Abstract
Background
Highthroughput technologies, such as DNA microarray, have significantly advanced biological and biomedical research by enabling researchers to carry out genomewide screens. One critical task in analyzing genomewide datasets is to control the false discovery rate (FDR) so that the proportion of false positive features among those called significant is restrained. Recently a number of FDR control methods have been proposed and widely practiced, such as the BenjaminiHochberg approach, the Storey approach and Significant Analysis of Microarrays (SAM).
Methods
This paper presents a straightforward yet powerful FDR control method termed miFDR, which aims to minimize FDR when calling a fixed number of significant features. We theoretically proved that the strategy used by miFDR is able to find the optimal number of significant features when the desired FDR is fixed.
Results
We compared miFDR with the BH approach, the Storey approach and SAM on both simulated datasets and public DNA microarray datasets. The results demonstrated that miFDR outperforms others by identifying more significant features under the same FDR cutoffs. Literature search showed that many genes called only by miFDR are indeed relevant to the underlying biology of interest.
Conclusions
FDR has been widely applied to analyzing highthroughput datasets allowed for rapid discoveries. Under the same FDR threshold, miFDR is capable to identify more significant features than its competitors at a compatible level of complexity. Therefore, it can potentially generate great impacts on biological and biomedical research.
Availability
If interested, please contact the authors for getting miFDR.
Background
FDR control is a statistical approach to correct multiple comparisons in dealing with multiple hypothesis testing problems. It has now been widely practiced in analyzing genomewide datasets generated by highthroughput technologies, such as DNA microarray and RNASeq, which allows users to simultaneously screen the activities of tens of thousands of genes. These highthroughput datasets require careful analysis to identify a subset of interesting molecular features for followup experiments. It is always desired to maximizing findings in data. In the meantime, it should be realized that followup experiments can be costly in both time and money. Therefore it is important to control the proportion of wrongly called features among those selected (i.e., FDR).
FDR was first introduced by Benjamini and Hochberg [1] and was later improved by the Storey procedure [2,3]. As two of the mainstream FDR controlling methods, the BH procedure fixes the error rate and then estimates its corresponding rejection region while the Storey procedure fixes the rejection region and then estimates its corresponding error rate. Efron and his colleagues framed the FDR control problem as a Bayesian problem, and showed that both the BH and Storey approaches are special cases [46]. Assuming that the same rejection region is used for each independent test, and the test statistics come from a random mixture of null and alternative distributions, the BH approach, the Storey approach and the Efron's Bayesian approach can be connected with a mixture model of null statistics and alternative statistics weighted by a factor representing the prior probability of getting true nulls. The BH approach simply assumes that the prior probability of true null is equal to 1, which makes it the most conservative one among the three. The Storey approach considers estimating the prior probability of true null. The Efron approach uses empirical Bayesian analysis to further estimate posterior probability of true null based on the prior probability. The BH, Storey and Efron approaches all estimate FDR by taking the pvalues of individual features calculated by some sorts of hypothesis tests. The ttest [7] and the Wilcoxon ranksum test [8] (also known as the MannWhitney U test, referred as ranksum test in the rest of the paper for conciseness) are two of the most wellknown tests for calculating the pvalues of individual features.
Significance Analysis of Microarrays (SAM) [9] is another widely applied technique for calling features that behave significantly differently between two conditions. Different from the BH, Storey and Efron approaches, SAM uses a nonparametric method to estimate FDR instead of relying on pvalues directly. SAM generates a large number of permutation controls, and the expected number of false positives can be estimated by counting the number of permuted statistics beyond a certain cutoff. Although SAM performed better than the BH and Storey approaches on many datasets in our practices, we found that SAM's results were not optimal in many cases. This is mainly because SAM decides the cutoffs based on the differences between the observed statistics (original statistics) and the expected statistics (averaged statistics from permuted measurements) instead of the estimated FDR, which does not guarantee the lowest FDR.
To address this problem, we developed miFDR  an advanced significance analysis method for optimizing FDR when the number of desired significant features is fixed. A preliminary version of miFDR was published in [10]. In this paper, we provide theoretical explanations and supports for miFDR, and generate more experimental results to demonstrate that miFDR empirically outperforms SAM, the BH approach and the Storey approach. In particular, the simulation test results showed that miFDR was capable of identifying more significant features with its true FDRs consistently bounded by the estimated FDRs. In addition, the true and estimated FDRs of miFDR were lower than those of the other three methods. Furthermore, when applied to real DNA microarray datasets, miFDR was able to identify more biologically relevant genes than other methods.
Methods
FDR under the Bayesian framework
FDR is defined as the expected proportion of incorrectly rejected null hypotheses among all rejected null hypotheses. It can be represented as a conditional probability P(H = 0d ∈ Γ ), where Γ denotes the rejection region for the statistic variable d. Applying the Bayes theorem, the above definition can be written as
