS7 CONCLUSIONREFERENCES |
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6.1 Small sample
We consider a small sample with n = 20, n1+ = 7, n+1 = 6 and n11 = 4, i.e. f1 = 0.57 and f2 = 0.15. The null distribution of N11
Hyper(n, n1+, n+1) is shown in Figure 1. The sample being very small, we consider only the tests based on this exact distribution.
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6.1.1 One-sided test
For illustration purposes, let us first consider a one-sided test (suppose one is interested in enrichments only). The corresponding one-sided P-value right equals:
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= 5%, the mid-P-value leads to reject H0, whereas the P-value does not: the use of a mid-P-value corresponds to a less conservative test. However, the actual significance level is no longer guaranteed to be smaller than the nominal significance level 5%.
6.1.2 Two-sided tests
The two-sided doubling P-value equals:
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The two-sided doubling mid-P-value equals:
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= 5%, a two-sided test does not reject H0.
The two-sided minimum-likelihood P-value equals:
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The two-sided minimum-likelihood mid-P-value equals:
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6.2 Large sample
We now consider a sample whose size is analogous to that of samples encountered when testing enrichment of GO categories among DE genes on dedicated microarrays. We have n = 800, n1+ = 40, n+1 = 100, and observe n11 = 10, i.e. f1 = 0.25 and f2 = 0.12. The alternative hypothesis is Ha = E/D (two-sided test).
- The exact two-sided doubling P-value obtained with the hypergeometric distribution is
= 3.95 x 10–2, and the two-sided mid-P-value is 
= 2.66 x 10–2. With the minimum-likelihood approach, 
= 2.39 x 10–2, and the two-sided mid-P-value is 
= 1.74 x 10–2. Note that, the null distribution being asymmetric, there is a noticeable difference between the two approaches, and, though the sample is quite large, between the P-values and the corresponding mid-P-values.
- The approximate binomial test leads to a doubling P-value of 4.54 x 10–2, and to a doubling mid-P-value of 3.11 x 10–2, to a minimum-likelihood P-value of 2.75 x 10–2, and to a minimum-likelihood mid-P-value of 2.03 x 10–2. Note that though the sample is not small, there is quite a difference with the exact distribution.
- The approximate test of equality of two probabilities leads to the value of an approximately normal statistic z = 2.45, and to a two-sided P-value of ptwo(z) = 1.42 x 10–2. This value is even less accurate than that obtained with the binomial approximation, because the DE sample is too small (n1+ = 40).
- The 2-test indeed leads to a statistic value d2 = 6.015 = z2, and hence to the same two-sided P-value.
In the case of larger samples, obtained with mouse or human pangenomic microrrays, typically with n of the order of 25 000:
- The approximate binomial test leads to (mid-) P-values that are very close to those of the exact hypergeometric test. However, with todays computing means, there is no decisive advantage in performing this approximation (see next section).
- The approximate test of equality of two probabilities becomes closer to the exact one only if the number of DE genes is large, which is not necessarily the case. There is thus no reason to use this test.
- This is hence also true for the equivalent 2 test.
6.3 Implementation with R and computational issues
All the exact tests can be implemented ‘by hand’ with the hypergeometric cumulative distribution function ‘phyper’ and the distribution function ‘dhyper’, and the binomial approximations with ‘pbinom’ and ‘dbinom’3.
The default implementation of the exact test with R provides the two-sided minimum-likelihood P-value. The corresponding instruction is ‘fisher.test(c)’, where the matrix c is the 2 x 2 contingency table [n11 n12; n21 n22]. The one-sided enrichment test is obtained with ‘fisher.test(c, alternative = "greater")’, the one-sided depletion test with ‘fisher.test(c, alternative = "less")’.
In order to evaluate the computation time of the two-sided tests, let us consider the case of a microarray with n = 25 000 genes, n1+ = 1000 DE genes, and 500 different GO categories. We take n+1 uniformly distributed in [0,n], and n11 uniformly distributed in [max(0, n+1+n1+–n), min(n1+, n+1)]. With R 2.1.0 running under Mac OS X on a 2 GHz two processor Macintosh (PowerPC 970 2.2), we obtain the following total elapsed times (mean and standard error on 20 runs) for the doubling approach:
- hypergeometric doubling P-values, computed with the functions ‘dhyper’ and ‘phyper’: 0.17 ± 0.02 s, and 0.20 ± 0.02s for the mid-P-values.
- binomial doubling P-values, computed with the functions ‘dbinom’ and ‘pbinom’: 0.16 ± 0.02s, and 0.19 ± 0.02s for the mid-P-values.
For the minimum-likelihood approach, the R function ‘fisher.test’, (which does not only compute a P-value) is much slower than a computation ‘by hand’:
- hypergeometric minimum-likelihood P-values, computed with the function ‘fisher.test’: 17.15 ± 0.21 s.
- hypergeometric minimum-likelihood P-values, computed with the functions ‘dhyper’ and ‘phyper’: 1.83 ± 0.04s and 2.10 ± 0.05s for the mid-P-values.
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7 CONCLUSION |
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TOPABSTRACT1 INTRODUCTION2 PROBLEM STATEMENT3 CANDIDATE FORMULATIONS4 TESTS AND P-VALUES5 SUMMARY AND DISCUSSION6 NUMERICAL ILLUSTRATIONS7 CONCLUSIONREFERENCES |
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The correct statement of the enrichment and/or depletion testing problem leads to a unique exact null distribution of the number of DE genes belonging to the GO category of interest, given the total gene number and the total number of genes belonging to the GO category. This distribution is the hypergeometric one, whose values are equivalently given by Fisher's formula for a 2 x 2 contingency table. Since both enrichments and depletions of GO categories are potentially of interest, two-sided tests are generally most appropriate. With the doubling or the popular minimum-likelihood definitions of the P-value, a loss of power due to the discreteness of the hypergeometric distribution is efficiently dealt with using mid-P-values, the doubling P-value involving lighter computations than the minimum-likelihood P-value. Finally, since many dedicated microarrays involve small data sets, and given the currently available algorithms and computing means, there is no strong argument in favor of the approximate large sample tests.
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Acknowledgments |
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Funding to pay the Open Access publication charges for this article was provided by the CNRS and the city of Paris.
Conflict of Interest: none declared.
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FOOTNOTES |
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Associate Editor: Jonathan Wren
1Random variables and their realizations are denoted respectively by uppercase and lowercase letters. 
2As a matter of fact, (Fisher, 1935) describes a one-sided test in the direction of the observed departure of the null hypothesis.
3The code of the R functions can be found at the R project site https://svn.r-project.org/R/trunk/src/nmath/. The best known and most complete software for contingency table methods in general is StatXact (Agresti, 2006).
Received on June 20, 2006; revised on December 11, 2006; accepted on December 11, 2006
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