 Methodology article
 Open Access
 Published:
Sample size calculation for microarray experiments with blocked oneway design
BMC Bioinformatics volume 10, Article number: 164 (2009)
Abstract
Background
One of the main objectives of microarray analysis is to identify differentially expressed genes for different types of cells or treatments. Many statistical methods have been proposed to assess the treatment effects in microarray experiments.
Results
In this paper, we consider discovery of the genes that are differentially expressed among K (> 2) treatments when each set of K arrays consists of a block. In this case, the array data among K treatments tend to be correlated because of block effect. We propose to use the blocked oneway ANOVA Fstatistic to test if each gene is differentially expressed among K treatments. The marginal pvalues are calculated using a permutation method accounting for the block effect, adjusting for the multiplicity of the testing procedure by controlling the false discovery rate (FDR). We propose a sample size calculation method for microarray experiments with a blocked oneway design. With FDR level and effect sizes of genes specified, our formula provides a sample size for a given number of true discoveries.
Conclusion
The calculated sample size is shown via simulations to provide an accurate number of true discoveries while controlling the FDR at the desired level.
Background
Clinical and translational medicine have benefited from genomewide expression profiling across two or more independent samples, such as various diseased tissues compared to normal tissue. DNA microarray is a high throughput biotechnology designed to measure simultaneously the expression level of tens of thousands of genes in cells. Microarray studies provide the means to understand the mechanisms of disease. However, various sources of error can influence microarray results [1]. Microarrays also present unique statistical problems because the data are high dimensional and are insufficiently replicated in many instances. Methods of adjustment for multiple testing therefore become extremely important. Multiple testing methods controlling the false discovery rate (FDR) [2] have been popularly used because they are easy to calculate and less strict in controlling the false positivity compared to the familywise error rate (FWER) control method [3].
Numerous sample size calculation methods have been proposed for comparing independent groups while controlling the FDR in designing microarray studies. Lee and Whitmore [4] considered comparing multiple groups using ANOVA models and derived the relationship between the effect sizes and the FDR using a Bayesian approach. Their power analysis does not address the multiple testing issue. Muller et al. [5] chose a pair of testing errors, including FDR, and minimized one while controlling the other at a specified level using a Bayesian decision rule. Jung [6] proposed a closed form sample size formula for a specified number of true rejections while controlling the FDR at a desired level. Pounds and Cheng [7] and Liu and Hwang [8] proposed similar sample size formulas which can be used for comparison of K independent samples. These methods are for the FDRcontrol methods based on independence or a weak dependency assumption among test statistics. Recently, Shao and Tseng [9] introduced an approach for calculating sample sizes for multiple comparisons accounting for dependency among test statistics.
In some studies, specimens for K treatments are collected from the same subject and means are compared across treatment groups. In this case, the gene expression data for the K treatments may be dependent since they share the same physiological conditions. For example, Feng et al. [10] conducted a study to discover the genes differentially expressed between center (C) and edge (E) of the uterine fibroid and the matched adjacent myometrium (M). In this study, specimens are taken from the three sites for each patient. The patients are blocks and the three sites (K = 3), C, E and M, are treatments (or groups) to be compared.
