Effects of subsampling on characteristics of RNA-seq data from triple-negative breast cancer patients
- Alexey Stupnikov^{1},
- Galina V Glazko^{2} and
- Frank Emmert-Streib^{1, 3}Email author
Received: 16 January 2015
Accepted: 3 July 2015
Published: 8 August 2015
Abstract
Background
Data from RNA-seq experiments provide a wealth of information about the transcriptome of an organism. However, the analysis of such data is very demanding. In this study, we aimed to establish robust analysis procedures that can be used in clinical practice.
Methods
We studied RNA-seq data from triple-negative breast cancer patients. Specifically, we investigated the subsampling of RNA-seq data.
Results
The main results of our investigations are as follows: (1) the subsampling of RNA-seq data gave biologically realistic simulations of sequencing experiments with smaller sequencing depth but not direct scaling of count matrices; (2) the saturation of results required an average sequencing depth larger than 32 million reads and an individual sequencing depth larger than 46 million reads; and (3) for an abrogated feature selection, higher moments of the distribution of all expressed genes had a higher sensitivity for signal detection than the corresponding mean values.
Conclusions
Our results reveal important characteristics of RNA-seq data that must be understood before one can apply such an approach to translational medicine.
Keywords
RNA-seq data Computational genomics Statistical robustness High-dimensional biology Triple-negative breast cancerBackground
In recent years, next-generation sequencing technology for generating RNA-seq data has gained considerable interest [1–4] in the biological [5, 6] and biomedical literature [7, 8]. Such data are frequently used, e.g., for identifying alternative splicing, finding differentially expressed genes, or detecting differentially expressed pathways [9–14]. The conventional analysis pipeline for RNA-seq data first maps the reads to genes for a given annotation, resulting in a high-dimensional count vector for each sample. Thereafter, these integer count vectors are normalized and further processed with statistical inference methods. Altering parameters of the preprocessing steps, e.g., aligning procedure, summarization of reads, choice of annotation, and normalization techniques, can change the output of a gene expression analysis drastically. This effect has been studied for different normalization procedures [15].
So far, a major focus has been placed on methods for identifying differentially expressed genes from RNA-seq data [16–18] because such analysis methods that are simpler than, e.g., network-based approaches yet provide meaningful insights into the basic biological functioning of different physiological conditions. Some of these methods assume that the count distribution of individual genes follows a Poisson distribution, whereas others assume a negative binomial distribution for their model. Interestingly, it has been argued that the negative binomial distribution does not perform well under specific conditions [18].
In this study, we carried out an analysis of RNA-seq count distributions for two biological conditions: triple-negative breast cancer (TNBC) samples and TNBC-free samples. The TNBC-free samples corresponded to the same cell types as TNBC samples but were from normal tissue; they formed a control group. For each biological sample, we repeatedly performed a subsampling of mapped reads and thus simulated new samples with a different sequencing depth. For these surrogate gene expression data sets, we studied and compared a variety of properties of their RNA-seq count distributions. We describe the biological data we used for our analysis and the preprocessing steps we applied, and we introduce a procedure, Depth of Sequencing Iterative Reduction Estimator (DESIRE), for subsampling RNA-seq data.
Methods
Dataset
The whole data set consists of 6 groups, including a total of 168 samples [19]. We randomly selected four samples of TNBC tumors from the primary tumor group and four samples of healthy breast tissues from TNBC-free group. This selection allowed us to estimate the main statistical entities under investigation. Other samples were not considered in our analysis.
Data preprocessing
To use RNA-seq data for a gene expression analysis, certain preprocessing steps must be performed. These include alignment of reads, count matrix computation, and normalization.
Depth of sequencing iterative reduction estimator (DESIRE)
It is commonly accepted that the depth of the sequencing can affect the results of an analysis [25–28]. However, these papers considered only results of a bioinformatics analysis and did not study the details of the count distributions. Another example is the study that addressed the question of the optimal sequencing depth [29].
To study the influence of the sequencing depth on a gene expression analysis, we developed a resampling procedure based on the subsampling of the data. By subsampling, we used only a fraction, f, of the total amount of available data in a systematic manner [30]. Another name for such a procedure used in the literature is m out of n bootstrap, whereas m < n and the bootstrap samples are drawn without replacement [31]. Our procedure, DESIRE, has the following underlying ideas.
