Volume 15 Supplement 10
Proceedings of the 25th International Conference on Genome Informatics (GIW/ISCB-Asia): Genomics
LFCseq: a nonparametric approach for differential expression analysis of RNA-seq data
- Bingqing Lin^{1, 3},
- Li-Feng Zhang^{1} and
- Xin Chen^{2}Email author
https://doi.org/10.1186/1471-2164-15-S10-S7
© Lin et al.; licensee BioMed Central Ltd. 2014
Published: 12 December 2014
Abstract
Background
With the advances in high-throughput DNA sequencing technologies, RNA-seq has rapidly emerged as a powerful tool for the quantitative analysis of gene expression and transcript variant discovery. In comparative experiments, differential expression analysis is commonly performed on RNA-seq data to identify genes/features that are differentially expressed between biological conditions. Most existing statistical methods for differential expression analysis are parametric and assume either Poisson distribution or negative binomial distribution on gene read counts. However, violation of distributional assumptions or a poor estimation of parameters often leads to unreliable results.
Results
In this paper, we introduce a new nonparametric approach called LFCseq that uses log fold changes as a differential expression test statistic. To test each gene for differential expression, LFCseq estimates a null probability distribution of count changes from a selected set of genes with similar expression strength. In contrast, the nonparametric NOISeq approach relies on a null distribution estimated from all genes within an experimental condition regardless of their expression levels.
Conclusion
Through extensive simulation study and RNA-seq real data analysis, we demonstrate that the proposed approach could well rank the differentially expressed genes ahead of non-differentially expressed genes, thereby achieving a much improved overall performance for differential expression analysis.
Keywords
Differential expression Nonparametric RNA-seqBackground
RNA sequencing (RNA-seq), which applies high-throughput DNA sequencing technologies to directly sequence complementary DNAs (cDNAs), has completely transformed the way in which transcriptomes are studied. In particular, it permits the quantitative analysis of gene expression and transcript variant discovery, which was not made possible with the previous microarray technologies [1, 2]. RNA-seq is increasingly being used to investigate a wide range of biological and medical questions, e.g., in genomics research [3, 4] and in clinic use [5, 6].
In RNA-seq experiments, millions of short fragments (reads) are sequenced from samples and aligned back to a reference genome. The expression level of a feature (gene, exon or transcript) is then measured by the read count which is the number of short reads that map to the feature. When RNA-seq measurements are made for multiple samples from different biological conditions, a question of particular interest is to identify genes/features that are differentially expressed across conditions. This is the primary aim of RNA-seq differential expression analysis.
There have been a number of statistical approaches proposed for differential expression analysis of RNA-seq data, and they broadly fall into two categories: parametric or nonparametric. In [7, 8], the over-dispersed RNA-seq data is transformed so that the Poisson distribution can be used to model read counts. edgeR [9, 10], DESeq [11], and sSeq [12] instead assume the negative binomial distribution on read counts--a flexible probability model allowing a larger variance than mean. The differences among these three approaches lie mainly in their different ways to estimate the dispersion parameter. EBSeq [13] and baySeq [14] also assume the negative binomial distribution, but they were cast within an empirical Bayesian framework. All the above parametric approaches are generally very efficient when the distributional assumption holds. However, violation of distributional assumptions or a poor estimation of parameters often leads to unreliable results. NOISeq [15] is a data-adaptive nonparametric approach that uses both log fold changes and absolute expression differences as test statistics. It is effective in controlling the false discovery rate (FDR) while being robust against the sequencing depth. SAMseq [16] is another nonparametric approach that utilizes a Wilcoxon statistic. It estimates the false discovery rate by a permutation plug-in procedure and thus is not sensitive to outliers in the data. Recently, an efficient algorithm based on a Markov random field model, called MRFSeq, was developed [17]. Different from previous methods, MRFSeq takes advantage of the additional gene co-expression data to effectively alleviate the selection bias of differentially expressed genes against genes with low read counts. For more discussions and comparisons of these differential expression analysis methods, we refer readers to [18] and [19].
In this paper, we propose a new data-driven nonparametric approach called LFCseq for differential expression analysis of RNA-seq data. Basically, it is based on a similar principle to NOISeq, but uses only log fold changes as the test statistic. To conduct a statistical test for each gene, LFCseq estimates a null or noise probability distribution by contrasting log fold changes for a selected set of genes at similar expression levels. In contrast, NOISeq relies on a null distribution estimated from all genes within an experimental condition regardless of their expression levels. However, as we shall demonstrate later, the null distribution of log fold changes varies considerably for genes at different expression levels, which makes the results from NOISeq less reliable.