Since a set of K specimens are collected from each patient, we require a much smaller number of patients than a regular unblocked design. Furthermore, the observations within each block tend to be positively correlated, so that a blocked design requires a smaller number of arrays than the corresponding unblocked design just as a paired twosample design with a positive pairwise correlation requires a smaller number of observations than a two independent sample design. The more heterogeneous the blocks are, the greater the savings in number of arrays for the blocked design.
In this paper, we consider a nonparametric blocked Ftest statistic to compare the gene expression level among K dependent groups. We adjust for multiple testing and control the FDR by employing a permutation method. We propose a sample size calculation method for a specified number of true rejections while controlling the FDR at a specified level. Through simulations, we show that the blocked Ftest accurately controls the FDR using the permutation resampling method and the calculated sample size provides an accurate number of true rejections while controlling the FDR at the desired level. For illustration, the proposed methods are applied to the fibroid study [10] mentioned above.
Methods
Nonparametric block Ftest statistic
Suppose that we want to discover genes that are differentially expressed among K sites (treatments or groups). For each of n patients (blocks), a specimen is collected from each site for a microarray experiment on m genes. In this case, the gene expression data from the K sites tend to be correlated. Let Y_{ ijk }denote the expression level of gene i (= 1,..., m) from treatment k (= 1,..., K) of block j (= 1,..., n). We consider the blocked oneway ANOVA model
where, for gene i, μ_{ i }is the population mean, δ_{ ik }is a fixed treatment effect and the primary interest, γ_{ ij }is a random block effect, and ε_{ ijk }is a random error term. We assume that , γ_{i 1},..., γ_{ in }are independent and identically distributed (IID) with mean 0 and variance v_{ i }, (ε_{ ijk }, 1 ≤ j ≤ n, 1 ≤ k ≤ K) are IID with mean 0 and variance , and error terms and block effects are independent. The standard ANOVA theory using parametric F distributions to test the treatment effect assumes a normal distribution for ε_{ ijk }. However, in this paper, we avoid the normality assumption by using a permutation resampling method in testing and a largesample approximation in sample size calculation.
For gene i(= 1,..., m), the hypotheses for testing the treatment effect are described as
against
We reject H_{ i }in favor of for a large value of Ftest statistic
where , and . If the error terms are normally distributed, F_{ i }marginally has the F_{K1, (K1)(n1)}distribution under H_{ i }. The normality assumption can be relaxed if n is large.
Without the normality assumption, the joint null distribution of the statistics can be approximated using a block permutation method, where the array data sets for K treatments are randomly shuffled within each block: the permuted data may be represented as , where is a random permutation of (1,..., K). Note that there are (K!)^{n}different permutations, among which (K!)^{n1}give different Fstatistic values. The R language package multtest [11] can be used to implement the permutationbased multiple testing procedure for blocked microarray data. We consider adjusting for the multiplicity of the testing procedure by controlling the FDR [12, 13].
Permutationbased multiple testing for FDRcontrol