For each biological sample, we drew a number of replicates of a smaller sequencing depth. To accomplish this, a particular portion, f, of reads, ranging from 10 to 90%, was randomly drawn from a biological sample without replacement. This process was repeated R times resulting in R simulated replicates for one simulated sequencing depth f. For our analysis, we used R = 24 resulting in a total of 240 subsampled data sets for a single biological sample for the 10 different sequencing depths, f = {0.1,…, 0.9, 1.0}.
The specific value of R is not crucial. However, if it is large, the computational complexity would increase without resulting in significant improvements in the statistical estimates of our analysis. On the other hand, values of R much lower than 24 potentially result in unstable results. The particular number of R = 24 considered the number of nodes in our computer cluster available for our analysis.
We calculated the count vectors using Entrez annotation from Bioconductor, database org.Hs.e.g.db2.9.0, which consisted of 23,648 (protein-coding and -noncoding) genes [32].
Results
The purpose of our study was to learn about the influence of the sequencing depth on inferred biological results. For this reason, we investigated 4 layers of complexity. First, we compared differences between an explicit subsampling of reads and a direct scaling of count matrices. The results from this analysis demonstrated that a subsampling via DESIRE was necessary to obtain realistic surrogates of sequencing experiments with a smaller sequencing depth. Second, we studied the absolute expression of genes and their growth. Third, we investigated the growth rate of the number of expressed genes. Fourth, we analyzed differences in the distributional shape of expressed genes between TNBC patients and TNBC-free patients. For each of these analysis steps, we used data generated by the DESIRE procedure.
Differences between subsampling of reads and direct scaling of count matrices
Our first analysis investigated differences between a subsampling of reads via the DESIRE procedure and a direct scaling of count matrices. The results of this analysis justified our approach for the following sections.
For all threshold values and all sequencing depths that we investigated, there were distinct differences between the two approaches (Fig. 4). Similar results were also observed in other patient samples (not shown). From these results, we concluded that the computationally efficient shortcut via a direct scaling of count matrices did not lead to the same results as the DESIRE procedure. Hence, the scaled count matrices did not correspond to sequencing experiments with a smaller sequencing depth but had an unclear biological interpretation. For this reason, the DESIRE procedure needs to be used for simulating realistic sequencing experiments because only in this way do the resulting data have a clear interpretation in biological terms. In the following sections, we used the DESIRE procedure for this purpose.
We would like to note that neither our statistic, the number of expressed genes, nor the specific threshold ⊝ was crucial for our conclusion, but other statistics led to similar results. For our following analysis, it was important only that there was a difference but not how each individual measure was affected. However, we thought that for particular measures that were used, e.g., as test statistic for hypothesis tests or distance metrics for clustering, it might be interesting to quantify these differences more specifically.
Absolute expression of genes
The first impression of the overall behavior was intuitive because the larger was the sequencing depth, the higher was the probability to obtain at least ⊝ reads for a gene, if it was expressed. Less intuitive was the fact that for all samples and all thresholds, there was no saturation in the number of expressed genes, but this number continued to grow, which suggests that even the maximally available sequencing depth was not sufficient to achieve a saturation of the measurements. In addition, this pointed to possible errors in either the sequencing or the alignment of reads because it was biologically implausible to assume that almost all 23,648 genes considered by our analysis were actually expressed for ⊝ = 1 (Fig. 5). This may open the possibility to quantify such errors statistically.
From the obtained results in Fig. 5 and the results from 6 further samples that looked qualitatively similar (not shown), we attempted to estimate the optimal sequencing depth in the following two ways using the available sequencing depth of the samples used for our analysis (TNBC samples: 34974017, 46677107, 17574408, and 24440340; TNBC-free samples: 25900791, 43454785, 31426867, and 33517581). Estimator (I)—average sequencing depth: the first estimator centers on average properties of our samples. Given that the average number of short reads per sample was 32,245,737 ± 9,710,593 (averaged over the 8 samples) and the fact that none of the growth curves saturated, we estimated that the average number of reads necessary for a saturation must be larger than 32,245,737. Estimator (II)—individual sequencing depth: the second estimator centers on the individual samples. The largest sequencing depth of our samples was 46,677,107, and even this sample did not lead to a saturating growth. Hence, a conservative estimate requires an individual sequencing depth larger than 46,677,107.
