Abstract
Background
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 largescale statistical analyses have been developed but most of them are applicable to twosample or twocondition data.
Results
We developed a largescale multiplegroup Ftest based method, named ranking analysis of Fstatistics (RAF), which is an extension of ranking analysis of microarray data (RAM) for twosample ttest. 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 twosimulation 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.
Conclusion
Our method is suitable to identify differentially expressed genes among multiple groups, in particular, when sample size is small.
Background
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 largescale gene expression experiments requires global statistical methods rather than traditional statistical methods based on single hypothesis testing. A variety of multipletesting procedures, such as the Bonferroni procedure, Holm procedure [5], Hochberg procedure [6], BenjaminiHochberg (BH) procedure [7], and BenjaminiLiu (BL) procedures [8] have already been developed for testing a large family of null hypotheses. The first three methods bound the familywiseerror 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 pvalues. 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 [1012] 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 Fdistribution based on permutation has a larger variance and a heavier tail compared to the true null Fdistribution, which leads to a potential loss of power. Similar phenomenon was also observed in comparison of the estimated null tdistribution to the true null tdistribution [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 multiclass microarray data, the analysis of variance (ANOVA) is useful to identify differentially expressed genes [4]. In ANOVA, the Ftest is a generalization of the ttest that allows for comparison of more than two samples [15]. However, due to small sample sizes, the classical Ftest is also subject to the same problems as the ttest: bias and unstable estimates of genespecific variances. To tackle this issue, many authors [1519] proposed modified Fstatistics. However, like the classical Ftest, these modified Ftests still suffer from high falsepositive rates because (i) the sample size is often so limited that the asymptotic F distribution is not accurate enough to obtain a valid pvalue and (ii) they appeal to multipletesting procedures such as the Bonferroni procedure or the BHprocedure. As mentioned above, these multipletesting procedures have a basic requirement that sample sizes are large enough for valid pvalues. 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 multiclass geneexpression data called Ranking Analysis of Fstatistics (RAF). RAF is a natural extension of our previous work, i.e., the ranking analysis of microarrary (RAM) for twoclass ttests [14]. It works on finding genes that are differentially expressed among multiple treatment groups by comparing the ordered real Fstatistics with the ordered estimated null Fstatistics and implementing a twosimulation strategy to estimate the false discovery rate (FDR).
Methods
Animal model and design
Studies were performed on male strokeresistant SHR/N (CRiv) (SHRSR) and strokeprone SHR/A3 (Heid) (SHRSP) rats from a breeding colony maintained by the investigators as previously described [20]. Agematched 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 "strokepermissive diet" high in sodium (HS) (0.63% potassium, 0.37% sodium) and 1% NaCl drinking solution; a "strokeprotective 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 12hour lightdark 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 oneway in statistical analysis instead of twoway, 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, HSSHRSPs, LSSHRSPs, HSSHRSRs, and LSSHRSRs 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 FTest
Let x_{gij }be the expression value for replicate j of gene g in group i where g = 1,..., N (number of genes), j = 1,..., r_{gi }(number of replicate observed values of gene g in group i) and i = 1,..., n (number of groups). The traditional Fstatistic in oneway ANOVA may be expressed as
where σ^{2 }(G_{g}) and σ^{2 }(e_{g}) are inter and intragroup variances of the expression values of gene g, respectively. In the conventional Ftests, 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 F_{1 }is the largest value, F_{2 }is the second largest value, etc., and F_{g* }is the g*th largest value where g* is a rank position of gene g. Thus, we have a ranking Ftest where
indicates that the expression variation of gene g among multiple groups (or multiple conditions) is significant. In Eq. (2), f_{g* }is the expectation of F_{g* }under the null hypothesis and Δ is a given threshold.
