Sample size calculation based on exact test for assessing differential expression analysis in RNA-seq data
- Chung-I Li^{1, 3},
- Pei-Fang Su^{2, 3} and
- Yu Shyr^{3}Email author
https://doi.org/10.1186/1471-2105-14-357
© Li et al.; licensee BioMed Central Ltd. 2013
Received: 3 June 2013
Accepted: 28 November 2013
Published: 6 December 2013
Abstract
Background
Sample size calculation is an important issue in the experimental design of biomedical research. For RNA-seq experiments, the sample size calculation method based on the Poisson model has been proposed; however, when there are biological replicates, RNA-seq data could exhibit variation significantly greater than the mean (i.e. over-dispersion). The Poisson model cannot appropriately model the over-dispersion, and in such cases, the negative binomial model has been used as a natural extension of the Poisson model. Because the field currently lacks a sample size calculation method based on the negative binomial model for assessing differential expression analysis of RNA-seq data, we propose a method to calculate the sample size.
Results
We propose a sample size calculation method based on the exact test for assessing differential expression analysis of RNA-seq data.
Conclusions
The proposed sample size calculation method is straightforward and not computationally intensive. Simulation studies to evaluate the performance of the proposed sample size method are presented; the results indicate our method works well, with achievement of desired power.
Keywords
Background
Next generation sequencing (NGS) technology has revolutionized genetic analysis; RNA-seq is a powerful NGS method that enables researchers to discover, profile, and quantify RNA transcripts across the entire transcriptome. In addition, unlike the microarray chip, which offers only quantification of gene expression level, RNA-seq provides expression level data as well as differentially spliced variants, gene fusion, and mutation profile data. Such advantages have gradually elevated RNA-seq as the technology of choice among researchers. Nevertheless, the advantages of RNA-seq are not without computational cost; as compared to microarray analysis, RNA-seq data analysis is much more complicated and difficult. In the past several years, the published literature has addressed the application of RNA-seq to multiple research questions, including abundance estimation [1-3], detection of alternative splicing [4-6], detection of novel transcripts [6, 7], and the biology associated with gene expression profile differences between samples [8-10]. With this rapid growth of RNA-seq applications, discussion of experimental design issues has lagged behind, though more recent literature has begun to address some of the relevant principles (e.g., randomization, replication, and blocking) to guide decisions in the RNA-seq framework [11, 12].
One of the principal questions in designing an RNA-seq experiment is: What is the optimal number of biological replicates to achieve desired statistical power? (Note: In this article, the term “sample size” is used to refer to the number of biological replicates or number of subjects.) Because RNA-seq data are counts, the Poisson distribution has been widely used to model the number of reads obtained for each gene to identify differential gene expression [8, 13]. Further, [12] used a Poisson distribution to model RNA-seq data and derive a sample size calculation formula based on the Wald test for single-gene differential expression analysis. It is worth noting that a critical assumption of the Poisson model is that the mean and variance are equal. This assumption may not hold, however, as read counts could exhibit variation significantly greater than the mean [14]. That is, the data are over-dispersed relative to the Poisson model. In such cases, one natural alternative to Poisson is the negative binomial model. Based on the negative binomial model, [14, 15] proposed a quantile-adjusted conditional maximum likelihood procedure to create a pseudocount which lead to the development of an exact test for assessing the differential expression analysis of RNA-seq data. Furthermore, [16] provided a Bioconductor package, edgeR, based on the exact test.
Sample size determination based on the exact test has not yet been studied, however. Therefore, the first goal of this paper is to propose a sample size calculation method based on the exact test.
In reality, thousands of genes are examined in an RNA-seq experiment; differential expression among those genes is tested simultaneously, requiring the correction of error rates for multiple comparisons. For the high-dimensional multiple testing problem, several such corrected measures have been proposed, such as family-wise error rate (FWER) and false discovery rate (FDR). In high-dimensional multiple testing circumstances, controlling FDR is preferable [17] because the Bonferroni correction for FWER is often too conservative [18]. Many methods have been proposed to control FDR in the analysis of high-dimensional data [17, 19, 20]. Those concepts have been extended to calculate sample size for microarray studies [21-25]. To our knowledge, however, the literature does not address determination of sample size while controlling FDR in RNA-seq data. Therefore, the second purpose of this paper is to propose a procedure to calculate sample size while controlling FDR for differential expression analysis of RNA-seq data.
