options(signif = 3, digits = 3) knitr::opts_chunk$set(tidy = FALSE, cache = TRUE, autodep = TRUE, fig.height = 5.5, message = FALSE, error = FALSE, warning = TRUE) set.seed(0xdada)

VSN is a method to preprocess microarray intensity data. This can be as simple as

library("vsn") data("kidney") xnorm = justvsn(kidney)

where `kidney`

is an `ExpressionSet`

object with unnormalised data and `xnorm`

the resulting `ExpressionSet`

with calibrated and glog$_2$-transformed data.

M = exprs(xnorm)[,1] - exprs(xnorm)[,2]

produces the vector of generalised log-ratios between the data in the first and second column.

VSN is a model-based method, and the more explicit way of doing the above is

fit = vsn2(kidney) ynorm = predict(fit, kidney)

stopifnot( max(abs(exprs(xnorm) - exprs(ynorm))) < 1e-16, max(abs(exprs(xnorm) - exprs(fit))) < 1e-16, identical(dim(exprs(xnorm)), dim(exprs(ynorm))), identical(dim(exprs(xnorm)), dim(exprs(fit))))

where `fit`

is an object of class `vsn`

that contains
the fitted calibration and transformation parameters, and the method
`predict`

applies the fit to the data.
The two-step protocol is useful when you want to fit the parameters on a
subset of the data, e.g. a set of control or spike-in features,
and then apply the model to the complete set of data
(see Section \@ref(sec:spike) for details). Furthermore, it
allows further inspection of the `fit`

object, e.g. for the
purpose of quality assessment.

Besides `ExpressionSet`

s, there are also `justvsn`

methods for `AffyBatch`

objects from the `r Biocpkg("affy")`

package and
`RGList`

objects from the `r Biocpkg("limma")`

package. They are described
in this vignette.

The so-called glog$_2$ (short for generalised logarithm) is a function that is like the logarithm (base 2) for large values (large compared to the amplitude of the background noise), but is less steep for smaller values. Differences between the transformed values are the generalised log-ratios. These are shrinkage estimators of the logarithm of the fold change. The usual log-ratio is another example for an estimator^[In statistics, the term estimator is used to denote an algorithm that calculates a value from measured data. This value is intended to correspond to the true value of a parameter of the underlying process that generated the data. Depending on the amount of the available data and the quality of the estimator, the intention may be more or less satisfied.] of log fold change. There is also a close relationship between background correction of the intensities and the variance properties of the different estimators. Please see Section \@ref(sec:shrink) for more explanation of these issues.

How does VSN work? There are two components: First, an affine transformation whose aim is to calibrate systematic experimental factors such as labelling efficiency or detector sensitivity. Second, a glog$_2$ transformation whose aim is variance stabilisation.

An affine transformation is simply a shifting and scaling of
the data, i.e. a mapping of the form $x\mapsto (x-a)/s$ with offset
$a$ and scaling factor $s$. By default, a different offset and a
different scaling factor are used for each column, but the same for
all rows within a column. There are two parameters of the function
`vsn2`

to control this behaviour: With the parameter
`strata`

, you can ask `vsn2`

to choose different
offset and scaling factors for different groups ("strata") of
rows. These strata could, for example, correspond to sectors on the
array^[See Section \@ref(sec:strata).]. With the parameter
`calib`

, you can ask `vsn2`

to choose the same
offset and scaling factor throughout^[See
Section \@ref(sec:nocalib).]. This can be useful, for example, if the
calibration has already been done by other means, e.g., quantile
normalisation.

Note that VSN's variance stabilisation only addresses the dependence of the variance on the mean intensity. There may be other factors influencing the variance, such as gene-inherent properties or changes of the tightness of transcriptional control in different conditions. These need to be addressed by other methods.

The dataset `kidney`

contains example data from a spotted cDNA
two-colour microarray on which cDNA from two adjacent tissue samples
of the same kidney were hybridised, one labeled in green (Cy3), one in
red (Cy5). The two columns of the matrix `exprs(kidney)`

contain the green and red intensities, respectively. A local
background estimate^[See Section \@ref(sec:shrink) for more
on the relationship between background correction and variance
stabilising transformations.] was calculated by the image analysis
software and subtracted, hence some of the intensities in
`kidney`

are close to zero or negative. In
Figure \@ref(fig:nkid-scp) you can see the scatterplot of the
calibrated and transformed data. For comparison, the scatterplot of
the log-transformed raw intensities is also shown. The code below involves
some data shuffling to move the data into datasframes for `ggplot`

.

library("ggplot2") allpositive = (rowSums(exprs(kidney) <= 0) == 0) df1 = data.frame(log2(exprs(kidney)[allpositive, ]), type = "raw", allpositive = TRUE) df2 = data.frame(exprs(xnorm), type = "vsn", allpositive = allpositive) df = rbind(df1, df2) names(df)[1:2] = c("x", "y") ggplot(df, aes(x, y, col = allpositive)) + geom_hex(bins = 40) + coord_fixed() + facet_grid( ~ type)