where P (d ∈ Γ) = P (H = 0) P (d ∈ ΓH = 0) + P (H = 1) P (d ∈ ΓH = 1).
The estimation of P (d ∈ ΓH = 0) is not straightforward. However, if the null distribution of d is known, the above term can be calculated by pvalue: P (p ≤ γ  H = 0) where p is the pvalue of d and γ is the corresponding pvalue cutoff to reject H = 0. The BH and Storey approaches are based on this idea. Assuming that the null distribution of pvalues is uniform between 0 and 1, we have
The term P (p ≤ γ) can be estimated in an empirical way as the proportion of features whose pvalues are bounded by the pvalue cutoff γ, namely , where #(p_{i }≤ γ) denotes the number of pvalues bounded by the cutoff γ, and #{p_{i}} denotes the total amount of pvalues which is equivalent to the total number of features M. The BH approach simply assumes that P(H = 0) = 1 while the Storey approach estimates P(H = 0) empirically [2]. SAM adopts the same method to estimate P(H = 0) as the Storey approach [11].
SAM
Different from the BH and Storey approaches, SAM does not assume the distributions of the test statistics. In addition, it introduces corrections to tstatistic and ranksum statistic so that the distribution of the corrected statistics is independent from the levels of feature values. Both tstatistic and ranksum statistic can be represented as a difference score r_{i }divided by the corresponding standard deviation s_{i }: r_{i}/s_{i}. In particular, let X and Y be two groups of samples with N_{X }and N_{Y }samples, respectively. The traditional tstatistic has and ; and the traditional ranksum statistic has and , where is the sum of the ranks of the ith feature from X (the measurements from X and Y are merged and then ranked from lowest to highest). They share one major drawback: the estimation of the standard deviation s_{i }is very unstable when the sample size is relatively small, which is very common in studies involving highthroughput technologies. In addition, the distributions of the test statistics vary with respect to the levels of feature values, which makes it difficult to compare features with different value levels. To address these problems, SAM adds a factor s_{0 }to the denominator s_{i }to reduce the variance of the corrected statistic d_{i }= r_{i}/(s_{i }+ s_{0}) (referred as dvalue in the rest of paper for conciseness). In practice, SAM chooses the value of s_{0 }from the pool of all {s_{i}} so that the variance of d_{i }is minimized. The goal is to make the variance of d_{i }independent to the expression level [11]. We found the corrected statistics useful in analyzing many datasets in our research.
Since the null distributions of the corrected statistics are unknown, SAM uses permutations of the replicates to estimate FDR. Given a particular rejection region Γ, SAM generates B permutations of the original measurements and estimates P (d ∈ ΓH = 0) as the median of , where denotes the dvalues in the bth permutation. SAM estimates P (H = 0) in the same way as the Storey approach does [11].
In SAM, the rejection region Γ is determined by a positive cutoff τ^{+ }> 0 and a negative cutoff τ^{ }< 0. The corresponding rejection regions are and , respectively. A feature is called "significant positive" if its dvalue is greater than τ^{+}, or "significant negative" if its dvalue is smaller than τ^{}. The total rejection region is
To decide τ^{+ }and τ^{}, SAM introduces a Δindex which is calculated as follows. First, the features are sorted in ascending order based on their original dvalues. Let denote the dvalues of the sorted features. Then the dvalues obtained from the permutated replicates are sorted and used to estimate . Finally, the Δvalue of the ith feature is calculated as . Given a userdefined threshold Δ_{0}, SAM searches in the ascending order of Δvalues and decides , where k is the index of the first feature satisfying Δ_{k }≥ Δ_{0}. Similarly, SAM searches in the descending order of Δvalue and decides the negative cutoff as where l is the index of the first feature satisfying Δ_{l }≤ Δ_{0}.
It is obvious that the behaviour of Δvalues has a great impact on SAM's results. We observed that Δvalues were not always monotonic with respect to dvalues, which can greatly limit SAM's performance. To illustrate this, we used a Gene Expression Omnibus (http://www.ncbi.nlm.nih.gov/geo/ webcite) dataset GDS3661 [12] as an example. Shown in Figure 1(a), the monotony does not hold at both ends of the curve, especially the lower end. In Figure 1(b), a group of features, whose dvalues ranged from 6.25 to 7, are circled by an ellipse. Let Δ^{* }denote the smallest Δvalues of these circled features. Given any small constant δ > 0, if we change the threshold from Δ*  δ to Δ* + δ, the negative cutoff τ− will jump significantly from 6.25 (marked by the white arrow) to 7.6 (marked by the black arrow). This means either all of those circled features will be called as significant simultaneously, or none of them will be called. No valid threshold allows a subset of them to be called significant even though FDR can be improved by doing so. Therefore it is not always reliable to determine dvalue cutoffs based on Δvalues. This inspired us to develop miFDR which relies only on dvalues and will be explained below.
Figure 1. The problem of relying on Δvalues to call features. An example showing the problem of using Δvalues to decide the dvalue cutoffs. The xaxis indicates dvalues and the yaxis indicates Δvalues. (b) is the blowout of the dashed rectangle region in (a). Two dvalue cutoffs are indicated by the solid arrow and the open arrow respectively in (b) (see main text for a detailed explanation). The solid arrow marks the negative dvalue cutoff τ^{− }corresponding to the Δvalue indicated by the black line.
Minimize FDR  miFDR
If we would like to select N > 0 significant features (N^{+ }positive significant features and N^{ }negative significant features), there are N + 1 different possible options for choosing (N^{+}, N^{}), i.e. (0, N), (1, N  1), (2, N  2), ···, (N, 0). These options assume: If two dvalues are of the same sign, the one with larger absolute value is more significant. SAM does not explore all of these options, and hence its results can be suboptimal. We designed a straightforward algorithm (pseudo codes in Algorithm 1) called miFDR to explore all N + 1 options and report the one with the lowest estimated FDR. Mathematically, this idea can be expressed as
