RUVSeq: Remove Unwanted Variation from RNA-Seq Data
Last modified: November 22, 2022; Compiled: October 24, 2023
Contents
1 Overview
In this document, we show how to conduct a differential expression (DE) analysis that controls for “unwanted variation”, e.g., batch, library preparation, and other nuisance effects, using the between-sample normalization methods proposed in Risso et al. (2014). We call this approach RUVSeq for remove unwanted variation from RNA-Seq data.
Briefly, RUVSeq works as follows. For \(n\) samples and \(J\) genes, consider the following generalized linear model (GLM), where the RNA-Seq read counts are regressed on both the known covariates of interest and unknown factors of unwanted variation,
\[\begin{equation} \log E[Y | W, X, O] = W \alpha + X \beta + O. \end{equation}\]
Here, \(Y\) is the \(n \times J\) matrix of observed gene-level read counts, \(W\) is an \(n \times k\) matrix corresponding to the factors of “unwanted variation” and \(\alpha\) its associated \(k \times J\) matrix of nuisance parameters, \(X\) is an \(n \times p\) matrix corresponding to the \(p\) covariates of interest/factors of “wanted variation” (e.g., treatment effect) and \(\beta\) its associated \(p \times J\) matrix of parameters of interest, and \(O\) is an \(n \times J\) matrix of offsets that can either be set to zero or estimated with some other normalization procedure (such as upper-quartile normalization).
The matrix \(X\) is a random variable, assumed to be known a priori. For instance, in the usual two-class comparison setting (e.g., treated vs. control samples), \(X\) is an \(n \times 2\) design matrix with a column of ones corresponding to an intercept and a column of indicator variables for the class of each sample (e.g., 0 for control and 1 for treated) (McCullagh and Nelder 1989). The matrix \(W\) is an unobserved random variable and \(\alpha\), \(\beta\), and \(k\) are unknown parameters.
The simultaneous estimation of \(W\), \(\alpha\), \(\beta\), and \(k\) is infeasible. For a given \(k\), we consider instead the following three approaches to estimate the factors of unwanted variation \(W\):
RUVguses negative control genes, assumed to have constant expression across samples;RUVsuses centered (technical) replicate/negative control samples for which the covariates of interest are constant;RUVruses residuals, e.g., from a first-pass GLM regression of the counts on the covariates of interest.
The resulting estimate of \(W\) can then be plugged into Equation , for the full set of genes and samples, and \(\alpha\) and \(\beta\) estimated by GLM regression. Normalized read counts can be obtained as residuals from ordinary least squares (OLS) regression of \(\log Y - O\) on the estimated \(W\).
Note that although here we illustrate the RUV approach using the GLM implementation of edgeR and DESeq2, all three RUV versions can be readily adapted to work with any DE method formulated within a GLM framework.
See Risso et al. (2014) for full details and algorithms for each of the three RUV procedures.
2 A typical differential expression analysis workflow
In this section, we consider the RUVg function to estimate the factors of unwanted variation using control genes. See the Sections below for examples using the RUVs and RUVr approaches.
We consider the zebrafish dataset of Ferreira et al. (2014), available through the Bioconductor package zebrafishRNASeq. The data correspond to RNA libraries for three pairs of gallein-treated and control embryonic zebrafish cell pools. For each of the 6 samples, we have RNA-Seq read counts for \(32{,}469\) Ensembl genes and \(92\) ERCC spike-in sequences. See Risso et al. (2014) and the zebrafishRNASeq package vignette for details.
library(RUVSeq)
library(zebrafishRNASeq)
data(zfGenes)
head(zfGenes)
#> Ctl1 Ctl3 Ctl5 Trt9 Trt11 Trt13
#> ENSDARG00000000001 304 129 339 102 16 617
#> ENSDARG00000000002 605 637 406 82 230 1245
#> ENSDARG00000000018 391 235 217 554 451 565
#> ENSDARG00000000019 2979 4729 7002 7309 9395 3349
#> ENSDARG00000000068 89 356 41 149 45 44
#> ENSDARG00000000069 312 184 844 269 513 243
tail(zfGenes)
#> Ctl1 Ctl3 Ctl5 Trt9 Trt11 Trt13
#> ERCC-00163 204 59 183 152 104 59
#> ERCC-00164 6 1 74 11 206 21
#> ERCC-00165 140 119 93 331 52 38
#> ERCC-00168 0 0 0 0 2 0
#> ERCC-00170 216 145 111 456 196 552
#> ERCC-00171 12869 6682 7675 47488 24322 261122.1 Filtering and exploratory data analysis
We filter out non-expressed genes, by requiring more than 5 reads in at least two samples for each gene.