The variability of all results, e.g., the interquartile range (IQR) of the box plots, was in general quite small. However, for larger ⊝ values, the IQR was even further decreased, which showed that the estimation for the number of expressed genes was even more stable for larger expression threshold values, corresponding to a more stringent filtering for expressed genes.
Results of two-sample t tests comparing the total number of expressed genes for various sequencing depths
Depth | P value, two-sided | P value, left-sided | P value, right-sided |
---|---|---|---|
0.1 | 0.096,85 | 0.048,45 | 0.951,57 |
0.2 | 0.125,02 | 0.062,51 | 0.937,49 |
0.3 | 0.124,18 | 0.062,10 | 0.937,91 |
0.4 | 0.123,61 | 0.061,81 | 0.938,19 |
0.5 | 0.118,90 | 0.059,45 | 0.940,55 |
0.6 | 0.128,56 | 0.064,28 | 0.935,72 |
0.7 | 0.145,83 | 0.072,92 | 0.927,08 |
0.8 | 0.161,76 | 0.080,88 | 0.919,12 |
0.9 | 0.155,24 | 0.077,62 | 0.922,38 |
This relation indicated that, on average, there were fewer genes expressed in TNBC patients than in the corresponding control samples, independent of the sequencing depth.
Growth rate of the number of expressed genes
Fitted growth factor values and standard deviations for the Gompertz functions
Depth | Sample | Condition | Growth rate (SD) |
---|---|---|---|
1 | SRR1313137 | TNBC | 18,645.168 (71.428) |
1 | SRR1313135 | TNBC | 19,218.431 (51.492) |
1 | SRR1313134 | TNBC | 18,885.949 (71.780) |
1 | SRR1313133 | TNBC | 18,993.399 (58.036) |
1 | SRR1313211 | TNBC-free | 18,615.636 (77.401) |
1 | SRR1313214 | TNBC-free | 18,726.438 (82.876) |
1 | SRR1313219 | TNBC-free | 18,286.856 (82.281) |
1 | SRR1313220 | TNBC-free | 18,930.056 (85.636) |
10 | SRR1313137 | TNBC | 14,904.344 (144.082) |
10 | SRR1313135 | TNBC | 16,096.457 (121.821) |
10 | SRR1313134 | TNBC | 15,053.704 (139.567) |
10 | SRR1313133 | TNBC | 15,592.711 (155.879) |
10 | SRR1313211 | TNBC-free | 14,756.491 (152.976) |
10 | SRR1313214 | TNBC-free | 14,740.701 (158.437) |
10 | SRR1313219 | TNBC-free | 13,971.554 (166.406) |
10 | SRR1313220 | TNBC-free | 15,280.019 (143.239) |
50 | SRR1313137 | TNBC | 11,532.85 (163.987) |
50 | SRR1313135 | TNBC | 12,782.861 (162.683) |
50 | SRR1313134 | TNBC | 11,211.735 (166.459) |
50 | SRR1313133 | TNBC | 12,170.85 (205.805) |
50 | SRR1313211 | TNBC-free | 11,336.443 (195.497) |
50 | SRR1313214 | TNBC-free | 11,514.378 (174.158) |
50 | SRR1313219 | TNBC-free | 10,577.339 (166.987) |
50 | SRR1313220 | TNBC-free | 11,654.075 (193.834) |
100 | SRR1313137 | TNBC | 9,983.466 (176.904) |
100 | SRR1313135 | TNBC | 11,235.174 (168.548) |
100 | SRR1313134 | TNBC | 9,634.957 (198.902) |
100 | SRR1313133 | TNBC | 10,398.413 (194.712) |
100 | SRR1313211 | TNBC-free | 9,654.874 (175.648) |
100 | SRR1313214 | TNBC-free | 9,898.436 (162.635) |
100 | SRR1313219 | TNBC-free | 9,089.057 (145.683) |
100 | SRR1313220 | TNBC-free | 9,947.798 (162.972) |
Results from comparing the growth rates of the fitted Gompertz functions for TNBC and TNBC-free patients
Depth | P value, two-sided | P value, left-sided | P value, right-sided |
---|---|---|---|
1 | 0.151,2 | 0.924,4 | 0.075,6 |
10 | 0.107,2 | 0.946,4 | 0.053,6 |
50 | 0.179,5 | 0.910,2 | 0.089,8 |
100 | 0.157,4 | 0.921,3 | 0.078,7 |
A normalization of the data does not remove the growth property observed in Fig. 5, but normalized data exhibit qualitatively the same behavior. For ⊝ = 1, this was obvious because the normalization led to a scaling of the data without changing the zero values. For ⊝ > 1, it was less intuitive but followed from our numerical analysis (results not shown).