where FDR(N^{+}, N^{}) indicates the FDR of choosing N^{+ }positive significant features and N^{ }negative significant features. Given a FDR cutoff Ψ ∈ (0,1), we can find the global optimal N*
Theorem 1: Given a FDR cutoff Ψ ∈ (0,1), miFDR always finds the maximum number of significant features.
Proof: Eq. (5) is equivalent to
According to eq. (4), for ∀N^{+}, N^{− }≥ 0 s.t. N^{+ }+ N^{− }= N, . This means: If , then ∀N^{+}, N^{− }≥ 0 s.t. N^{+ }+ N^{− }= N, we have . Therefore, eq. (6) can be written as
This is equal to
Eq. (8) indicates that miFDR always finds the maximum number of features given a specific FDR cutoff.
The maximum number of features that SAM is able to call under the FDR cutoff Ψ can be written as:
where and respectively are the numbers of positive and negative features called significant by the positive and negative dvalue cutoffs decided by Δ_{0}. It should be noted that users need to manually try several Δvalue cutoffs to find the best Δ_{0}. It is obvious that is a special case of in eq. (8). Hence SAM only explores a subset of options considered by miFDR mainly because SAM does not directly tune dvalue cutoffs. Instead, SAM control dvalue cutoffs via Δ_{0}. Thus the best result of SAM is bounded by the best result of miFDR.
Algorithm 1: [fdr, feature^{+}, feature^{}] = miFDR
Input: {d_{i}}  the original dvalues;  the dvalues in permutations; N  the number of desired significant features.
1) Initialize fdr ← ∞, N^{+ }← 0, and N^{ }← 0.
2) Sort {d_{i}} in the ascending order to obtain
3) For (n = 0; n ≤ N; n++)
a) Select n positive significant features and N  n negative significant features.
b) Define the corresponding rejection region
c) Estimate FDR "cFDR" for above rejection region using .
d) If cFDR <fdr, then fdr = cFDR, and update N^{+ }← n and N^{ }← N  n.
4) Let feature^{+ }= the indexes of N^{+ }features with the largest dvalues; and feature^{ }as the indexes of N^{ }features with the smallest dvalues.
Output: fdr  the estimated FDR; feature^{+ } the indexes of positive significant features; feature^{ } the indexes of negative significant features.
Computational complexity of miFDR
Assume that a dataset is composed of W samples, each sample has M features, and the samples are permutated P times to generate the control. It takes O(WMP) for miFDR to compute the permuted statistics (the computation time for one feature in one permutation is proportional to the sample size W), and the computation time for the original statistics can be ignored because . Once the original and permuted statistics are computed, miFDR can be applied to achieve two typical goals:
• Minimize FDR when finding N significant features: In this case, miFDR needs to explore N + 1 options. For each option, namely a given (N^{+}, N^{}) pair, the expected computation time for miFDR to estimate FDR is O(MP) because it has to go through the entire permutation matrix. Thus, the computational complexity for miFDR applied to this goal is O(NMP).
• Find the maximal number of significant features given a FDR cutoff: In this case, miFDR needs to examine up to a certain number M_{ρ }(0 <M_{ρ }≤ M) in order to find the best result yielding the required FDR cutoff. In the worstcase scenario, miFDR has to estimate FDRs for all possible (N^{+}, N^{}) airs that satisfy N^{+ }+ N^{− }= 1,2, ..., M_{ρ}. If not well implemented, the worst computational complexity for doing this is , which is much worse than that of the first goal. However, we noticed that both N^{+ }and N^{ }can only be chosen from 0, 1, 2, ...M_{ρ}. Thus, miFDR can be implemented in a very efficient way as below. We can use the permutated measurements to calculate in advance the onesided false positives for N^{+ }= 0,1,2, ..., M_{ρ }and N^{− }= 0,1,2, ..., M_{ρ }in each permutation. This will take O(M_{ρ}MP) in total. Then, to evaluate a given (N^{+}, N^{}) pair, all we need to do is simply combine the precomputed false positives for N^{+ }and those for N^{ }in each permutation, and then calculate the median of the combined false positives in O(P). Hence, it will take miFDR to cover all options. Since M_{ρ }≤ M, miFDR has a computational complexity of O(M_{ρ}MP) to check up to M_{ρ }features, which is comparable to that of the first goal if the desired number of features N ~ M_{ρ}. Based on the above idea, we designed Algorithm 2. In practice, the value of M_{ρ }can be easily tuned by users. By default, we set it to 1000, which worked well in practice so far, and the calculation can be finished in a few minutes. Nevertheless, we theoretically proved that M_{ρ }can be determined automatically and efficiently (see Theorem S1 in Additional File 1 Appendix C).
Algorithm 2: [N, feature^{+}, feature^{}] = miFDR2
Input: {d_{i}}  the original dvalues;  the dvalues in permutations; Ψ  target FDR cutoff; M_{ρ } the expected number of significant features to be tested.
1) Initialize fdr ← ∞, N^{+ }← 0, and N^{ }← 0.
2) Calculate P(H = 0).
3) Sort {d_{i}} in the ascending order to obtain
4) For (k = 0; k ≤ M_{ρ}; k++)
a) the number of the original dvalues larger than
b) the number of the original dvalues smaller than
c) For (b = 0; k ≤ P; b++)
i. the number of the dvalues in bth permutation that are larger than
ii. = the number of the dvalues in bth permutation that are smaller than
5) For (k =M_{ρ}; k ≥ 0; k −−)
b) For (n = 0; n ≤ k; n++)
i. The estimated FDR can be obtained efficiently as follows:
c) , then , , and N ← k. Jump to (6).
6) Let feature^{+ }= the indexes of N^{+ }features with the largest dvalues; and feature^{ }= the indexes of N^{ }features with the smallest dvalues.
Output: N  the maximum number of significant features satisfying the FDR cutoff Ψ; feature^{+ } the indexes of positive significant features; feature^{ } the indexes of negative significant features.
Results