Estimation of f_{g*}
In the ANOVA framework, we have
where and ; τ_{gi }and are treatment effect and average random error in group i, respectively; and are the overall observed mean and the overall average error for gene g, respectively, and μ_{g }is the overall expected mean for gene g and r_{gi }is the number of replicate observed values of gene g in group i. Therefore, the intergroup variance σ^{2 }(G_{g}) consists of two parts: variance of treatment effects on expression of gene g, σ^{2 }(τ_{g}), and average error variance, σ^{2 }(). Thus, Fstatistic can be rewritten as
Therefore, the null hypothesis is equivalent to F_{g }= f_{g }because σ^{2 }(τ_{g}) = 0 under the null hypothesis. Note that σ^{2 }(τ_{g}) = 0 means the treatment effects τ_{gi }= ... = τ_{gn }= μ_{g}. In order to do a ranking Ftest, it is necessary to obtain a good estimate of f_{g* }. In the twogroup scenario, Tusher et al. [9] employed a permutation approach to estimate the expected tstatistics. The permutation process cannot completely clear the treatment effect in the ranked dstatistics so that the estimated ranked dstatistics 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 tstatistics. In this study, we extended the RS approach to estimating the null distribution of Fstatistics.
In the RS approach, one sample consisting of r_{gi }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 μ_{g }and the same treatment effect τ_{gi }on expression of gene g except for expression noises. A sample of r_{gi }replicate values for gene g is randomly split into two subsamples denoted by s = 1 and s = 2. If let be the mean of subsample s of sample i for gene g at split J(J = 1,..., M), then can be expressed as
where and and are replicate number and noise in the observed expression value j in subsample s in group i for gene g at split J, respectively. is estimated by the difference between two subsample means in sample i for gene g at split J,
It can be seen from Eq. (6) that μ_{g }and τ_{gi }are cleared in difference between two subsample means, which is unrelated to sample size. Thus, the average random error variance σ^{2 }() in Eq.s (3) and (4) can be estimated by
where is estimate of mean () of expression noise of gene g among groups at split J. Variance σ^{2}() is an estimate of expectation (σ^{2}()) of intergroup variance (σ^{2 }(G_{g})) under the null hypothesis at split J. We therefore have
Note that since treatment effect is completely removed from the difference between two subsample means, the difference is pure noise. We rank across all g and let denote the value in ordered position g* at split J. After running M splits, we have M values of for position g*. Thus f_{g* }in Eq. (2) can be estimated by the average of over all M splits, i.e., .
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 twosimulation approach for FDR estimation [14]. Consider a series of threshold values Δ_{k}(k = 1,...L) and let N_{k }be the number of genes that are claimed as significant by RAF at threshold Δ_{k}. N_{k }comprises two parts: the number N_{k}(t) of the true positives and the number N_{k}(f) of the false positives, i.e., N_{k }= N_{k}(t) + N_{k}(f). Thus, given a threshold Δ_{k}, FDR is defined as λ_{k }= N_{k}(f)/N_{k}. N_{k}(f) is unknown, hence λ_{k }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 permutationbased 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 Fstatistics. 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 () and a set of sample error variances [s^{2 }(e_{g1}),..., s^{2 }(e_{gn})]. Here we set 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., σ^{2 }(G_{g}) = 0 and for each of the other half with unknown effect that the group variance is larger than or equal to zero, i.e., σ^{2 }(G_{g}) ≥ 0. s^{2 }(e_{gi}) is set to be equal to σ^{2 }(e_{gi}) where and σ^{2 }(e_{gi}) 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 (b = 1,..., B). Here those that are called significant by comparing to at a given threshold Δ_{k }are counted as across all ranking positions. Let Given the fact that a small part of genes have unequal means in the samples, the simulation data set produces a partially null Fdistribution. In other words, it may produce , possibly leading to λ_{k }= N_{1k}/N_{k }> 1. To avoid this situation, suppose N_{1k}(k = 1,..., L) takes the maximum value N(m) at Δ_{k = m}, we define
as a function of threshold Δ_{k }for estimating FDR where we set N_{1k }= N(m) for all Δ_{k }< Δ_{m}(k = 1,..., L). Obviously, λ_{1k }is a decreasing function of the threshold and bounded between 0 and 1. For example, λ_{1k }= 1 when N_{1k }= N(m), and λ_{1k }= 2/3 when N_{1k }= N(m)/2. Results from the simulation study in Figure 1 indicate that λ_{1k }≥ λ_{k }(true value of FDR at threshold Δ_{k}) when threshold Δ_{k }is smaller than a value Δ* but λ_{1k }≤ λ_{k }when Δ_{k }> Δ* (see Figure 1).
