The impact of quantile and rank normalization procedures on the testing power of gene differential expression analysis
 Xing Qiu^{1},
 Hulin Wu^{1} and
 Rui Hu^{1}Email author
https://doi.org/10.1186/1471210514124
© Qiu et al.; licensee BioMed Central Ltd. 2013
Received: 14 September 2012
Accepted: 7 February 2013
Published: 11 April 2013
Abstract
Background
Quantile and rank normalizations are two widely used preprocessing techniques designed to remove technological noise presented in genomic data. Subsequent statistical analysis such as gene differential expression analysis is usually based on normalized expressions. In this study, we find that these normalization procedures can have a profound impact on differential expression analysis, especially in terms of testing power.
Results
We conduct theoretical derivations to show that the testing power of differential expression analysis based on quantile or rank normalized gene expressions can never reach 100% with fixed sample size no matter how strong the gene differentiation effects are. We perform extensive simulation analyses and find the results corroborate theoretical predictions.
Conclusions
Our finding may explain why genes with well documented strong differentiation are not always detected in microarray analysis. It provides new insights in microarray experimental design and will help practitioners in selecting proper normalization procedures.
Background
Microarray technology has been widely adopted in many genomic related studies in the past decade. Despite its popularity, it is well known that various technical noises exist in microarray experiments [1, 2] due to the limitation of technology. As a remedy, many normalization procedures have been proposed to remove these systematic noises, thus improving the detection of differentially expressed genes. Some efforts have been made to evaluate different normalization procedures [36]. Interested readers are referred to [7, 8] for background and more detailed reviews of normalization procedures.
Quantile normalization is perhaps the most widely adopted method for analyzing microarray data generated by Affymetrix GeneChip platform. Motivated by quantilequantile plot, it makes the empirical distribution of gene expressions pooled from each array to be the same [3]. It is the default option of BioConductor [9], which is a very popular open source software for analyzing microarray data implemented in R[10], the de facto standard statistical computing language in the statistical research community. This algorithm is also used for normalizing Affymetrix exon arrays [11, 12], Illumina BeadChip arrays [1315], Illumina transcriptome sequencing (mRNASeq) data [16], Illumina Infinium whole genome genotyping (WGG) arrays [17], and Solexa/Illumina deep sequencing technology [18], etc. In addition, several other popular normalization procedures are variants of quantile normalization, such as the enhanced quantile normalization [19] and subset quantile normalization [20] designed for microarrays, and the conditional quantile normalization [21] designed primarily for normalizing RNAseq data.
Rank normalization is an alternative to quantile normalization. It replaces each observation by its fractional rank (the rank divided by the total number of genes) within array [22, 23]. This normalization procedure achieves robustness to nonadditive noise at the expense of losing some parametric information of expressions.
After normalization, a pertinent statistical test such as Student’s ttest [24] is applied to these normalized gene expression levels. The resulting pvalues are adjusted by a multiple testing procedure (MTP) in order to control certain quantity of perfamily Type I error, such as familywise error rate (FWER) [2528] and false discovery rate (FDR) [29]. Differentially expressed genes are identified based on a prespecified threshold of adjusted pvalues. More detailed introduction of statistical methods for detecting differentially expressed genes can be found in [3033].
Without compromising the control of type I error, better testing power can be achieved by either increasing sample size or improving the strength of gene differentiation effect (fold changes between different phenotypes). Sometimes large expected differential effects based on biological considerations are invoked as a reason to justify a microarray study with very small sample sizes.
In this study, we find that one cannot “trade” differentiation effects with sample size. When the sample size is small, the statistical power for a gene differentiation analysis will not reach 100% even when the effect size approaches to infinity. This counterintuitive phenomenon is due to the nature of the normalization procedures, which alters both sample mean difference and pooled sample standard deviation of the normalized expressions. As a result, they both grow at most linearly as functions of effect size and their effects cancel out. Our findings provide new insights into microarray experimental design which may help practitioners in selecting appropriate normalization procedures.
Methods
Notations and biological data
Notations
We assume that all expression levels are logtransformed. For convenience, the words “gene” and “gene expression” are used interchangeably to refer to these logtransformed random variables. These genes are indexed by i = 1,2,…,m, where m is the total number of genes.