filter <- apply(zfGenes, 1, function(x) length(x[x>5])>=2)
filtered <- zfGenes[filter,]
genes <- rownames(filtered)[grep("^ENS", rownames(filtered))]
spikes <- rownames(filtered)[grep("^ERCC", rownames(filtered))]After the filtering, we are left with 20806 genes and 59 spike-ins.
We store the data in an object of S4 class SeqExpressionSet from the EDASeq package. This allows us to make full use of the plotting and normalization functionality of EDASeq. Note, however, that all the methods in RUVSeq are implemented for both SeqExpressionSet and matrix objects. See the help pages for details.
x <- as.factor(rep(c("Ctl", "Trt"), each=3))
set <- newSeqExpressionSet(as.matrix(filtered),
phenoData = data.frame(x, row.names=colnames(filtered)))
set
#> SeqExpressionSet (storageMode: lockedEnvironment)
#> assayData: 20865 features, 6 samples
#> element names: counts, normalizedCounts, offset
#> protocolData: none
#> phenoData
#> sampleNames: Ctl1 Ctl3 ... Trt13 (6 total)
#> varLabels: x
#> varMetadata: labelDescription
#> featureData: none
#> experimentData: use 'experimentData(object)'
#> Annotation:The boxplots of relative log expression (RLE = log-ratio of read count to median read count across sample) and plots of principal components (PC) reveal a clear need for betwen-sample normalization.
library(RColorBrewer)
colors <- brewer.pal(3, "Set2")
plotRLE(set, outline=FALSE, ylim=c(-4, 4), col=colors[x])
plotPCA(set, col=colors[x], cex=1.2)
We can use the betweenLaneNormalization function of EDASeq to normalize the data using upper-quartile (UQ) normalization (Bullard et al. 2010).
set <- betweenLaneNormalization(set, which="upper")
plotRLE(set, outline=FALSE, ylim=c(-4, 4), col=colors[x])
plotPCA(set, col=colors[x], cex=1.2)
After upper-quartile normalization, treated sample Trt11 still shows extra variability when compared to the rest of the samples. This is reflected by the first principal component, that is driven by the difference between Trt11 and the other samples.
2.2 RUVg: Estimating the factors of unwanted variation using control genes
To estimate the factors of unwanted variation, we need a set of negative control genes, i.e., genes that can be assumed not to be influenced by the covariates of interest (in the case of the zebrafish dataset, the Gallein treatment). In many cases, such a set can be identified, e.g., housekeeping genes or spike-in controls. If a good set of negative controls is not readily available, one can define a set of “in-silico empirical” controls.
Here, we use the ERCC spike-ins as controls and we consider \(k=1\) factors of unwanted variation. See Risso et al. (2014) and Gagnon-Bartsch and Speed (2012) for a discussion on the choice of \(k\).
set1 <- RUVg(set, spikes, k=1)
pData(set1)
#> x W_1
#> Ctl1 Ctl -0.04539413
#> Ctl3 Ctl 0.50347642
#> Ctl5 Ctl 0.40575319
#> Trt9 Trt -0.30773479
#> Trt11 Trt -0.68455406
#> Trt13 Trt 0.12845337
plotRLE(set1, outline=FALSE, ylim=c(-4, 4), col=colors[x])
plotPCA(set1, col=colors[x], cex=1.2)
The RUVg function returns two pieces of information: the estimated factors of unwanted variation (added as columns to the phenoData slot of set) and the normalized counts obtained by regressing the original counts on the unwanted factors. The normalized values are stored in the normalizedCounts slot of set and can be accessed with the normCounts method. These counts should be used only for exploration. It is important that subsequent DE analysis be done on the original counts (accessible through the counts method), as removing the unwanted factors from the counts can also remove part of a factor of interest (Gagnon-Bartsch, Jacob, and Speed 2013).
Note that one can relax the negative control
gene assumption by requiring instead the identification of a set of
positive or negative controls, with a priori known
expression fold-changes between samples, i.e., known \(\beta\).