We compared the miFDR approach to SAM (v4.0), the BH approach and the Storey approach on both simulation test and real microarray data analysis. We selected twosided ttest pvalues for the BH and Storey approaches (implemented in MATLAB Bioinformatics Toolbox v4.2) because we found twosided ttest has better performance than onesided ttest, onesided ranksum test and twosided ranksum test. The results showed that miFDR outperformed the other three methods in a wide range of FDR cutoffs.
Simulation test
We ran the simulation test 1000 times, which were designed to have enough complexity to extensively test different FDR controlling methods. In each run, a dataset was simulated according to the distributions in Table 1. Each simulation dataset contains 16 samples in total, 8 samples in each group. Each sample has 10400 features: 10000 null hypothesis features + 400 alternative hypothesis features. Out of 10000 null hypothesis features, 5000 features follow standard normal distribution and the rest follow uniform distribution in range And 400 alternative hypothesis features follow a mixture of multiple distributions described in Table 1.
Table 1. Null and alternative hypotheses in simulated datasets
In each simulation, every approach produced a curve describing the estimated FDR vs. the number of significant features. Those 1000 curves were then averaged with respect to the number of significant features. Since the groundtruth was known, we were able to calculate the true FDR and derive the averaged curve to show true FDR vs. the number of significant features for each approach.
As expected, miFDR consistently called more significant features than SAM at the same estimated FDR levels (see Figure 2a). In particular, at FDR cutoff level 0.05, miFDR identified 19.64 features on average, 17.61% more than the average 16.18 features identified by SAM. Paired ttest showed that the results of miFDR was significantly better than those of SAM (pvalue = 6.04e73). In addition, the true FDR curve of miFDR was consistently bounded by that of SAM (see Figure 2b). This means miFDR made less false calls than SAM did. Finally, the true FDR curve of miFDR was well bounded by its estimated FDR curve (see Figure 2c).
Figure 2. Simulation result I. Compare the average performance of BH, Storey, SAM and miFDR on 1000 simulation runs. In each plot, a blowout of the curve segment in the dashed rectangle is shown at the bottomright corner for clearer illustration. (a) Compare the estimated FDRs among four approaches. The performance of miFDR is the best. (b) Compare the true FDRs. The performance of miFDR is the best. (c) Compare the estimated FDRs and the true FDRs of miFDR. The true FDR curve of miFDR is well bounded by its estimated FDR curve, which indicates that miFDR does not underestimate the number of falsely called significant features.
The BH and Storey approaches were also included in the comparison. But their performance was much worse than miFDR and SAM (Figure 2a &2b), with two reasons: Firstly, they consistently identified fewer significant features than miFDR and SAM did at the same FDR levels. Secondly, their true FDRs are much higher than those of miFDR and SAM when calling the same numbers of significant features. Such worse performance can be because 50% null features follow uniform distributions. However the BH and Storey approaches used ttest pvalues, which assume Gaussian distributions. When ranksum pvalues were used in the BH and Storey approaches, the results were even worse (see Figure 3).
Figure 3. Simulation result II. Compare the average performance of four approaches: 1) BH using ttest, 2) Storey using ttest, 3) BH using ranksum and 4) Storey using ranksum on 1000 simulation runs. The performance of BH using ttest and Storey using ttest are much better than those of BH using ranksum and Storey using ranksum.
We also ran the simulation test with sample size 6 vs. 6 and 10 vs. 10. The results resonated the above findings (see the Additional File 1 Appendix A & B).
Additional file 1. The supplementary materials contain Appendices A, B, and C. Appendices A and B show the simulation test results using different sample sizes: 6 vs. 6 and 10 vs. 10, respectively. Appendix C proves a theorem that allows us to automatically decide the upperbound of M_{ρ }in Algorithm 2 for a given FDR cutoff.
Format: PDF Size: 1.6MB Download file
This file can be viewed with: Adobe Acrobat Reader
Analyze DNA microarray datasets
To further demonstrate that miFDR has high performance in practice, we compared miFDR, SAM, the BH and Storey approaches on a couple of public DNA microarray gene expression datasets obtained from Gene Expression Omnibus (GEO, http://www.ncbi.nlm.nih.gov/geo/ webcite). The results (see Table 2) clearly showed that miFDR significantly outperformed the other three approaches.
Table 2. Numbers of detected significant genes under FDR cutoff = 0.05
Two of those datasets happen to be related to hypertension: GDS3661 and GDS3689, which caught our attentions. Hypertension accounts for about 25% of heart failures [13]. If uncontrolled, hypertension can cause various changes in myocardial structure, conduction system and coronary vasculature of the heart, which can further cause the development of left ventricular hypertrophy, atherosclerosis and other complications. It has been proved by both experimental animal and clinical studies that left ventricular hypertrophy could induce myocardial ischemia [3], and eventually result in largescale programmed cell death and heart failure. We therefore want to see if genes called by miFDR in these datasets are indeed biologically relevant. GDS3661 was generated to investigate the molecular activity underlying the onset hypertensive heart failure, by profiling left ventricular samples from rats with spontaneously hypertension [14]. It used Affymetrix Rat Genome 230 2.0 Array to compare the gene expression levels of 6 heart failure rats (HFrats) with those of 6 rats without heart failure (Controlrats). Hypertension has been proved by many studies to be highly related to environmental pollution, especially diesel exhaust exposure [1519]. To discover molecular links between hypertension and diesel exhaust exposure, GDS3689 was generated by profiling samples from rats exposed to diesel exhaust particles [20] using Affymetrix Gene Chip Rat 230A microarray. It compared 4 rats exposed to diesel exhaust particles (DErats) with 4 rats without exposure (Controlrats). GDS3689 contains samples of both hypertensive rats and healthy rats. In this paper, we only analyzed the samples of healthy rats.
