Sample size calculation based on exact test for assessing differential expression analysis in RNA-seq data

M E T H O D O L O G Y A R T I C L E

Open Access

Sample size calculation based on exact test for

assessing differential expression analysis in

RNA-seq data

Chung-I Li

, Pei-Fang Su

and Yu Shyr

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.

Background

Next generation sequencing (NGS) technology has revo-lutionized 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 pro-vides expression level data as well as differentially spliced variants, gene fusion, and mutation profile data. Such advantages have gradually elevated RNA-seq as the tech-nology 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 ques-tions, including abundance estimation [1-3], detection of

*Correspondence: [email protected]

3Center for Quantitative Sciences, Vanderbilt University, 571 Preston Building Nashville, TN, USA

Full list of author information is available at the end of the article

alternative splicing [4-6], detection of novel transcripts [6,7], and the biology associated with gene expression pro-file differences between samples [8-10]. With this rapid growth of RNA-seq applications, discussion of experi-mental design issues has lagged behind, though more recent literature has begun to address some of the rel-evant 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 bio-logical 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 num-ber 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 dif-ferential 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

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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 expres-sion analysis of RNA-seq data. Furthermore, [16] pro-vided 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 correc-tion 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, con-trolling 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, how-ever, 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 pro-pose a procedure to calculate sample size while control-ling 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 multi-ple gene comparisons, what is the suitable sammulti-ple size while controlling FDR? The article is organized as fol-lows. 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 compar-ison test issue. Performance comparcompar-isons via numerical studies are described in the Results section. Two real RNA-seq data sets are used to illustrate sample size cal-culation. Finally, discussion follows in the Conclusions section.

Method Exact test

In an RNA-seq experiment, the total number of reads, also referred to as library size, mapped to the genome are dif-ferent among the samples. In such cases, the counts in each group are not identically distributed, and it is diffi-cult to develop an exact test for assessing the differential expression analysis of RNA-seq data. To handle this issue, [14,15] proposed a quantile-adjusted conditional maxi-mum likelihood procedure to create pseudocounts which are approximately identically distributed and which lead to the development of an exact test. In the following, the proposed sample size calculation method is based the exact test for a single-gene comparison. Let Yijbe the ran-dom variable corresponding to the pseudocount, with yij being the observed value of Yij, of the jth (j= 1, 2, . . . , ni) sample of the ith (i = 0, 1) group where n0and n1 are

the numbers of samples from the control and treatment group, respectively. Assume pseudocount Yijcan be mod-eled as a negative binomial (NB) distribution, NB(dijγi, φ). Here, γirepresents the normalized gene expression level of group i, dij represents a normalization factor for the total number of reads mapped in the jth sample of the ith group, and φ is the dispersion. We use the NB parame-terization where the mean is μij = dijγi and variance is μ_ij(1+ μ2_ijφ). Because the question of interest is to iden-tify the differential gene expression between two groups, the corresponding testing hypothesis is

H0: γ1= γ0 vs. H1: γ1= γ0. (1)

Because the pseudocounts in each group have an approximately identical negative binomial distribution [14,15], the sum of pseudocounts of each group, Yi = nj=1i Yij, has a negative binomial distribution NB(nid∗iγi, φ/ni) where d∗i is the geometric mean of nor-malization factors in group i. Under the null hypothesis (1), the sum of the total pseudocount, Y1+ Y0, follows

a negative binomial distribution. In analogy with Fisher’s exact test, [14,15] proposed an exact test for replac-ing the hypergeometric probabilities with negative bino-mial 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 In this section, we focus on sample size calculation based on the exact test for a single-gene comparison as described in the test statistics section. For simplicity, we assume the RNA-seq experiment uses a balanced design (i.e., n0 = n1 = n), which is a special but common case.