Distributional shape of expressed genes
Last, we studied the distributional shape of expressed gene values (and not of their numbers) by estimating individually for each parameter configuration its mean value, variance, skewness, and kurtosis. Here, we mean the distribution over all genes within a sample, and not the count distribution of individual genes across samples. Because every distribution with existing moments was fully characterized by all of its moments, either via its moment-generating function or via its probability generating function [38, 39], our analysis was an approximation of the distributional shape because we limited our focus to 4 dimensions.
Moments for TNBC-free patients
Depth | Mean (SD) | Variance (SD) | Skewness (SD) | Kurtosis (SD) |
---|---|---|---|---|
0.1 | 42.29 (0) | 29,616.28 (2,817.52) | 18.51 (2.75) | 599.44 (222.29) |
0.2 | 42.29 (0) | 29,588.5 (2,790.57) | 18.48 (2.71) | 597.09 (219.38) |
0.3 | 42.29 (0) | 29,600.99 (2,803.23) | 18.50 (2.71) | 598.63 (219.27) |
0.4 | 42.29 (0) | 29,602.40 (2,819.85) | 18.51 (2.72) | 598.92 (220.32) |
0.5 | 42.29 (0) | 29,592.22 (2,807.72) | 18.49 (2.71) | 597.70 (218.81) |
0.6 | 42.29 (0) | 29,597.76 (2,805.76) | 18.50 (2.71) | 598.30 (219.30) |
0.7 | 42.29 (0) | 29,599.37 (2,807.41) | 18.51 (2.72) | 599.09 (220.33) |
0.8 | 42.29 (0) | 29,595.86 (2,805.56) | 18.51 (2.72) | 598.86 (220.13) |
0.9 | 42.29 (0) | 29,593.61 (2,806.58) | 18.50 (2.71) | 598.45 (219.59) |
Moments for TNBC patients
Depth | Mean (SD) | Variance (SD) | Skewness (SD) | Kurtosis (SD) |
---|---|---|---|---|
0.1 | 42.29 (0) | 23,176.53 (5446.28) | 17.99 (7.33) | 759.6 (618.96) |
0.2 | 42.29 (0) | 23,172.85 (5454.44) | 18.01 (7.33) | 760.33 (618.23) |
0.3 | 42.29 (0) | 23,155.44 (5437.48) | 17.99 (7.34) | 758.8 (620.44) |
0.4 | 42.29 (0) | 23,169.84 (5448.77) | 18.02 (7.35) | 761.69 (621.23) |
0.5 | 42.29 (0) | 23,165.79 (5448.29) | 18.01 (7.34) | 760.62 (620.49) |
0.6 | 42.29 (0) | 23,166.48 (5449.83) | 18.01 (7.35) | 760.84 (621.57) |
0.7 | 42.29 (0) | 23,168.39 (5451.98) | 18.02 (7.35) | 761.56 (621.74) |
0.8 | 42.29 (0) | 23,165.7 (5448.48) | 18.02 (7.35) | 761.45 (621.81) |
0.9 | 42.29 (0) | 23,164.67 (5448.95) | 18.01 (7.35) | 761.16 (621.53) |
Results from pooled (across different sequencing depths) two-sample t tests for the 4 moments of the gene expression distributions
Moment | P value, two-sided | P value, left-sided | P value, right-sided |
---|---|---|---|
Mean | 1.0 | 1.0 | 1.0 |
Variance | 2.238800e−11 | 1.000000e + 00 | 1.119400e−11 |
Kurtosis | 1.966589e−04 | 9.832945e−05 | 9.999017e−01 |
Skewness | 6.808568e−03 | 3.404284e−0.3 | 9.965957e−01 |
Discussion
- 1.