Let c = A,B be two different phenotypic groups. For simplicity we assume that the number of arrays in both groups are the same and denoted by n. Without loss of generality, phenotypic group A is set to represent the phenotype of interest (usually the disease or the treatment group) and group B the normal phenotype. So up (down) regulation of a gene refers to its over (under) expression in group A. We denote by ${y}_{\mathit{\text{ij}}}^{c}$ the observed expression level of the ith gene recorded on the jth array sampled from the cth phenotypic group. The normalized counterpart of ${y}_{\mathit{\text{ij}}}^{c}$ is written as ${y}_{\mathit{\text{ij}}}^{\ast c}$.
The mean and standard deviation of ${y}_{\mathit{\text{ij}}}^{c}$ are denoted by $\mathrm{E}\left({y}_{\mathit{\text{ij}}}^{c}\right)={\mu}_{i}^{c}$ and $\text{var}\left({y}_{\mathit{\text{ij}}}^{c}\right)={\sigma}_{\mathit{\text{ic}}}^{2}$, respectively. Their normalized sample counterparts are denoted by ${\stackrel{\u0304}{y}}_{i\xb7}^{\ast c}=\frac{1}{n}\sum _{k=1}^{n}{y}_{\mathit{\text{ik}}}^{\ast c}$ and ${\left({\widehat{\sigma}}_{i}^{\ast c}\right)}^{2}=\frac{1}{n1}\sum _{j=1}^{n}{({y}_{\mathit{\text{ij}}}^{\ast c}{\stackrel{\u0304}{y}}_{i\xb7}^{\ast c})}^{2}$, respectively.
In practice, the true level of gene differentiation is not a constant. It depends on the biological settings. The variance of gene expressions is nor constant either — it depends on the accuracy of measuring instruments and the homogeneity of biological subjects, just to name a few factors. In terms of statistical power, the decrease of gene expression variance is equivalent to the increase of mean difference. For simplicity, we consider gene expression variance to be fixed and define the effect size, our analysis tuning parameter, to be the expected mean difference of the ith gene expression between two phenotypes ${e}_{i}:={\mu}_{i}^{A}{\mu}_{i}^{B}$.
We divide genes into three sets:

G_{0}, the set of nondifferentially expressed genes (abbreviated as NDEGs). For all i ∈ G_{0}, ${e}_{i}:={\mu}_{i}^{A}{\mu}_{i}^{B}=0$.

, the set of upregulated genes. For all $i\in {G}_{1}^{+}$, e_{ i }> 0.${G}_{1}^{+}$

G1, the set of downregulated genes. For all i ∈ G 1, e_{ i }< 0.
The set of differentially expressed genes (abbreviated as DEGs) is the union of both upregulated and downregulated genes, which is denoted by ${G}_{1}={G}_{1}^{+}\cup {G}_{1}^{}$. We write the size of these gene sets by m_{0} = G_{0}, ${m}_{1}^{+}=\left{G}_{1}^{+}\right$, m1 = G1, and m_{1} = G_{1}. Apparently ${m}_{1}={m}_{1}^{+}+{m}_{1}^{}$ and m_{0} + m_{1} = m.
Biological data
The biological dataset used in this study is the childhood leukemia dataset from the St. Jude Children’s Research Hospital database [34]. We select three groups of data: 88 patients (arrays) with hyperdiploid acute lymphoblastic leukemia (HYPERDIP), 79 patients (arrays) with a special translocation type of acute lymphoblastic leukemia (TEL) and 45 patients (arrays) with a T lineage leukemia (TALL). Each patient is represented by an array reporting the logarithm (base 2) of expression level on the set of 9005 genes.
Analytic analysis of the impact of normalization procedures on differential expression analysis
In this section, we evaluate the impact of quantile and rank normalization on ttest. We are especially interested in studying the asymptotic property of the tstatistic as the effect size of differentiation approaches infinity while other parameters such as n and ${\sigma}_{i}^{2}$ are fixed. Empirical evidences in Section “Results and discussion” show that our findings are also valid for other statistical tests such as Wilcoxon ranksum test and permutation based test.