Figure 1. Profile of estimates of FDRs for a series of thresholds. λ_{1k }and λ_{2k }are two threshold functions from simulations 1 and 2 and were used to construct an estimation interval for estimate of FDR at threshold Δ_{k}. λ_{k }and are true and estimated FDRs at threshold Δ_{k}, 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, and a set of sample variances s^{2 }(e_{g1}) = σ^{2 }(e_{g1}),..., s^{2 }(e_{gn}) = σ^{2 }(e_{gn}).
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 Fvalue denoted as (b = 1,..., B). Let . The positives found by comparing to F_{g*2 }at a given threshold Δ_{k }are counted as across all ranking positions. Here let . Unlike the first simulation, here the simulation data sets produce B null Fdistributions, so N_{2k }should be approximate to the true number of false positives N_{k}(f). However, when threshold Δ_{k }is large, it is possible to have N_{k }= 0 so that N_{2k}/N_{k }is undefined. To avoid this situation, we define
as the second function of threshold (see Figure 1). In particular, we let λ_{2k }= 1 if N_{k }= N_{2k }= 0 because λ_{2k }= 1 when N_{k }= 0 and N_{2k }> 0.
Thus, an interval for FDR estimation at threshold Δ_{k }can be constructed between λ_{1k }and λ_{2k }. The third function of threshold for FDR estimation is given as
λ_{3k }= α_{k}λ_{1k }+ β_{k}λ_{2k}
where α_{k }= min(λ_{1k}, λ_{2k})/(λ_{1k }+ λ_{2k}) and β_{k }= 1  a_{k}. λ_{3k }plays the role of weight in balancing λ_{1k }and λ_{2k }. Therefore, at threshold Δ_{k}, a putative probability that a false discovery is found in the genes called significant by RAF is
Note that as shown in the simulation result section, λ_{2k }is an underestimate of λ_{k }and λ_{1k }is an overestimate of FDR when the threshold Δ_{k }< Δ*. However, the situation is reversed when threshold Δ_{k }> Δ*. This is because N_{1k }becomes very small when Δ_{k }> Δ* so that λ_{1k }becomes very small whereas, from Eq. (10), λ_{2k }slowly decreases if N_{k }> N_{2k }or increases if N_{k }<N_{2k }as threshold increases. In addition, when the microarray data have no treatment effects for all the genes detected, then λ_{1k }= λ_{2k }= λ_{3k }= 1, leading to = 1
In order to smooth between thresholds Δ_{k }and Δ_{k+1}, we define a recursive formula modifying the probability as
where p_{k }= (N_{k } N_{k+1})/(1 + N_{k } N_{k+1}) and q_{k }= 1  p_{k}. Eq. (13) suggests that λ_{k+1 }= λ_{k }if N_{k }= N_{k+1}. The number of false discoveries among those found to be significant at threshold Δ_{k }in the observed data is estimated by . Figure 1 shows that the curve of agrees well with that of λ_{k}.
Results
Estimation of the null distribution of Fstatistics
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 F_{k }values was computed from the simulated data set. We applied our RS approach to this simulated data set to generate over 50 splits. This set of 3770 F_{k }values formed null distribution of Fstatistics, which is called fdistribution. To display the profile that our estimate of fdistribution is approximate to the null fdistribution, we plotted the ranked F_{k }versus ranked . The result displays in Figure 2 where all ranked F  dots roughly fall on a diagonal line as expected by two sets of the same ranked distributions. These results suggest that the distribution is indeed an approximate estimate of fdistribution.