(i)
Compute the Ftest statistics (F_{1},..., F_{ m }) from the original data, (f_{1},..., f_{ m }).

(ii)
From the bth permutation data (b = 1,..., B), compute the Ftest statistics .

(iii)
For gene i, estimate the marginal pvalue by
where I(A) is an indicator function of event A.

(iv)
For a chosen constant λ ∈ (0, 1), estimate the qvalue by

(v)
For a specified FDR level q*, discover gene i (or reject H_{ i }) if q_{ i } < q*.
Sample size calculation
Let ℳ_{0} and ℳ_{1} denote the sets of indices of genes that are equally and differentially expressed, respectively, in K treatments, and { = δ_{ ik }/σ_{ i }, i ∈ ℳ_{1}, 1 ≤ k ≤ K} denote the standardized effect sizes for the differentially expressed genes. Let m_{0} and m_{1} = m  m_{0} denote the cardinalities of ℳ_{0} and ℳ_{1}, respectively.
Suppose that we want to discover gene i (or reject H_{ i }) if the marginal pvalue p_{ i }is smaller than α ∈ (0, 1). For large m and under the independence assumption or weak dependence among the Ftest statistics, the FDR corresponding to the cutoff value α is approximated by
where β_{ i }(α) = P(p_{ i }≤ α) is the marginal power of a single αtest applied to gene i ∈ ℳ_{1} and denotes the expected number of true rejections when we reject H_{ i }for p_{ i }<α, see Jung [6].
Now, we derive β_{ i }(α) for gene i ∈ ℳ_{1}. By the standard blocked oneway ANOVA theory under the normality assumption for ε_{ ijk },
and
are independent, where is the noncentral χ^{2}distribution with ν degrees of freedom and noncentrality parameter η, and . Hence, for the Ftest statistic (2), we have
where is the noncentral Fdistribution with ν_{1} and ν_{2} degrees of freedom, and noncentrality parameter η. Note that, for i ∈ ℳ_{0}, and F_{ i }~F_{(K1),(K1)(n1)}(0) = F_{(K1),(K1)(n1)}, the central Fdistribution.
The marginal powers are expressed as
where denotes the 100(1  α) percentile of distribution. The marginal powers can be calculated using R, SAS or some other packages. Suppose we want r_{1} true rejections while controlling the FDR at q*. By combining this with (3) and (4), we obtain two equations
and
Note that r_{1}/m_{1} denotes the probability of true rejection. At the design stage of a study, m is given by the number of genes included in the chips to be used for microarray experiment, m_{1} and {, i ∈ ℳ_{1}, 1 ≤ k ≤ K} are projected based on biological knowledge or estimated from pilot data, and K, r_{1} (or r_{1}/m_{1}) and q* are prespecified. The only unknown variables in (5) and (6) are α and n. By solving (6) with respect to α, we obtain α* = r_{1} q*/{m_{0} (1  q*)} and, by plugging this in (5), we obtain an equation for r_{1} depending only on n,
The marginal power function (4) includes n in the degrees of freedom of the denominator as well as the noncentrality parameter of the Fdistributions. The impact of the degrees of freedom of the denominator of the Fstatistic on the marginal power is much weaker than that of the noncentrality parameter, so that β_{ i }(α) is a monotone increasing function of n, and consequently equation (7) has a unique solution. Figure 1 demonstrates the relationship between n and β_{ i }(α) with α = 0.05; = {k  (K + 1)/2}/K for 1 ≤ k ≤ K; K = 3, 4 or 5. This monotone relationship becomes clear for large n as shown by an approximate sample size formula given below. Note that the variance of block effect v_{ i }has no impact on the sample size and power of the test statistic for treatment effect.
In summary, the sample size (i.e., number of blocks) n for r_{1} (≤ m_{1}) true rejections is calculated as follows, assuming that the error terms in model (1) are normally distributed.
Sample size calculation based on the noncentral Fdistribution

(i)
Specify the input variables:

K = number of treatments;

m = total number of genes for testing;

m_{1} = number of genes differentially expressed in K treatments (m_{0} = m  m_{1});

{, i ∈ ℳ_{1}, 1 ≤ k ≤ K} = standardized effect sizes for prognostic genes;

q* = FDR level;

r_{1} = number of true rejections

(ii)
Using the bisection method, solve
with respect to n, where α* = r_{1}q*/{m_{0}(1  q*)}.

