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Open Access Highly Accessed Methodology article

Ranking analysis of F-statistics for microarray data

Yuan-De Tan12, Myriam Fornage2 and Hongyan Xu3*

Author Affiliations

1 College of Life Sciences, Hunan Normal University, Changsha, 410081, China

2 Institute of Molecular Medicine, the University of Texas-Houston, Houston, Texas 77030, USA

3 Department of Biostatistics, Medical College of Georgia, Augusta, Georgia, 30912, USA

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BMC Bioinformatics 2008, 9:142  doi:10.1186/1471-2105-9-142

The electronic version of this article is the complete one and can be found online at:

Received:23 November 2007
Accepted:6 March 2008
Published:6 March 2008

© 2008 Tan et al; licensee BioMed Central Ltd.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.



Microarray technology provides an efficient means for globally exploring physiological processes governed by the coordinated expression of multiple genes. However, identification of genes differentially expressed in microarray experiments is challenging because of their potentially high type I error rate. Methods for large-scale statistical analyses have been developed but most of them are applicable to two-sample or two-condition data.


We developed a large-scale multiple-group F-test based method, named ranking analysis of F-statistics (RAF), which is an extension of ranking analysis of microarray data (RAM) for two-sample t-test. In this method, we proposed a novel random splitting approach to generate the null distribution instead of using permutation, which may not be appropriate for microarray data. We also implemented a two-simulation strategy to estimate the false discovery rate. Simulation results suggested that it has higher efficiency in finding differentially expressed genes among multiple classes at a lower false discovery rate than some commonly used methods. By applying our method to the experimental data, we found 107 genes having significantly differential expressions among 4 treatments at <0.7% FDR, of which 31 belong to the expressed sequence tags (ESTs), 76 are unique genes who have known functions in the brain or central nervous system and belong to six major functional groups.


Our method is suitable to identify differentially expressed genes among multiple groups, in particular, when sample size is small.


Microarray gene expression technology, which profiles the expression of multiple genes in parallel [1,2], affords the means for globally exploring physiological and pathological processes [3] regulated by the coordinated expression of thousands of genes [4]. However, identification of genes that are differentially expressed in large-scale gene expression experiments requires global statistical methods rather than traditional statistical methods based on single hypothesis testing. A variety of multiple-testing procedures, such as the Bonferroni procedure, Holm procedure [5], Hochberg procedure [6], Benjamini-Hochberg (BH) procedure [7], and Benjamini-Liu (BL) procedures [8] have already been developed for testing a large family of null hypotheses. The first three methods bound the family-wise-error rate (FWER) that is the probability of at least one false positive over all tests and hence remain too stringent and have lower power for finding genes from the real data sets. The last two methods have an upper bound for the false discovery rate (FDR) with both strong and weak controls [9] and require a large sample size for valid p-values. Tusher et al. [9] has proposed a ranking statistic approach based on permutation for resampling. However, permutation is not a desirable approach to estimating null distribution in microarray data [10-12] because in general a microarray dataset has a large number of genes but small sample sizes [13] due to cost. Permutation fails to remove treatment effect and due to small sample sizes the difference of treatment effects between permutated groups may become a main component in differences between group means so that the estimated null distribution is not well approximate to the true null distribution ([13] and also see Appendix in Tan et al. [14]). For example, Xie et al. [12] found that the estimated null F-distribution based on permutation has a larger variance and a heavier tail compared to the true null F-distribution, which leads to a potential loss of power. Similar phenomenon was also observed in comparison of the estimated null t-distribution to the true null t-distribution [14]. To remove the group or treatment effects on the estimated null distribution, Tan et al. [14] developed a random splitting (RS) approach. Since treatment effects are completely eliminated, the estimated null distribution obtained by the RS method is smooth, stable and approximate true null distribution well.

For the multi-class microarray data, the analysis of variance (ANOVA) is useful to identify differentially expressed genes [4]. In ANOVA, the F-test is a generalization of the t-test that allows for comparison of more than two samples [15]. However, due to small sample sizes, the classical F-test is also subject to the same problems as the t-test: bias and unstable estimates of gene-specific variances. To tackle this issue, many authors [15-19] proposed modified F-statistics. However, like the classical F-test, these modified F-tests still suffer from high false-positive rates because (i) the sample size is often so limited that the asymptotic F distribution is not accurate enough to obtain a valid p-value and (ii) they appeal to multiple-testing procedures such as the Bonferroni procedure or the BH-procedure. As mentioned above, these multiple-testing procedures have a basic requirement that sample sizes are large enough for valid p-values. In microarray data, the requirement is not realistic. Based on consideration of these problems, we here propose a novel statistical method for the analysis of multi-class gene-expression data called Ranking Analysis of F-statistics (RAF). RAF is a natural extension of our previous work, i.e., the ranking analysis of microarrary (RAM) for two-class t-tests [14]. It works on finding genes that are differentially expressed among multiple treatment groups by comparing the ordered real F-statistics with the ordered estimated null F-statistics and implementing a two-simulation strategy to estimate the false discovery rate (FDR).


Animal model and design

Studies were performed on male stroke-resistant SHR/N (CRiv) (SHRSR) and stroke-prone SHR/A3 (Heid) (SHRSP) rats from a breeding colony maintained by the investigators as previously described [20]. Age-matched male rats from each strain (N = 12 SHRSP and 12 SHRSR) were fed a standard rat chow and water ad libitum until age 8 weeks. Subsequently, animals from each strain were randomized to one of 2 dietary regimens (N = 6 in each strain × diet group): a "stroke-permissive diet" high in sodium (HS) (0.63% potassium, 0.37% sodium) and 1% NaCl drinking solution; a "stroke-protective diet" low in sodium and high in potassium (LS) (1.3% potassium, 0.35% sodium) and regular drinking water. All animals were housed at 23°C on a 12-hour light-dark cycle. Rats were sacrificed at 12 weeks of age, and brain tissue was collected for RNA extraction and subsequent microarray analysis. The study protocols were approved by the Animal Care Committee of the University of Texas – Houston. Since strain and dietary factor each have only two levels, we here treated them as one-way in statistical analysis instead of two-way, that is, we are neither interested in strain effects alone nor in dietary effects alone but focus on their combined contributions to gene expression. Thus, HS-SHRSPs, LS-SHRSPs, HS-SHRSRs, and LS-SHRSRs are viewed as four treatment groups for the purpose of the analyses.