In sum, in this article, we address the following two questions: (i) For a single-gene comparison, what is the minimum number of biological replicates needed to achieve a specified power for identifying differential gene expression between two groups? (ii) For multiple gene comparisons, what is the suitable sample size while controlling FDR? The article is organized as follows. In the Method section, a sample size calculation method is proposed for a single-gene comparison. We then extend the method to address the multiple comparison test issue. Performance comparisons via numerical studies are described in the Results section. Two real RNA-seq data sets are used to illustrate sample size calculation. Finally, discussion follows in the Conclusions section.
Method
Exact test
Because the pseudocounts in each group have an approximately identical negative binomial distribution [14, 15], the sum of pseudocounts of each group, ${Y}_{i}=\sum _{j=1}^{{n}_{i}}{Y}_{\mathit{\text{ij}}}$, has a negative binomial distribution NB(${n}_{i}{d}_{i}^{\ast}{\gamma}_{i},\varphi /{n}_{i}$) where ${d}_{i}^{\ast}$ is the geometric mean of normalization factors in group i. Under the null hypothesis (1), the sum of the total pseudocount, Y_{1}+Y_{0}, follows a negative binomial distribution. In analogy with Fisher’s exact test, [14, 15] proposed an exact test for replacing the hypergeometric probabilities with negative binomial probabilities. Because [16] developed a Bioconductor software package edgeR which is an implementation of methodology developed by [14, 15], the p-value can be easily calculated for conducting the exact test.
In the following simulation and application sections, we used edgeR version 3.0.6 for estimating model parameters and performing the exact test.
Sample size calculation for controlling type I error rate
Thus, the required sample size n to attain the given power 1-β at level of significance α can then be calculated by solving (2) through a numerical approach, such as a gradient-search or bisection procedure.
Sample size calculation for controlling false discovery rate
In reality, thousands of genes are examined in an RNA-seq experiment, and those genes are tested simultaneously for significance of differential expression. In such cases, the sample size calculation for a single-gene comparison discussed above cannot be applied directly. Jung, 2005 [23] incorporated FDR controlling based on a two-sample t-test under the Gaussian distribution assumption. In this section, we borrowed their concept to incorporate FDR controlling based on the test statistics described in the test statistics section.
where R_{0} is the number of false discoveries and R is the number of results declared significant (i.e., rejections of the null hypothesis).
Then, by solving g_{1}(n) = 0 via a numerical approach, the required sample size for controlling FDR at level f can be obtained.
To calculate the sample size, we have to estimate all of the fold changes ρ_{ g }, dispersions ϕ_{ g }, and average read counts μ_{0g} of gene g for the set of prognostic genes g∈M_{1} prior to the RNA-seq experiment. However, we may not have enough information to estimate all of those parameters in practice. To address this issue, we propose the following method to obtain a conservative estimate of the required sample size. Because the power increases as | log2(ρ_{ g })| or μ_{0g} increases and ϕ_{ g } decreases, we suggest using a common ${\rho}^{\ast}=arg\underset{g\in {M}_{1}}{min}\left\{\right|\underset{2}{log}\left({\rho}_{g}\right)\left|\right\}$ minimum fold change, ${\mu}_{0}^{\ast}=\underset{g\in {M}_{1}}{min}\left\{{\mu}_{0g}\right\}$ minimum average read count, and ${\varphi}^{\ast}=\underset{g\in {M}_{1}}{max}\left\{{\varphi}_{g}\right\}$ maximum dispersion to estimate each ρ_{ g }, μ_{0g}, and ϕ_{ g }, respectively. In such cases, it gives a more conservative estimate of the required sample size.