(iii)
The required sample size is n blocks, or nK array chips.
In the sample size formula based on the noncentral Fdistribution, the relationship between n and the marginal power functions based on the Fdistribution is complicated and a normal distribution assumption of the error terms is required. In the large sample case, we can loosen the normality assumption and simplify this relationship. If the error terms have a finite 4th moment, then, for large n, the distribution of F_{ i }is approximated by
A proof is given in the Appendix. Similarly, for large n, the F_{(K1),(K1)(n1)}distribution can be approximated by (K  1)^{1} , so that F_{(K1),(K1)(n1),α}≈ (K  1)^{1} , where is the 100(1  α) percentile of the χ^{2} distribution with ν degrees of freedom. Hence, the marginal power for F_{ i }is approximated by
and a sample size based on the χ^{2}distribution approximation is obtained by solving
with respect to n, where α* = r_{1}q*/{m_{0}(1  q*)}. In this equation, n appears only in the noncentrality parameter of the χ^{2} distributions.
Equation (8) is especially useful when we want to compare the powers between a blocked oneway design and an unblocked oneway design. Using similar approximations, it is easy to show that an approximate sample size N = nK for a study with unblocked oneway design with a balanced allocation is obtained by solving
with respect to n, where . The only difference between (8) and (9) is the standardized effect sizes, = δ_{ ik }/σ _{ i }and . The latter is always smaller than the former because of the variance among blocks, v_{ i }. If v_{ i }is large compared to the variance of experimental errors, , then a blocked oneway design requires much smaller number of arrays than an unblocked oneway design. Let n_{ u }and n_{ b }denote the sample sizes n calculated under an unblocked and a blocked design, respectively. If are constant f among the prognostic genes, then from (8) and (9), we have n_{ u }= (1 + f)n_{ b }. As an example, consider the design of the fibrosis study as discussed in Background Section and suppose that the variance of the block effects is half of that of measurement errors for the prognostic genes, i.e. f = 0.5. In this case, if a blocked design requires n_{ b }= 100 patients and 3n_{ b }= 300 array chips, then the corresponding unblocked design with a balanced allocation requires n_{ u }= 150 patients per group or a total 450 patients. For an unblocked design, the number of array chips is identical to that of patients, and compared to the blocked design, the unblocked design requires 1.5 times more chips and 4.5 times more patients.
Results and discussion
Simulations
First, we investigate the accuracy of the FDR control based on blocked oneway ANOVA tests and the sample size formulas via simulations. For the simulations on FDR control, we consider blocked oneway designs with K = 3 treatments and n = 10, 30, or 50 blocks. For gene i (= 1,..., m) from treatment k (= 1,..., K) of block j (= 1,..., n), block effect γ_{ ij }and error terms ϵ_{ ijk }are generated from N (0, 0.5^{2}) and N(0,1), respectively. For differentially expressed genes i ∈ ℳ_{1}, the standardized treatment effects are set at = (1, 0, 1) or (1, 2, 1). We set the total number of genes m = 4000; the number of differentially expressed genes m_{1} = 40 or 200; and the nominal FDR level q* = 0.05, 0.1, 0.2, 0.3, 0.4, or 0.5. We conducted N = 1000 simulations under each setting, and the null distribution of the test statistics is approximated from B = 1000 permutations for each simulation sample. In simulation l(= 1,..., N), the FDRcontrol multiple testing method is applied to the simulated data using tuning parameter λ = 0.95 [12] to count the numbers of total rejections and false rejections and to estimate the FDR, . Then the empirical FDR is obtained as
Table 1 reports the simulation results. The testing procedure controls the FDR accurately, i.e. ≈ q*, when m_{1} is large (m_{1} = 200), but tends to be anticonservative, i.e. > q*, when m_{1} is small (m_{ i }= 40). Jung and Jang [13] made similar observations for twosample ttests and Cox regression.
For the simulations on sample size calculation, we set m = 4000; m_{1} = 40 or 200; number of treatment K = 3; treatment effects = (1/4, 0, 1/4) or (1/4, 1/2, 1/4) for i ∈ ℳ_{1}; γ_{ ij }~N (0, 0.5^{2}) and ϵ_{ ijk }~N (0. 1). We want the number of true rejections r_{1} to be 30%, 60% or 90% of m_{1} while controlling the FDR level at q* = 1%, 5% or 10%. For each design setting, we first calculate the sample size n based on the Fdistribution or the chisquare approximation, and then generate N = 1000 samples of size n under the same setting. From each simulation sample, the number of true rejections are counted while controlling the FDR at the specified level using λ = 0.95. The first, second and third quartiles, Q_{1}, Q_{2} and Q_{3}, of the observed true rejections, , are estimated from the 1000 simulation samples.