Microarray experiment

Microarray analysis was performed as described by Lockhart et al. [21]. Briefly, 10 μg total RNA extracted from each of the 24 rats was used to synthesize cDNA, which was then used as a template to generate biotinylated cRNA. cRNA was fragmented and hybridized to a Test2 chip to verify quality and quantity of the samples. Each sample was then hybridized to a RGU34A array (Affymetrix, Santa Clara, CA). After hybridization, each array was washed and scanned, and fluorescence values were measured and normalized using the Affymetrix Microarray Suite v. 5.0 software.

Ranking F-Test

Let xgij be the expression value for replicate j of gene g in group i where g = 1,..., N (number of genes), j = 1,..., rgi (number of replicate observed values of gene g in group i) and i = 1,..., n (number of groups). The traditional F-statistic in one-way ANOVA may be expressed as

F g = σ 2 ( G g ) σ 2 ( e g ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemOray0aaSbaaSqaaiabdEgaNbqabaGccqGH9aqpjuaGdaWcaaqaaiabeo8aZnaaCaaabeqaaiabikdaYaaacqGGOaakcqWGhbWrdaWgaaqaaiabdEgaNbqabaGaeiykaKcabaGaeq4Wdm3aaWbaaeqabaGaeGOmaidaaiabcIcaOiabdwgaLnaaBaaabaGaem4zaCgabeaacqGGPaqkaaaaaa@3ED5@ (1)

where σ2 (Gg) and σ2 (eg) are inter- and intra-group variances of the expression values of gene g, respectively. In the conventional F-tests, for example, significance of p = 0.01 in the context of the standard F distribution is for a single hypothesis to be tested; therefore, it is unsuitable to microarray data because in a microarray experiment for 10,000 genes we would expect to identify 100 genes by chance [9]. To address this problem, an alternative approach is to rank genes by magnitude of their F values so that F1 is the largest value, F2 is the second largest value, etc., and Fg* is the g*th largest value where g* is a rank position of gene g. Thus, we have a ranking F-test where

Fg* - fg* > Δ(2)

indicates that the expression variation of gene g among multiple groups (or multiple conditions) is significant. In Eq. (2), fg* is the expectation of Fg* under the null hypothesis and Δ is a given threshold.

Estimation of fg*

In the ANOVA framework, we have

σ 2 ( G g ) = i = 1 n r g i ( x ¯ g i x ¯ g ) 2 n 1 = i = 1 n r g i [ ( τ g i + e ¯ g i ) ( μ g + e ¯ g ) ] 2 n 1 = i = 1 n r g i ( τ g i μ g ) + i = 1 n r g i ( e ¯ g i e ¯ g ) 2 n 1 = σ 2 ( τ g ) + σ 2 ( e ¯ g ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@B8FF@ (3)

where x ¯ g i = τ g i + e ¯ g i MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmiEaGNbaebadaWgaaWcbaGaem4zaCMaemyAaKgabeaakiabg2da9iabes8a0naaBaaaleaacqWGNbWzcqWGPbqAaeqaaOGaey4kaSIafmyzauMbaebadaWgaaWcbaGaem4zaCMaemyAaKgabeaaaaa@3B2C@ and x ¯ g = μ g i + e ¯ g MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmiEaGNbaebadaWgaaWcbaGaem4zaCgabeaakiabg2da9iabeY7aTnaaBaaaleaacqWGNbWzcqWGPbqAaeqaaOGaey4kaSIafmyzauMbaebadaWgaaWcbaGaem4zaCgabeaaaaa@3867@ ; τgi and e ¯ g i MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmyzauMbaebadaWgaaWcbaGaem4zaCMaemyAaKgabeaaaaa@301E@ are treatment effect and average random error in group i, respectively; x ¯ g MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmiEaGNbaebadaWgaaWcbaGaem4zaCgabeaaaaa@2EE9@ and e ¯ g MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmyzauMbaebadaWgaaWcbaGaem4zaCgabeaaaaa@2EC3@ are the overall observed mean and the overall average error for gene g, respectively, and μg is the overall expected mean for gene g and rgi is the number of replicate observed values of gene g in group i. Therefore, the inter-group variance σ2 (Gg) consists of two parts: variance of treatment effects on expression of gene g, σ2 (τg), and average error variance, σ2 ( e ¯ g MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmyzauMbaebadaWgaaWcbaGaem4zaCgabeaaaaa@2EC3@ ). Thus, F-statistic can be rewritten as

F g = σ 2 ( G g ) σ 2 ( e g ) = σ 2 ( τ g ) + σ 2 ( e ¯ g ) σ 2 ( e g ) = σ 2 ( τ g ) σ 2 ( e g ) + σ 2 ( e ¯ g ) σ 2 ( e g ) = σ 2 ( τ g ) σ 2 ( e g ) + f g . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemOray0aaSbaaSqaaiabdEgaNbqabaGccqGH9aqpjuaGdaWcaaqaaiabeo8aZnaaCaaabeqaaiabikdaYaaacqGGOaakcqWGhbWrdaWgaaqaaiabdEgaNbqabaGaeiykaKcabaGaeq4Wdm3aaWbaaeqabaGaeGOmaidaaiabcIcaOiabdwgaLnaaBaaabaGaem4zaCgabeaacqGGPaqkaaGccqGH9aqpjuaGdaWcaaqaaiabeo8aZnaaCaaabeqaaiabikdaYaaacqGGOaakcqaHepaDdaWgaaqaaiabdEgaNbqabaGaeiykaKIaey4kaSIaeq4Wdm3aaWbaaeqabaGaeGOmaidaaiabcIcaOiqbdwgaLzaaraWaaSbaaeaacqWGNbWzaeqaaiabcMcaPaqaaiabeo8aZnaaCaaabeqaaiabikdaYaaacqGGOaakcqWGLbqzdaWgaaqaaiabdEgaNbqabaGaeiykaKcaaOGaeyypa0tcfa4aaSaaaeaacqaHdpWCdaahaaqabeaacqaIYaGmaaGaeiikaGIaeqiXdq3aaSbaaeaacqWGNbWzaeqaaiabcMcaPaqaaiabeo8aZnaaCaaabeqaaiabikdaYaaacqGGOaakcqWGLbqzdaWgaaqaaiabdEgaNbqabaGaeiykaKcaaOGaey4kaSscfa4aaSaaaeaacqaHdpWCdaahaaqabeaacqaIYaGmaaGaeiikaGIafmyzauMbaebadaWgaaqaaiabdEgaNbqabaGaeiykaKcabaGaeq4Wdm3aaWbaaeqabaGaeGOmaidaaiabcIcaOiabdwgaLnaaBaaabaGaem4zaCgabeaacqGGPaqkaaGccqGH9aqpjuaGdaWcaaqaaiabeo8aZnaaCaaabeqaaiabikdaYaaacqGGOaakcqaHepaDdaWgaaqaaiabdEgaNbqabaGaeiykaKcabaGaeq4Wdm3aaWbaaeqabaGaeGOmaidaaiabcIcaOiabdwgaLnaaBaaabaGaem4zaCgabeaacqGGPaqkaaGccqGHRaWkcqWGMbGzdaWgaaWcbaGaem4zaCgabeaakiabc6caUaaa@8E77@ (4)