The subsampling of RNA-seq data gave biologically realistic simulations of next-generation sequencing experiments with smaller sequencing depth, but a direct scaling of count matrices did not. This is an important finding because, first of all, it demonstrated that the conceptually simpler and computationally more efficient approach of a direct scaling of count matrices led to data sets with an unclear biological interpretation. This is of course a major problem because whatever results were obtained from such data sets, e.g., using them for identifying differentially expressed genes, the meaning is at best unclear and possibly even uninterruptable in the sense that replicated next-generation sequencing experiments would not result in data with such a characteristic.
- 2.
To obtain saturating results, we estimated an average sequencing depth of >32 million reads and an individual sequencing depth of >46 million reads. The literature gives context-specific suggestions. For instance, for detecting rare transcripts in human, >200 million paired-end reads should be used, and for the accurate quantification of genes across the entire expression range, >80 million reads per sample should be used [29, 40]. However, for the identification of differentially expressed genes, 36 million reads per sample may be sufficient [29].
For future studies, it would be interesting to derive improved bounds for optimal sequencing depths with respect to two complementary aspects. The first aspect involves distinguishing different application domains because the optimal sequencing depth is likely to depend on the bioinformatics analysis. For gene expression data from DNA microarray experiments, such differences have already been known for, e.g., methods identifying differentially expressed genes and methods for identifying differentially expressed gene sets [41–43]. Second, in this study, we considered only simple statistical estimators for the optimal sequencing depth; however, more elaborate approaches are possible, e.g., by exploiting the results from the fitted growth curves.
- 3.
For an abrogated feature selection, i.e., using all expressed genes that have read counts of ⊝ = 1 or larger, the higher moments of the distribution of expressed genes showed a much better sensitivity for the signal detection of differing phenotypic conditions than the corresponding mean values (Table 6). This could be further explored by designing statistical tests that use such higher moments as a test statistic. A potential advantage of such tests over, e.g., the conventional mean-based tests such as a t test or ANOVA could be a reduced need in sample size, as suggested by our results. However, this requires a further detailed analysis.
Conclusions
The subsampling of RNA-seq data allows us to explore important aspects of gene expression data. These must be understood before such high-throughput data types can be used for applications in translational medicine.
Declarations
Author’s contributions
AS performed the analysis, interpreted the results, and wrote the article. GVG interpreted the results and wrote the article. FES conceived the study, performed the analysis, interpreted the results, and wrote the article. All authors read and approved the final manuscript.
Acknowledgements
We would like to thank Manuel Salto-Tellez, Ricardo de Matos Simoes, and Shaliesh Tripathi for fruitful discussions. GVG was supported in part by the Arkansas Biosciences Institute under Grant (No. UL1TR000039) and the IDeANetworks of Biomedical Research Excellence (INBRE) Grant (No. P20RR16460).
Compliance with ethical guidelines
Competing interests The authors declare that they have no competing interests.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. 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.