Table 2 summarizes the simulation results by the two methods. As expected, sample size increases in r_{1} and decreases in m_{1} and q*. Since the standardized effect sizes for the differentially expressed genes influence the sample size through their sum of squares, the combination of effect sizes (1/4, 0, 1/4) requires a larger sample size than (1/4, 1/2, 1/4). The sample size based on the chisquare approximation is always smaller than that based on the Fdistribution. The median (Q_{2}) of the empirical true rejections is smaller than the nominal r_{1} for the sample size based on the chisquare approximation, especially with a small n, while the sample size based on the Fdistribution is always accurately powered, i.e. Q_{2} ≈ r_{1}.
Example
We applied the permutationbased blocked oneway ANOVA and the sample size calculation method to the fibroid study discussed in the Background Section. From each patient, specimens are taken from two sites of fibroid tissue, center (C) and edge (E), and one normal myometrium (M). Five patients are accrued to the study. We regard the three sites as treatments (K = 3) and the patients as blocks (n = 5). mRNA was amplified and hybridized onto HGU133 GeneChips according to the protocols recommended by Affymetrix (Santa Clara, CA), and m = 54675 probe sets on the array were analyzed. Expression values were calculated using the Robust Multichip Average (RMA) method [14]. RMA estimates are based upon a robust average of background corrected PM intensities. Normalization was done using quantile normalization [15]. We filtered out all "AFFX" genes and genes for which there were 4 or fewer present calls (based on Affymetrix's present/marginal/absent (PMA) calls using mismatch probe intensity, the ratio of PM to MM). That is, a gene is included only if there are at least 3 present calls among the 15 PMA calls. Filtering yielded 30711 genes to be used in the subsequent analyses.
In order to group the samples according to the degree of similarity present in the gene expression data, we first applied a hierarchical clustering analysis to the filtered 30711 gene expression data and generated a dendrogram (Figure 2). We used the Complete Linkage method [16] and Pearson's correlation coefficient as a measure of similarity. In the dendrogram, the height of each branch point indicates the similarity level at which each cluster was generated. We obtained the same clustering using the L_{2} norm as a measure of similarity. Except for patient 2, E and C are clustered together for each patient. In spite of the block effect, M is clustered separately from E and C regardless of patient assignment. We conclude that C and E have similar gene expression profiles, but M has a different gene expression profile from either C or E. While the clustering analysis investigates the genome wide expression profile, blocked oneway ANOVA helps us identify individual genes differentially expressed among the three sites. Using the blocked oneway ANOVA method, we selected the top 50 genes in terms of parametric pvalues (Table 3). The expression patterns of six genes that are identified as differentially expressed are presented in Figure 3. The expression levels of each patients are connected among three sites. These genes are similarly expressed between C and E, but differentially expressed in M. Further, 220273_at, 210255_at, 229160_at, 204620_s _at and 217287_s _at are underexpressed in M while 1553194_at is overexpressed in M.
The results of our analysis of the two sites of fibroid tissue, center and edge, compared to the normal myometrium using a blocked oneway design suggest that reduced FDR provides an enhanced approach to clinical microarray studies. Our findings are consistent with previously reported genomewide profiling studies [17, 18]. We believe that these results support the hypothesis that uterine fibroids develop through altered wound healing signaling pathways leading to tissue fibrosis [19, 20]. Using the method described in this paper, genes differentially overexpressed in the fibroid tissue compared to myometrium are related to extracellular matrix (ECM) and ECM regulation such as collagen IV, alpha 1, versican (chondroitin sulfated 2) and IL17β [21]. IL17β, a cellcell signaling transducer has been reported to enhance MMP secretion and to rapidly induce phosphorylation of the extracellular signalrelated kinases (ERK) 1/2 and p38MAPK in colonic myofibroblasts and has been shown to stimulate MMP1 expression in cardiac fibroblasts through ERK 1/2 and p38 MAPK [22, 23]. Thus IL17β is important in remodelling of the extracellular matrix. According to our analysis, RAD51like 1, a recombinational repair gene, is also overexpressed in fibroids, which is consistent with a report that RAD51B is the preferential translocation partner of high mobility group protein gene (HMGIC) in uterine leiomyomas [24]. HMGIC codes for a protein that is a nonhistone DNA binding factor that is expressed during development in embryonic tissue and is an important regulator of cell growth, differentiation and transformation as well as apoptosis [25]. Arrest of apoptosis appears to be a hallmark of uterine fibroids, a finding that is characteristic of altered wound healing as well [19]. HMGIC appears to play a role in the development of uterine fibroids [19, 26, 27].