To verify the variance stabilisation, there is the function `meanSdPlot`

. For each
feature $k=1,\ldots,n$ it shows the empirical standard deviation
$\hat{\sigma}_k$ on the $y$-axis versus the rank of the average
$\hat{\mu}_k$ on the $x$-axis.

\begin{equation}
\hat{\mu}*k =\frac{1}{d} \sum*{i=1}^d h_{ki}\quad\quad
\hat{\sigma}*k^2=\frac{1}{d-1}\sum*{i=1}^d (h_{ki}-\hat{\mu}_k)^2
\end{equation}

meanSdPlot(xnorm, ranks = TRUE)

meanSdPlot(xnorm, ranks = FALSE)

The red dots, connected by lines, show the running median of the standard
deviation^[The parameters used were: window width 10%, window
midpoints 5%, 10%, 15%, ... . It should be said that the proper
way to do is with quantile regression such as provided by the
`r CRANpkg("quantreg")`

package---what is done here for these plots is
simple, cheap and should usually be good enough due to the abundance
of data.]. The aim of these plots is to see whether there is a
systematic trend in the standard deviation of the data as a function
of overall expression. The assumption that underlies the usefulness of
these plots is that most genes are not differentially expressed, so
that the running median is a reasonable estimator of the standard
deviation of feature level data conditional on the mean. After
variance stabilisation, this should be approximately a horizontal
line. It may have some random fluctuations, but should not show an
overall trend. If this is not the case, that usually indicates a data
quality problem, or is a consequence of inadequate prior data
preprocessing. The rank ordering distributes the data evenly along
the $x$-axis. A plot in which the $x$-axis shows the average
intensities themselves is obtained by calling the `plot`

command with the argument `ranks=FALSE`

; but this is
less effective in assessing variance and hence is not the default.

The histogram of the generalized log-ratios:

hist(M, breaks = 100, col = "#d95f0e")

The package includes example data from a series of 8 spotted cDNA arrays on which cDNA samples from different lymphoma were hybridised together with a reference cDNA [@Alizadeh].

data("lymphoma") dim(lymphoma)

The 16 columns of the `lymphoma`

object contain the red and
green intensities, respectively, from the 8 slides.

```
pData(lymphoma)
```

We can call `justvsn`

on all of them at once:

lym = justvsn(lymphoma)

```
meanSdPlot(lym)
```

We see that the variance stabilisation worked. As above, we can obtain the generalised log-ratios for each slide by subtracting the common reference intensities from those for the 8 samples:

iref = seq(1, 15, by=2) ismp = seq(2, 16, by=2) M = exprs(lym)[,ismp]-exprs(lym)[,iref] A =(exprs(lym)[,ismp]+exprs(lym)[,iref])/2 colnames(M) = lymphoma$sample[ismp] colnames(A) = colnames(M) j = "DLCL-0032" smoothScatter(A[,j], M[,j], main=j, xlab="A", ylab="M", pch=".") abline(h=0, col="red")

The package `r Biocpkg("affy")`

provides excellent functionality for
reading and processing Affymetrix genechip data, and you are encouraged
to refer to the documentation of the package `r Biocpkg("affy")`

for more information about data structures and methodology.
%
The preprocessing of Affymetrix genechip data involves
the following steps:
(i) background correction,
(ii) between-array normalization,
(iii) transformation and (iv) summarisation.
The VSN method addresses steps (i)--(iii).
For the summarisation, I recommend to use the RMA method [@RMA],
and a simple wrapper that provides all of these is provided through
the method `vsnrma`

.

library("affydata") data("Dilution") d_vsn = vsnrma(Dilution)

For comparison, we also run `rma`

.

d_rma = rma(Dilution)

par(mfrow = c(1, 3)) ax = c(2, 16) plot(exprs(d_vsn)[,c(1,3)], main = "vsn: array 1 vs 3", asp = 1, xlim = ax, ylim = ax, pch = ".") plot(exprs(d_rma)[,c(1,3)], main = "rma: array 1 vs 3", asp = 1, xlim = ax, ylim = ax, pch = ".") plot(exprs(d_rma)[,1], exprs(d_vsn)[,1], xlab = "rma", ylab = "vsn", asp = 1, xlim = ax, ylim = ax, main = "array 1", pch = ".") abline(a = 0, b =1, col = "#ff0000d0")

Both methods control the variance at low intensities, but we see that VSN does so more strongly. See also Section \@ref(sec:shrink) for further discussion on the VSN shrinkage.