The following method could be easily extended to the unbalanced case (i.e. let n0 = n and n1 = kn where k

is a predetermined ratio of the sample size of the control group to the treatment group). In order to perform sample size calculations, it is necessary to construct a power func-tion for the testing described above. The power of a test is the probability that the null hypothesis is rejected when the alternative hypothesis is true. Since the distribution of the exact test statistic under the alternative hypothesis is unknown, however, it is difficult to derive a closed-form expression of the power function. Instead of deriving the distribution of test statistic under the alternative hypothe-sis, [26] proposed a method to calculate the power for the exact test based on a given p-value. Here, we borrow their concept to calculate power. For a given p-value, p(y1, y0)

where y0and y1are the observed pseudo-sums, described

in the previous section, the power can be expressed as ξ(n,ρ, μ0, φ, w, α)= ∞  y0=0 ∞  y1=0 f  nwρμ0, φ n  f  nμ0, φ n  I( p( y1, y0) < α),

where w = d₁∗/d∗₀is the ratio of the geometric means of normalization factors between two groups, ρ = γ1/γ0is

the fold change, μ0= d∗₀γ0is the average number of reads

in the control group, f (μ, φ) is the probability mass func-tion of the negative binomial distribufunc-tion with mean μ as well as dispersion φ, α is the the level of significance, and I(.) denotes the indicator function. For a given desired power 1− β, the power of the test can be represented as the function of sample size in the form

1− β = ξ(n,ρ, μ0, φ, w, α). (2)

Thus, the required sample size n to attain the given power 1− β at level of significance α can then be calcu-lated 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.

For the multiple testing problem, [19] suggested the use of false discovery rate (FDR) which is defined as the expected proportion of false discoveries among rejected

null hypotheses. Storey, 2002 [17] further proposed an improvement to FDR to achieve higher power, in the form

FDR= E  R0 R   R > 0,

where R0is the number of false discoveries and R is the

number of results declared significant (i.e., rejections of the null hypothesis).

To calculate the sample size for microarray data analysis, [23] proposed an FDR-controlled method which is based on the expression of FDR under independence (or weak dependence) among test statistics, as

FDR= m0α

m0α+ E(R1)

,

[17,27], where m0is the number of true null hypotheses

and E(R1)is the expected number of true rejections. By

borrowing their concepts, the expected number of true rejections for RNA-seq data can be calculated as

E(R1)=

 g∈M1

ξ(n, ρg, μ0g, φg, w, α),

where ρg is the fold change, φgis the dispersion, and μ0g

is the average read count in the control group for gene g ∈ M1(the set of prognostic genes), respectively. Thus,

to guarantee an expected number of true rejections, say r1, and control FDR at a specified level f, we have

f = m0α m0α+ r1 (3) and r1=  g∈M1 ξ(n, ρg, μ0g, φg, w, α). (4)

By solving equation (3) with respect to α, we have α∗= r1f

m0(1− f )

,

where α∗is the marginal type I error level for the expected number of true rejections r1at a given FDR f. Replacing α

with α∗in (4), we have the function with respect to n as g1(n)=

 g∈M1

ξ(n, ρg, μ0g, φg, w, α∗)− r1.

Then, by solving g1(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 ρg, dispersions φg, and average read counts μ0gof gene g for the set of prognostic genes g∈ M1

prior to the RNA-seq experiment. However, we may not have enough information to estimate all of those param-eters 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(ρg)| or μ0g increases and φg decreases, we suggest

using a common ρ∗= arg ming∈M1{| log2(ρg)|} minimum fold change, μ∗₀ = ming∈M1{μ0g} minimum average read

count, and φ∗ = maxg∈M1{φg} maximum dispersion to

estimate each ρg, μ0g, and φg, respectively. In such cases, it gives a more conservative estimate of the required sample size.

When we use ρ∗, μ∗₀, and φ∗ to estimate each ρg, μ0g,

and φg, g∈ M1, in the multiple testing context, α∗and β∗

can be calculated as r1f /(m0(1−f )) and 1−r1/m1,

respec-tively, where m1 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 αand β with α∗and β∗.

The procedures for sample size calculation detailed in this section can be summarized as follows:

1. Specify the following parameters: m : total number genes for testing; m1 : number of prognostic genes; r1 : number of true rejections;

f : FDR level;

w : ratio of normalization factors between two groups;

{μ0g, g∈ M1} : average read counts for prognostic

geneg in control group;

{ρg, g∈ M1} : fold changes for prognostic genes g in

control group;

{φg, g∈ M1} : dispersion for prognostic genes g in

control group; 2. Calculate sample size:

(a) If all the parameters μ0g, ρg, and φgfor each prognostic geneg are known, use a

numerical approach to solve the equation below with respect ton.

r1=  g∈M1 ξ(n, ρg, μ0g, φg, w, α∗), where α∗= r1f /(m0(1− f )) and m0= m − m1; (b) Otherwise,

(I) specify a desired minimum fold change ρ∗, a minimum average read count μ∗₀, and a maximum dispersion φ∗; (II) replace ρ= ρ∗, μ0= μ∗0, φ= φ∗,

α= r₁f /(m0(1− f )), and

β= 1 − r₁/m1in equation (2) and

solve it with respect ton.