Methods
Notation
Although biological experimental designs may vary greatly, RNA-seq data generated for differential expression analysis can all be written into a matrix N, whose element N_{ ij } is the number of reads mapped to gene i in sample j from an experimental condition A or B. Without ambiguity, we also use A (and B, respectively) to denote the set of samples j under the condition A (and B, respectively). That is, if $j\in A$, it indicates that sample j is under the experimental condition A rather than condition B. Let x_{ i } be a binary random variable indicating whether gene i is differentially expressed between two conditions A and B. We have x_{ i } = 1 if gene i is differentially expressed (DE) and x_{ i } = 0 if gene i is not differentially expressed (non-DE).
Typically, only a few samples are available in current RNA-seq experimental data; however, there could instead have up to tens of thousands of genes under examination. In the present study, we limit our discussions to two experimental conditions only, although our proposed approach can be extended to three or more conditions.
Normalization
Since different samples may have different sequencing depths, the read counts N_{ ij } are not directly comparable across samples before being properly normalized [20, 21]. A simple normalization scheme is to divide the read counts by the sample library size and gene length [20]. However, this total-count normalization was shown to be problematic, as the normalized read count of a gene is adversely affected by expression levels of all the other genes [11, 21, 22].
Many sophisticated normalization procedures have been proposed, including the trimmed means of M values (TMM) normalization in edgeR [22], quantile normalization [21], a 'median' normalization method in DESeq [11] and a goodness-of-fit method in PoissonSeq [7]. In our experiments below, we use the goodness-of-fit method to normalize read counts. It defines the sequencing depth for sample j as ${\hat{d}}_{j}={\sum}_{i\in S}{N}_{ij}\mathsf{\text{/}}{\sum}_{i\in S}{\sum}_{j}{N}_{ij}$, where S is a half set of genes that are least differentially expressed between two conditions as estimated by a Poisson goodness-of-fit statistic [7]. The normalized read count n_{ ij } is subsequently computed as ${n}_{ij}={N}_{ij}\mathsf{\text{/}}{\hat{d}}_{j}$.
LFCseq
as the statistic to test differential expression. Because there are usually only a small number of samples under one condition, no read counts could be reliably identified as outliers. Therefore, we choose to use the mean instead of the median of read counts in the above definition (as NOISeq did). However, when there is obvious evidence that the outliers of read count exist or when the number of samples is large enough, median may be a better choice than mean. On the other hand, to avoid the division by zero, genes with zero read counts in all samples are removed from the analysis, and the zero counts are replaced by 0.5 for the rest of genes, as in [15].
When |A| ≤ 7, we may compute the log fold change value ${L}_{i}^{{A}_{1}\cup {A}_{2}}$ for all the possible partitions of A into A_{1} and A_{2}. However, when |A| >7, we compute it only for 20 random partitions in order to reduce the computational cost. Finally, we pool all these log fold change values together, and denote the resulting collection by ${L}_{i}^{A}$. By applying the same procedure as above, we can obtain a collection LB of log fold change values of read counts for gene i within condition B.
Note that this null distribution is gene-specific, as it takes into account only genes from the neighborhood of gene i. A special case of the above proposed approach is obtained when the neighborhood of a gene includes all the genes in a sample under investigation.
With the log fold change L_{ i } of read counts of gene i between two conditions and a null fold change distribution L_{ i }, we approximate the probability of gene i being not differentially expressed as the fraction of points from L_{ i } that correspond to a larger absolute fold change value than |L_{ i }|. Therefore, we may write this probability as
$P\left({x}_{i}=0|{n}_{ij},\forall j\right)=\frac{\left|\left\{l:\left|l\right|>\left|{L}_{i}\right|,l\in {L}_{i}\right\}\right|}{\left|{L}_{i}\right|}$.
LFCseq is implemented in R and publicly available at http://www1.spms.ntu.edu.sg/~chenxin/LFCseq/.