Suppose that we want to design a new fibroid study using the data analyzed above as pilot data. In the sample size calculation, we set m = 30, 000. We assume that the m_{1} = 50 genes which were selected as the top 50 genes in terms of parametric pvalue are differentially expressed in the three sites (K = 3). From the pilot data, we estimate the standardized treatment effect δ_{ ik }. For illustration, the effect sizes of these m_{1} = 50 genes are taken to be δ_{ ik }= 0.1 . We need n = 15 patients (blocks) to discover 90% of the prognostic genes, i.e. r_{1} = [0.9 × 50] = 45, while controlling the FDR at q* = 5% level. In a simulation study, we generated N = 1000 microarray data sets of size n = 15 under this design setting. With q* = 0.05, we observed the quartiles Q_{2}(Q_{1}, Q_{3}) = 46(45, 47) from the empirical distribution of the observed true rejections.
Conclusion
We have considered studies where microarray data for K treatment groups are collected from the same subjects (blocks). We discover the genes differentially expressed among K groups using nonparametric Fstatistics for blocked oneway ANOVA while controlling the FDR. We employ a permutation method to generate the null distribution of the Fstatistics without a normal distribution assumption for the gene expression data. The permutationbased multiple testing procedure can be easily modified for controlling the familywise error rate, see e.g. Westfall and Young [28] and Jung et al. [29].
We propose a simple sample size calculation method to estimate the required number of subjects (blocks) given the total number of genes m, number of differentially expressed genes m_{1} and their standardized effect sizes (, 1 ≤ i ≤ m_{1}, 1 ≤ k ≤ K) and the number of true rejections r_{1} at a specified FDR level q*. Through simulations and analysis of a real data set, we found that the permutationbased analysis method controls the FDR accurately and the sample size formula performs accurately. While we specify the individual effect sizes for the prognostic genes, some investigators [30, 31] use a mixture model for the marginal pvalues by specifying a distribution for the effect sizes among m genes.
Glueck et al. [32] propose an exact calculation of average power for the BenjaminiHochberg [2] procedure for controlling the FDR. Their formula may is useful for deriving sample sizes when the test statistics are independent and the number of hypotheses m is small. However, it is not appropriate for designing a microarray study with a large number of dependent test statistics.
A sample size calculation program in R is available from http://www.duke.edu/~is29/BlockANOVA/.
Appendix
We want to prove that F_{ i }converges to in distribution regardless of the normal distribution assumption on ϵ_{ ijk }and γ_{ ij }. We only assume that . The following is one of key lemmas used to derive the distribution of the Fstatistics in the standard ANOVA theory, see e.g. Section 3b.4 of Rao [33].
Lemma: Suppose that, for k = 1,..., K, z_{ k }are independent N (μ_{ k }, 1) random variables and A is an idempotent K × K matrix with rank ν. Let z= (z_{1},..., z_{ K })^{T}and μ= (μ_{1},..., μ_{ K })^{T}. Then, .
We have
where and . By the strong law of large numbers, we have , and almost surely (a.s.).
Hence,
Let and . Then, z_{1},..., z_{ K }are independent and, by the central limit theorem, z_{ k }is approximately . Let I be the K × K identity matrix, 1 = (1,..., 1)^{T}the K × 1 vector with components 1, z= (z_{1},..., z_{ K })^{T}A = I  K^{1} 11^{T}. Note that A is an idempotent matrix with rank K  1 and , where . Then, is approximately distributed as by the lemma. Since , is approximately distributed as . By combining this result with (A.1) using the Slutsky's theorem, we complete the proof.
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Acknowledgements
We are grateful to Holly Dressler and the staff of the Duke Microarray Facility for their assistance in the conduct of the genomewide expression profiling. Funding in part was provided by NIH 1UL1RR024128 and Department of Obstetrics and Gynecology Duke University.
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SJ proposed the research project and wrote the manuscript. IS performed statistical analysis. SLG supported the research and participated in the writing of the manuscripted. PCL was responsible for the study design, conduct and oversight of the experiments and interpretation of results. She contributed to the preparation of the manuscript. LF was responsible for preparing the tissue samples for microarray analysis and interpretation of results and in manuscript preparation. The authors are solely responsible for the content of this study. All authors read and approved the final manuscript.
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Jung, S., Sohn, I., George, S.L. et al. Sample size calculation for microarray experiments with blocked oneway design. BMC Bioinformatics 10, 164 (2009). https://doi.org/10.1186/1471210510164
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Keywords
 False Discovery Rate
 Uterine Fibroid
 Standardize Effect Size
 Prognostic Gene
 Noncentrality Parameter