Results

Numerical studies

In this section, we conducted simulation studies to eval-uate 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 prognos-tic genes m1 = 100. We wanted to detect the expected

number of true rejections r1= 80, which corresponds to

a power of 80% (i.e. β∗ = 0.2). All parameters μ0g, ρg, and φg( g = 1, . . . , 10000) were assumed to be unknown.

Thus, we used a minimum fold change ρ∗ and a

mini-mum average read count μ∗₀and a maximum dispersion φ∗ to estimate each ρg, μ0g, and φg, g = 1 . . . , 10000. We varied μ∗₀ = 1 or 5; log2-fold changes log2(ρ∗) =

0.5, 1.0, 1.5, 2.0 or 2.5; and φ∗= 0.1, or 0.5. With these set-tings, 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 sam-ples 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 μ∗₀and dispersion φ∗; for the treatment group, the count of each gene is generated from a negative binomial distribution with mean ρ∗μ∗₀and 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 (ˆr1).

In Table 1, we showed the calculated sample size with corresponding ˆr1in parentheses under the case w = 1.

For a fixed log2-fold change, dispersion, and FDR,

sam-ple size increases when μ0 decreases. This result is as

expected; a small average read count provides less infor-mation, such that a larger sample size is required to detect the difference. For a fixed μ∗₀, φ∗, and FDR, sample size increases when log₂(ρ∗) decreases (i.e. the smaller log2

-fold changes requires greater sample sizes with all else being equal). This result is as expected; a larger sample size is required for detecting a smaller difference. For a fixed μ∗₀, log₂(ρ∗), and FDR, sample size increases when φ∗increases. This result, also, is as expected; the variation increases when dispersion increases, such that a larger sample size is required to detect the difference. Note that all ˆr1in Table 1 are close to the pre-specified number of

true rejections (r1= 80); thus, the proposed method

esti-mated a sample size that achieves correct power at the specified FDR level.

Table 1 Sample size calculation for simulation study (andˆr1) with r1= 80 at FDR = 1%, 5% and 10% when w= 1, m = 10000, m1= 100 μ∗₀= 1 μ∗₀= 5 FDR FDR log2( 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

To identify differentially expressed genes between human liver and kidney RNA samples, [8] explored an RNA-seq data set containing 5 human kidney samples and 5 human liver samples. In the following, we used this data set as pilot data for designing a new study with the same study objective. For the purpose of demonstration, we assumed that the human kidney is the control group. After filtering genes with no more than 5 total reads in liver samples or kidney samples, there were 17306 genes left. We assumed that the top 175 (≈ 1% of 17306) genes are prognostic. From the pilot data, the minimum average read counts among the prognostic genes in the control group were estimated as μ∗₀ = 5, the maximum dispersion was esti-mated as φ∗ = 0.0029, and the ratio of the geometric mean of normalization factors between the two groups was estimated as w = 0.9 using edgeR. Suppose we want to identify 80% of the prognostic genes (i.e. r1 = 0.8 ×

175 = 140), while controlling FDR at 1% (i.e. f = 0.01). Based on the pilot data, we set m = 17306, m1 = 175,

m0 = 17131, r1 = 140, and f = 0.01. In this case,

we have α∗= r1f m0(1− f ) = 8.2549 × 10−5 and β∗= 1 − r1 m1 = 0.2.

After substituting those parameters into equation (2) and solving it with respect to n, the required sample size can be obtained. In the second column from the left

of Table 2, we report the sample size while controlling FDR at 1% under various desired minimum fold changes ρ∗ = 0.10, 0.25, 0.50, 0.75, 1.25, 1.50, 2.00, 2.50, and 3.0. From Table 2, we found that the original RNA-seq experi-ment described in [8] with sample size 5 in each group can identify 80% of the prognostic genes at FDR= 1% if the desired minimum fold change ρ∗is 3.0.