We must point out that all these assumptions are made only for the simplification of the theoretical derivations. Our findings essentially do not depend on these assumptions. This has been confirmed in our biological simulation study in Section “Results and discussion” (SIMUBIO).
where ${\widehat{\sigma}}_{i}^{\ast}=\sqrt{\frac{{\left({\widehat{\sigma}}_{i}^{\ast A}\right)}^{2}+{\left({\widehat{\sigma}}_{i}^{\ast B}\right)}^{2}}{2}}$ is called the pooled sample standard deviation.
The testing power of a twosided ttest is determined by the absolute value of tstatistic. Based on Equation (3), it is clear that the testing power converges to 100% when n approaches infinity. For a fixed n (which also implies a fixed number of degrees of freedom), the testing power is determined by the absolute sample mean difference, ${\stackrel{\u0304}{y}}_{i\xb7}^{\ast A}{\stackrel{\u0304}{y}}_{i\xb7}^{\ast B}$, and the pooled sample variance, ${\left({\widehat{\sigma}}_{i}^{\ast}\right)}^{2}$. Below we study the asymptotic properties of these two quantities for quantile and rank normalized expressions separately.
Quantile normalization
We refer the reader to [3] for more details.
In group A, over(under)expressed genes tend to have high (low) ranks in each array. When the effect size is small, the ranks of DEGs in group A are mixed with those of NDEGs and the downstream testing power will be low. When the effect size is large, the DEGs in group A effectively take up all the top and bottom ranks, so the NDEGs in group A can only compete for ranks between m1 + 1 and $m{m}_{1}^{+}$. We assume that the ${m}_{1}^{+}$ upregulated genes almost always take the top ${m}_{1}^{+}$ ranks with equal chances and the m1 downregulated genes almost always take the bottom m1 ranks with equal chances. We will show that the Student’s tstatistic of quantile normalized gene expressions follows a mixture distribution in which the doubly noncentral part converges to a distribution with finite all order moments instead of infinity when the true effect size becomes large.
Detailed derivations can be found in Section 3 in the Additional file 1.
More detailed derivations can be found in Section 3 in the Additional file 1.
Empirical evidences in Section “Results and discussion” also show that the statistical power converges to a fixed number strictly less than 1.0; and this convergence is independent of the hypothesis testing methods and MTPs being applied. Heuristically speaking, QUANT “borrows” information from both NDEGs and DEGs to reduce data variation, and as a result the normalized expressions are complex mixture of both NDEGs and DEGs with possibly very high true group differences. Consequently, the variances of normalized DEGs are asymptotically dominated by the differences between the NDEGs and DEGs and become increasing functions of effect sizes. Asymptotically, the increased variances cancel out the contributions of the increased effect sizes to the testing power.
Rank normalization
This method was proposed by [22] and discussed further in [23].
It is easy to check that the pooled standard deviation is also independent of the effect size. As a result, the testing power with rank normalization converges to a constant strictly less than 1.0 as the effect size increases. More details can be found in Section 5 in the Additional file 1.
Simulation studies
Extensive simulations are conducted to verify above theoretical predictions. We document these simulation studies in this section.
Simulation data
Two sets of simulated data are used in this study. Each set of data has two groups of n arrays representing gene expressions under two phenotypic groups (group A and group B). The numbers of up and down regulated genes are denoted by ${m}_{1}^{+}$ and m1, respectively. Without loss of generality, group B is set to represent the normal phenotype, so up (down) regulation of a gene refers to its over (under) expression in group A.

SIMU: Each array has m = 1000 genes. The number of differentially expressed genes (DEGs) is set to be 100, which implies that the number of nondifferentially expressed genes (NDEGs) is m_{0} = 900. For both groups, all genes are normally distributed with standard deviation σ = 0.35 which is estimated from the biological data. Every two distinct genes have correlation coefficient 0.9 which is estimated from the biological data. As a reference, the sample Pearson correlation coefficient averaged over all pairs of genes for biological data used in this study are: 0.91 for HYPERDIP, 0.93 for TEL, and 0.91 for TALL. The algorithm used to generate these correlated observations is stated in [36] and is similar to the method used in [37]. This high correlation between nonnormalized gene expressions can introduce high correlation between the test statistics [38] and result in high instability of the list of DEGs. This phenomenon was documented and discussed in [39]. We also conduct simulations with nonhomogeneous gene correlation structure and the results are similar to that of SIMU. Details can be found in Section 6 of the Additional file 1.