Therefore, the null hypothesis is equivalent to Fg = fg because σ2 (τg) = 0 under the null hypothesis. Note that σ2 (τg) = 0 means the treatment effects τgi = ... = τgn = μg. In order to do a ranking F-test, it is necessary to obtain a good estimate of fg* . In the two-group scenario, Tusher et al. [9] employed a permutation approach to estimate the expected t-statistics. The permutation process cannot completely clear the treatment effect in the ranked d-statistics so that the estimated ranked d-statistics distribution is biased against its null distribution and unstable, in particular, when sample sizes are small (see Appendix A in Tan et al., [14]). Tan et al. [14] developed a "Randomly Splitting" (RS) approach to estimate the null distribution of t-statistics. In this study, we extended the RS approach to estimating the null distribution of F-statistics.

In the RS approach, one sample consisting of rgi replicates is drawn from group i. Since only one sample is drawn from a group, sample i represents group i. Within a sample all the observed expression values of gene g come from the same group. These values have the same overall expected mean μg and the same treatment effect τgi on expression of gene g except for expression noises. A sample of rgi replicate values for gene g is randomly split into two sub-samples denoted by s = 1 and s = 2. If let x ¯ g i s J MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmiEaGNbaebadaqhaaWcbaGaem4zaCMaemyAaKMaem4CamhabaGaemOsaOeaaaaa@32D1@ be the mean of sub-sample s of sample i for gene g at split J(J = 1,..., M), then x ¯ g i s J MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmiEaGNbaebadaqhaaWcbaGaem4zaCMaemyAaKMaem4CamhabaGaemOsaOeaaaaa@32D1@ can be expressed as

X ¯ g i s J = μ g + τ g i + e ¯ g i s J MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmiwaGLbaebadaqhaaWcbaGaem4zaCMaemyAaKMaem4CamhabaGaemOsaOeaaOGaeyypa0JaeqiVd02aaSbaaSqaaiabdEgaNbqabaGccqGHRaWkcqaHepaDdaWgaaWcbaGaem4zaCMaemyAaKgabeaakiabgUcaRiqbdwgaLzaaraWaa0baaSqaaiabdEgaNjabdMgaPjabdohaZbqaaiabdQeakbaaaaa@4479@ (5)

where e ¯ g i s J = j = 1 m g i s J e g i s j J / m g i s J MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmyzauMbaebadaqhaaWcbaGaem4zaCMaemyAaKMaem4CamhabaGaemOsaOeaaOGaeyypa0ZaaabmaeaacqWGLbqzdaqhaaWcbaGaem4zaCMaemyAaKMaem4CamNaemOAaOgabaGaemOsaOeaaOGaei4la8IaemyBa02aa0baaSqaaiabdEgaNjabdMgaPjabdohaZbqaaiabdQeakbaaaeaacqWGQbGAcqGH9aqpcqaIXaqmaeaacqWGTbqBdaqhaaadbaGaem4zaCMaemyAaKMaem4CamhabaGaemOsaOeaaaqdcqGHris5aaaa@4FAD@ and m g i s J MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemyBa02aa0baaSqaaiabdEgaNjabdMgaPjabdohaZbqaaiabdQeakbaaaaa@32A3@ and e g i s j J MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemyzau2aa0baaSqaaiabdEgaNjabdMgaPjabdohaZjabdQgaQbqaaiabdQeakbaaaaa@33F0@ are replicate number and noise in the observed expression value j in sub-sample s in group i for gene g at split J, respectively. e ¯ g i MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmyzauMbaebadaWgaaWcbaGaem4zaCMaemyAaKgabeaaaaa@301E@ is estimated by the difference between two sub-sample means in sample i for gene g at split J,

e ¯ g i J = 1 2 ( x ¯ g i 1 J x ¯ g i 2 J ) = 1 2 [ ( μ g + τ g + e ¯ g i 1 J ) ( μ g + τ g + e ¯ g i 2 J ) ] . = 1 2 ( e ¯ g i 1 J e ¯ g i 2 J ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@7F5D@ (6)

It can be seen from Eq. (6) that μg and τgi are cleared in difference between two sub-sample means, which is unrelated to sample size. Thus, the average random error variance σ2 ( e ¯ g MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmyzauMbaebadaWgaaWcbaGaem4zaCgabeaaaaa@2EC3@ ) in Eq.s (3) and (4) can be estimated by

σ 2 ( e ¯ g J ) = i = 1 n r g i ( e ¯ g i J e ¯ g J ) 2 n 1 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeq4Wdm3aaWbaaSqabeaacqaIYaGmaaGccqGGOaakcuWGLbqzgaqeamaaDaaaleaacqWGNbWzaeaacqWGkbGsaaGccqGGPaqkcqGH9aqpjuaGdaWcaaqaamaaqadabaGaemOCai3aaSbaaeaacqWGNbWzcqWGPbqAaeqaaiabcIcaOiqbdwgaLzaaraWaa0baaeaacqWGNbWzcqWGPbqAaeaacqWGkbGsaaGaeyOeI0IafmyzauMbaebadaqhaaqaaiabdEgaNbqaaiabdQeakbaacqGGPaqkdaahaaqabeaacqaIYaGmaaaabaGaemyAaKMaeyypa0JaeGymaedabaGaemOBa4gacqGHris5aaqaaiabd6gaUjabgkHiTiabigdaXaaaaaa@51AD@ (7)