Authors’ Affiliations
References
- McGettigan PA. Transcriptomics in the RNA-seq era. Curr Opin Chem Biol. 2013;17(1):4–11.View ArticlePubMedGoogle Scholar
- Marguerat S, Bähler J. RNA-seq: from technology to biology. Cell Mol Life Sci. 2010;67(4):569–79.PubMed CentralView ArticlePubMedGoogle Scholar
- Metzker ML. Sequencing technologies–the next generation. Nat Rev Genet. 2009;11(1):31–46.View ArticlePubMedGoogle Scholar
- Wang Z, Gerstein M, Snyder M. RNA-Seq: a revolutionary tool for transcriptomics. Nat Rev Genet. 2009;10(1):57–63.PubMed CentralView ArticlePubMedGoogle Scholar
- Mortazavi A, Williams BA, McCue K, Schaeffer L, Wold B. Mapping and quantifying mammalian transcriptomes by RNA-seq. Nat Methods. 2008;5(7):621–8.View ArticlePubMedGoogle Scholar
- Peng Z, Cheng Y, Tan BC, Kang L, Tian Z, Zhu Y, et al. Comprehensive analysis of RNA-Seq data reveals extensive RNA editing in a human transcriptome. Nat Biotechnol. 2012;30(3):253–60.View ArticlePubMedGoogle Scholar
- Beane J, Vick J, Schembri F, Anderlind C, Gower A, Campbell J, et al. Characterizing the impact of smoking and lung cancer on the airway transcriptome using RNA-Seq. Cancer Prev Res. 2011;4(6):803–17.View ArticleGoogle Scholar
- Sinicropi D, Qu K, Collin F, Crager M, Liu ML, Pelham RJ, et al. Whole transcriptome RNA-Seq analysis of breast cancer recurrence risk using formalin-fixed paraffin-embedded tumor tissue. PLoS One. 2012;7(7):e40092.PubMed CentralView ArticlePubMedGoogle Scholar
- Anders S, Huber W. Differential expression analysis for sequence count data. Genome Biol. 2010;11(10):R106.PubMed CentralView ArticlePubMedGoogle Scholar
- Rahmatallah Y, Emmert-Streib F, Glazko G. Comparative evaluation of gene set analysis approaches for RNA-Seq data. BMC Bioinform. 2014;15:397.View ArticleGoogle Scholar
- Nicolae M, Mangul S, Mandoiu II, Zelikovsky A. Estimation of alternative splicing isoform frequencies from RNA-Seq data. Algorithm Mol Biol. 2011;6(1):9.View ArticleGoogle Scholar
- Robinson MD, Oshlack A. A scaling normalization method for differential expression analysis of RNA-seq data. s. 2010;11(3):R25.Google Scholar
- Trapnell C, Pachter L, Salzberg SL. TopHat: discovering splice junctions with RNA-Seq. Bioinformatics. 2009;25(9):1105–11.PubMed CentralView ArticlePubMedGoogle Scholar
- Wang L, Feng Z, Wang X, Wang X, Zhang X. DEGseq: an R package for identifying differentially expressed genes from RNA-seq data. Bioinformatics. 2010;26(1):136–8.View ArticlePubMedGoogle Scholar
- Dillies MA, Rau A, Aubert J, Hennequet-Antier C, Jeanmougin M, Servant N, et al. A comprehensive evaluation of normalization methods for illumina high-throughput RNA sequencing data analysis. Brief Bioinform. 2013;14(6):671–83.View ArticlePubMedGoogle Scholar
- Wu H, Wang C, Wu Z. A new shrinkage estimator for dispersion improves differential expression detection in RNA-seq data. Biostatistics. 2013;14(2):232–43.PubMed CentralView ArticlePubMedGoogle Scholar
- Robinson MD, McCarthy DJ, Smyth GK. edgeR: a Bioconductor package for differential expression analysis of digital gene expression data. Bioinformatics. 2010;26(1):139–40.PubMed CentralView ArticlePubMedGoogle Scholar
- Law C, Chen Y, Shi W, Smyth G. Voom: precision weights unlock linear model analysis tools for RNA-seq read counts. Genome Biol. 2014;15(2):R29.PubMed CentralView ArticlePubMedGoogle Scholar
- Varley KE, Gertz J, Roberts BS, Davis NS, Bowling KM, Kirby MK, et al. Recurrent read-through fusion transcripts in breast cancer. Breast Cancer Res Treat. 2014;146(2):287–97.PubMed CentralView ArticlePubMedGoogle Scholar
- Leinonen R, Sugawara H, Shumway M. International Nucleotide Sequence Database Collaboration. The sequence read archive. Nucleic Acids Res. 2010;39:D19–21.PubMed CentralView ArticlePubMedGoogle Scholar