When we use ρ^{∗}, ${\mu}_{0}^{\ast}$, and ϕ^{∗} to estimate each ρ_{ g }, μ_{0g}, and ϕ_{ g }, g∈M_{1}, in the multiple testing context, α^{∗} and β^{∗} can be calculated as r_{1}f/(m_{0}(1-f)) and 1-r_{1}/m_{1}, respectively, where m_{1} is the number of prognostic genes. In other words, the power function (2) can be applied in the case of multiple gene comparison, with the replacement of α and β with α^{∗} and β^{∗}.
- 1.
Specify the following parameters: m : total number genes for testing; m_{1} : number of prognostic genes; r_{1} : number of true rejections; f : FDR level; w : ratio of normalization factors between two groups; {μ_{0g},g∈M_{1}} : average read counts for prognostic gene g in control group; {ρ_{ g },g∈M_{1}} : fold changes for prognostic genes g in control group; {ϕ_{ g },g∈M_{1}} : dispersion for prognostic genes g in control group;
- 2.Calculate sample size:
- (a)
If all the parameters μ_{0g}, ρ_{ g }, and ϕ_{ g } for each prognostic gene g are known, use a numerical approach to solve the equation below with respect to n.
${r}_{1}=\sum _{g\in {M}_{1}}\xi (n,{\rho}_{g},{\mu}_{0g},{\varphi}_{g},w,{\alpha}^{\ast}),$where α^{∗} = r_{1}f/(m_{0}(1-f)) and m_{0} = m-m_{1};
- (b)Otherwise,
- (I)
specify a desired minimum fold change ρ^{∗}, a minimum average read count ${\mu}_{0}^{\ast}$, and a maximum dispersion ϕ^{∗};
- (II)
replace ρ = ρ^{∗}, ${\mu}_{0}={\mu}_{0}^{\ast}$, ϕ = ϕ^{∗}, α = r_{1}f/(m_{0}(1-f)), and β = 1-r_{1}/m_{1} in equation (2) and solve it with respect to n.
- (I)
- (a)
Results
Numerical studies
In this section, we conducted simulation studies to evaluate the accuracy of the proposed sample size formula. The parameter settings in simulation studies are based on empirical data sets.
We set the total number of genes for testing to be m = 10000 and the number of statistically significant prognostic genes m_{1} = 100. We wanted to detect the expected number of true rejections r_{1} = 80, which corresponds to a power of 80% (i.e. β^{∗} = 0.2). All parameters μ_{0g}, ρ_{ g }, and ϕ_{ g } (g = 1,…,10000) were assumed to be unknown. Thus, we used a minimum fold change ρ^{∗} and a minimum average read count ${\mu}_{0}^{\ast}$ and a maximum dispersion ϕ^{∗} to estimate each ρ_{ g }, μ_{0g}, and ϕ_{ g }, g = 1…,10000. We varied ${\mu}_{0}^{\ast}=1$ or 5; log_{2}-fold changes log2(ρ^{∗}) = 0.5,1.0,1.5,2.0 or 2.5; and ϕ^{∗} = 0.1, or 0.5. With these settings, α^{∗} = 8.162×10^{-5},4.253×10^{-4}, and 8.979×10^{-4}, which correspond to controlling FDR at level 1%, 5%, and 10%, respectively.
Then, we substituted α^{∗} and β^{∗} into the formulas (2) and calculated sample size by solving this equation. In addition, for each design setting, we generated 5000 samples from independent negative binomial distributions based on the calculated sample size n; for the control group, the count of each gene is generated by R program from a negative binomial distribution with mean ${\mu}_{0}^{\ast}$ and dispersion ϕ^{∗}; for the treatment group, the count of each gene is generated from a negative binomial distribution with mean ${\rho}^{\ast}{\mu}_{0}^{\ast}$ and dispersion ϕ^{∗}. Then, edgeR is used to estimate model parameters and perform the exact test. The number of true rejections was counted using the q-value procedure proposed by [20]. The expected number of true rejections was estimated as the sample mean of the number of rejections of the 5000 simulation samples (${\widehat{r}}_{1}$).