GDS3661
We set the FDR cutoff as 0.05, and compared the number of probe sets called significant by the BH approach, the Storey approach, SAM, and miFDR. Using either ttest or ranksum test, both the BH approach and the Storey approach failed to identify any significant probe set. At the same FDR cutoff, miFDR identified 210 probe sets versus 129 probe sets identified by SAM.
The probe set lists detected by SAM and miFDR were submitted to DAVID [21,22] for gene ontology (GO) enrichment analysis. The result showed that miFDR was better than SAM in identifying genes with functions closely related to phenotypic changes from compensated hypertrophy to systolic heart failure. Some typical GO categories are: GO0007179: transforming growth factor beta receptor signaling pathway (miFDR matched 4 genes vs. SAM matched 3 genes), GO0010647: positive regulation of cell communication (miFDR 10 genes vs. SAM 5 genes), GO0012501: programmed cell death (miFDR 9 genes vs. SAM 7 genes), GO0033554: cellular response to stress (miFDR 10 genes vs. SAM 5 genes), and GO0040008: regulation of growth (miFDR 9 genes vs. SAM 3 genes).
We found in literature that several genes identified only by miFDR may shed new light on the molecular mechanism underlying the deterioration of cardio function and remodeling associated with hypertensive heart failure. Figure 4 illustrates the potential roles of three genes (Mmp2, Rtn4 and Pdlim5) in the context of hypertensive heart failure.
Figure 4. Literature supports for the results of analyzing GDS3661. Three genes (Mmp2, Rtn4 and Pdlim5, marked in blue) identified only by miFDR may perform important roles in the context of hypertensive heart failure. See main text for the detailed explanations of literature evidence labeled as L16 (marked in yellow).
• Mmp2 (upregulated in HFrats by 1.88 folds) may play a critical role in preventing hypertensive heart failure. On the one hand, it was reported that classic preconditioning can inhibit ischemia/reperfusion induced release and therefore offer cardio protection [23] (Figure 4, L1). On the other hand, the suppression of Mmp2 activity by angiotensinconverting enzyme inhibitors can prevent left ventricular remodeling in a rat model of heart failure [24] (Figure 4, L2). Thus, we hypothesize that inhibiting Mmp2 may help prevent heart failure from hypertension.
• Rtn4 (upregulated in HFrats by 1.59 folds) may have great effects on hypertensive heart failure. Programmed cell death of cardiomyocytes following myocardial ischemia imposes a biomechanical stress on the remaining myocardium, leading to myocardial dysfunction that may cause heart failure or sudden death. It was shown that knocking down Rtn4 inhibits the loss of cardiomyocytes following ischemic/hypoxic injury [25]. It was also reported that Rtn4 expression was significantly increased in cardiac tissue from patients with dilated cardiomyopathy and from patients who have experienced an ischemic event [26,27] (Figure 4, L6). These evidences suggest that myocardial ischemia may trigger Rtn4mediated large scale programmed cell death of cardiomyocytes, which eventually leads to heart failure.
• Pdlim5 (upregulated in HFrats by 2.96 folds) is a heart and skeletal musclespecific protein that may perform an essential role in heart development [28]. Pdlim5 are related to hypertensive HF in three ways. Firstly, it was reported that Pdlim5 promoted the expression of hypertrophy markers and increased cell volume when overexpressed in rat neonatal cardiomyocytes [29] (Figure 4, L3). Secondly, Pdlim5 protein was found to preferentially interact with protein kinase C beta (PKCB) which is markedly activated in the cardiac hypertrophic signaling [30] (Figure 4, L4). Finally, it was suggested that the protein of Pdlim5 scaffolded protein kinase D1 (PKD1), a key enzyme in the response to stress signals in cardiomyocytes, to regulate the cardiac Ltype voltagegated calcium channels [31] (Figure 4, L5). There are several drugs for treating hypertension, myocardial ischemia and cardiac arrhythmias by targeting at this channel.
Besides the three genes (Mmp2, Rtn4 and Pdlim5) mentioned above, we also found several other interesting genes in literature, such as Ptgs1 (upregulated in HFrats with 2.06 folds) and Glrx2 (upregulated in HFrats with 1.35 folds). The human homology of Ptgs1 regulates the physiological process involving the growth of new blood vessels from preexisting vessels in endothelial cells. Ptgs1 can mediate endothelial dysfunction under oxidative stress in chronic heart failure [32]. Therefore, Ptgs1 may have a strong effect on the onset of hypertensive heart failure.
Mitochondrial Glrx2 plays a crucial role in cardioprotection [33]. It was shown that Doxorubicininduced cardiac injury is reduced in transgenic mice expressing the human Glrx2 when compared to nontransgenic mice [34]. Overexpression of human Glrx2 in transgenic mice reduces myocardial cell death by preventing both apoptosis and necrosis [33]. We think that the upregulation of Glrx2 is most likely due to the autoadjustment of the heart system to compensate for heart failure. However, the endogenous mechanisms in those heart failure rats were not able to raise Glrx2 up to a level high enough to prevent the onset from happening.
GDS3689
At FDR < 0.05, the BH and Storey approaches using ranksum pvalues failed to identify any significant probe set. If ttest pvalues were used, the BH and Storey approaches identified 18 and 249 significant probe sets, respectively. At the same FDR level, miFDR identified 640 probe sets while SAM only identified 388. We submitted the probe set lists identified by miFDR and SAM for GO enrichment analysis. The result showed that miFDR outperformed SAM in identifying genes in those functional categories closely related to the response to diesel exhaust exposure and hypertension, such as GO0006952: defense response (miFDR identified 17 genes vs. SAM identified 8 genes), GO0006954: inflammatory response (miFDR 12 genes vs. SAM 6 genes), GO0009967: positive regulation of signal transduction (miFDR 11 genes vs. SAM 4 genes), GO0009968: negative regulation of signal transduction (miFDR 8 genes vs. SAM 7 genes), GO0030198: extracellular matrix organization (miFDR 3 genes vs. SAM 0 genes) and GO0033554: cellular response to stress (miFDR 21 genes vs. SAM 10 genes).