There is a `justvsn`

method for `RGList`

objects.
Usually, you will produce an `RGList`

from your own data using the
`read.maimages`

from the `r Biocpkg("limma")`

package. Here,
for the sake of demonstration, we construct an `RGList`

from
`lymphoma`

.

library("limma") wg = which(lymphoma$dye=="Cy3") wr = which(lymphoma$dye=="Cy5") lymRG = new("RGList", list( R=exprs(lymphoma)[, wr], G=exprs(lymphoma)[, wg])) lymNCS = justvsn(lymRG)

The `justvsn`

method for `RGList`

converts its argument into an `NChannelSet`

, using a copy
of the coercion method from Martin Morgan in the package `r Biocpkg("convert")`

.
It then passes this on to
the `justvsn`

method for `NChannelSet`

.
The return value is an `NChannelSet`

, as shown below.

lymNCS

Note that, due to the flexibility in the amount and
quality of metadata that is in an `RGList`

, and due to
differences in the implementation of these classes,
the transfer of the metadata into the `NChannelSet`

may not always produce the expected results, and that some
checking and often further dataset-specific postprocessing
of the sample metadata and the array feature annotation
is needed. For the current example, we
construct the `AnnotatedDataFrame`

object `adf`

and assign it into the `phenoData`

slot of `lymNCS`

.

vmd = data.frame( labelDescription = I(c("array ID", "sample in G", "sample in R")), channel = c("_ALL", "G", "R"), row.names = c("arrayID", "sampG", "sampR")) arrayID = lymphoma$name[wr] stopifnot(identical(arrayID, lymphoma$name[wg])) ## remove sample number suffix sampleType = factor(sub("-.*", "", lymphoma$sample)) v = data.frame( arrayID = arrayID, sampG = sampleType[wg], sampR = sampleType[wr]) v adf = new("AnnotatedDataFrame", data = v, varMetadata = vmd) phenoData(lymNCS) = adf

Now let us combine the red and green values from each array
into the glog-ratio M
and use the linear modeling tools from `r Biocpkg("limma")`

to find
differentially expressed genes (note that it is often suboptimal
to only consider M, and that taking into account absolute
intensities as well can improve analyses).

lymM = (assayData(lymNCS)$R - assayData(lymNCS)$G)

design = model.matrix( ~ lymNCS$sampR) lf = lmFit(lymM, design[, 2, drop=FALSE]) lf = eBayes(lf)

The following plots show the resulting $p$-values and the expression profiles of the genes corresponding to the top 5 features.

par(mfrow=c(1,2)) hist(lf$p.value, breaks = 100, col="orange") pdat = t(lymM[order(lf$p.value)[1:5],]) matplot(pdat, lty = 1, type = "b", lwd = 2, col=hsv(seq(0,1,length=5), 0.7, 0.8), ylab = "M", xlab = "arrays")

Many image analysis programmes for microarrays provide local background estimates, which are typically calculated from the fluorescence signal outside, but next to the features. These are not always useful. Just as with any measurement, these local background estimates are also subject to random measurement error, and subtracting them from the foreground intensities will lead to increased random noise in the signal. On the other hand side, doing so may remove systematic artifactual drifts in the data, for example, a spatial gradient.

So what is the optimal analysis strategy, should you subtract local background estimates or not? The answer depends on the properties of your particular data. VSN itself estimates and subtracts an over-all background estimate (per array and colour, see Section \@ref(sec:calib)), so an additional local background correction is only useful if there actually is local variability across an array, for example, a spatial gradient.

Supposing that you have decided to subtract the local background
estimates, how is it done?
When called with the argument `backgroundsubtract=TRUE`

^[
Note that the default value for this parameter is `FALSE`

.],
the `justvsn`

method will subtract local background estimates in
the `Rb`

and `Gb`

slots of the incoming `RGList`

.
To demonstrate this, we construct an `RGList`

object `lymRGwbg`

.

rndbg = function(x, off, fac) array(off + fac * runif(prod(dim(x))), dim = dim(x)) lymRGwbg = lymRG lymRGwbg$Rb = rndbg(lymRG, 100, 30) lymRGwbg$Gb = rndbg(lymRG, 50, 20)

In practice, of course, these values will be read from the image
quantitation file with a function such as `read.maimages`

that produces the `RGList`

object. We can call `justvsn`

lymESwbg = justvsn(lymRGwbg[, 1:3], backgroundsubtract=TRUE)

Here we only do this for the first 3 arrays to save compute time.

By default, VSN computes one normalisation transformation with a common set of parameters
for all features of an array (separately for each colour if it is a multi-colour
microarray), see Section \@ref(sec:calib). Sometimes, there is a need for stratification
by further variables of the array manufacturing process, for example, print-tip groups
(sectors) or microtitre plates. This can be done with the `strata`

parameter of
`vsn2`

.

The example data that comes with the package does not directly provide the information which print-tip each feature was spotted with, but we can easily reconstruct it:

ngr = ngc = 4L nsr = nsc = 24L arrayGeometry = data.frame( spotcol = rep(1:nsc, times = nsr*ngr*ngc), spotrow = rep(1:nsr, each = nsc, times=ngr*ngc), pin = rep(1:(ngr*ngc), each = nsr*nsc))

and call

EconStr = justvsn(lymRG[, 1], strata = arrayGeometry$pin)

To save CPU time, we only call this on the first array. We compare the
result to calling `justvsn`

without `strata`

,

EsenzaStr = justvsn(lymRG[, 1])

A scatterplot comparing the transformed red intensities, using the two models, is shown in Figure \@ref(fig:figstrata2).

j = 1 plot(assayData(EsenzaStr)$R[,j], assayData(EconStr)$R[,j], pch = ".", asp = 1, col = hsv(seq(0, 1, length=ngr*ngc), 0.8, 0.6)[arrayGeometry$pin], xlab = "without strata", ylab = "print-tip strata", main = sampleNames(lymNCS)$R[j])