Sample size calculation for simulation study (and${\widehat{r}}_{\mathbf{1}}$) with r_{ 1 } = 8 0 at FDR = 1 %, 5 % and 1 0 % when w = 1 , m = 1 0 0 0 0 , m_{ 1 } = 1 0 0
${\mu}_{0}^{\ast}=1$ | ${\mu}_{0}^{\ast}=5$ | ||||||
---|---|---|---|---|---|---|---|
FDR | FDR | ||||||
log_{2}( ρ^{∗}) | ϕ ^{∗} | 1% | 5% | 10% | 1% | 5% | 10% |
0.5 | 0.1 | 365 (81) | 305 (84) | 278 (88) | 104 (81) | 87 (84) | 79 (88) |
0.5 | 518 (81) | 433 (84) | 394 (88) | 257 (81) | 215 (84) | 196 (89) | |
1.0 | 0.1 | 79 (81) | 67 (84) | 61 (87) | 24 (82) | 20 (84) | 19 (91) |
0.5 | 119 (81) | 99 (83) | 91 (88) | 63 (82) | 53 (85) | 48 (89) | |
1.5 | 0.1 | 31 (82) | 26 (83) | 24 (86) | 10 (83) | 9 (90) | 8 (91) |
0.5 | 49 (81) | 41 (83) | 38 (88) | 28 (83) | 23 (84) | 21 (86) | |
2.0 | 0.1 | 16 (85) | 13 (84) | 12 (86) | 6 (90) | 5 (92) | 4 (86) |
0.5 | 26 (82) | 22 (84) | 20 (86) | 16 (84) | 13 (85) | 12 (89) | |
2.5 | 0.1 | 8 (85) | 7 (89) | 6 (87) | 3 (78) | 3 (81) | 3 (98) |
0.5 | 14 (83) | 12 (87) | 11 (84) | 10 (82) | 9 (90) | 8 (91) |
Applications
Liver and kidney RNA-seq data set
Sample size calculation for liver and kidney RNA-seq data set under various desired minimum fold changes ( ρ ^{ ∗ } ) for r _{ 1 } = 1 4 0 at F D R = 1 % when m = 1 7 3 6 0 and m _{ 1 } = 1 7 5
NB | Poisson | ||||||||
---|---|---|---|---|---|---|---|---|---|
ρ ^{∗} | n | n _{ w } | n _{ s } | n _{ lw } | n _{ ls } | n _{ tp } | n _{ lr } | ||
0.10 | 7 | 7 | 7 | 11 | 5 | 5 | 7 | ||
0.25 | 11 | 11 | 11 | 13 | 9 | 9 | 10 | ||
0.50 | 30 | 29 | 30 | 31 | 28 | 27 | 29 | ||
0.75 | 139 | 134 | 136 | 137 | 133 | 132 | 135 | ||
1.25 | 178 | 175 | 173 | 174 | 174 | 177 | 181 | ||
1.50 | 50 | 49 | 48 | 49 | 48 | 50 | 50 | ||
2.00 | 15 | 15 | 15 | 15 | 14 | 16 | 15 | ||
2.50 | 8 | 8 | 8 | 8 | 7 | 8 | 8 | ||
3.00 | 5 | 5 | 5 | 6 | 5 | 6 | 5 |
Li, 2013 [28] proposed several sample size calculation methods for RNA-seq data under the Poisson model. To compare the difference in sample size calculation between the negative binomial method and Poisson method, in the last six right columns of Table 2 we report the sample size calculation based on Poisson model (i.e. the sample size based on the Wald test n_{ w }, score test n_{ s }, log transformation of Wald statistic n_{ lw }, log transformation of score statistic n_{ ls }, transformation of Poisson n_{ tp }, and likelihood ratio test n_{ lr }) with the same settings as the negative binomial method. As we can see, the sample size calculation based on the negative binomial and Poisson methods are similar. This result is as expected since the data set explored by [8] has technical and not biological replicates (i.e. the maximum dispersion estimated from the liver and kidney RNA-seq data set is close to zero). Thus, it is not surprising that the results of the negative binomial and Poisson methods are similar when the dispersion parameter is close to zero. Moreover, in Table 2, the estimated sample size is about the same size for very small fold changes (ρ^{∗} = 0.10) and very large fold changes (ρ^{∗} = 3.0). This result is expected since it tends to the same conclusion no matter what statistical model is used when the treatment effect is very large (i.e. the fold change is very large or small).