Literature evidences also suggested that several genes identified only by miFDR can elucidate new molecular connections between diesel exhaust exposure and hypertension, in particular through atherosclerosis. Atherosclerosis is one of the most serious hypertensionrelated health problems. The arteries of hypertensive animals have greater mass of vascular smooth muscle than normotensive ones, and the alteration in the differentiated state (e.g. increased proliferation, enhanced migration and downregulation of vascular smooth muscle differentiation marker genes) of vascular smooth muscle cells is known to perform a key function in the development of atherosclerosis. In addition, diesel exhaust particles greatly promote atherosclerosis [3537]. One study showed that the synergy between diesel exhaust particles and oxidized phospholipids affect the expression profiles of several gene modules corresponding to the pathways relevant to vascular inflammatory processes such as atherosclerosis [38].
Here we focused on five genes (Tgfbr1, Zeb1, Hdac2, Rab5a, and Ets1), which were all identified by miFDR alone, and discussed their potential roles in the context of diesel particle exposure and atherosclerosis (see Figure 5).
Figure 5. Literature supports for the results of analyzing GDS3689. The miFDR approach identified five genes (Tgfbr1, Zeb1, Hdac2, Rab5a, and Ets1, marked in blue) that may shed new light on the molecular connections between diesel exhaust exposure and hypertension, in particular through atherosclerosis. See main text for detailed explanations of literature evidence labeled as L17 (marked in yellow).
• Tgfbr1 (upregulated in DErats by 2.05 folds) acts as the upstream of p38 in MAPK signaling pathway (http://www.genome.jp/keggbin/show_pathway?hsa04010). It was shown that diesel exhaust particles activate p38 to produce interleukin 8 and RANTES by human bronchial epithelial cells [39]. Thus, we suggest that diesel exhaust particles trigger p38 by activating Tgfbr1 (Figure 5, L1). Tgfbr1 also forms a heteromeric receptor complex with TGFbeta type II receptor that mediates TGFbeta signaling (Entrez Gene summary as of May 30th, 2013: http://www.ncbi.nlm.nih.gov/gene/29591 webcite) (Figure 5, L2).
• Zeb1 (upregulated in DErats by 1.43 folds) mediates TGFbeta signaling in vascular disease and vascular smooth muscle cell differentiation during development [40] (Figure 5, L3), which eventually leads to atherosclerosis.
• Hdac2 (upregulated in DErats by 1.43 folds) was reported to mediate the suppression of vascular smooth muscle cell differentiation marker genes by POVPC [1palmitoyl2(5oxovaleroyl)snglycero3phosphocholine] [41]. POVPC is concentrated within atherosclerotic lesions and contributes to the pathogenesis of atherosclerosis by inducing profound suppression of vascular smooth muscle cell differentiation marker genes via a transcription factor KLF4 [42] (Figure 5, L5).
• Rab5a (upregulated in DErats by 1.83 folds) was shown to have a strong effect on vascular smooth muscle cell proliferation and migration, which can cause intimal hyperplasia and restenosis. And RNAimediated Rab5a suppression can inhibit proliferation and migration of vascular smooth muscle cells [43] (Figure 5, L4).
• Ets1 (upregulated in DErats with 1.69 folds) may be related to atherosclerosis in two ways. On the one hand, Ets1 was reported to activate plateletderived growth factor (PDGF) Achain [44] and PDGF Dchain [45] (Figure 5, L6). PDGF has been implicated in the pathogenesis of vascular occlusive disorders such as atherosclerosis and restenosis in part due to its regulation of vascular smooth muscle cell phenotype. On the other hand, Ets1 is also involved in the signaling mechanisms whereby angiotensin II, a potent upregulator of osteopontin, increases osteopontin expression in vascular smooth muscle cells [46] (Figure 5, L5). Several recent studies have revealed that osteopontin performed multiple roles in the progression of atherosclerotic plaques [4751].
Conclusions
We presented a new powerful FDR control method  miFDR, which minimizes the estimated FDR when calling a fixed number of significant features. We showed theoretically that the search strategy of miFDR maximizes findings given any certain FDR cutoff. We validated this idea by showing that miFDR outperformed the other three widely accepted FDR control methods (SAM, BH and Storey) in simulation tests and DNA microarray analysis. Literature evidences support that several genes identified only by miFDR are indeed relevant to the underlying biology of interest. Controlling FDR is critical in analyzing genomewide datasets. Therefore, miFDR is an important innovation that will benefit projects utilizing highthroughput technologies and make a broad impact in the future.
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
Both authors contributed equally to this work.
Acknowledgements
Yuanzhe Bei's work is partially supported by the Alexander and Shirley Leaderman President's Scholarship and Fellowship at Brandeis University.
Based on "Significance analysis by minimizing false discovery rate", by Yuanzhe Bei and Pengyu Hong, which appeared in Bioinformatics and Biomedicine (BIBM), 2012 IEEE International Conference on. © 2012 IEEE [10].
Declarations
The publication costs for this article were funded by the corresponding author.
This article has been published as part of BMC Systems Biology Volume 7 Supplement 4, 2013: Selected articles from the IEEE International Conference on Bioinformatics and Biomedicine 2012: Systems Biology. The full contents of the supplement are available online at http://www.biomedcentral.com/bmcsystbiol/supplements/7/S4.
References