The parameter estimation algorithm of VSN is able to deal with
missing values in the input data. To demonstrate this, we generate an
`ExpressionSet`

`lym2`

in which about 10% of all intensities
are randomly missing,

lym2 = lymphoma nfeat = prod(dim(lym2)) wh = sample(nfeat, nfeat/10) exprs(lym2)[wh] = NA table(is.na(exprs(lym2)))

and call `vsn2`

on it.

fit1 = vsn2(lymphoma, lts.quantile=1) fit2 = vsn2(lym2, lts.quantile=1)

The resulting fitted parameters are not identical, but very similar, see Figure \@ref(fig:miss). %

par(mfrow=c(1,2)) for(j in 1:2){ p1 = coef(fit1)[,,j] p2 = coef(fit2)[,,j] d = max(abs(p1-p2)) stopifnot(d < c(0.05, 0.03)[j]) plot(p1, p2, pch = 16, asp = 1, main = paste(letters[j], ": max diff=", signif(d,2), sep = ""), xlab = "no missing data", ylab = "10% of data missing") abline(a = 0, b = 1, col = "blue") }

Note that `p1`

and `p2`

would differ more if we used a different value than 1 for the `lts.quantile`

argument in the above calls of `vsn2`

. This is because the outlier removal algorithm will, for this dataset, identify different sets of features as outliers for `fit1`

and `fit2`

and consequently the optimisation result will be slightly different; this difference is arguably negligible compared to the noise level in the data.

Normally, VSN uses all features on the array to fit the calibration
and transformation parameters, and the algorithm relies, to a certain
extent, on the assumption that most of the features' target genes are
not differentially expressed (see also Section \@ref(sec:mostunch)).
If certain features are known to correspond to, or not to
correspond to, differentially expressed targets, then we can help the
algorithm by fitting the calibration and transformation parameters
only to the subset of features for which the "not
differentially expressed" assumption is most appropriate, and then
applying the calibration and transformation to **all**
features. For example, some experimental designs provide "spike-in"
control spots for which we know that their targets' abundance is the
same across all arrays (and/or colours).

For demonstration, let us assume that in the `kidney`

data,
features 100 to 200 are spike-in controls. Then we can obtain a
normalised dataset `nkid`

as follows.

spikeins = 100:200 spfit = vsn2(kidney[spikeins,], lts.quantile=1) nkid = predict(spfit, newdata=kidney)

Note that if we are sufficiently confident that the `spikeins`

subset is really not differentially expressed, and also has no
outliers for other, say technical, reasons, then we can set the
robustness parameter `lts.quantile`

to 1. This corresponds no
robustness (least sum of squares regression), but makes most use of
the data, and the resulting estimates will be more precise, which may
be particularly important if the size of the `spikeins`

set is
relatively small.

Not that this explicit subsetting strategy is designed for features
for which we have a priori knowledge that their normalised
intensities should be unchanged. There is no need for you to devise
data-driven rules such as using a first call to VSN to get a
preliminary normalisation, identify the least changing features, and
then call VSN again on that subset. This strategy is already built
into the VSN algorithm and is controlled by its `lts.quantile`

parameter. Please see Section \@ref(sec:mostunch) and
reference [@Huber:2003:SAGMB] for details.

So far, we have considered the joint normalisation of a set of arrays to each other. What happens if, after analysing a set of arrays in this fashion, we obtain some additonal arrays? Do we re-run the whole normalisation again for the complete, new and bigger set of arrays? This may sometimes be impractical.

Suppose we have used a set of training arrays for setting up a classifier that is able to discriminate different biological states of the samples based on their mRNA profile. Now we get new test arrays to which we want to apply the classifier. Clearly, we do not want to re-run the normalisation for the whole, new and bigger dataset, as this would change the training data; neither can we normalise only the test arrays among themselves, without normalising them towards the reference training dataset. What we need is a normalisation procedure that normalises the new test arrays towards the existing reference dataset without changing the latter.

To simulate this situation with the available example data,
pretend that the Cy5 channels of the
`lymphoma`

dataset can be treated as 8
single-colour arrays, and fit a model to the first 7.

ref = vsn2(lymphoma[, ismp[1:7]])

Now we call `vsn2`

on the 8-th array, with the output
from the previous call as the reference.

f8 = vsn2(lymphoma[, ismp[8]], reference = ref)

We can compare this to what we get if we fit the model to all 8 arrays,

fall = vsn2(lymphoma[, ismp])

coefficients(f8)[,1,] coefficients(fall)[,8,]

and compare the resulting values in the scatterplot shown in Figure \@ref(fig:ref): they are very similar.

plot(exprs(f8), exprs(fall)[,8], pch = ".", asp = 1) abline(a = 0, b = 1, col = "red")

stopifnot(length(ismp)==8L) maxdiff = max(abs(exprs(f8) - exprs(fall)[,8])) if(maxdiff>0.3) stop(sprintf("maxdiff is %g", maxdiff))

More details on this can be found in the vignettes
*Verifying and assessing the performance with simulated data*
and
*Likelihood Calculations for vsn*
that come with this package.