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 sam-ple size based on the Wald test nw, score test ns, log transformation of Wald statistic nlw, log transformation of score statistic nls, transformation of Poisson ntp, and likelihood ratio test n_lr) with the same settings as the neg-ative 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 dis-persion 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

Blekhman, 2010 [29] used RNA-seq to study transcript regulation in humans, chimpanzees, and rhesus macaques Table 2 Sample size calculation for liver and kidney RNA-seq data set under various desired minimum fold changes ( ρ∗) for r1= 140 at FDR = 1% when m = 17360 and m1= 175 NB Poisson ρ∗ n nw ns nlw nls ntp nlr 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

using liver RNA samples from three males and three females from each species. For the purpose of demon-stration, we assumed that the goal of the study is to identify differential gene expression between male and female in humans and that the female is consid-ered the control group. There were 13267 genes in the data set after performing quality control analyses. Suppose that the top 133 (≈ 1% of 13267) genes are prog-nostic. After filtering genes with no more than 5 total reads in male samples or female samples, there were 7658 genes left. Those genes are considered pilot data, and we assessed the differential expression by using edgeR. From the pilot data, the minimum average read counts among the prognostic genes in the control group were estimated as μ∗₀ = 1.67, φ∗ = 0.6513, and the ratio of the geometric mean of normalization factors between the two groups was estimated as w = 1.08. Suppose we want to identify 80% of the prognostic genes (i.e. r1 =

0.8 × 133 = 107), while controlling the FDR at 10%.

Based on the pilot data, we set m = 13267, m1 = 133, m0 = 13134, r1 = 107 and f = 0.1. In this case, we

have α∗ = 9.0512 × 10−4 and β∗ = 0.2. In the second column from the left of Table 3, we report the required sample sizes under various desired minimum fold changes while controlling the FDR at 10% under the negative bino-mial distribution. We also report the required sample size based on the Poisson model proposed by [28] under the same settings in the last six columns on the right of Table 3. As we can see, the required sample size based on the negative binomial method is greater than the Poisson method. In the transcript regulation data set, the max-imum dispersion was estimated as φ∗ = 0.6513 > 0. This indicates that the read counts in this data set exhibit over-dispersion. In such a situation, it is inappropriate to model this data set based on the Poisson, and the sample size calculation based on the Poisson model will

Table 3 Sample size calculation for transcript regulation data set under various desired minimum fold changes (ρ∗) for r1= 107 at FDR = 10% when m = 13267 and m1=133

NB Poisson ρ∗ n nw ns nlw nls ntp nlr 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

be underestimated due to underestimation of variance (i.e. the study based on the corresponding sample size will be underpowered).

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 bino-mial distribution for correlated high-dimensional data will be a challenge. Moreover, most of the major R pack-ages dedicated to RNA-seq differential analyses (edgeR, DESeq, etc.) are now starting to enable multi-group com-parisons. However, the proposed method is developed for comparing two-group means. Thus, the sample size cal-culation for multi-group comparisons would be an inter-esting research topic for us in the future. In addition, it has already been noted that typical RNA-seq differ-ential 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 con-dition). 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 vari-ation [32,33]. With a large sample size, RNA-seq can become costly; on the other hand, insufficient sam-ple 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 crit-ical issue for RNA-seq experimental design. For com-paring 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 mul-tiple testing (i.e., mulmul-tiple 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.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Authors CIL, and PFS were involved in the development of the models. CIL and PFS wrote the manuscript. SY generated the original idea and guided and supervised the research. All authors read and approved the final version of this manuscript.

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.

Author details

1_{Department of Applied Mathematics, National Chiayi University, Chiayi,} Taiwan.2Department of Statistics, National Cheng Kung University, Tainan, Taiwan.3_{Center for Quantitative Sciences, Vanderbilt University, 571 Preston} Building Nashville, TN, USA.

Received: 3 June 2013 Accepted: 28 November 2013 Published: 6 December 2013

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Cite this article as: Li et al.: Sample size calculation based on exact

test for assessing differential expression analysis in RNA-seq data. BMC