---
title: "Introduction to vsn (HTML version)"
author: "Wolfgang Huber"
date: "`r doc_date()`"
package: "`r pkg_ver('vsn')`"
output: 
  BiocStyle::html_document2:
    toc: true
bibliography: vsn.bib
vignette: >
  %\VignetteIndexEntry{Introduction to vsn (HTML version)}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding[utf8]{inputenc}
---

```{r setup, echo = FALSE, results = "hide"}
options(signif = 3, digits = 3, width = 42)
knitr::opts_chunk$set(tidy = FALSE, cache = TRUE, autodep = TRUE,
    dev = "png", dpi = 180, message = FALSE, error = FALSE, warning = TRUE)
set.seed(0xdada)
```

<span style="color:red">This is an unfinished port of the package's "old" Sweave/PDF
vignette into Rmarkdown/HTML. There are still bugs in here, like unevaluated LaTeX markup
(incl. footnotes), unmapped cross-references, figure formatting. Please bear with
me. In doubt, please consult the [existing PDF vignette](A-vsn.pdf).</span>
  
# Getting started{#sec:overv}

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

```{r overv1, results = "hide"}
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.

```{r overv2}
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 

```{r overv3, results = "hide"}
fit = vsn2(kidney)
ynorm = predict(fit, kidney)
```
```{r overv4, echo = FALSE, results = "hide"}
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\footnote{%
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\footnote{See Section~\ref{sec:strata}.}. With the parameter
`calib`, you can ask `vsn2` to choose the same
offset and scaling factor throughout\footnote{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.

# Running VSN on data from a single two-colour array

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\footnote{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.

```{r nkid-scp, results = "hide", fig.cap = paste0("Scatterplots of the kidney example data, which were obtained from a two-color cDNA array by quantitating spots and subtracting a local background estimate. a) unnormalised and $\\log_2$-transformed. b) normalised and transformed with VSN. Panel b shows the data from the complete set of ", nrow(kidney), " spots on the array, Panel a only the ", sum(0==rowSums(exprs(kidney)<=0)), " spots for which both red and green net intensities were greater than 0. Those spots which are missing in Panel a are coloured in orange in Panel b."), fig.width = 6, fig.height = 3, fig.wide = TRUE}
library("ggplot2")
allpositive = (rowSums(exprs(kidney) <= 0) == 0)

## some data shuffling to bring data into the right shape and data.frame for ggplot
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}

```{r nkid-sdmean1, fig.height = 3, fig.width = 3, fig.cap = "Standard deviation versus rank of the mean"}
meanSdPlot(xnorm, ranks = TRUE)
```
```{r nkid-sdmean2, fig.height = 3, fig.width = 3, fig.cap = "Standard deviation versus mean"}
meanSdPlot(xnorm, ranks = FALSE) 
```

The red dots, connected by lines, show the running median of the standard
deviation\footnote{The parameters used were: window width 10\%, window
midpoints 5\%, 10\%, 15\%, \ldots.  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:

```{r nkid-histM, fig.cap = "Histogram of generalised log-ratios for the kidney example data.", fig.width = 3.5, fig.height = 3.5}
hist(M, breaks = 100, col = "#d95f0e")
```

# Running VSN on data from multiple arrays ("single colour normalisation"){#sec:lymphoma} 

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].

```{r lymphoma}
data("lymphoma")
dim(lymphoma)
```

The 16 columns of the `lymphoma` object contain the red and
green intensities, respectively, from the 8 slides.

```{r pDatalym}
pData(lymphoma)
``` 

We can call `justvsn` on all of them at once:

```{r lymjustvsn, results = "hide"}
lym = justvsn(lymphoma)
``` 
```{r lym-sdmean, fig.height = 3, fig.width = 3, fig.cap = "Standard deviation versus rank of the mean for the lymphoma example data"}
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:

```{r lym-M, fig.cap = "MA-plot for slide DLCL-0032 from the lymphoma example data. A false-colour representation of the data point density is used, in addition the 100 data points in the least dense regions are plotted as dots.", fig.width = 4.5, fig.height = 4.5}
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")
```


# Running VSN on Affymetrix genechip data{#sec:affy}

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`.

```{r loadaffy1, results = "hide", warning = FALSE}
library("affydata")
data("Dilution")
d_vsn = vsnrma(Dilution)
``` 
For comparison, we also run `rma`.
```{r loadaffy2, results = "hide"}
d_rma = rma(Dilution)
``` 

```{r affy, fig.width = 8, fig.height = 3, fig.wide = TRUE, fig.cap = paste0("Results of `vsnrma` and `rma` on the Dilution data. Array 1 was hybridised with ", pData(Dilution)$liver[1], "$\\mu$g RNA from liver, Array 3 with ", pData(Dilution)$liver[3], "$\\mu$g of the same RNA.")}
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.

# Running VSN on RGList objects{#sec:limma}

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`.

```{r loadlimma, results = "hide"} 
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.

```{r lymNCS}
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`.

```{r addmeta}
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).

```{r lymM}
lymM = (assayData(lymNCS)$R - 
        assayData(lymNCS)$G)
```

```{r design}
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.

```{r limma, fig.width = 8, fig.height = 3.5, fig.wide = TRUE, fig.cap = paste("Left: histogram of p-values from the moderated $t$-test between the",  levels(lymNCS$sampR)[1], "and",  levels(lymNCS$sampR)[2], "groups on the `lymM` values. Right: M-values for the 5 genes with the smallest p-values.")}
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")
``` 


## Background subtraction

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`\footnote{%
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`.

```{r makebg}
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`

```{r justvsnwbg, message = FALSE}
lymESwbg = justvsn(lymRGwbg[, 1:3], backgroundsubtract=TRUE)
``` 

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


## Print-tip groups{#sec:strata}

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:

```{r pinId}
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

```{r strata, results = "hide"}
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`,

```{r nostrata, results = "hide"}
EsenzaStr = justvsn(lymRG[, 1])
```

A scatterplot comparing the transformed red intensities,
using the two models, is shown in \figphrase{vsn-figstrata}.

```{r figstrata2, fig.width = 4, fig.height = 4.5, fig.cap = "Scatterplot of normalised and transformed intensities for the red channel of Array 1. Values on the $x$-axis correspond to normalisation without strata (`EsenzaStr`), values on the $y$-axis to normalisation with strata (`EconStr`). The different colours correspond to the 16 different strata."}
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])
```

# Missing values{#sec:miss}

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,

```{r miss1}
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.

```{r miss2, results = "hide"}
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}.
%
```{r miss, fig.width = 6, fig.height = 3.5, fig.cap="Scatterplots of fitted parameters, values on the $x$-axis correspond to normalisation without missing data (`fit1`), values on the $y$-axis to normalisation with around 10\\% missing data (`fit2`)."}
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.


# Normalisation with 'spike-in' probes{#sec:spike}

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.

```{r spikein, warning = FALSE, message = FALSE}
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.


# Normalisation against an existing reference dataset{#sec:ref}

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.

```{r ref1, results = "hide"}
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.

```{r ref2, results = "hide"}
f8 = vsn2(lymphoma[, ismp[8]], reference = ref)
```

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

```{r ref3, results = "hide"}
fall = vsn2(lymphoma[, ismp])
```
```{r ref4}
coefficients(f8)[,1,]
coefficients(fall)[,8,]
```
 
and compare the resulting values in the scatterplot shown in Figure \@ref{fig:ref}: they are very similar. 

```{r ref, fig.width = 3, fig.height = 3.7, fig.cap = "Scatterplot of normalised intensities after normalisation by reference ($x$-axis, `f8`) and joint normalisation ($y$-axis, `fall`). There is good agreement."}
plot(exprs(f8), exprs(fall)[,8], pch = ".", asp = 1)
abline(a = 0, b = 1, col = "red")
``` 

```{r hiddenchecks, echo = FALSE}
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.

# The calibration parameters{#sec:calib}

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}\label{eq.cal}
y_{ki}' = \lambda_{si}y_{ki}+\alpha_{si}
\end{equation}
where $s\equiv s(k)$ is the so-called \textit{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) \label{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}\label{eq.transf}
h_{ki} = \text{arsinh}\left(e^{b_{si}}\cdot y_{ki}+a_{si}\right). 
\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
```{r nkid-calib1}
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 $b_{si}$. 

## The calibration parameters and the additive-multiplicative error model 

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})\footnote{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}}\label{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.

1.) 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}.\label{eq:trsf2errmod}
\end{eqnarray}

2.) 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].


## More on calibration 

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 $0.25$:

```{r nkid-calib2}
bkid = kidney
exprs(bkid)[,1] = 0.25*(500+exprs(bkid)[,1])
```

We can again call `vsn2` on these data

```{r nkid-calib3, results = "hide"}
bfit = vsn2(bkid)
```
 
```{r nkid-calib4, fig.width = 3, fig.height = 3.7, warning = FALSE, fig.cap = "Scatterplot for the badly biased `bkid` data: log-log scale."}
plot(exprs(bkid), main = "raw", pch = ".", log = "xy")
```
```{r nkid-calib5, fig.width = 3, fig.height = 3.7, warning = FALSE, fig.cap = "Scatterplot for the badly biased `bkid` data: after calibration and transformation with vsn."}
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.  


# Variance stabilisation without calibration{#sec:nocalib}

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.

```{r vsnQ1, results = "hide", fig.width = 3, fig.height = 3.7, fig.cap = "Scatterplot between the red and green intensities of the array of the lymphoma dataset after quantile normalisation followed by VSN variance stabilisation without calibration. "} 
lym_q = normalizeQuantiles(exprs(lymphoma))
lym_qvsn = vsn2(lym_q, calib="none")
plot(exprs(lym_qvsn)[, 1:2], pch=".", main="lym_qvsn")
```
```{r vsnQ2, results = "hide", fig.width = 3, fig.height = 3.7, fig.cap = "Comparison of the values calculated in this section, for CH1 of the first array, to those of VSN variance stabilisation with affine calibration (`lym` was computed in Section \\@ref{sec:lymphoma})."} 
plot(exprs(lym)[,1], exprs(lym_qvsn)[, 1], 
  main="lym_qvsn vs lym", pch=".", 
  ylab="lym_qvsn[,1]", xlab="lym[,1]")
```

# Assessing the performance of VSN

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](http://affycomp.biostat.jhsph.edu).  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.

# VSN, shrinkage and background correction{#sec:shrink}

```{r calcshrink, echo = FALSE, results = "hide"}
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
``` 

```{r shrink, results = "hide", fig.width = 5, fig.height = 4, echo = FALSE, fig.cap = "The shrinkage property of the generalised log-ratio.  Blue diamonds and error bars correspond to mean and standard deviation of the generalised log-ratio $h$, as obtained from VSN, and black dots and error bars to the standard log-ratio $q$ (both base 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 figgraph, results = "hide", fig.width = 3, fig.height = 3, echo = FALSE, fig.cap = "Graphs of the functions. $y=\\log_2(x)$, $y=$glog$_2(x,c)=\\log_2(x+\\sqrt{x^2+c^2})-1$, and $y=\\log_2(x+x_{\\text{off}})$, where $c=50$ and $x_{\\text{off}}=50$."}
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 = log[2](x)), expression(y = glog[2](x,c)), expression(y = log[2](x+x[off]))), 
       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 \figphrase{vsn-figshrink}.  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 lymbox, fig.width = 10, fig.height = 4.5, fig.cap = "Boxplot of A values of array CLL-13 stratified by within-sector row. The features in rows 22 and 23 are all very dim."}
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 \figphrase{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
(\figphrase{vsn-lymquscp}), we see that these features not only have
low $A$-values, but fall systematically away from the $M=0$ line.

```{r lymquscp2, fig.width = 4, fig.height = 3, fig.cap = "MA-plot of the data from array `r j`. Dots coloured in orange are from rows 22 and 23. A possible explanation may be as follows (although I do not know for sure that this is the right explanation): The PCR product (cDNA) that is spotted on these arrays is put on by a print head that sucks cDNA out of microtitre plates and deposits them in spots one after another, row by row. If the content of one plate is faulty, this results in a set of subsequent spots that are faulty. Because the 16 print-tip sectors are spotted in parallel, this affects all sectors in the same way."}
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.


## Stratifying factors such as print-tip, PCR plate, reporter-sequence

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 the $i$-th array, or 
on the $i$-th colour channel:
\begin{equation}\label{assumption.1} 
y_{ik}\approx \alpha_i + \lambda_i \lambda_k c_{ik}.  
\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\footnote{% 
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{assumption.1}). 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{assumption.1} can be generalised to
\begin{equation}\label{assumption.2} 
y_{ik}\approx \alpha_{is} + \lambda_{is} \lambda_k c_{ik}.  
\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, however, the optimal stratification
of reporters based on their sequence is an active area of research.

Nevertheless, there are situations in which either 
assumption~(\ref{assumption.1}) or ~(\ref{assumption.2}) 
is violated, and these include:

\begin{description}
\item[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.

\item[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.
\end{description}

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]. 

## Most genes unchanged assumption{#sec:mostunch}

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\footnote{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.

# Acknowledgements

I acknowledge helpful comments and feedback from Anja von Heydebreck, 
Richard Bourgon, Martin Vingron, Ulrich Mansmann and Robert Gentleman.


# Session info

```{r sessionInfo}
sessionInfo()
```

# References