where e ¯ g J = i = 1 n e ¯ g i J / n MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmyzauMbaebadaqhaaWcbaGaem4zaCgabaGaemOsaOeaaOGaeyypa0ZaaabmaeaacuWGLbqzgaqeamaaDaaaleaacqWGNbWzcqWGPbqAaeaacqWGkbGsaaGccqGGVaWlcqWGUbGBaSqaaiabdMgaPjabg2da9iabigdaXaqaaiabd6gaUbqdcqGHris5aaaa@3F65@ is estimate of mean ( e ¯ g MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmyzauMbaebadaWgaaWcbaGaem4zaCgabeaaaaa@2EC3@ ) of expression noise of gene g among groups at split J. Variance σ2( e ¯ g MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmyzauMbaebadaWgaaWcbaGaem4zaCgabeaaaaa@2EC3@ ) is an estimate of expectation (σ2( e ¯ g MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmyzauMbaebadaWgaaWcbaGaem4zaCgabeaaaaa@2EC3@ )) of inter-group variance (σ2 (Gg)) under the null hypothesis at split J. We therefore have

f g J = σ 2 ( e ¯ g J ) σ 2 ( e g ) . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemOzay2aa0baaSqaaiabdEgaNbqaaiabdQeakbaakiabg2da9KqbaoaalaaabaGaeq4Wdm3aaWbaaeqabaGaeGOmaidaaiabcIcaOiqbdwgaLzaaraWaa0baaeaacqWGNbWzaeaacqWGkbGsaaGaeiykaKcabaGaeq4Wdm3aaWbaaeqabaGaeGOmaidaaiabcIcaOiabdwgaLnaaBaaabaGaem4zaCgabeaacqGGPaqkaaGccqGGUaGlaaa@4293@ (8)

Note that since treatment effect is completely removed from the difference between two sub-sample means, the difference is pure noise. We rank f g J MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemOzay2aa0baaSqaaiabdEgaNbqaaiabdQeakbaaaaa@2FCB@ across all g and let f g J MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemOzay2aa0baaSqaaiabdEgaNjabgEHiQaqaaiabdQeakbaaaaa@30BA@ denote the value in ordered position g* at split J. After running M splits, we have M values of f g J MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemOzay2aa0baaSqaaiabdEgaNjabgEHiQaqaaiabdQeakbaaaaa@30BA@ for position g*. Thus fg* in Eq. (2) can be estimated by the average of f g J MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemOzay2aa0baaSqaaiabdEgaNjabgEHiQaqaaiabdQeakbaaaaa@30BA@ over all M splits, i.e., f ¯ g = J = 1 M f g J / M MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmOzayMbaebadaWgaaWcbaGaem4zaCMaey4fIOcabeaakiabg2da9maaqadabaGaemOzay2aa0baaSqaaiabdEgaNjabgEHiQaqaaiabdQeakbaakiabc+caViabd2eanbWcbaGaemOsaOKaeyypa0JaeGymaedabaGaemyta0eaniabggHiLdaaaa@3DF4@ .

Estimation of FDR

To identify genes whose expression is significantly changed among multiple conditions, it is necessary to estimate the FDR for a given threshold [7,22]. Here we propose a two-simulation approach for FDR estimation [14]. Consider a series of threshold values Δk(k = 1,...L) and let Nk be the number of genes that are claimed as significant by RAF at threshold Δk. Nk comprises two parts: the number Nk(t) of the true positives and the number Nk(f) of the false positives, i.e., Nk = Nk(t) + Nk(f). Thus, given a threshold Δk, FDR is defined as λk = Nk(f)/Nk. Nk(f) is unknown, hence λk must be estimated. Many approaches such as BH procedure [7,22], BL procedure [8], Storey's procedure [23,24], and Pounds and Cheng's procedure [25] have been proposed to estimate the FDR. These approaches, however, are based on the assumption that the tests are independent. As mentioned previously, this assumption may not be met in practice. Therefore, these methods may not be suitable to our ranking test. Based on the fact that sampling distribution fluctuates around the expected distribution via permutation, Tusher et al. [9] developed a permutation-based estimator to estimate FDR in the ranking tests. It has been proved, however, in theory and in simulation that when the sample sizes are small, the number of permutations is very limited so that the treatment effects cannot be removed in the permutated data [14]. As a result, the estimator is biased for a given threshold. Here we extend the interval approach by Tan et al. [14] to the ranking analysis of F-statistics. In this approach, we first construct an estimated interval of the true FDR, and then we find a reasonable estimate of FDR. This interval is based on the complete and partial null distributions given by two simulations.

In simulation 1, for each gene, n samples (groups) each having r replicates are generated from normal distributions with a set of sample means ( y ¯ g 1 , ... , y ¯ g n MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmyEaKNbaebadaWgaaWcbaGaem4zaCMaeGymaedabeaakiabcYcaSiabc6caUiabc6caUiabc6caUiabcYcaSiqbdMha5zaaraWaaSbaaSqaaiabdEgaNjabd6gaUbqabaaaaa@38CC@ ) and a set of sample error variances [s2 (eg1),..., s2 (egn)]. Here we set y ¯ g 1 = .... = y ¯ g n = x ¯ g i MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmyEaKNbaebadaWgaaWcbaGaem4zaCMaeGymaedabeaakiabg2da9iabc6caUiabc6caUiabc6caUiabc6caUiabg2da9iqbdMha5zaaraWaaSbaaSqaaiabdEgaNjabd6gaUbqabaGccqGH9aqpcuWG4baEgaqeamaaBaaaleaacqWGNbWzcqWGPbqAaeqaaaaa@3F7B@ and i is a randomly chosen group from the observed data, for each of a half of the genes with the null effect that the group variance is zero, i.e., σ2 (Gg) = 0 and y ¯ g i = x ¯ g i MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmyEaKNbaebadaWgaaWcbaGaem4zaCMaemyAaKgabeaakiabg2da9iqbdIha4zaaraWaaSbaaSqaaiabdEgaNjabdMgaPbqabaaaaa@35C5@ for each of the other half with unknown effect that the group variance is larger than or equal to zero, i.e., σ2 (Gg) ≥ 0. s2 (egi) is set to be equal to σ2 (egi) where x ¯ g i MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmiEaGNbaebadaWgaaWcbaGaem4zaCMaemyAaKgabeaaaaa@3044@ and σ2 (egi) are the observed values from the real microarray data set.