Figure 2. The dotplot of Fvalues versus values. Fvalues and values obtained from the simulated microarray data of 3770 genes were ranked where values were yielded by the random splitting approach. All ranked F  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 and for the 4 groups where τ = 100U, 0 <U ≤ 1, 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 BenjaminiHochberg (BH) procedure are conventional multipletesting procedures based on a series of pvalues obtained from the classical Ftest; SAM is a ranking method using the Fisher linear discrimination [9]. Our method also is a ranking method but based on the classical Ftest. Although the Ftest based on the hierarchical error model (HEM) proposed by Cho and Lee [26] also is suitable to multiplesample 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 HSSHRSPs, LSSHRSPs, HSSHRSRs, and LSSHRSRs. This set of microarray data is readily applicable to our ranking Ftest analysis. Figure 3 shows a scattereddot plot of Fvalues versus values obtained by the RS approach.
Figure 3. The scatter plot of Fvalues versus values. Fvalues were observed from real microarray data set and values yielded by random splitting approach are an estimate of null fdistribution.
Figure 4 compares the observed plot of ranked F  to the simulated one. One can see from Figure 4 that the observed F  plot begins to deviate from the simulated F  plot at about = 2.1, suggesting that a part of the Fstatistics deviates from the 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.
Figure 4. Comparison between the observed (red) and simulated (blue) plots of Fvalues versus values. Fvalues were observed from real (red) and simulated (blue) microarray data sets of 3770 genes and 6 replicates. value yielded by randomly splitting approach is an estimate in null fdistribution. Fdistribution from simulated data set without treatment effects is null distribution. Ranked Fvalues corresponds to ranked values.
The numbers of genes whose expression is significantly different among the four groups HSSHRSPs, LSSHRSPs, HSSHRs, and LSSHRs, 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 IIIalpha, Neurodap1, nonneuronal enolase (NNE), beta isoform of catalytic subunit of cAMPdependent protein kinase; (b) cell signaling and transportation such as transGolgi network integral membrane protein (TGN38), glutamate transporter, alternatively spliced GTPbinding protein alpha subunit intracellular, signal regulatory protein alpha, synaptic vesicle protein 2B (SV2B), Ltype amino acid transporter 1 (LAT1), Nethylmaleimidesensitive factor (NSF); (c) cell proliferation, differentiation, and apoptosis antiproliferative factor (BTG1), thyroid hormone receptor a1 (cerb A α1); (d) metabolism such as stearoylCoA desaturase 2, beta isoform of catalytic subunit of cAMPdependent protein kinase, ATPcitrate lyase. (e) RNA transcript and regulation such as Zinc finger gene, JunD gene, and ribosomal protein genes encoding larger ribosomal subunits L13, L8, and L22; (f) ion channel/pump such as potassium channelKv2, electrogenic Na+ bicarbonate cotransporter (NBC), type II Ca2+/calmodulindependent protein kinase beta subunit, and protein kinase Cregulated 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 HSSHRSPs, LSSHRSPs, HSSHRSRs, and LSSHRSRs.
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), glycerol3phosophate acyltransterase (Gpan), and inositol 1,4,5triphosphate 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 HSSHRSP, LSSHRSP, HSSHR, and LSSHR strains. Interestingly, Tropea et al [27] also found the genes encoding glutamate receptor (GluRA) and GABA receptor had significant expression difference between two groups of mice treated by dark rearing and monocular deprivation.
Discussion
To our knowledge, the ranking analysis of Fstatistics 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 Fdistribution 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 [1013] 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 distribution obtained by applying the RS approach [14] does not contain treatment effects and hence is a desirable estimate of the null Fdistribution, 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 Fdistribution. Simulation results showed that 50 splits are enough to obtain a stable and smooth distribution. In addition, since the distribution is generated from all the genes detected on microarrays and does not contain treatment effect residences, impact of sample size on the distribution is very weak. However, we also noted that the distribution would underestimate the null Fdistribution when sample sizes are smaller than 4. In this situation, Eq. (7) should be changed to .
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 twosimulation 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 BrownForsythe test statistic [28], Welch test statistic [29], and Cochran test statistic [30] by replacing Fstatistic with the respective statistics.
Conclusion
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.
Acknowledgements
This research was supported by grants from the U.S. National Institutes of Health (HL69126) to MF.
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