If $y_{ki}$ is the matrix of uncalibrated data, with $k$ indexing the rows and $i$ the columns, then the calibrated data $y_{ki}'$ is obtained through scaling by $\lambda_{si}$ and shifting by $\alpha_{si}$:

\begin{equation} y_{ki}' = \lambda_{si}y_{ki}+\alpha_{si} (#eq:cal) \end{equation}

where $s\equiv s(k)$ is the so-called *stratum* for feature $k$. In the
simplest case, there is only one stratum, i.e. the index $s$ is always
equal to 1, or may be omitted altogether. This amounts to assuming that
the data of all features on an array were subject to the same
systematic effects, such that an array-wide calibration is sufficient.

A model with multiple strata per array may be useful for spotted arrays. For these, stratification may be according to print-tip [@Dudoit578] or PCR-plate [@Huber:2003:HSG]. For oligonucleotide arrays, it may be useful to stratify the features by physico-chemical properties, e.g., to assume that features of different sequence composition attract systematically different levels of unspecific background signal.

The transformation to a scale where the variance of the data is approximately independent of the mean is

\begin{eqnarray} h_{ki} &=& \text{arsinh}(\lambda_0y_{ki}'+\alpha_0) (#eq:vs) \ &=& \log\left( \lambda_0y_{ki}'+\alpha_0+ \sqrt{\left(\lambda_0y_{ki}'+\alpha_0\right)^2+1}\right),\nonumber \end{eqnarray}

with two parameters $\lambda_0$ and $\alpha_0$. Equations \@ref(eq:cal) and \@ref(eq:vs) can be combined, so that the whole transformation is given by

\begin{equation} h_{ki} = \text{arsinh}\left(e^{b_{si}}\cdot y_{ki}+a_{si}\right). (#eq:transf) \end{equation}

Here, $a_{si}=\alpha_{si}+\lambda_0\alpha_{si}$ and $b_{si}=\log(\lambda_0\lambda_{si})$ are the combined calibation and transformation parameters for features from stratum $s$ and sample $i$. Using the parameter $b_{si}$ as defined here rather than $e^{b_{si}}$ appears to make the numerical optimisation more reliable (less ill-conditioned).

We can access the calibration and transformation parameters through

```
coef(fit)[1,,]
```

For a dataset with $d$ samples and $s$ strata,
`coef(fit)`

is a numeric array with dimensions $(s, d, 2)$.
For the example data that was used in Section \@ref(sec:overv) to generate
`fit`

, $d=2$ and $s=1$.
`coef(fit)[s, i, 1]`

, the first line in the results of the above code chunk,
is what was called $a_{si}$ in Eqn. \@ref(eq:transf), and
`coef(fit)[s, i, 2]`

, the second line, is $b_{si}$.

VSN is based on the additive-multiplicative error model [@RockeDurbin2001,@Huber:2004:Encyc], which predicts a quadratic variance-mean relationship of the form [@Huber:2002:Bioinformatics]

\begin{equation} v(u)=(c_1u+c_2)^2+c_3. \end{equation}

This is a general parameterization of a parabola with three parameters $c_1$, $c_2$ and $c_3$. Here, $u$ is the expectation value (mean) of the signal, and $v$ the variance. $c_1$ is also called the coefficient of variation, since for large $u$, $\sqrt{v}/u\approx c_1$. The minimum of $v$ is $c_3$, this is the variance of the additive noise component. It is attained at $u=-c_2/c_3$, and this is the expectation value of the additive noise component, which ideally were zero ($c_2=0$), but in many applications is different from zero. Only the behaviour of $v(u)$ for $u\ge -c_2/c_3$ is typically relevant.

The parameters $a$ and $b$ from Equation \@ref(eq:transf)^[I drop the indices $s$, $k$ and $i$, since for the purpose of this section, they are passive] and the parameters of the additive-multiplicative error model are related by [@Huber:2002:Bioinformatics]

\begin{eqnarray} a&=&\frac{c_2}{\sqrt{c_3}}\nonumber\ e^b&=&\frac{c_1}{\sqrt{c_3}}(#eq:errmod2trsf) \end{eqnarray}

This relationship is not 1:1, and it has a divergence at $c_3\to0$; both of these observations have practical consequences, as explained in the following.

- The fact that Equations \@ref(eq:errmod2trsf) do not constitute a 1:1
relationship means that multiple parameter sets
of the additive-multiplicative error model can lead to the same
transformation. This can be resolved, for example, if the coefficient of variation $c_1$ is
obtained by some other means than the
`vsn2`

function. For example, it can be estimated from the standard deviation of the VSN-transformed data, which is, in the approximation of the delta method, the same as the coefficient of variation [@Huber:2002:Bioinformatics,@Huber:2003:SAGMB]. Then,

\begin{eqnarray} c_3&=& c_1^2\, e^{-2b}\nonumber\ c_2&=& c_1 \, a e^{-b}.(#eq:trsf2errmod) \end{eqnarray}

- The divergence for $c_3\to0$ can be a more serious problem. In some datasets, $c_3$ is in fact very small. This is the case if the size of the additive noise is negligible compared to the multiplicative noise throughout the dynamic range of the data, even for the smallest intensities. In other words, the additive-multiplicative error model is overparameterized, and a simpler multiplicative-only model would be good enough. VSN is designed to still produce reasonable results in these cases, in the sense that the transformation stabilizes the variance (it turns essentially into the usual logarithm transformation), but the resulting fit coefficients can be unstable.