B sets of simulation data are obtained from this procedure. Each is subject to the ranking analysis described in the previous section. For simulation data set b, every ranked position has thus its corresponding F value that is denoted by F g 1 b MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemOray0aa0baaSqaaiabdEgaNjabgEHiQiabigdaXaqaaiabdYeambaaaaa@316E@ (b = 1,..., B). Here those that are called significant by comparing F g 1 b MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemOray0aa0baaSqaaiabdEgaNjabgEHiQiabigdaXaqaaiabdkgaIbaaaaa@319A@ to f ¯ g MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmOzayMbaebadaWgaaWcbaGaem4zaCMaey4fIOcabeaaaaa@2FB4@ at a given threshold Δk are counted as N 1 k b MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemOta40aa0baaSqaaiabigdaXiabdUgaRbqaaiabdkgaIbaaaaa@30C3@ across all ranking positions. Let N 1 k = max b = 1 B N 1 k b MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemOta40aaSbaaSqaaiabigdaXiabdUgaRbqabaGccqGH9aqpcyGGTbqBcqGGHbqycqGG4baEdaqhaaWcbaGaemOsaOKaeyypa0JaeGymaedabaGaemOqaieaaOGaemOta40aa0baaSqaaiabigdaXiabdUgaRbqaaiabdkgaIbaaaaa@3DF0@ Given the fact that a small part of genes have unequal means in the samples, the simulation data set produces a partially null F-distribution. In other words, it may produce N 1 k b > N k ( f ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemOta40aa0baaSqaaiabigdaXiabdUgaRbqaaiabdkgaIbaakiabg6da+iabd6eaonaaBaaaleaacqWGRbWAaeqaaOGaeiikaGIaemOzayMaeiykaKcaaa@3796@ , possibly leading to λk = N1k/Nk > 1. To avoid this situation, suppose N1k(k = 1,..., L) takes the maximum value N(m) at Δk = m, we define

λ 1 k = 2 N 1 k N ( m ) + N 1 k MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeq4UdW2aaSbaaSqaaiabigdaXiabdUgaRbqabaGccqGH9aqpjuaGdaWcaaqaaiabikdaYiabd6eaonaaBaaabaGaeGymaeJaem4AaSgabeaaaeaacqWGobGtcqGGOaakcqWGTbqBcqGGPaqkcqGHRaWkcqWGobGtdaWgaaqaaiabigdaXiabdUgaRbqabaaaaaaa@3F38@ (9)

as a function of threshold Δk for estimating FDR where we set N1k = N(m) for all Δk < Δm(k = 1,..., L). Obviously, λ1k is a decreasing function of the threshold and bounded between 0 and 1. For example, λ1k = 1 when N1k = N(m), and λ1k = 2/3 when N1k = N(m)/2. Results from the simulation study in Figure 1 indicate that λ1k λk (true value of FDR at threshold Δk) when threshold Δk is smaller than a value Δ* but λ1k λk when Δk > Δ* (see Figure 1).

thumbnailFigure 1. Profile of estimates of FDRs for a series of thresholds. λ1k and λ2k are two threshold functions from simulations 1 and 2 and were used to construct an estimation interval for estimate of FDR at threshold Δk. λk and λ ˆ k MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafq4UdWMbaKaadaWgaaWcbaGaee4AaSgabeaaaaa@2F22@ are true and estimated FDRs at threshold Δk, respectively, where k = 1, 2,...,L.

The second simulation for estimating FDR is carried out in the following fashion. n samples (groups) each having r replicates for each gene are generated from normal distributions with a set of sample means, y ¯ g 1 = .... = y ¯ g n = x ¯ g i MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmyEaKNbaebadaWgaaWcbaGaem4zaCMaeGymaedabeaakiabg2da9iabc6caUiabc6caUiabc6caUiabc6caUiabg2da9iqbdMha5zaaraWaaSbaaSqaaiabdEgaNjabd6gaUbqabaGccqGH9aqpcuWG4baEgaqeamaaBaaaleaacqWGNbWzcqWGPbqAaeqaaaaa@3F7B@ and a set of sample variances s2 (eg1) = σ2 (eg1),..., s2 (egn) = σ2 (egn).

We also produce B sets of data from simulation 2. As in simulation 1, for each simulation data set, every ranked position also has its corresponding F-value denoted as F g 2 b MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemOray0aa0baaSqaaiabdEgaNjabgEHiQiabikdaYaqaaiabdkgaIbaaaaa@319C@ (b = 1,..., B). Let F g 2 = min b = 1 B F g 2 b MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemOray0aaSbaaSqaaiabdEgaNjabgEHiQiabikdaYaqabaGccqGH9aqpcyGGTbqBcqGGPbqAcqGGUbGBdaqhaaWcbaGaemOyaiMaeyypa0JaeGymaedabaGaemOqaieaaOGaemOray0aa0baaSqaaiabdEgaNjabgEHiQiabikdaYaqaaiabdkgaIbaaaaa@3FCE@ . The positives found by comparing F g 2 b MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemOray0aa0baaSqaaiabdEgaNjabgEHiQiabikdaYaqaaiabdkgaIbaaaaa@319C@ to Fg*2 at a given threshold Δk are counted as N 2 k b MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemOta40aa0baaSqaaiabikdaYiabdUgaRbqaaiabdkgaIbaaaaa@30C5@ across all ranking positions. Here let N 2 k = b = 1 B N 2 k b / B MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemOta40aaSbaaSqaaiabikdaYiabdUgaRbqabaGccqGH9aqpdaaeWaqaaiabd6eaonaaDaaaleaacqaIYaGmcqWGRbWAaeaacqWGIbGyaaGccqGGVaWlcqWGcbGqaSqaaiabdkgaIjabg2da9iabigdaXaqaaiabdkeacbqdcqGHris5aaaa@3DC6@ . Unlike the first simulation, here the simulation data sets produce B null F-distributions, so N2k should be approximate to the true number of false positives Nk(f). However, when threshold Δk is large, it is possible to have Nk = 0 so that N2k/Nk is undefined. To avoid this situation, we define

λ 2 k = N 2 k N k + N 2 k MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeq4UdW2aaSbaaSqaaiabikdaYiabdUgaRbqabaGccqGH9aqpjuaGdaWcaaqaaiabd6eaonaaBaaabaGaeGOmaiJaem4AaSgabeaaaeaacqWGobGtdaWgaaqaaiabdUgaRbqabaGaey4kaSIaemOta40aaSbaaeaacqaIYaGmcqWGRbWAaeqaaaaaaaa@3CB7@ (10)

as the second function of threshold (see Figure 1). In particular, we let λ2k = 1 if Nk = N2k = 0 because λ2k = 1 when Nk = 0 and N2k > 0.