The expectations of DEGs in group A (${y}_{\mathit{\text{ij}}}^{A}$, i = 1, 2,…,m1+ + m1, j = 1,2,…,n) are set to be a constant e for overexpressed genes (i = 1,…,m1+) and e for underexpressed genes (i= m1 + 1,…, 100). Here the effect size e takes value in {0.2,0.4,⋯, 3.4,3.6}. $({m}_{1}^{+},{m}_{1}^{})$ is set to be either (60,40) (balanced differential expression structure) or (90,10) (unbalanced differential expression structure). For all genes in group B and NDEGs in group A, their expectations are set to be 0. The sample size in each group is set to be n, taking values in {5,10}.

SIMUBIO: To match the statistical properties of real gene expression more closely and mimic other noise sources such as nonadditive noise, we apply resampling method to the biological data to construct an additional set of data.

We apply ttest to HYPERDIP and TEL (79 arrays chosen from each set) without any normalization procedure or multiple testing adjustment. At significance level 0.05, 734 genes are detected as DEGs with an unbalanced differential expression structure (677 upregulated and 57 downregulated). We record the mean difference across HYPERDIP and TEL for each DEG as its effect size (e_{ i }). Then we combine HYPERDIP and TEL data and randomly permute the arrays. After that we randomly choose 2n arrays and divide them into two groups A and B of n arrays each, mimicking two biological conditions without differentially expressed genes. Here the sample size n takes value in {5,10}. We add the recorded effect sizes to 734 genes (identified earlier) in group A. We also add addition effect size e to 677 upregulated genes and e to 57 downregulated genes in group A where e takes value in {0,0.2,0.4, ⋯, 3.4,3.6}. These 734 genes are defined as our DEGs in this simulation. Similarly, we apply this resampling procedure to TALL and TEL (45 arrays chosen from each set) and 546 genes are defined to be DEGs with a balanced differential expression structure (259 upregulated and 287 downregulated). The sample size n takes value in {5,10} and the additional effect size e takes value in {0,0.2,0.4,⋯,3.4,3.6}.
Hypothesis testing methods
We use Student’s ttest to compute unadjusted pvalues and then apply the Bonferroni multiple testing adjustment to compute the adjusted pvalues and control the familywise error rate (FWER) at 0.05 level.
Two alternative tests, namely the Wilcoxon ranksum test and permutation Ntest are also used in this study. The results are largely consistent with those obtained from the ttest and can be found in Section 6 in the Additional file 1. The Ntest is a multivariate nonparametric test which has been used to successfully select differentially expressed genes and gene combinations in microarray data analysis [23, 4042]. A brief introduction of this test can be found in Section 1 in the Additional file 1.
Results and discussion
By removing the noise from the observed gene expressions, quantile and rank normalization procedures improve the statistical power of the subsequent differential expression analyses when effect size is small. However, when e becomes large, the testing powers based on the normalized expressions converge to fixed numbers strictly less than 1.0. This confirms our previous theoretical derivations.
Conclusions
Microarray technology has been used in many areas of biomedical research. Biomedical researchers rely on this technology to identify differentially expressed genes. Due to the “large p, small n” nature of the microarray data, multiple testing correction must be applied in differentially expression analysis. As we all know, stringent control of Type I error invariably comes with the price of reduced testing power. However, the success of most microarray studies depends critically on the ability of differential expression analysis to identify the “right genes” and researchers cannot afford to miss many these targets.
 1.
An adequate sample size. Clearly, this is a reliable way to increase statistical power. Everyone seems to agree on it but not everyone practices it. Many years ago this was due to the high cost of conducting microarray experiments. Currently it only costs a fraction to obtain the same number of arrays. In a sense, the myth that “five arrays per group should be good enough” only reflects the fact that it takes a long time to change old, perhaps even anachronic habits.
 2.
Small variance. It is well known that a large proportion of the variance of gene expression is induced by undesirable systematic variations and various technical noise. Microarray technology has been evolving very fast in the past years and we think it is not unreasonable to assume that the technical noise level is getting lower. However, variance induced by biological heterogeneity will not be affected by the advances of technology. For certain data, using a normalization procedure, such as QUANT or RANK, can reduce this variance and help detect DEGs. We must point out that these elegant variance reduction procedures can also alter the mean expression and increase sample variance when the true effect size is large. This biasvariance tradeoff is common in different branches of statistics and should not be conveniently ignored.