Relation to NOISeq
Results and discussion
Datasets
We test the performance of LFCseq on two simulated and three real RNA-seq datasets, and compare it with six existing parametric and nonparametric approaches, including NOISeq, SAMseq, edgeR, DESeq, sSeq and EBSeq (see Additional file 1 for their running R codes).
where µ_{ ij } and ${\sigma}_{ij}^{2}$ are the mean and variance, respectively. As in [10], we further let µ_{ ij } = E{N_{ ij } } = q_{ iA } d_{ j } under condition A and µ_{ ij } = E{N_{ ij } } = q_{ iB } d_{ j } under condition B, where q_{ iA } and q_{ iB } represent the true expression values of gene i under condition A and B, respectively, and where dj represents the sequencing library size of sample j. For the variance, we let ${\sigma}_{ij}^{2}={\mu}_{ij}+{\varphi}_{i}.{\mu}_{ij}^{2}$, where φ_{ i } is the dispersion parameter of the negative binomial distribution. As a typical setting, 30% of the genes are simulated to be differentially expressed, among which 70% are set to be up-regulated. The library size factors are generated from the uniform distribution d_{ j } ~ U (0.5, 1.5). We consider three different sample sizes |A| = |B| = 2, 5 and 8 under each condition.
Simulation 2. We generate read counts for 20,000 genes using the same procedure as above in Simulation 1, except that the parameter values of q_{ iA }, q_{ iB } , and φ_{ i } are randomly sampled with replacement from the experimental Bottomly's dataset [23]. Thus we expect this setting is more realistic than the previous one in Simulation 1.
MAQC dataset. MAQC dataset [24] contains two RNA sample types, Stratagene's human universal reference RNA (UHR) and Ambion's human brain reference RNA (brain). Each sample type has seven replicates. In this dataset, 844 genes have been assayed by the quantitative real-time polymerase chain reaction (qRT-PCR). As in [21], a gene is considered as differentially expressed if the log fold change ratio of its cycle threshold values exceeds 2 or as non-differentially expressed if this log fold change ratio is smaller than 0.2. As a result, we obtain 235 DE genes and 53 non-DE genes from the qRT-PCR gold-standard to assess the performance of the proposed approach.
Griffith's dataset. Gene expression is compared between two human colorectal cell lines [25], MIP101 and MIP/5-FU, of the fluorouracil (5-FU)-resistant and -nonresistant phenotype, respectively. qRT-PCR measurements were made for 94 genes. A two-tailed t-test was applied to identify DE and non-DE genes with a cutoff point 0.05, which left 83 DE genes and 11 non-DE genes for performance evaluation.
Sultan's Dataset. Gene expression of two human cell lines, Ramos B and HEK 293T, were compared using RNA-seq [26]. In this dataset, there are two replicates for each cell line. See Additional file 1 for further details of these testing datasets.
Evaluation criteria
We evaluate the performance of LFCseq from the following two aspects. First, we evaluate its ability to discriminate between DE genes and non-DE genes by ranking genes in order of significance for differential expression between conditions. With the gene ranking list, we plot a receiver operating characteristic (ROC) curve and compute the area under the curve (AUC) to measure the overall discriminating ability. Then, LFCseq is compared with six other approaches in terms of AUC without imposing any arbitrary cutoffs. For LFCseq, we rank genes in increasing order of the probability P (x_{ i } = 0 | n_{ ij } , ∀j). For three parametric approaches that assumed the negative binomial distribution (edgeR, DESeq, sSeq), genes are ranked according to their estimated nominal p-values. For SAMseq, we use the false discovery rates (FDR) estimated by a permutation plug-in method and, for NOISeq and EBSeq, we use their estimated probabilities of genes being differentially expressed for ranking. Second, we evaluate the experimental results of LFCseq in terms of precision, sensitivity, and F-score. These evaluation metrics are defined as follows: PRE (precision) = TP/(TP+FP), SEN (sensitivity) =TP/(TP+FN), and FS (F-score) =2 × PRE × SEN/(PRE + SEN), where TP, FP, and FN are the number of true positives, the number of false positives and the number of false negatives, respectively. Note that the metric F-score is the harmonic mean of sensitivity and precision and thus measure the overall differential expression inference performance of a method. In general, the higher the F-score, the better the inference performance. In order to compute precision and sensitivity scores, all the approaches used their respective default settings to call a list of DE genes. Specifically, LFCseq, NOISeq, and EBSeq used a probability cutoff of 0.1, 0.8, and 0.95, respectively. SAMseq used a FDR cutoff of 0.05, while edgeR, DESeq, and sSeq all used a p-value cutoff of 0.05 after adjusted for multiple testing.