Transcript regulation data set
Sample size calculation for transcript regulation data set under various desired minimum fold changes ( ρ ^{ ∗ } ) for r _{ 1 } = 1 0 7 at F D R = 1 0 % when m = 1 3 2 6 7 and m _{ 1 = 1 3 3 }
NB | Poisson | ||||||||
---|---|---|---|---|---|---|---|---|---|
ρ ^{∗} | n | n _{ w } | n _{ s } | n _{ lw } | n _{ ls } | n _{ tp } | n _{ lr } | ||
0.10 | 19 | 15 | 14 | 21 | 10 | 10 | 14 | ||
0.25 | 35 | 23 | 23 | 26 | 19 | 19 | 21 | ||
0.50 | 109 | 62 | 60 | 62 | 58 | 56 | 59 | ||
0.75 | 558 | 284 | 281 | 282 | 280 | 273 | 281 | ||
1.25 | 821 | 316 | 363 | 366 | 360 | 371 | 381 | ||
1.50 | 240 | 100 | 102 | 103 | 99 | 105 | 105 | ||
2.00 | 79 | 30 | 31 | 32 | 29 | 32 | 32 | ||
2.50 | 44 | 16 | 16 | 18 | 15 | 17 | 16 | ||
3.00 | 30 | 10 | 11 | 12 | 9 | 11 | 10 |
Discussion
In this research, we assume independent gene expression levels; however, this assumption may not hold in reality. For correlated RNA-seq gene expression data, evaluation of the accuracy of our method is an important future research question; however, generating a negative binomial distribution for correlated high-dimensional data will be a challenge. Moreover, most of the major R packages dedicated to RNA-seq differential analyses (edgeR, DESeq, etc.) are now starting to enable multi-group comparisons. However, the proposed method is developed for comparing two-group means. Thus, the sample size calculation for multi-group comparisons would be an interesting research topic for us in the future. In addition, it has already been noted that typical RNA-seq differential analyses have very low power; see for example the simulation studies in [30], where power for edgeR was always less than 60%, or [31], where power ranged from about 45% to 55% (both with 10 samples per condition). In our simulation and application sections, the minimum sample sizes required to achieve 80% power would be prohibitively large for RNA-seq experiments in practice, given their current cost. In such situations, the findings in [30, 31] can provide useful information for specifying achievable power. It is well known that low study power will decrease the reproducibility of scientific research. We hope that this paper can benefit researchers by allowing them to understand their study power.
Conclusions
In recent years, RNA-seq technology has emerged as an attractive alternative to microarray studies, due to its ability to produce digital signals (counts) rather than analog signals (intensities), and to produce more highly reproducible results with relatively little technical variation [32, 33]. With a large sample size, RNA-seq can become costly; on the other hand, insufficient sample size may lead to unreliable answers to the research question of interest. To manage the trade-off between cost and accuracy, sample size determination is a critical issue for RNA-seq experimental design. For comparing the differential expression of a single gene, we have proposed a sample size calculation method based on an exact test proposed by [14, 15]. To address multiple testing (i.e., multiple genes), we further extended our proposed method to incorporate FDR control. Our methods are not computationally intensive for pilot data or other relevant data with a specified desired minimum fold change, minimum average read count, and maximum dispersion. To facilitate implementation of the sample size calculation, R code is available from the corresponding author.
Declarations
Acknowledgements
This work was partly supported by NIH grants P30CA068485, P50CA095103, P50CA098131, and U01CA163056. The authors wish to thank Margot Bjoring for editorial work on this manuscript.
Authors’ Affiliations
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