 3.
Strong true effect size. Based on our experience, this is often invoked as a reason to justify the use of small sample size in a study a priori. In our study, we demonstrate that one cannot simply “trade” sample size by effect size. Both our theoretical derivations and simulation studies indicate that as long as the sample size is small, the testing power of a typical gene differential expression analysis based on quantile or rank normalized data never reaches 100% no matter how large the effect size is. A large n is still critical for finding informative genes in this situation.
One main motivation of our study is to dismiss the dangerous idea that “five arrays pergroup ought to be good enough for my study”. Our somewhat counterintuitive findings suggest that if data with dramatic gene differentiation have only limited sample size (e.g., less than 10 per group), rank and quantile normalizations may not be able to improve testing power as one expects. For such a scenario we recommend conducting an additional differential expression analysis with other normalization procedure or even without normalization first, and then compare/combine the results with the original analysis with quantile or rank normalization.
Although we choose to focus on the Affymetrix GeneChip platform throughout this paper, we believe our conclusions should be valid for other array platforms which require/recommend normalization, such as Affymetrix exon arrays, Illumina BeadChip arrays and many others. We hope this study can help biological researchers choose an appropriate normalization procedure in their experiments or even develop novel normalization procedures with better downstream testing power when the gene differential expression is dramatic.
Declarations
Acknowledgements
This research is supported by the University of Rochester CTSA award number UL1 RR024160 from the National Center for Research Resources and the National Center for Advancing Translational Sciences of the National Institutes of Health; NIH/NIAID HHSN272201000055C/N01AI50020 from the National Institutes of Health; NIH 5 R01 AI08713502 from the National Institutes of Health; and NIH 2 R01 HL06282609A2 from the National Institutes of Health. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Center for Research Resources or the National Institutes of Health. We appreciate Ms. Christine Brower’s technical assistance with computing. In addition, we would like to thank Ms. Malora Zavaglia and Ms. Jing Che for their proofreading effort.
Authors’ Affiliations
References
 Hartemink AJ, Gifford DK, Jaakkola TS, Young RA: Maximum likelihood estimation of optimal scaling factors for expression array normalization. Proc SPIE BIOS. 2001, 132: Article4266Google Scholar
 Scherer A: Batch Effects and Noise in Microarray Experiments: Sources and Solutions. 2009, Chichester: WileyView ArticleGoogle Scholar
 Bolstad B, Irizarry R, Astrand M, Speed T: A comparison of normalization methods for high density oligonucleotide array data based on variance and bias. Bioinformatics. 2003, 19: 185193. 10.1093/bioinformatics/19.2.185.View ArticlePubMedGoogle Scholar
 Park T, Yi S, Kang S, Lee S, Lee Y, Simon R: Evaluation of normalization methods for microarray data. BMC Bioinformatics. 2003, 4: 3310.1186/14712105433.PubMed CentralView ArticlePubMedGoogle Scholar
 Rao Y, Lee Y, Jarjoura D, Ruppert AS, Liu CG, Hsu JC, Hagan JP: A comparison of normalization techniques for microRNA microarray data. Stat Appl Genet Mol Biol. 2008, 7: Article2210.2202/15446115.1287.PubMedGoogle Scholar
 Pradervand S, Weber J, Thomas J, Bueno M, Wirapati P, Lefort K, Dotto GP, Harshman K: Impact of normalization on miRNA microarray expression profiling. RNA. 2009, 15 (3): 493501. 10.1261/rna.1295509.PubMed CentralView ArticlePubMedGoogle Scholar