Performance on simulated data
Precision, sensitivity and F-score for Simulation 1.
Methods | PRE | SEN | FS | PRE | SEN | FS | PRE | SEN | FS |
---|---|---|---|---|---|---|---|---|---|
|A|=|B|=2 | |A|=|B|=5 | |A|=|B|=8 | |||||||
LFCseq | 0.88 | 0.42 | 0.57 | 0.93 | 0.59 | 0.72 | 0.93 | 0.68 | 0.78 |
NOISeq | 0.91 | 0.29 | 0.44 | 1.00 | 0.29 | 0.45 | 1.00 | 0.29 | 0.45 |
SAMseq | NA | 0.00 | NA | 0.96 | 0.37 | 0.53 | 0.96 | 0.62 | 0.75 |
DESeq | 0.98 | 0.20 | 0.34 | 0.99 | 0.47 | 0.63 | 0.98 | 0.58 | 0.73 |
Edger | 0.96 | 0.32 | 0.48 | 0.94 | 0.55 | 0.70 | 0.93 | 0.63 | 0.75 |
sSeq | NA | 0.01 | NA | 0.97 | 0.52 | 0.68 | 0.94 | 0.63 | 0.76 |
EBSeq | 0.72 | 0.37 | 0.49 | 0.94 | 0.46 | 0.62 | 0.97 | 0.53 | 0.69 |
In addition, we compared LFCseq with a simple hypergeometric test (SHGT) when the numbers of replicates per condition are 5 and 8 in Simulation 1. In the simple hypergeometric test, the null distribution for gene i is built on the randomly permuted samples of gene i between conditions A and B, instead of using the neighborhood N (i). From Figure S8 and Table S2 in Additional file 1, it can be seen that LFCseq performs better than SHGT in the terms of both AUC values and F-scores.
0.1 Performance on real data
On Griffith's dataset, the corresponding curves of precision, sensitivity and Fscores are presented in Figures S4-S6 in Additional file 1. Overall, we observed a similar pattern of performance to that observed on MAQC dataset. One noticeable difference is that although LFCseq still achieved the best overall performance in terms of F-score, there are only marginal improvements over the two parametric approaches edgeR and DESeq. Recall that only 11 truly non-DE genes were identified from Griffith's limited RT-PCR data for the validation of prediction results. Such a small true negative dataset is hardly sufficient to fully characterize the performance behavior of a method.
On Sultan's dataset, no gold-standard is available for performance validation. Instead of computing precision and sensitivity scores, we plotted in Figure S7 in Additional file 1 the fold changes of genes against their mean expression levels on the logarithmic scale. In those scatter plots, each red dot represents a gene being called DE while each black dot represents a gene being called non-DE. As we can see, LFCseq called DE genes at both high and low expression ranges. However, NOISeq called few DE genes at low expression ranges, which might suggest that NOISeq is biased against genes with low read counts and that its sensitivity could still be very low as we observed earlier. We also notice that sSeq called a considerably less number of DE genes than other approaches, which indicates that it is very conservative when calling DE genes.
Conclusions
In this paper we proposed a new nonparametric approach for differential expression analysis of RNA-seq data. It relies on the statistical tests of log fold changes of gene read counts between and within biological conditions. Following the observation that the standard errors of log fold changes vary considerably with gene expression levels, we choose to create a gene-specific null probability distribution for each gene rather than a common null probability distribution for all genes. This is done by considering the gene neighborhood, which is defined as a set of genes at similar expression levels. As a result, the estimated probability of a gene being DE depends only on the read counts of genes from its neighborhood.
Our experimental results demonstrate that the proposed approach LFCseq outperforms its competitors in better ranking the truly DE genes ahead of non-DE genes. It has the best overall performance as it achieved the highest F-scores in almost all our tests (except a few tests on Griffith's dataset). The improvements over other methods are especially remarkable when the number of replicates is small. In such cases, those parametric methods based on negative binomial distribution, such as edgeR, DESeq and sSeq, could not estimate the distributional parameters accurately, while for the nonparametric SAMseq method, its Wilcoxon statistic has a relatively low testing power.
In this study, we applied a pre-specified probability cutoff of 0.1 for our approach LFCseq. This cutoff generally works well, as shown in our experiments on both simulated data and real RNA-seq data. However, it is certainly of interest to develop a data-driven cutoff selection method for a wide applicability of the approach. In addition, it is also interesting to formulate a framework to control the false discovery rate [27] for our approach. We will explore these in future work.