One can then use the centered counts for these genes (\(\log Y - X\beta\)) for normalization purposes.
2.3 Differential expression analysis
Now, we are ready to look for differentially expressed genes, using the negative binomial GLM approach implemented in edgeR (see the edgeR package vignette for details). This is done by considering a design matrix that includes both the covariates of interest (here, the treatment status) and the factors of unwanted variation.
design <- model.matrix(~x + W_1, data=pData(set1))
y <- DGEList(counts=counts(set1), group=x)
y <- calcNormFactors(y, method="upperquartile")
y <- estimateGLMCommonDisp(y, design)
y <- estimateGLMTagwiseDisp(y, design)
fit <- glmFit(y, design)
lrt <- glmLRT(fit, coef=2)
topTags(lrt)
#> Coefficient: xTrt
#> logFC logCPM LR PValue FDR
#> ENSDARG00000000906 -13.139608 1.3700668 41.83052 9.953755e-11 2.076851e-06
#> ENSDARG00000086720 12.005547 0.6073574 37.82485 7.738980e-10 6.212838e-06
#> ENSDARG00000040816 -15.428901 -0.3395098 37.28917 1.018489e-09 6.212838e-06
#> ENSDARG00000052057 -8.753811 6.0407525 36.98395 1.191054e-09 6.212838e-06
#> ENSDARG00000026651 9.170089 3.2087681 32.92258 9.590304e-09 4.002034e-05
#> ENSDARG00000091974 -11.869688 1.5354803 32.09091 1.471239e-08 5.116235e-05
#> ENSDARG00000094510 8.992629 1.6823082 30.89969 2.717146e-08 8.099035e-05
#> ENSDARG00000069293 -10.447662 0.2042986 30.47915 3.374746e-08 8.801759e-05
#> ENSDARG00000088842 9.993295 0.9907429 30.01999 4.276157e-08 9.913557e-05
#> ENSDARG00000041516 7.747856 7.8305044 28.67513 8.559599e-08 1.785960e-042.4 Empirical control genes
If no genes are known a priori not to be influenced by the covariates of interest, one can obtain a set of “in-silico empirical” negative controls, e.g., least significantly DE genes based on a first-pass DE analysis performed prior to RUVg normalization.
design <- model.matrix(~x, data=pData(set))
y <- DGEList(counts=counts(set), group=x)
y <- calcNormFactors(y, method="upperquartile")
y <- estimateGLMCommonDisp(y, design)
y <- estimateGLMTagwiseDisp(y, design)
fit <- glmFit(y, design)
lrt <- glmLRT(fit, coef=2)
top <- topTags(lrt, n=nrow(set))$table
empirical <- rownames(set)[which(!(rownames(set) %in% rownames(top)[1:5000]))]Here, we consider all but the top \(5{,}000\) genes as ranked by edgeR \(p\)-values.
set2 <- RUVg(set, empirical, k=1)
pData(set2)
#> x W_1
#> Ctl1 Ctl -0.10879677
#> Ctl3 Ctl 0.23066424
#> Ctl5 Ctl 0.19926266
#> Trt9 Trt 0.07672121
#> Trt11 Trt -0.83540924
#> Trt13 Trt 0.43755790
plotRLE(set2, outline=FALSE, ylim=c(-4, 4), col=colors[x])
plotPCA(set2, col=colors[x], cex=1.2)
2.5 Differential expression analysis with DESeq2
In alternative to edgeR, one can perform differential expression analysis with DESeq2. The approach is very similar, namely, we will use the same design matrix, but we need to specify it within the DESeqDataSet object.
library(DESeq2)
dds <- DESeqDataSetFromMatrix(countData = counts(set1),
colData = pData(set1),
design = ~ W_1 + x)
dds <- DESeq(dds)
#> estimating size factors
#> estimating dispersions
#> gene-wise dispersion estimates
#> mean-dispersion relationship
#> final dispersion estimates
#> fitting model and testing
res <- results(dds)
res
#> log2 fold change (MLE): x Trt vs Ctl
#> Wald test p-value: x Trt vs Ctl
#> DataFrame with 20865 rows and 6 columns
#> baseMean log2FoldChange lfcSE stat pvalue
#> <numeric> <numeric> <numeric> <numeric> <numeric>
#> ENSDARG00000000001 232.866 0.8786748 1.389381 0.6324219 0.527111
#> ENSDARG00000000002 499.660 0.8482861 1.250605 0.6783007 0.497581
#> ENSDARG00000000018 467.170 0.9994368 0.926668 1.0785271 0.280799
#> ENSDARG00000000019 6959.358 0.0483696 1.061977 0.0455468 0.963671
#> ENSDARG00000000068 108.518 0.1188108 1.411262 0.0841876 0.932907
#> ... ... ... ... ... ...
#> ERCC-00163 137.567 -1.0762878 1.189237 -0.9050240 0.3654527
#> ERCC-00164 80.339 -1.1169190 2.035970 -0.5485930 0.5832848
#> ERCC-00165 142.686 0.0418648 1.385185 0.0302233 0.9758890
#> ERCC-00170 314.887 1.8354418 1.027683 1.7859999 0.0740993
#> ERCC-00171 26113.478 1.7594758 0.943496 1.8648473 0.0622028
#> padj
#> <numeric>
#> ENSDARG00000000001 0.902319
#> ENSDARG00000000002 0.891360
#> ENSDARG00000000018 0.768956
#> ENSDARG00000000019 0.996814
#> ENSDARG00000000068 0.992634
#> ... ...
#> ERCC-00163 0.830022
#> ERCC-00164 0.920096
#> ERCC-00165 0.998801
#> ERCC-00170 0.495916
#> ERCC-00171 0.462520Note that this will perform by default a Wald test of significance of the last variable in the design formula, in this case \(x\). If one wants to perform a likelihood ratio test, she needs to specify a reduced model that includes \(W\) (see the DESeq2 vignette for more details on the test statistics).
dds <- DESeq(dds, test="LRT", reduced=as.formula("~ W_1"))
res <- results(dds)3 RUVs: Estimating the factors of unwanted variation using replicate samples}
As an alternative approach, one can use the RUVs method to estimate the factors of unwanted variation using replicate/negative control samples for which the covariates of interest are constant.
First, we need to construct a matrix specifying the replicates. In the case of the zebrafish dataset, we can consider the three treated and the three control samples as replicate groups. The function makeGroups can be used.
differences <- makeGroups(x)
differences
#> [,1] [,2] [,3]
#> [1,] 1 2 3
#> [2,] 4 5 6Although in principle one still needs control genes for the estimation of the factors of unwanted variation, we found that RUVs is robust to that choice and that using all the genes works well in practice (Risso et al. 2014).
set3 <- RUVs(set, genes, k=1, differences)
pData(set3)
#> x W_1
#> Ctl1 Ctl 0.1860825
#> Ctl3 Ctl 0.4917394
#> Ctl5 Ctl 0.4926805
#> Trt9 Trt 0.4279274
#> Trt11 Trt -0.5784460
#> Trt13 Trt 0.72891454 RUVr: Estimating the factors of unwanted variation using residuals
Finally, a third approach is to consider the residuals (e.g., deviance residuals) from a first-pass GLM regression of the counts on the covariates of interest. This can be achieved with the RUVr method.
First, we need to compute the residuals from the GLM fit, without RUVg normalization, but possibly after normalization using a method such as upper-quartile normalization.
design <- model.matrix(~x, data=pData(set))
y <- DGEList(counts=counts(set), group=x)
y <- calcNormFactors(y, method="upperquartile")
y <- estimateGLMCommonDisp(y, design)
y <- estimateGLMTagwiseDisp(y, design)
fit <- glmFit(y, design)
res <- residuals(fit, type="deviance")Again, we can use all the genes to estimate the factors of unwanted variation.
set4 <- RUVr(set, genes, k=1, res)5 Session info
sessionInfo()
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Gagnon-Bartsch, J., L. Jacob, and T. P. Speed. 2013. “Removing Unwanted Variation from High Dimensional Data with Negative Controls.” 820. Department of Statistics, University of California, Berkeley.
Gagnon-Bartsch, J., and T. P. Speed. 2012. “Using Control Genes to Correct for Unwanted Variation in Microarray Data.” Biostatistics 13 (3): 539–52.
McCullagh, P, and JA Nelder. 1989. Generalized Linear Models. New York: Chapman; Hall.
Risso, D., J. Ngai, T. P. Speed, and S. Dudoit. 2014. “Normalization of RNA-seq Data Using Factor Analysis of Control Genes or Samples.” Nature Biotechnology.