Benjamini Y, Hochberg Y: Controlling the False Discovery Rate: A Practical and Powerful Approach to Multiple Testing.
Journal of the Royal Statistical Society 1995, 57(1):289300.
Series B (Methodological)

Storey JD: A direct approach to false discovery rates.
Journal of the Royal Statistical Society Series B 2002, 64(3):479498. Publisher Full Text

Storey JD, Tibshirani R: Statistical significance for genomewide studies.
Proc Natl Acad Sci USA 2003, 100(16):94405. PubMed Abstract  Publisher Full Text  PubMed Central Full Text

Efron B: LargeScale Simultaneous Hypothesis Testing: The Choice of a Null Hypothesis.
Journal of the American Statistical Association 2004, 99:96104. Publisher Full Text

Efron B, Tibshirani R: Empirical bayes methods and false discovery rates for microarrays.
Genet Epidemiol 2002, 23(1):7086. PubMed Abstract  Publisher Full Text

Efron B, et al.: Empirical Bayes Analysis of a Microarray Experiment.
Journal of the American Statistical Association 2001, 96(456):11511160. Publisher Full Text

Wilcoxon F: Individual Comparisons by Ranking Methods.
Biometrics Bulletin 1945, 1(6):8083. Publisher Full Text

Tusher VG, Tibshirani R, Chu G: Significance analysis of microarrays applied to the ionizing radiation response.
Proc Natl Acad Sci USA 2001, 98(9):511621. PubMed Abstract  Publisher Full Text  PubMed Central Full Text

Bei Y, Hong P: Significance analysis by minimizing false discovery rate.
Bioinformatics and Biomedicine (BIBM), 2012 IEEE International Conference on: 47 October 2012 2012, 16. Publisher Full Text

Chu G, et al.: Significance Analysis of Microarrays  Users guide and technical document ver 4.0.
2011.

Brooks WW, et al.: Transition from compensated hypertrophy to systolic heart failure in the spontaneously hypertensive rat: Structure, function, and transcript analysis.
Genomics 2010, 95(2):8492. PubMed Abstract  Publisher Full Text

Edgar R, Domrachev M, Lash AE: Gene Expression Omnibus: NCBI gene expression and hybridization array data repository.
Nucleic Acids Res 2002, 30(1):20710. PubMed Abstract  Publisher Full Text  PubMed Central Full Text

McGoron A, Li C, Lin WC: 25th Southern Biomedical Engineering Conference 2009. Miami, Florida, USA. 2009: Springer; 2009.
15  17 May

IbaldMulli A, et al.: Effects of air pollution on blood pressure: a populationbased approach.
Am J Public Health 2001, 91(4):5717. PubMed Abstract  PubMed Central Full Text

IbaldMulli A, et al.: Effects of particulate air pollution on blood pressure and heart rate in subjects with cardiovascular disease: a multicenter approach.
Environ Health Perspect 2004, 112(3):36977. PubMed Abstract  PubMed Central Full Text

Brook RD, et al.: Inhalation of fine particulate air pollution and ozone causes acute arterial vasoconstriction in healthy adults.
Circulation 2002, 105(13):15346. PubMed Abstract  Publisher Full Text

Pope CA III, et al.: Cardiovascular mortality and longterm exposure to particulate air pollution: epidemiological evidence of general pathophysiological pathways of disease.
Circulation 2004, 109(1):717. PubMed Abstract  Publisher Full Text

Zanobetti A, et al.: Ambient pollution and blood pressure in cardiac rehabilitation patients.
Circulation 2004, 110(15):21849. PubMed Abstract  Publisher Full Text

Lin D, et al.: Significance Analysis of DoseResponse Microarray Data. In Modeling DoseResponse Microarray Data in Early Drug Development Experiments Using R. Edited by D. Lin, et al.. Springer Berlin Heidelberg; 2012:123133.

Huang da W, Sherman BT, Lempicki RA: Systematic and integrative analysis of large gene lists using DAVID bioinformatics resources.
Nat Protoc 2009, 4(1):4457. PubMed Abstract  Publisher Full Text

Huang da W, Sherman BT, Lempicki RA: Bioinformatics enrichment tools: paths toward the comprehensive functional analysis of large gene lists.
Nucleic Acids Res 2009, 37(1):113. PubMed Abstract  Publisher Full Text  PubMed Central Full Text

Lalu MM, et al.: Preconditioning decreases ischemia/reperfusioninduced release and activation of matrix metalloproteinase2.
Biochem Biophys Res Commun 2002, 296(4):93741. PubMed Abstract  Publisher Full Text

Brower GL, Levick SP, Janicki JS: Inhibition of matrix metalloproteinase activity by ACE inhibitors prevents left ventricular remodeling in a rat model of heart failure.
Am J Physiol Heart Circ Physiol 2007, 292(6):H305764. PubMed Abstract  Publisher Full Text

Sarkey JP, et al.: NogoA knockdown inhibits hypoxia/reoxygenationinduced activation of mitochondrialdependent apoptosis in cardiomyocytes.
J Mol Cell Cardiol 2011, 50(6):104455. PubMed Abstract  Publisher Full Text  PubMed Central Full Text

Bullard TA, et al.: Identification of Nogo as a novel indicator of heart failure.
Physiol Genomics 2008, 32(2):1829. PubMed Abstract  Publisher Full Text

Gramolini AO, et al.: Comparative proteomics profiling of a phospholamban mutant mouse model of dilated cardiomyopathy reveals progressive intracellular stress responses.
Mol Cell Proteomics 2008, 7(3):51933. PubMed Abstract  Publisher Full Text