Thus, an interval for FDR estimation at threshold Δk can be constructed between λ1k and λ2k . The third function of threshold for FDR estimation is given as

                         λ3k = αkλ1k + βkλ2k

where αk = min(λ1k, λ2k)/(λ1k + λ2k) and βk = 1 - ak. λ3k plays the role of weight in balancing λ1k and λ2k . Therefore, at threshold Δk, a putative probability that a false discovery is found in the genes called significant by RAF is

λ ˆ k = 1 3 ( λ 1 k + λ 2 k + λ 3 k ) . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafq4UdWMbaKaadaWgaaWcbaGaem4AaSgabeaakiabg2da9KqbaoaalaaabaGaeGymaedabaGaeG4mamdaaOGaeiikaGIaeq4UdW2aaSbaaSqaaiabigdaXiabdUgaRbqabaGccqGHRaWkcqaH7oaBdaWgaaWcbaGaeGOmaiJaem4AaSgabeaakiabgUcaRiabeU7aSnaaBaaaleaacqaIZaWmcqWGRbWAaeqaaOGaeiykaKIaeiOla4caaa@4419@ (12)

Note that as shown in the simulation result section, λ2k is an underestimate of λk and λ1k is an overestimate of FDR when the threshold Δk < Δ*. However, the situation is reversed when threshold Δk > Δ*. This is because N1k becomes very small when Δk > Δ* so that λ1k becomes very small whereas, from Eq. (10), λ2k slowly decreases if Nk > N2k or increases if Nk <N2k as threshold increases. In addition, when the microarray data have no treatment effects for all the genes detected, then λ1k = λ2k = λ3k = 1, leading to λ ˆ k MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafq4UdWMbaKaadaWgaaWcbaGaem4AaSgabeaaaaa@2F24@ = 1

In order to smooth λ ˆ k MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafq4UdWMbaKaadaWgaaWcbaGaem4AaSgabeaaaaa@2F24@ between thresholds Δk and Δk+1, we define a recursive formula modifying the probability λ ˆ k MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafq4UdWMbaKaadaWgaaWcbaGaem4AaSgabeaaaaa@2F24@ as

λ ˆ k = λ k p k + λ k + 1 q k MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafq4UdWMbaKaadaWgaaWcbaGaem4AaSgabeaakiabg2da9iabeU7aSnaaBaaaleaacqWGRbWAaeqaaOGaemiCaa3aaSbaaSqaaiabdUgaRbqabaGccqGHRaWkcqaH7oaBdaWgaaWcbaGaem4AaSMaey4kaSIaeGymaedabeaakiabdghaXnaaBaaaleaacqWGRbWAaeqaaaaa@3FBC@ (13)

where pk = (Nk - Nk+1)/(1 + Nk - Nk+1) and qk = 1 - pk. Eq. (13) suggests that λk+1 = λk if Nk = Nk+1. The number of false discoveries among those found to be significant at threshold Δk in the observed data is estimated by N ˆ k ( f ) = λ ˆ k N k MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmOta4KbaKaadaWgaaWcbaGaem4AaSgabeaakiabcIcaOiabdAgaMjabcMcaPiabg2da9iqbeU7aSzaajaWaaSbaaSqaaiabdUgaRbqabaGccqWGobGtdaWgaaWcbaGaem4AaSgabeaaaaa@38B5@ . Figure 1 shows that the curve of λ ˆ k MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafq4UdWMbaKaadaWgaaWcbaGaem4AaSgabeaaaaa@2F24@ agrees well with that of λk.


Estimation of the null distribution of F-statistics

To examine if the empirical distributions obtained by the RS approach are appropriate for the analysis of the expression data, we simulated a microarray data set consisting of 3770 genes and four groups each having 6 replicates using one group mean and error variance for each gene. Thus, the simulation without treatment effect generated a set of pure noise data.

A set of 3770 Fk values was computed from the simulated data set. We applied our RS approach to this simulated data set to generate f ¯ k MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmOzayMbaebadaWgaaWcbaGaem4AaSgabeaaaaa@2ECD@ over 50 splits. This set of 3770 Fk values formed null distribution of F-statistics, which is called f-distribution. To display the profile that our estimate of f-distribution is approximate to the null f-distribution, we plotted the ranked Fk versus ranked f ¯ k MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmOzayMbaebadaWgaaWcbaGaem4AaSgabeaaaaa@2ECD@ . The result displays in Figure 2 where all ranked F - f ¯ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmOzayMbaebaaaa@2D42@ dots roughly fall on a diagonal line as expected by two sets of the same ranked distributions. These results suggest that the f ¯ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmOzayMbaebaaaa@2D42@ -distribution is indeed an approximate estimate of f-distribution.

thumbnailFigure 2. The dot-plot of F-values versus f ¯ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmOzayMbaebaaaa@2D42@ -values. F-values and f ¯ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmOzayMbaebaaaa@2D42@ -values obtained from the simulated microarray data of 3770 genes were ranked where f ¯ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmOzayMbaebaaaa@2D42@ -values were yielded by the random splitting approach. All ranked F - f ¯ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmOzayMbaebaaaa@2D42@ dots roughly fall on a diagonal line as expected by two sets of the same ranked distributions.

Estimation of FDR

Since it is in general unknown if a given gene expresses differently among multiple conditions, it is not necessarily best to use real data of gene expression to evaluate an FDR estimator. But simulation is a useful approach to doing such a task. Therefore, we also conducted a computer simulation for comparing expression status (significant or not) of a gene identified by a method with its real status. This simulation was also based on our real data set of 3770 genes. Treatment effect τ on expression variation was set for 30 % of the genes and assigned in 4 groups. The mean expression value of gene g was set y ¯ g 1 = x ¯ g + 2 τ , y ¯ g 2 = x ¯ g + τ , y ¯ g 3 = x ¯ g τ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmyEaKNbaebadaWgaaWcbaGaem4zaCMaeGymaedabeaakiabg2da9iqbdIha4zaaraWaaSbaaSqaaiabdEgaNbqabaGccqGHRaWkcqaIYaGmcqaHepaDcqGGSaalcuWG5bqEgaqeamaaBaaaleaacqWGNbWzcqaIYaGmaeqaaOGaeyypa0JafmiEaGNbaebadaWgaaWcbaGaem4zaCgabeaakiabgUcaRiabes8a0jabcYcaSiqbdMha5zaaraWaaSbaaSqaaiabdEgaNjabiodaZaqabaGccqGH9aqpcuWG4baEgaqeamaaBaaaleaacqWGNbWzaeqaaOGaeyOeI0IaeqiXdqhaaa@4F29@ and y ¯ g 4 = x ¯ g 2 τ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmyEaKNbaebadaWgaaWcbaGaem4zaCMaeGinaqdabeaakiabg2da9iqbdIha4zaaraWaaSbaaSqaaiabdEgaNbqabaGccqGHsislcqaIYaGmcqaHepaDaaa@37B3@ for the 4 groups where τ = 100U, 0 <U ≤ 1, x ¯ g MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmiEaGNbaebadaWgaaWcbaGaem4zaCgabeaaaaa@2EE9@ is overall observed average for gene g, and each group has 6 replicates. Obviously, treatment effect τ on expression changes randomly with genes in our simulation, which would make it more difficult to identify differentially expressed genes than the simulations with a fixed treatment effect. Figure 1 displays a comparison between RAF estimated and true FDRs. One can see that the RAF estimate curve is very close to the true FDR curve given a series of thresholds.

