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
The mean and standard deviation of
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
We divide genes into three sets:
• G_{0}, the set of nondifferentially expressed genes (abbreviated as NDEGs). For all i ∈ G_{0},
•
• 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
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
To simplify theoretical derivation, we assume that the mean expression levels in the
normal phenotype (group B) are zeros (
Therefore, the expected group differences of nonnormalized gene expression data are
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).
For the ith normalized gene expression, its tstatistic is defined as
where
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,
Quantile normalization
With quantile normalization (QUANT), a reference array of empirical quantiles, denoted as q = (q_{1},q_{2},…,q_{m}), is first computed by taking the average across all ordered arrays. Let
The original expressions are replaced by the entries of the reference array with the
same rank. Denote
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
We first investigate the asymptotic properties of sample mean difference
Similarly for downregulated DEGs (i ∈ G1),
Detailed derivations can be found in Section 3 in the Additional file 1.
Additional file 1. Supplementary material.
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Similarly,
More detailed derivations can be found in Section 3 in the Additional file 1.
According to Equations (6), (7) and (8), the sample mean difference and pooled sample
standard deviation both grow at most linearly as functions of e^{+ }(e^{}). As a result, the (absolute values of) tstatistics
Similarly, the tstatistics
To see this mixture under the normality assumption, we assume that all observed gene
expressions
Here f_{t}, f_{T(γ) }and f_{T(γ,λ) }are the density functions of central, noncentral and doubly noncentral tdistributions, respectively, with ν = 2 n  2 degrees of freedom. γ ∝ O(e^{+},e^{}) is the numerator noncentrality parameter and λ ∝ O((e^{+})^{2},(e^{})^{2}) is the denominator noncentrality parameter (from noncentral χ^{2}) [35]. Similarly, according to Equation (10), the tstatistics
In microarray analysis it is reasonable to assume m_{1 }≪ m, i.e., the proportion of DEGs is small (m1 ≪ m and
Figure 1. Empirical density estimates of the tstatistics before and after quantile normalization. Empirical density estimates of the tstatistics before and after quantile normalization. Gene expression are simulated
by using normal random numbers with standard deviation 0.35 and genegene correlation
0.9. Total number of genes is m = 1000. Total numbers of truly differentially expressed genes are
Figure 2. Medians of the tstatistic absolute values. Medians of the absolute values of the tstatistics for data with and without quantile normalization. Gene expression are
simulated by using normal random numbers with standard deviation 0.35 and genegene
correlation 0.9. Total number of genes is m = 1000. Total numbers of truly differentially expressed genes are
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
With rank normalization (RANK), we replace each entry in one array by its position (rank) in the ordered array
counted from the smallest value divided by the total number of genes. Denote
This method was proposed by [22] and discussed further in [23].
Compared with QUANT, RANK goes even further in the nonparametric direction. It removes the noise by only preserving
the ordering of observations. We know m is usually very large in a typical microarray study. If the effect size is large such
that the overexpressed genes always take up the top
Here for simplicity, again we assume that the genes take the specified ranks with equal chances within each group. Therefore, the normalized gene expressions no longer depend on the effect size. The expected group differences for rank normalized genes are
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
• 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 (
• 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
We randomly generate 20 sets of data per tuning parameter for SIMU and SIMUBIO. We apply normalization procedures first and then conduct hypothesis tests to obtain raw pvalues. After that, we apply the Bonferroni multiple testing adjustment to get adjusted pvalues. We declare a gene to be differentially expressed if its adjusted pvalue is less than a prespecified significance level 0.05. The estimated mean and standard deviation of the true positives are reported in Figures 3 and 4. Various results with additional tests (Wilcoxon ranksum test and permutation Ntest), sample sizes (n = 15,20) and nonhomogeneous gene correlation structure including false positive plots can be found in Section 6 in the Additional file 1.
Figure 3. Simulation results (SIMU). Average number of true positives as functions of effect size for SIMU. The error bar represents one standard deviation above and below average. Total number of truly differentially expressed genes is 100 with m1+ upregulated and m1 downregulated genes, respectively. Data replicates: 20.
Figure 4. Simulation results (SIMUBIO). Average number of true positives as functions of effect size for SIMUBIO. The error bar represents one standard deviation above and below average. Total number
of truly differentially expressed genes is
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.
High statistical power can be achieved in a study with the following properties.
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.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All three authors have equal contribution to this paper including the original idea, study design, theoretical derivations, simulations and summary of the findings. All authors read and approved the final manuscript.
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.
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