Nakagawa N, et al.: ENH, containing PDZ and LIM domains, heart/skeletal musclespecific protein, associates with cytoskeletal proteins through the PDZ domain.
Biochem Biophys Res Commun 2000, 272(2):50512. PubMed Abstract  Publisher Full Text

Yamazaki T, et al.: Splice variants of enigma homolog, differentially expressed during heart development, promote or prevent hypertrophy.
Cardiovasc Res 2010, 86(3):37482. PubMed Abstract  Publisher Full Text  PubMed Central Full Text

Kuroda S, et al.: Proteinprotein interaction of zinc finger LIM domains with protein kinase C.
J Biol Chem 1996, 271(49):3102932. PubMed Abstract  Publisher Full Text

Maturana AD, et al.: Enigma homolog 1 scaffolds protein kinase D1 to regulate the activity of the cardiac Ltype voltagegated calcium channel.
Cardiovasc Res 2008, 78(3):45865. PubMed Abstract  Publisher Full Text  PubMed Central Full Text

Miller JD, et al.: MnSOD protects against COX1mediated endothelial dysfunction in chronic heart failure.
Am J Physiol Heart Circ Physiol 2010, 298(5):16007. Publisher Full Text

Nagy N, et al.: Overexpression of glutaredoxin2 reduces myocardial cell death by preventing both apoptosis and necrosis.
J Mol Cell Cardiol 2008, 44(2):25260. PubMed Abstract  Publisher Full Text

Diotte NM, et al.: Attenuation of doxorubicininduced cardiac injury by mitochondrial glutaredoxin 2.
Biochim Biophys Acta 2009, 1793(2):42738. PubMed Abstract  Publisher Full Text

Bhatnagar A: Environmental cardiology: studying mechanistic links between pollution and heart disease.
Circ Res 2006, 99(7):692705. PubMed Abstract  Publisher Full Text

Brook RD: Cardiovascular effects of air pollution.
Clin Sci (Lond) 2008, 115(6):17587. PubMed Abstract  Publisher Full Text

Brook RD, et al.: Air pollution and cardiovascular disease: a statement for healthcare professionals from the Expert Panel on Population and Prevention Science of the American Heart Association.
Circulation 2004, 109(21):265571. PubMed Abstract  Publisher Full Text

Gong KW, et al.: Airpollutant chemicals and oxidized lipids exhibit genomewide synergistic effects on endothelial cells.
Genome Biol 2007, 8(7):R149. PubMed Abstract  BioMed Central Full Text  PubMed Central Full Text

Hashimoto S, et al.: Diesel exhaust particles activate p38 MAP kinase to produce interleukin 8 and RANTES by human bronchial epithelial cells and Nacetylcysteine attenuates p38 MAP kinase activation.
Am J Respir Crit Care Med 2000, 161(1):2805. PubMed Abstract  Publisher Full Text

Nishimura G, et al.: DeltaEF1 mediates TGFbeta signaling in vascular smooth muscle cell differentiation.
Dev Cell 2006, 11(1):93104. PubMed Abstract  Publisher Full Text

Yoshida T, Gan Q, Owens GK: Kruppellike factor 4, Elk1, and histone deacetylases cooperatively suppress smooth muscle cell differentiation markers in response to oxidized phospholipids.
Am J Physiol Cell Physiol 2008, 295(5):C117582. PubMed Abstract  Publisher Full Text  PubMed Central Full Text

Pidkovka NA, et al.: Oxidized phospholipids induce phenotypic switching of vascular smooth muscle cells in vivo and in vitro.
Circ Res 2007, 101(8):792801. PubMed Abstract  Publisher Full Text

Ma Z, et al.: RNAimediated Rab5a suppression inhibits proliferation and migration of vascular smooth muscle cells.
Acta Cardiol 2010, 65(5):50714. PubMed Abstract

Santiago FS, Khachigian LM: Ets1 stimulates plateletderived growth factor Achain gene transcription and vascular smooth muscle cell growth via cooperative interactions with Sp1.
Circ Res 2004, 95(5):47987. PubMed Abstract  Publisher Full Text

Liu MY, et al.: Inducible plateletderived growth factor Dchain expression by angiotensin II and hydrogen peroxide involves transcriptional regulation by Ets1 and Sp1.
Blood 2006, 107(6):23229. PubMed Abstract  Publisher Full Text

Abe K, et al.: Angiotensin IIinduced osteopontin expression in vascular smooth muscle cells involves Gq/11, Ras, ERK, Src and Ets1.
Hypertens Res 2008, 31(5):98798. PubMed Abstract  Publisher Full Text

Scott JA, et al.: Osteopontin inhibits inducible nitric oxide synthase activity in rat vascular tissue.
Am J Physiol 1998, 275(6 Pt 2):H225865. PubMed Abstract  Publisher Full Text

Wada T, et al.: Calcification of vascular smooth muscle cell cultures: inhibition by osteopontin.
Circ Res 1999, 84(2):16678. PubMed Abstract  Publisher Full Text

Li G, et al.: Fibroblast growth factor receptor1 signaling induces osteopontin expression and vascular smooth muscle celldependent adventitial fibroblast migration in vitro.
Circulation 2002, 106(7):8549. PubMed Abstract  Publisher Full Text

Chen NX, et al.: Phosphorus and uremic serum upregulate osteopontin expression in vascular smooth muscle cells.
Kidney Int 2002, 62(5):172431. PubMed Abstract  Publisher Full Text

Bidder M, et al.: Osteopontin transcription in aortic vascular smooth muscle cells is controlled by glucoseregulated upstream stimulatory factor and activator protein1 activities.
J Biol Chem 2002, 277(46):4448596. PubMed Abstract  Publisher Full Text