 Quackenbush J: Microarray data normalization and transformation. Nat Genet. 2002, 32 (Suppl): 496501. 10.1038/ng1032.View ArticlePubMedGoogle Scholar
 Bilban M, Buehler LK, Head S, Desoye G, Quaranta V: Normalizing DNA microarray data. Curr Issues Mol Biol. 2002, 4 (2): 5764.PubMedGoogle Scholar
 Gentleman RC, Carey VJ, Bates DM, Bolstad B, Dettling M, Dudoit S, Ellis B, Gautier L, Ge Y, Gentry J, Hornik K, Hothorn T, Huber W, Iacus S, Irizarry R, Leisch F, Li C, Maechler M, Rossini AJ, Sawitzki G, Smith C, Smyth G, Tierney L, Yang JYH, Zhang J: Bioconductor: open software development for computational biology and bioinformatics. Genome Biol. 2004, 5 (10): R8010.1186/gb2004510r80.PubMed CentralView ArticlePubMedGoogle Scholar
 R Development Core Team: R: A Language and Environment for Statistical Computing. 2006, Vienna: R Foundation for Statistical Computing, [http://www.Rproject.org] [ISBN 3900051070].Google Scholar
 Okoniewski M, Miller C: Comprehensive analysis of affymetrix exon arrays using BioConductor. PLoS Comput Biol. 2008, 4: e610.1371/journal.pcbi.0040006.PubMed CentralView ArticlePubMedGoogle Scholar
 Robinson MD, Speed TP: A comparison of Affymetrix gene expression arrays. BMC Bioinformatics. 2007, 8: 44910.1186/147121058449.PubMed CentralView ArticlePubMedGoogle Scholar
 Du P, Kibbe WA, Lin SM: Lumi: a pipeline for processing Illumina microarray. Bioinformatics. 2008, 24 (13): 15471548. 10.1093/bioinformatics/btn224.View ArticlePubMedGoogle Scholar
 Schmid R, Baum P, Ittrich C, FundelClemens K, Huber W, Brors B, Eils R, Weith A, Mennerich D, Quast K: Comparison of normalization methods for Illumina BeadChip HumanHT12 v3. BMC Genomics. 2010, 11: 34910.1186/1471216411349.PubMed CentralView ArticlePubMedGoogle Scholar
 Dunning MJ, Smith ML, Ritchie ME, Tavaré S: beadarray: R classes and methods for Illumina beadbased data. Bioinformatics. 2007, 23 (16): 21832184. 10.1093/bioinformatics/btm311.View ArticlePubMedGoogle Scholar
 Bullard JH, Purdom E, Hansen KD, Dudoit S: Evaluation of statistical methods for normalization and differential expression in mRNASeq experiments. BMC Bioinformatics. 2010, 11: 9410.1186/147121051194.PubMed CentralView ArticlePubMedGoogle Scholar
 Staaf J, VallonChristersson J, Lindgren D, Juliusson G, Rosenquist R, Höglund M, Borg A: Normalization of Illumina Infinium wholegenome SNP data improves copy number estimates and allelic intensity ratios. BMC Bioinformatics. 2008, 9: 40910.1186/147121059409.PubMed CentralView ArticlePubMedGoogle Scholar
 ’t Hoen P, Ariyurek Y, Thygesen H, Vreugdenhil E, Vossen R, De Menezes R, Boer J, Van Ommen G, Den Dunnen J: Deep sequencingbased expression analysis shows major advances in robustness, resolution and interlab portability over five microarray platforms. Nucleic Acids Res. 2008, 36 (21): e14110.1093/nar/gkn705.PubMed CentralView ArticlePubMedGoogle Scholar
 Hu J, He X: Enhanced quantile normalization of microarray data to reduce loss of information in gene expression profiles. Biometrics. 2007, 63: 5059. 10.1111/j.15410420.2006.00670.x.View ArticlePubMedGoogle Scholar
 Wu Z, Aryee M: Subset quantile normalization using negative control features. Int J Comput Biol. 2010, 17 (10): 13851395. 10.1089/cmb.2010.0049.View ArticleGoogle Scholar
 Hansen K, Irizarry R, Wu Z: Removing technical variability in RNAseq data using conditional quantile normalization. Biostatistics. 2011, 13 (2): 204216.View ArticleGoogle Scholar
 Tsodikov A, Szabo A, Jones D: Adjustments and measures of differential expression for microarray data. Bioinformatics. 2002, 18 (2): 251260. 10.1093/bioinformatics/18.2.251.View ArticlePubMedGoogle Scholar
 Szabo A, Boucher K, Carroll W, Klebanov L, Tsodikov A, Yakovlev A: Variable selection and pattern recognition with gene expression data generated by the microarray technology. Math Biosci. 2002, 176: 7198. 10.1016/S00255564(01)001031.View ArticlePubMedGoogle Scholar