- Langmead B, Salzberg SL. Fast gapped-read alignment with Bowtie 2. Nat Methods. 2012;9(4):357–9.PubMed CentralView ArticlePubMedGoogle Scholar
- Karolchik D, Barber GP, Casper J, Clawson H, Cline MS, Diekhans M, et al. The UCSC genome browser database: 2014 update. Nucleic Acids Res. 2014;42:D764–70.PubMed CentralView ArticlePubMedGoogle Scholar
- Liao Y, Smyth GK, Shi W. featureCounts: an efficient general purpose program for assigning sequence reads to genomic features. Bioinformatics. 2013;30(7):923–30.View ArticlePubMedGoogle Scholar
- Mortazavi A, Williams BA, McCue K, Schaeffer L, Wold B. Mapping and quantifying mammalian transcriptomes by RNA-Seq. Nat Methods. 2008;5(7):621–8.View ArticlePubMedGoogle Scholar
- Fumagalli M. Assessing the effect of sequencing depth and sample size in population genetics inferences. PLoS One. 2013;8(11):e79667.PubMed CentralView ArticlePubMedGoogle Scholar
- Rapaport F, Khanin R, Liang Y, Pirun M, Krek A, Zumbo P, et al. Comprehensive evaluation of differential gene expression analysis methods for RNA-seq data. Genome Biol. 2013;14(9):R95.PubMed CentralView ArticlePubMedGoogle Scholar
- Robinson DG, Storey JD. subSeq: determining appropriate sequencing depth through efficient read subsampling. Bioinformatics. 2014;30(23):3424–6.PubMed CentralView ArticlePubMedGoogle Scholar
- Liu Y, Zhou J, White KP. RNA-seq differential expression studies: more sequence or more replication? Bioinformatics. 2014;30(3):301–4.PubMed CentralView ArticlePubMedGoogle Scholar
- Sims D, Sudbery I, Ilott NE, Heger A, Ponting CP. Sequencing depth and coverage: key considerations in genomic analyses. Nat Rev Genet. 2014;15(2):121–32.View ArticlePubMedGoogle Scholar
- Politis DN, Romano JP, Wolf M. Subsampling Springer series in statistics. Berlin: Springer; 1999.Google Scholar
- Bickel PJ, Gotze F, van Zwet W. Resampling fewer than n observations: gains, losses and remedies for losses. Statist Sinica. 1997;7(1):1–31.Google Scholar
- Gentleman RC, Carey VJ, Bates DM, Bolstad B, Dettling M, Dudoit S, et al. Bioconductor: open software development for computational biology and bioinformatics. Genome Biol. 2004;5(10):80.View ArticleGoogle Scholar
- Hanahan D, Weinberg RA. The hallmarks of cancer. Cell. 2000;100(1):57–70.View ArticlePubMedGoogle Scholar
- Ashburner M, Ball CA, Blake JA, Botstein D, Butler H, Cherry JM, et al. Gene ontology: tool for the unification of biology. Gene ontology consortium. Nat Genet. 2000;25(1):25–9.PubMed CentralView ArticlePubMedGoogle Scholar
- Laird AK. Dynamics of tumour growth. Br J Cancer. 1964;18(3):490.PubMed CentralView ArticleGoogle Scholar
- Emmert-Streib F. Structural properties and complexity of a new network class: Collatz step graphs. PLoS One. 2013;8(2):e56461.PubMed CentralView ArticlePubMedGoogle Scholar
- Harrell FE. Regression modeling strategies. New York: Springer; 2001.View ArticleGoogle Scholar
- Casella G, Berger RL. Statistical inference. Belmont: Duxbury Press; 2002.Google Scholar
- Feller W. An introduction to probability theory and its applications. New York: Wiley; 1968.Google Scholar
- Tarazona S, Garcia-Alcalde F, Dopazo J, Ferrer A, Conesa A. Differential expression in RNA-seq: a matter of depth. Genome Res. 2011;21(12):2213–23.PubMed CentralView ArticlePubMedGoogle Scholar
- Emmert-Streib F, Tripathi S, de Matos Simoes R. Harnessing the complexity of gene expression data from cancer: from single gene to structural pathway methods. Biol Direct. 2012;7:44.PubMed CentralView ArticlePubMedGoogle Scholar
- Hung JH, Yang TH, Hu Z, Weng Z, DeLisi C. Gene set enrichment analysis: performance evaluation and usage guidelines. Brief Bioinform. 2012;13(3):281–91.PubMed CentralView ArticlePubMedGoogle Scholar
- Steinhoff C, Vingron M. Normalization and quantification of differential expression in gene expression microarrays. Brief Bioinform. 2006;7(2):166–77.View ArticlePubMedGoogle Scholar