The assessment of the precision of the estimated values of $a$ and $b$ (e.g. by resampling, or by using replicate data) is therefore usually not very relevant; what is relevant is an assessment of the precision of the estimated transformation, i.e. how much do the transformed values vary [@Huber:2003:SAGMB].

Now suppose the kidney example data were not that well measured, and the red channel had a baseline that was shifted by 500 and a scale that differed by a factor of $0.25$:

bkid = kidney exprs(bkid)[,1] = 0.25*(500+exprs(bkid)[,1])

We can again call `vsn2`

on these data

bfit = vsn2(bkid)

plot(exprs(bkid), main = "raw", pch = ".", log = "xy")

plot(exprs(bfit), main = "vsn", pch = ".") coef(bfit)[1,,]

Notice the change in the parameter $b$ of the red channel: it is now larger by about $\log(4)\approx 1.4$, and the shift parameter $a$ has also been adjusted.

It is possible to force $\lambda_{si}=1$ and $\alpha_{si}=0$ for all $s$ and $i$ in Equation \@ref(eq:cal) by setting `vsn2`

's parameter `calib`

to `none`

. Hence, only the global variance stabilisation transformation \@ref(eq:vs) will be applied, but no column- or row-specific calibration.

Here, I show an example where this feature is used in conjunction with quantile normalisation.

lym_q = normalizeQuantiles(exprs(lymphoma)) lym_qvsn = vsn2(lym_q, calib="none") plot(exprs(lym_qvsn)[, 1:2], pch=".", main="lym_qvsn")

plot(exprs(lym)[,1], exprs(lym_qvsn)[, 1], main="lym_qvsn vs lym", pch=".", ylab="lym_qvsn[,1]", xlab="lym[,1]")

VSN is a parameter estimation algorithm that fits the parameters for a
certain model. In order to see how good the estimator is, we can look
at bias, variance, sample size dependence, robustness against model
misspecificaton and outliers. This is done in the vignette
*Verifying and assessing the performance with simulated data*
that comes with this package.

Practically, the more interesting question is how different microarray calibration and data transformation methods compare to each other. Two such comparisons were made in reference [@Huber:2002:Bioinformatics], one with a set of two-colour cDNA arrays, one with an Affymetrix genechip dataset. Fold-change estimates from VSN led to higher sensitivity and specificity in identifying differentially expressed genes than a number of other methods.

A much more sophisticated and wider-scoped approach was taken by the
Affycomp benchmark study. It uses two benchmark
datasets: a *Spike-In* dataset, in which a small number of cDNAs
was spiked in at known concentrations and over a wide range of
concentrations on top of a complex RNA background sample; and a
*Dilution* dataset, in which RNA samples from heart and brain
were combined in a number of dilutions and proportions. The design of
the benchmark study, which has been open for anyone to submit their method, was
described in [@Affycomp]. A discussion of its results
was given in [@Affycomp2]. One of the results that emerged
was that VSN compares well with the background correction and
quantile normalization method of RMA; both methods place a high
emphasis on *precision* of the expression estimate, at the price
of a certain *bias* (see also Section \@ref(sec:shrink)). Another
result was that reporter-sequence specific effects (e.g., the effect
of GC content) play a large role in these data and that substantial
improvements can be achieved when they are taken into account
(something which VSN does not do).

Of course, the two datasets that were used in Affycomp were somewhat artificial: they had fewer differentially expressed genes and were probably of higher quality than in most real-life applications. And, naturally, in the meanwhile the existence of this benchmark has led to the development of new processing methods where a certain amount of overfitting may have occured.

I would also like to note the interaction between normalization/preprocessing and data quality. For data of high quality, one can argue that any decent preprocessing method should produce more or less the same results; differences arise when the data are problematic, and when more or less successful measures may be taken by preprocessing methods to correct these problems.

log2.na = function(x){ w = which(x>0) res = rep(as.numeric(NA), length(x)) res[w] = log2(x[w]) return(res) } fc = 0.5 ## true fold change x2 = seq(0.5, 15, by=0.5) ## 'true values' in sample 1 x1 = x2/fc ## 'true values' in sample 2 m = s = numeric(length(x1)) n = 10000 sa = 1 b = 1 sb = 0.1 sdeta = 0.1 for(i in 1:length(x1)){ z1 = sa*rnorm(n)+x1[i]*b*exp(sb*rnorm(n)) z2 = sa*rnorm(n)+x2[i]*b*exp(sb*rnorm(n)) if(i%%2==1) { q = log2.na(z1/z2) m[i] = mean(q, na.rm=TRUE) s[i] = sd(q, na.rm=TRUE) } else { h = (asinh(z1/(sa*b/sb))-asinh(z2/(sa*b/sb)))/log(2) m[i] = mean(h) s[i] = sd(h) } } colq = c("black", "blue") lty = 1 pch = c(20,18) cex = 1.4 lwd = 2