 Tusher VG, Tibshirani R, Chu G: Significance analysis of microarrays applied to the ionizing radiation response. Proc Natl Acad Sci USA. 2001, 98 (9): 51165121. 10.1073/pnas.091062498.PubMed CentralView ArticlePubMedGoogle Scholar
 Sidak Z: Rectangular confidence regions for the means of multivariate normal distributions. J Am Stat Assoc. 1967, 62: 626633.Google Scholar
 Holm S: A simple sequentially rejective multiple test procedure. Scand J Stat. 1979, 6: 6570.Google Scholar
 Simes R: An improved Bonferroni procedure for multiple tests of significance. Biometrika. 1986, 73 (3): 75110.1093/biomet/73.3.751.View ArticleGoogle Scholar
 Westfall PH, Young SS: ResamplingBased Multiple Testing. 1993, New York: WileyGoogle Scholar
 Benjamini Y, Hochberg Y: Controlling the false discovery rate: A practical and powerful approach to multiple testing. J R Stat Soc Ser B. 1995, 57: 289300.Google Scholar
 Dudoit S, Yang YH, Callow MJ, Speed TP: Statistical methods for identifying differentially expressed genes in replicated cDNA microarray experiments. Stat Sin. 2002, 12: 111139.Google Scholar
 Lee MLT: Analysis of Microarray Gene Expression Data. 2004, New York: SpringerGoogle Scholar
 Bremer M, Himelblau E, Madlung A: Introduction to the statistical analysis of twocolor microarray data. Methods Mol Biol. 2010, 620: 287313. 10.1007/9781607615804_9.View ArticlePubMedGoogle Scholar
 Yakovlev AY, Klebanov L, Gaile D: Statistical Methods for Microarray Data Analysis. 2010, New York: SpringerGoogle Scholar
 Yeoh EJ, Ross ME, Shurtleff SA, Williams WK, Patel D, Mahfouz R, Behm FG, Raimondi SC, Relling MV, Patel A, Cheng C, Campana D, Wilkins D, Zhou X, Li J, Liu H, Pui CH, Evans WE, Naeve C, Wong L, Downing JR: Classification, subtype discovery, and prediction of outcome in pediatric acute lymphoblastic leukemia by gene expression profiling. Cancer Cell. 2002, 1 (2): 133143. 10.1016/S15356108(02)000326.View ArticlePubMedGoogle Scholar
 Johnson NL, Kotz S, Balakrishnan N: Continuous Univariate Distributions, Volumn 2, second edition. 1995, New York: John Wiley & SOns Inc.Google Scholar
 Hu R, Qiu X, Glazko G, Klebanov L, Yakovlev A: Detecting intergene correlation changes in microarray analysis: a new approach to gene selection. BMC Bioinformatics. 2009, 10: 2010.1186/147121051020.PubMed CentralView ArticlePubMedGoogle Scholar
 Tripathi S, EmmertStreib F: Assessment method for a power analysis to identify differentially expressed pathways. PloS one. 2012, 7 (5): e3751010.1371/journal.pone.0037510.PubMed CentralView ArticlePubMedGoogle Scholar
 Qiu X, Hu R: Correlation between the true and false discoveries in a positively dependent multiple comparison problem. IMS Andrei Yakovlev Collection. 2010, Beachwood, Ohio, USA: Institute of Mathematical StatisticGoogle Scholar
 Qiu X, Brooks AI, Klebanov L, Yakovlev A: The effects of normalization on the correlation structure of microarray data. BMC Bioinformatics. 2005, 6: 12010.1186/147121056120.PubMed CentralView ArticlePubMedGoogle Scholar
 Szabo A, Boucher K, Jones D, Tsodikov AD, Klebanov LB, Yakovlev AY: Multivariate exploratory tools for microarray data analysis. Biostatistics. 2003, 4 (4): 555567. 10.1093/biostatistics/4.4.555.View ArticlePubMedGoogle Scholar
 Xiao Y, Frisina R, Gordon A, Klebanov L, Yakovlev A: Multivariate search for differentially expressed gene combinations. BMC Bioinformatics. 2004, 5: 16410.1186/147121055164.PubMed CentralView ArticlePubMedGoogle Scholar
 Klebanov L, Gordon A, Xiao Y, Land H, Yakovlev A: A permutation test motivated by microarray data analysis. Comput Stat and Data Anal. 2006, 50 (12): 36193628. 10.1016/j.csda.2005.08.005.View ArticleGoogle Scholar
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