Declarations
Acknowledgements
The authors would like to thank Ei-Wen Yang for his assistance in data analysis and Prof Tao Jiang for his helpful comments. This work was partially supported by the Singapore National Medical Research Council grant (CBRG11nov091) and the Ministry of Education Academic Research Fund (MOE2012-T2-1-055).
Declarations
The publication cost for this article was funded by a grant from the Singapore Ministry of Education Academic Research Fund (MOE2012-T2-1-055).
This article has been published as part of BMC Genomics Volume 15 Supplement 10, 2014: Proceedings of the 25th International Conference on Genome Informatics (GIW/ISCB-Asia): Genomics. The full contents of the supplement are available online at http://www.biomedcentral.com/bmcgenomics/supplements/15/S10.
Authors’ Affiliations
References
- Wang Z, Gerstein M, Snyder M: Rna-seq: a revolutionary tool for transcriptomics. Nature Reviews Genetics. 2009, 10: 57-63. 10.1038/nrg2484.PubMedPubMed CentralView ArticleGoogle Scholar
- Ozsolak F, Milos PM: Rna sequencing: advances, challenges and opportunities. Nature Reviews Genetics. 2011, 12: 87-98. 10.1038/nrg2934.PubMedPubMed CentralView ArticleGoogle Scholar
- Nagalakshmi U, Wang Z, Waern K, Shou C, Raha D, Gerstein M, Snyder M: The transcriptional landscape of the yeast genome defined by rna sequencing. Science. 2008, 320: 1344-1349. 10.1126/science.1158441.PubMedPubMed CentralView ArticleGoogle Scholar
- Trapnell C, Williams BA, Pertea G, Mortazavi A, Kwan G, van Baren MJ, Salzberg SL, Wold BJ, Pachter L: Transcript assembly and quantification by rna-seq reveals unannotated transcripts and isoform switching during cell differentiation. Nature Biotechnology. 2010, 28: 511-515. 10.1038/nbt.1621.PubMedPubMed CentralView ArticleGoogle Scholar
- Berger MF, Levin JZ, Vijayendran K, Sivachenko A, Adiconis X, Maguire J, Johnson LA, Robinson J, Verhaak RG, Sougnez C, Onofrio RC, Ziaugra L, Cibulskis K, Laine E, Barretina J, Winckler W, Fisher DE, Getz G, Meyerson M, Jaffe DB, Lander SBGES, Dummerv R, Gnirke A, Nusbaum C, Garraway LA: Integrative analysis of the melanoma transcriptome. Genome Research. 2010, 20: 413-427. 10.1101/gr.103697.109.PubMedPubMed CentralView ArticleGoogle Scholar
- Biesecker LG, Burke W, Kohane I, Plon SE, Zimmern R: Next-generation sequencing in the clinic: are we ready?. Nature Reviews Genetics. 2012, 13: 818-824. 10.1038/nrg3357.PubMedPubMed CentralView ArticleGoogle Scholar
- Li J, Witten DM, Johnstone IM, Tibshirani R: Normalization, testing, and false discovery rate estimation for rna-sequencing data. Biostatistics. 2011, 13: 523-538.PubMedPubMed CentralView ArticleGoogle Scholar
- Witten DM: Classification and clustering of sequencing data using a poisson model. Annals of Applied Statistics. 2011, 5: 2265-2687. 10.1214/11-AOAS529.View ArticleGoogle Scholar
- Robinson MD, Smyth GK: Moderated statistical tests for assessing differences in tag abundance. Bioinformatics. 2007, 23: 2881-2887. 10.1093/bioinformatics/btm453.PubMedView ArticleGoogle Scholar
- Robinson MD, McCarthy DJ, Smyth GK: edger: a bioconductor package for differential expression analysis of digital gene expression data. Bioinformatics. 2010, 26: 139-140. 10.1093/bioinformatics/btp616.PubMedPubMed CentralView ArticleGoogle Scholar
- Anders S, Huber W: Differential expression analysis for sequence count data. Genome Biology. 2010, 11: 106-10.1186/gb-2010-11-2-106.View ArticleGoogle Scholar
- Yu D, Huber W, Vitek O: Shrinkage estimation of dispersion in negative binomial models for rna-seq experiments with small sample size. Bioinformatics. 2013, 29: 1275-1282. 10.1093/bioinformatics/btt143.PubMedPubMed CentralView ArticleGoogle Scholar