mai = par("mai"); mai[3]=0; par(mai) plot(x2, m, ylim=c(-2, 3.5), type="n", xlab=expression(x[2]), ylab="") abline(h=-log2(fc), col="red", lwd=lwd, lty=1) abline(h=0, col="black", lwd=1, lty=2) points(x2, m, pch=pch, col=colq, cex=cex) segments(x2, m-2*s, x2, m+2*s, lwd=lwd, col=colq, lty=lty) legend(8.75, -0.1, c("q","h"), col=colq, pch=pch, lty=lty, lwd=lwd)

```r)-1$, and $y=\log_2(x+x_{\text{off}})$, where $c=50$ and $x_{\text{off}}=50$.", fig.height = 5, fig.small = TRUE} par(mai = c(0.8, 0.6, 0.01, 0.01)) x = seq(-70.5, 170.5, by=1) cols = c("black", "blue", "darkgrey") xoff = cc = 50 ymat = cbind(log2.na(x), log2( (x+sqrt(x^2+cc^2))/2 ), log2.na(x+xoff)) ylim = range(ymat, na.rm=TRUE) matplot(x, ymat, lty=c(1,1,2), col=cols, lwd=3, type="l", ylim=ylim, ylab="") abline(v = 0, col = "#80808080", lty = 2) legend(x = par("usr")[2], y = par("usr")[3], legend = c(expression(y = log2), expression(y = glog2), expression(y = log2)), fill=cols, density = c(NA, NA, 50), xjust=1.1, yjust=-0.1)

Generalised log-ratios can be viewed as a *shrinkage estimator*: for low intensities either in the numerator and denominator, they are smaller in absolute value than the standard log-ratios, whereas for large intensities, they become equal. Their advantage is that they do not suffer from the variance divergence of the standard log-ratios at small intensities: they remain well-defined and have limited variance when the data come close to zero or even become negative. An illustration is shown in Figure \@ref(fig:shrink). Data were generated from the additive-multiplicative error model [@RockeDurbin2001,@Huber:2003:SAGMB,@Huber:2004:Encyc]. The horizontal line corresponds to the true $\log_2$-ratio $1$ (corresponding to a factor of 2). For intensities $x_2$ that are larger than about ten times the additive noise level $\sigma_a$, generalised log-ratio $h$ and standard log-ratio $q$ coincide. For smaller intensities, we can see a *variance-bias trade-off*: $q$ has almost no bias, but a huge variance, thus an estimate of the fold change based on a limited set of data can be arbitrarily off. In contrast, $h$ keeps a constant variance -- at the price of systematically underestimating the true fold change. This is the main argument for using a variance stabilising transformation. Note that there is also some bias in the behaviour of $q$ for small $x_2$, particularly at $x_2=0.5$. This results from the occurence of negative values in the data, which are discarded from the sampling when the (log-)ratio is computed. Please consult the references for more on the mathematical background [@Huber:2002:Bioinformatics,@Huber:2003:HSG,@Huber:2003:SAGMB]. It is possible to give a Bayesian interpretation: our prior assumption is the conservative one of no differential expression. Evidence from a feature with high overall intensity is taken strongly, and the posterior results in an estimate close to the empirical intensity ratio. Evidence from features with low intensity is downweighted, and the posterior is still strongly influenced by the prior. # Quality assessment{#sec:qc} Quality problems can often be associated with physical parameters of the manufacturing or experimental process. Let us look a bit closer at the `lymphoma` data. Recall that M is the `r nrow(M)` times `r ncol(M)` matrix of generalized log-ratios and `A` a matrix of the same size with the average glog$_2$-transformed intensities. The dataframe `arrayGeometry` (from Section \@ref(sec:strata)) contains, for each array feature, the identifier of the print-tip by which it was spotted and the row and column within the print-tip sector. Figure \@ref(fig:lymbox) shows the boxplots of A values of array CLL-13 stratified by row. ```r colours = hsv(seq(0,1, length = nsr), 0.6, 1) j = "CLL-13" boxplot(A[, j] ~ arrayGeometry$spotrow, col = colours, main = j, ylab = "A", xlab = "spotrow")

You may want to explore similar boxplots for other stratifying factors such as column within print-tip sector or print-tip sector and look at these plots for the other arrays as well.

In Figure \@ref(fig:lymbox), we see that the features in rows 22 and 23 are all very dim. If we now look at these data in the $M$-$A$-plot (Figure \@(fig:lymquscp)), we see that these features not only have low $A$-values, but fall systematically away from the $M=0$ line.

plot(A[,j], M[,j], pch = 16, cex = 0.3, col = ifelse(arrayGeometry$spotrow%in%(22:23), "orange", "black")) abline(h = 0, col = "blue")

Hence, in a naive analysis the data from these features would be interpreted as contributing evidence for differential expression, while they are more likely just the result of a quality problem. So what can we do? There are some options:

- Flag the data of the affected features as unreliable and set them aside from the subsequent analysis.
- Use a more complex, stratified normalisation method that takes into account the different row behaviours, for example, VSN with strata (see Section \@ref(sec:strata)).
- It has also been proposed to address this type of problem by using
a non-linear regression on the $A$-values, for example the
`loess`

normalization of reference [@YeeNorm] that simply squeezes the $M$-$A$-plot to force the centre of the distribution of $M$ to lie at 0 along the whole $A$-range.

An advantage of Option 3 is that it works without knowing the real underlying stratifying factor. However, it assumes that the stratifying factor is strongly confounded with $A$, and that biases that it causes can be removed through a regression on $A$.

In the current example, if we believe that the real underlying stratifying factor is indeed row within sector, this assumption means that (i) few of the data points from rows 22 and 23 have high $A$-values, and that (ii) almost all data points with very low $A$ values are from these rows; while (i) appears tenable, (ii) is definitely not the case.

By default, the VSN method assumes that the measured signal $y_{ik}$ increases, to sufficient approximation, proportionally to the mRNA abundance $c_{ik}$ of gene $k$ on array $i$ (or in colour channel $i$):

\begin{equation} y_{ik}\approx \alpha_i + \lambda_i \lambda_k c_{ik}. (#eq:assumption1) \end{equation}

For a series of $d$ single-colour arrays, $i=1,\ldots,d$, and the different factors $\lambda_i$ reflect the different initial amounts of sample mRNA or different overall reverse transcription, hybridisation and detection efficiencies. The feature affinity $\lambda_k$ contains factors that affect all measurements with feature $k$ in the same manner, such as sequence-specific labelling efficiency. The $\lambda_k$ are assumed to be the same across all arrays. There can be a non-zero overall offset $\alpha_i$. For a two-colour cDNA array, $i=1,2$, and the $\lambda_i$ take into account the different overall efficiencies of the two dyes^[ It has been reported that for some genes the dye bias is different from gene to gene, such that the proportionality factor does not simply factorise as in \@ref(eq:assumption1). As long as this only occurs sporadically, this will not have much effect on the estimation of the calibration and variance stabilisation parameters. Further, by using an appropriate experimental design such as colour-swap or reference design, the effects of gene-specific dye biases on subsequent analyses can be reduced.].

Equation \@ref(eq:assumption1) can be generalised to

\begin{equation} y_{ik}\approx \alpha_{is} + \lambda_{is} \lambda_k c_{ik}. (#eq:assumption2) \end{equation}

that is, the background term $\alpha_{is}$ and the gain factor $\lambda_{is}$ can
be different for different groups $s$ of features on an array. The
VSN methods allows for this option by using the `strata`

argument of the function `vsn2`

. We have seen an example above
where this could be useful. For Affymetrix genechips, one can find
systematic dependences of the affinities $\lambda_{is}$ and the background
terms $\alpha_{is}$ on the reporter sequence.

Nevertheless, there are situations in which either assumption \@ref(eq:assumption1) or \@ref(eq:assumption2) is violated, and these include:

Saturation : The biochemical reactions and/or the photodetection can be run in such a manner that saturation effects occur. It may be possible to rescue such data by using non-linear transformations. Alternatively, it is recommended that the experimental parameters are chosen to avoid saturation.

Batch effects : The feature affinities $\lambda_k$ may differ between different manufacturing batches of arrays due, e.g., to different qualities of DNA amplification or printing. VSN cannot be used to simultaneously calibrate and transform data from different batches.

How to reliably diagnose and deal with such violations is beyond the scope of this vignette; see the references for more [@Huber:2003:HSG,@Dudoit578].

With respect to the VSN model fitting, data from
differentially transcribed genes can act as outliers
(but they do not necessarily need to do so in all cases).
The maximal number of outliers that do not gravely affect the model
fitting is controlled by the parameter `lts.quantile`

.
Its default value is 0.9, which allows for 10% outliers. The value of
`lts.quantile`

can be reduced down to 0.5, which allows for up
to 50% outliers. The maximal value is 1, which results in a
least-sum-of-squares estimation that does not allow for any outliers.

So why is this parameter `lts.quantile`

user-definable and why
don't we just always use the most "robust" value of 0.5?
The answer is that the precision of the estimated VSN
parameters is better the more data points go into the estimates, and
this may especially be an issue for arrays with a small number of
features^[more precisely, number of features per stratum].
So if you are confident that the number of outliers is not that large,
using a high value of `lts.quantile`

can be justified.

There has been confusion on the role of the ``most genes unchanged assumption'', which presumes that only a minority of genes on the arrays is detectably differentially transcribed across the experiments. This assumption is a sufficient condition for there being only a small number of outliers, and these would not gravely affect the VSN model parameter estimation. However, it is not a necessary condition: the parameter estimates and the resulting normalised data may still be useful if the assumption does not hold, but if the effects of the data from differentially transcribed genes balance out.

I acknowledge helpful comments and feedback from Anja von Heydebreck, Richard Bourgon,
Martin Vingron, Ulrich Mansmann and Robert Gentleman. The active development of the
`r Biocpkg("vsn")`

package was from 2001 to 2003, and most of this document was written
during that time (although the current layout is more recent).

```
sessionInfo()
```

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