Efficiencies of different methods in finding genes differentially expressed among multiple groups

To evaluate different methods, we generated 30 simulation data sets of 3770 genes with the same simulation procedure described above where treatment effect τ was randomly assigned to 10% of the genes in 4 groups, each with 6 replicates. We compared four typical methods with these simulation datasets, of which the Bonferroni (B) procedure and Benjamini-Hochberg (BH) procedure are conventional multiple-testing procedures based on a series of p-values obtained from the classical F-test; SAM is a ranking method using the Fisher linear discrimination [9]. Our method also is a ranking method but based on the classical F-test. Although the F-test based on the hierarchical error model (HEM) proposed by Cho and Lee [26] also is suitable to multiple-sample data, the HEM method has consistent performance with the SAM and has no estimate of FDR. Therefore we did not take the HEM method into account of our comparisons among methods. Table 1 summarizes the results obtained by applying these four methods to the 30 simulation datasets where efficiency of a method in finding genes differentially expressed among multiple groups was comprehensively evaluated by averaging number of called significances (NCS), estimated number of false positives (ENFP), true number of false positives (TNFP), and differences (d values) between ENFPs and TNFPs within a given range of FDR over these 30 simulation data sets. Here we measure the conservativeness of a method by the conservative degree C(d ≥ 0), defined as the proportion of simulations with d ≥ 0. In Table 1, as expected, the B procedure gave the most conservative findings and the lowest power among these methods. Similarly, the BH procedure also yielded a very conservative result in which 96.7 percent of ENFPs were larger than TNFPs, in other words, conservation degree reached 96.7%. For the two ranking methods, Table 1 displays the results in the cases of FDR at 6 levels 0.04 <λ ≤ 0.05, 0.03 <λ ≤ 0.04, 0.02 <λ ≤ 0.03, 0.01 <λ ≤ 0.02, 10-4 <λ ≤ 0.01, and λ < 10-4 . It is clear that RAF has slightly larger means of NCS at all these 6 FDR levels than SAM. In RAF, the means of ENFP are all higher than the means of TNFP while in SAM the means of ENFP are all less than the means of TNFP. Table 1 also shows that RAF has 75~86.2% conservation degree in estimates of false positives under 5% of FDR whereas SAM has 23~66% of conservation degrees. These results suggest that RAF has the highest efficiency in finding genes differentially expressed among these four methods.

Table 1. Efficiencies of different methods in identifying genes differentially expressed among four groups each with 6 replicates in 30 simulated datasets

We also generated a simulated data set of 3770 genes where treatment effect τ was randomly assigned to 10% of genes but sample size for each group was changed from 6 replicates to 4. Table 2 displays the results obtained by SAM and RAF from this data set. It can be seen that SAM has very high FDRs while RAF still works well and detects 9 genes without false positives.

Table 2. Comparison between SAM and RAF in finding genes differentially expressed among four classes in a simulated data set of small sample size (n = 4)

Array findings by RAF

We obtained a set of the observed data in which expression of 3770 genes was measured among four treatment groups HS-SHRSPs, LS-SHRSPs, HS-SHRSRs, and LS-SHRSRs. This set of microarray data is readily applicable to our ranking F-test analysis. Figure 3 shows a scattered-dot plot of F-values versus f ¯ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmOzayMbaebaaaa@2D42@ -values obtained by the RS approach.

thumbnailFigure 3. The scatter plot of F-values versus f ¯ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmOzayMbaebaaaa@2D42@ -values. F-values were observed from real microarray data set and f ¯ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmOzayMbaebaaaa@2D42@ -values yielded by random splitting approach are an estimate of null f-distribution.

Figure 4 compares the observed plot of ranked F - f ¯ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmOzayMbaebaaaa@2D42@ to the simulated one. One can see from Figure 4 that the observed F - f ¯ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmOzayMbaebaaaa@2D42@ plot begins to deviate from the simulated F - f ¯ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmOzayMbaebaaaa@2D42@ plot at about f ¯ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmOzayMbaebaaaa@2D42@ = 2.1, suggesting that a part of the F-statistics deviates from the f ¯ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmOzayMbaebaaaa@2D42@ -distribution. This result underscores that these genotypes and diet feeds significantly impact on expression of a portion of genes in rat with respect to stroke.

thumbnailFigure 4. Comparison between the observed (red) and simulated (blue) plots of F-values versus f ¯ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmOzayMbaebaaaa@2D42@ -values. F-values were observed from real (red) and simulated (blue) microarray data sets of 3770 genes and 6 replicates. f ¯ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmOzayMbaebaaaa@2D42@ -value yielded by randomly splitting approach is an estimate in null f-distribution. F-distribution from simulated data set without treatment effects is null distribution. Ranked F-values corresponds to ranked f ¯ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmOzayMbaebaaaa@2D42@ -values.

The numbers of genes whose expression is significantly different among the four groups HS-SHRSPs, LS-SHRSPs, HS-SHRs, and LS-SHRs, are found to be 392, 145, and 107 by our RAF under estimates of FDR of 4.8, 0.7, and <7.0% (see Table 3), respectively. These 107 genes with FDR<0.7% are listed in the Additional file 1. Among these 107 identified probes, 31 belong to the expressed sequence tags (ESTs), 76 are unique genes who have known functions in the brain or central nervous system and belong to six major functional classes: (a) neurotransmission such as neurexin III-alpha, Neurodap-1, non-neuronal enolase (NNE), beta isoform of catalytic subunit of cAMP-dependent protein kinase; (b) cell signaling and transportation such as trans-Golgi network integral membrane protein (TGN38), glutamate transporter, alternatively spliced GTP-binding protein alpha subunit intracellular, signal regulatory protein alpha, synaptic vesicle protein 2B (SV2B), L-type amino acid transporter 1 (LAT1), N-ethylmaleimide-sensitive factor (NSF); (c) cell proliferation, differentiation, and apoptosis anti-proliferative factor (BTG1), thyroid hormone receptor a1 (c-erb A α1); (d) metabolism such as stearoyl-CoA desaturase 2, beta isoform of catalytic subunit of cAMP-dependent protein kinase, ATP-citrate lyase. (e) RNA transcript and regulation such as Zinc finger gene, Jun-D gene, and ribosomal protein genes encoding larger ribosomal subunits L13, L8, and L22; (f) ion channel/pump such as potassium channel-Kv2, electrogenic Na+ bicarbonate cotransporter (NBC), type II Ca2+/calmodulin-dependent protein kinase beta subunit, and protein kinase C-regulated chloride channel.