- Leng N, Dawson JA, Thomson JA, Ruotti V, Rissman AI, Smits BMG, Haag JD, Gould MN, Stewart RM, Kendziorski C: Ebseq: an empirical bayes hierarchical model for inference in rna-seq experiments. Bioinformatics. 2013, 29: 1035-1043. 10.1093/bioinformatics/btt087.PubMedPubMed CentralView ArticleGoogle Scholar
- Hardcastle TJ, Kelly KA: bayseq: Empirical bayesian methods for identifying differential expression in sequence count data. BMC Bioinformactics. 2010, 11: 422-10.1186/1471-2105-11-422.View ArticleGoogle Scholar
- Tarazona S, Carcía-Alcalde F, Dopazo J, Ferrer A, Conesa A: Differential expression in rna-seq: A matter of depth. Genome Research. 2011, 21: 2213-2223. 10.1101/gr.124321.111.PubMedPubMed CentralView ArticleGoogle Scholar
- Li J, Tibshirani R: Finding consistent patterns: A nonparametric approach for identifying differential expression in rna-seq data. Statistical Methods in Medical Research. 2013, 22: 519-536. 10.1177/0962280211428386.PubMedPubMed CentralView ArticleGoogle Scholar
- Yang EW, Girke T, Jiang T: Differential gene expression analysis using coexpression and rna-seq data. Bioinformatics. 2013, 29: 2153-2161. 10.1093/bioinformatics/btt363.PubMedView ArticleGoogle Scholar
- Soneson C, Delorenzi M: A comparison of methods for differential expression analysis of rna-seq data. BMC Bioinformatics. 2013, 14: 91-10.1186/1471-2105-14-91.PubMedPubMed CentralView ArticleGoogle Scholar
- Rapaport F, Khanin R, Liang Y, Pirun M, Krek A, Zumbo P, Mason CE, Soccil ND, Betel D: Comprehensive evaluation of differential gene expression analysis methods for rna-seq data. Genome Biology. 2013, 14: 95-10.1186/gb-2013-14-9-r95.View ArticleGoogle Scholar
- Mortazavi A, Williams BA, McCue K, Schaeffer L, Wold B: Mapping and quantifying mammalian transcriptomes by rna-seq. Nature Methods. 2008, 5: 621-628. 10.1038/nmeth.1226.PubMedView ArticleGoogle Scholar
- Bullard JH, Purdom E, Hansen KD, Dudoit S: Evaluation of statistical methods for normalization and differential expression in mrna-seq experiments. BMC Bioinformatics. 2010, 11: 94-10.1186/1471-2105-11-94.PubMedPubMed CentralView ArticleGoogle Scholar
- Robinson MD, Oshlack A: A scaling normalization method for differential expression analysis of rna-seq data. Genome Biology. 2010, 11: 25-View ArticleGoogle Scholar
- Bottomly D, Walter NAR, Huner JE: Evaluating gene expression in c57bl/6j and dba/2j mouse striatum using rna-seq and microarrays. Plos One. 2011, 6: 17820-10.1371/journal.pone.0017820.View ArticleGoogle Scholar
- Shi L, Reid L, Jones W: The microarray quality control (maqc) project shows interand intraplatform reproducibility of gene expression measurements. Nature Biotechnology. 2006, 24: 1151-1161. 10.1038/nbt1239.PubMedView ArticleGoogle Scholar
- Griffith M, Griffith O, Mwenifumbo J: Alternative expression analysis by rna sequencing. Nature Methods. 2010, 7: 843-847. 10.1038/nmeth.1503.PubMedView ArticleGoogle Scholar
- Sultan M, Schulz MH, Richard H, Magen A, Klingenhoff A, Scherf M, Seifert M, Borodina T, Soldatov A, Parkhomchuk D, Schmidt D, O'Keeffe S, Haas S, Lehrach MVH, Yaspo ML: A global view of gene activity and alternative splicing by deep sequencing of the human transcriptome. Science. 2008, 321: 956-960. 10.1126/science.1160342.PubMedView ArticleGoogle Scholar
- Benjamini Y, Hochberg Y: Controlling the false discovery rate: a practical and powerful approach to multiple testing. Journal of the Royal Statistical Society. 1995, Series B 57: 289-300.Google Scholar
Copyright
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The Creative Commons Public Domain Dedication waiver (http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated.