Smooth quantile normalization is a generalization of quantile normalization, which is average of the two types of assumptions about the data generation process: quantile normalization and quantile normalization between groups.

qsmooth 1.20.0

Global normalization methods such as quantile normalization [@Bolstad2003], [@Irizarry2003] have become a standard part of the analysis pipeline for high-throughput data to remove unwanted technical variation. These methods and others that rely solely on observed data without external information (e.g. spike-ins) are based on the assumption that only a minority of genes are expected to be differentially expressed (or that an equivalent number of genes increase and decrease across biological conditions [@aanes2014normalization]). This assumption can be interpreted in different ways leading to different global normalization procedures. For example, in one normalization procedure, the method assumes the mean expression level across genes should be the same across samples [@Robinson2010]. In contrast, quantile normalization assumes the only difference between the statistical distribution of each sample is technical variation. Normalization is achieved by forcing the observed distributions to be the same and the average distribution, obtained by taking the average of each quantile across samples, is used as the reference [@Bolstad2003].

While these assumptions may be reasonable in certain experiments, they may not always be appropriate [@Loven2012], [@Hicks2015]. For example, mRNA content has been shown to fluctuate significantly during zebrafish early developmental stages [@aanes2014normalization]. Similarly, cells expressing high levels of c-Myc undergo transcriptional amplification causing a 2 to 3 fold change in global gene expression compared to cells expressing low c-Myc [@Loven2012].

Recently, an R/Bioconductor package (`quantro`

) [@Hicks2015]
has been developed to test for global differences between groups of
distributions to evaluate whether global normalization methods such
as quantile normalization should be applied. If global differences
are found between groups of distributions, these changes may be of technical
or biological of interest. If these changes are of technical interest
(e.g. batch effects), then global normalization methods should be applied.
If these changes are related to a biological covariate (e.g. normal/tumor or
two tissues), then global normalization methods should not be applied
because the methods will remove the interesting biological variation
(i.e. differentially expressed genes) and artifically induce differences
between genes that were not differentially expressed. In the cases
with global differences between groups of distributions
between biological conditions, quantile normalization is
not an appropriate normalization method. In
these cases, we can consider a more relaxed assumption about the data,
namely that the statistical distribution of each sample should be the
same within biological conditions or groups (compared to the more
stringent assumption of quantile normalization, which states the
statistical distribution is the same across all samples).

In this vignette we introduce a generalization of quantile
normalization, referred to as **smooth quantile normalization**
(**qsmooth**), which is a weighted average of the two
types of assumptions about the data. The `qsmooth`

R-package
contains the `qsmooth()`

function, which computes a weight at
every quantile that compares the variability between groups relative
to within groups. In one extreme, quantile normalization is applied
and in the other extreme quantile normalization within each
biological condition is applied. The weight shrinks the group-level
quantile normalized data towards the overall reference quantiles
if variability between groups is sufficiently smaller than the
variability within groups. The algorithm is described in
Figure 1 below.

Let `gene(g)`

denote the \({g}^{th}\) row after sorting
each column in the data. For each row, `gene(g)`

, we
compute the weight \(w_{(g)} \in [0, 1]\), where a weight of 0 implies
quantile normalization within groups is applied and
a weight of 1 indicates quantile normalization is applied.
The weight at each row depends on the between group sum of squares
\(\hbox{SSB}_{(g)}\) and total sum of squares \(\hbox{SST}_{(g)}\),
as follows:

\[ w_{(g)} = \hbox{median} \{1 - \hbox{SSB}_{(i)} / \hbox{SST}_{(i)} | ~i = g-k, \dots, g, \dots, g+k \} \] where \(k=\) floor(Total number of genes * 0.05). The number 0.05 is a flexible parameter that can be altered to change the window of the number of genes considered. In this way, we can use a rolling median to borrow information from neighboring genes in the weight.

`knitr::include_graphics("qsmooth_algo.jpg")`

Load the package in R

`library(qsmooth)`

The **bodymapRat** package contains an
`SummarizedExperiment`

of 652 RNA-Seq samples
from a comprehensive rat transcriptomic BodyMap study.
This data was derived from the raw FASTQ files obtained
from Yu et al. (2014) . It contains expression
levels from 11 organs in male and female rats at 4
developmental stages. We will use a subset of this data in
this vignette.

This example is based a dataset which contains brain and liver tissue samples from 21 week old male and female rats. eight samples are from males and eight samples are from females.

```
library(SummarizedExperiment)
library(bodymapRat)
bm_dat <- bodymapRat()
# select brain and liver samples, stage 21 weeks, and only bio reps
keepColumns = (colData(bm_dat)$organ %in% c("Brain", "Liver")) &
(colData(bm_dat)$stage == 21) & (colData(bm_dat)$techRep == 1)
keepRows = rowMeans(assay(bm_dat)) > 10 # Filter out low counts
bm_dat_e1 <- bm_dat[keepRows,keepColumns]
bm_dat_e1
```

```
## class: SummarizedExperiment
## dim: 18629 16
## metadata(0):
## assays(1): counts
## rownames(18629): ENSRNOG00000000007 ENSRNOG00000000010 ... ERCC-00170
## ERCC-00171
## rowData names(0):
## colnames(16): SRR1169975 SRR1169977 ... SRR1170275 SRR1170277
## colData names(22): sraExperiment sraRun ... xtile ytile
```

`qsmooth()`

function`qsmooth()`

The `qsmooth()`

function must have two objects as input:

`object`

: a data frame or matrix with observations (e.g. probes or genes) on the rows and samples as the columns.`qsmooth()`

accepts objects which are a data frame or matrix with observations (e.g. probes or genes) on the rows and samples as the columns.`group_factor`

: a continuous or categorial covariate that represents the group level biological variation about each sample. For example if the samples represent two different tissues, provide`qsmooth()`

with a covariate representing which columns in the`object`

are different tissue samples.`batch`

:**optional**batch covariate (multiple batches are not allowed). If batch covariate is provided,`ComBat()`

from`sva`

is used prior to qsmooth normalization to remove batch effects. See`ComBat()`

for more details.`norm_factors`

:**optional**scaling normalization factors. Default is`NULL`

. If`norm_factors`

is not equal to`NULL`

, the user can provide a vector of scaling factors that will be used to modify the expression data set prior to applying the`qsmooth`

algorithm.`window`

: window size for running median (defined as a fraction of the number of rows of the data object. Default is 0.05.

`qsmooth()`

Here, the groups we are interested in comparing
are the two types of tissues in the 21 week old
male and female rats. The groups we are interested
in comparing is contained in the `organ`

column in the `colData(bm_dat_e1)`

dataset. To run the
`qsmooth()`

function, input the data object and the
object containing the phenotypic data.

The first row shows the boxplots and density plot
of the raw data that has been transformed on the
`log2()`

scale and added a pseudo-count of 1 (i.e.
`log2(counts+1)`

).

The second row shows the boxplots and density plot of the qsmooth normalized data.

```
library(quantro)
par(mfrow=c(2,2))
pd1 <- colData(bm_dat_e1)
counts1 <- assay(bm_dat_e1)[!grepl("^ERCC",
rownames( assay(bm_dat_e1))), ]
pd1$group <- paste(pd1$organ, pd1$sex, sep="_")
matboxplot(log2(counts1+1), groupFactor = factor(pd1$organ),
main = "Raw data", xaxt="n",
ylab = "Expression (log2 scale)")
axis(1, at=seq_len(length(as.character(pd1$organ))),
labels=FALSE)
text(seq_len(length(pd1$organ)), par("usr")[3] -2,
labels = pd1$organ, srt = 90, pos = 1, xpd = TRUE)
matdensity(log2(counts1+1), groupFactor = pd1$organ,
main = "Raw data", ylab= "density",
xlab = "Expression (log2 scale)")
legend('topright', levels(factor(pd1$organ)),
col = 1:2, lty = 1)
qs_norm_e1 <- qsmooth(object = counts1, group_factor = pd1$organ)
qs_norm_e1
```

```
## qsmooth: Smooth quantile normalization
## qsmoothWeights (length = 18581 ):
## 0.933 0.933 0.933 ... 0.887 0.887 0.887
## qsmoothData (nrows = 18581 , ncols = 16 ):
## SRR1169975 SRR1169977 SRR1169979 SRR1169981 SRR1170009
## ENSRNOG00000000007 2174.512012 1801.779468 1713.209769 1947.183795 2009.479744
## ENSRNOG00000000010 408.563196 340.283829 438.121141 495.806792 488.897971
## ENSRNOG00000000012 2.280523 2.118090 1.471557 3.036711 1.431085
## ENSRNOG00000000017 5.506024 6.860026 5.087043 1.605515 6.692905
## ENSRNOG00000000021 93.035004 109.028768 98.549182 113.522437 89.975354
## ENSRNOG00000000024 20.060899 30.947871 35.232957 37.354217 28.396052
## SRR1170011 SRR1170013 SRR1170015 SRR1170239 SRR1170241
## ENSRNOG00000000007 1753.942781 1830.5599808 1836.751451 3.5632320 3.9628398
## ENSRNOG00000000010 533.290776 479.6564095 500.205500 0.6699115 1.5252205
## ENSRNOG00000000012 1.821102 0.7560356 2.969323 0.6699115 2.7658111
## ENSRNOG00000000017 1.193240 5.8792008 6.034787 2.5774451 0.2429941
## ENSRNOG00000000021 100.365576 92.8453323 88.179494 27.8363393 37.4890104
## ENSRNOG00000000024 19.681690 25.2376300 35.361946 178.5266277 197.0539613
## SRR1170243 SRR1170245 SRR1170271 SRR1170273 SRR1170275
## ENSRNOG00000000007 4.5519846 3.7718210 6.134075 3.780457 6.2792639
## ENSRNOG00000000010 0.2404039 0.2911755 1.949617 2.618445 0.1800509
## ENSRNOG00000000012 2.5981055 1.5737439 1.949617 0.236370 0.1800509
## ENSRNOG00000000017 2.5981055 1.5737439 1.949617 2.618445 0.1800509
## ENSRNOG00000000021 32.4504495 44.2178479 32.026870 40.259346 32.0963011
## ENSRNOG00000000024 192.9318750 206.2424800 199.157984 179.002150 135.9229221
## SRR1170277
## ENSRNOG00000000007 0.1944369
## ENSRNOG00000000010 1.3573017
## ENSRNOG00000000012 0.1944369
## ENSRNOG00000000017 4.8631190
## ENSRNOG00000000021 48.9281491
## ENSRNOG00000000024 171.2001833
## .......
```

```
matboxplot(log2(qsmoothData(qs_norm_e1)+1),
groupFactor = pd1$organ, xaxt="n",
main = "qsmooth normalized data",
ylab = "Expression (log2 scale)")
axis(1, at=seq_len(length(pd1$organ)), labels=FALSE)
text(seq_len(length(pd1$organ)), par("usr")[3] -2,
labels = pd1$organ, srt = 90, pos = 1, xpd = TRUE)
matdensity(log2(qsmoothData(qs_norm_e1)+1), groupFactor = pd1$organ,
main = "qsmooth normalized data",
xlab = "Expression (log2 scale)", ylab = "density")
legend('topright', levels(factor(pd1$organ)), col = 1:2, lty = 1)
```