Additional File 1. Genes with significant expression changes among four groups. The data provided are the tables containing detailed information of the 107 genes with significant expression changes among the four groups with FDR < 0.7%.

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Table 3. The results of RAF identifying genes differentially expressed among HS-SHRSPs, LS-SHRSPs, HS-SHRSRs, and LS-SHRSRs.

Independent verification of array findings

Fornage et al (2003) used TagMan assay to measure the relative expressions of 7 genes encoding atrial natriuretic peptide (Anp), the neurotrophin receptor protein tyrosine kinase (TrkB, short), casein kinase 2 (Ck2), complexin 2 (Cplx2), stearoyl CoA desaturase 2 (Scd2), glycerol-3-phosophate acyltransterase (Gpan), and inositol 1,4,5-triphosphate receptor (Itpr1). They found these 7 genes had significantly differentially expressed between SHRSP and SHR strains with p < 0.05. Except that genes Anp and Gpan were out of our data, genes for TrkB (short), Cplx2, and Scd2 called significant differential expressions at FDR<0.7%, and for CK2 and Itpr1 at FDR = 0.7% were found among HS-SHRSP, LS-SHRSP, HS-SHR, and LS-SHR strains. Interestingly, Tropea et al [27] also found the genes encoding glutamate receptor (GluR-A) and GABA receptor had significant expression difference between two groups of mice treated by dark rearing and monocular deprivation.


To our knowledge, the ranking analysis of F-statistics for identifying differentially expressed genes among multiple groups (classes) has not been reported. There are two main difficulties to be overcome: (a) estimate of the null F-distribution and (b) estimate of FDR. In conventional statistical methods, permutation is very popular to generate empirical distributions as estimates of the null distributions. However, the permutation approach may not be suitable for microarray data [10-13] because in general microarray experiments have a small sample size due to cost, as a result, treatment effect residues that cannot be removed are amplified in permutation distribution and resulting estimated null distribution has a heavier tail compared to true null distribution [12]. This would results in two consequences: (a) the estimated null distribution is not stable, which, as seen in Table 3, leads to low conservativeness of estimate of FDR, and (b) low power. Our RAF method is successful because the f ¯ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmOzayMbaebaaaa@2D42@ -distribution obtained by applying the RS approach [14] does not contain treatment effects and hence is a desirable estimate of the null F-distribution, which is supported by the fact that the observed and simulated results agree well with those expected by theory. Therefore, the number (M) of splits is much smaller than that of permutations for estimate of the null F-distribution. Simulation results showed that 50 splits are enough to obtain a stable and smooth f ¯ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmOzayMbaebaaaa@2D42@ -distribution. In addition, since the f ¯ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmOzayMbaebaaaa@2D42@ -distribution is generated from all the genes detected on microarrays and does not contain treatment effect residences, impact of sample size on the f ¯ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmOzayMbaebaaaa@2D42@ -distribution is very weak. However, we also noted that the f ¯ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmOzayMbaebaaaa@2D42@ -distribution would underestimate the null F-distribution when sample sizes are smaller than 4. In this situation, Eq. (7) should be changed to σ 2 ( e ¯ g J ) = i = 1 n 4 ( e ¯ g i J e ¯ g J ) 2 / ( n 1 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeq4Wdm3aaWbaaSqabeaacqaIYaGmaaGccqGGOaakcuWGLbqzgaqeamaaDaaaleaacqWGNbWzaeaacqWGkbGsaaGccqGGPaqkcqGH9aqpdaaeWaqaaiabisda0iabcIcaOiqbdwgaLzaaraWaa0baaSqaaiabdEgaNjabdMgaPbqaaiabdQeakbaakiabgkHiTiqbdwgaLzaaraWaa0baaSqaaiabdEgaNbqaaiabdQeakbaakiabcMcaPmaaCaaaleqabaGaeGOmaidaaOGaei4la8IaeiikaGIaemOBa4MaeyOeI0IaeGymaeJaeiykaKcaleaacqWGPbqAcqGH9aqpcqaIXaqmaeaacqWGUbGBa0GaeyyeIuoaaaa@5066@ .

FDR is often used to control the error rate in the BH procedure [7], the BL procedure [8], and in SAM [9]. In practice, for a ranking test, it is necessary to obtain an accurate estimate of FDR. In SAM, FDR is estimated through the permutation approach in which fluctuations around expectation occur among permutated samples. The fluctuations would be dependent on the data itself, i.e., sample size, treatment effect, and data noise. In addition, as indicated above, permutation fails to remove the treatment effects in the data permuted from the microarray data with a small sample size so that the fluctuations are not purely due to random errors. Thus, this approach may give a biased estimate of FDR for a given threshold. The RAF estimator is based on a two-simulation strategy and hence avoids these problems of the SAM estimator, that is, its accuracy is not affected by sample size, treatment effect, and noise. As a result, the number (B) of simulations is also relatively small. Our simulation study indicates that more than 40 simulated data sets (B ≥ 40) would produce stable estimates of FDR across all given thresholds.

Our current RAF method can be readily extended to other test statistics such as Brown-Forsythe test statistic [28], Welch test statistic [29], and Cochran test statistic [30] by replacing F-statistic with the respective statistics.


We developed a new statistical method that is suitable for analyzing microarray data to identify differentially expressed genes among multiple groups, especially, when sample size is small.

Authors' contributions

YDT participated in the method development, performed the statistical analysis and drafted the manuscript. MF participated in the design of the study, carried out the microarray experiment. HX conceived of the study, and participated in its design and coordination and helped to draft the manuscript. All authors read and approved the final manuscript.


This research was supported by grants from the U.S. National Institutes of Health (HL69126) to MF.


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