%\VignetteIndexEntry{Analysing RNA-Seq data with the "DESeq" package}
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\newcommand{\thetitle}{Differential expression of RNA-Seq data at the gene level -- the DESeq package}

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\author{Simon Anders$^1$, Wolfgang Huber\\[1em]European Molecular Biology Laboratory (EMBL),\\ Heidelberg, Germany\\
\texttt{$^1$sanders@fs.tum.de}}

\title{\textsf{\textbf{\thetitle}}}

% The following command makes use of SVN's 'Date' keyword substitution
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\date{\StrMid{$Date: 2012-09-12 04:59:54 -0700 (Wed, 12 Sep 2012) $}{8}{18}}

\begin{document}

\maketitle

\begin{abstract}
A basic task in the analysis of count data from RNA-Seq is the detection of differentially expressed genes.
The count data are presented as a table which reports, for each sample, the number of reads that have been
assigned to a gene. Analogous analyses also arise for other assay types, such as comparative ChIP-Seq.
The package \Rpackage{DESeq} provides methods to test for differential
expression by use of the negative binonial distribution and a shrinkage estimator for the distribution's
 variance\footnote{Other Bioconductor packages with similar aims are \Rpackage{edgeR} and \Rpackage{baySeq}.}.
This vignette explains the use of the package. For an exposition of
the statistical method, please see our paper~\cite{Anders:2010:GB} and the additional information in
Section~\ref{sec:changes}.
\end{abstract}

<<options,results=hide,echo=FALSE>>=
options(digits=3, width=100)
library("pasilla") # make sure this is installed, since we need it in the next section
@

\tableofcontents

%--------------------------------------------------
\section{Input data and preparations} \label{sec:prep}
%--------------------------------------------------
\subsection{The count table}
%--------------------------------------------------
As input, the \Rpackage{DESeq} package expects count data as
obtained, e.\,g., from RNA-Seq or another high-throughput
sequencing experiment, in the form of a rectangular table of integer values.
The table cell in the $i$-th row and the
$j$-th column of the table tells how many reads have been mapped to
gene $i$ in sample $j$. Analogously, for other types of assays, the
rows of the table might correspond e.\,g.\ to binding regions (with
ChIP-Seq) or peptide sequences (with quantitative mass spectrometry).

The count values must be raw counts of sequencing
reads. This is important for \Rpackage{DESeq}'s statistical model to
hold, as only the actual counts allow assessing the measurement
precision correctly. Hence, please do do not supply other quantities,
such as (rounded) normalized counts, or counts of covered base pairs
-- this will only lead to nonsensical results.

Furthermore, it is important that each column stems from an
independent biological replicate.  For technical replicates (e.\,g.\
when the same library preparation was distributed over multiple lanes
of the sequencer), please sum up their counts to get a single column,
corresponding to a unique biological replicate.  This is needed in
order to allow \Rpackage{DESeq} to estimate variability in the
experiment correctly.

To obtain such a count table for your own data, you will need to
create it from your sequence alignments and suitable
annotation. In Bioconductor, you can use the function
\Rfunction{summarizeOverlaps} in the \Rpackage{GenomicRanges}
package. See the vignette, reference~\cite{summarizeOverlaps}, for a
worked example.  Another possibility is provided by the Bioconductor
package \Rpackage{easyRNASeq}.

Another easy way to produce such a table from the output of the
aligner is to use the \texttt{htseq-count} script distributed with the
\emph{HTSeq} Python package. Even though \emph{HTSeq} is written in
Python, you do not need to know any Python to use
\texttt{htseq-count}. See
\url{http://www-huber.embl.de/users/anders/HTSeq/doc/count.html}.
HTSeq-count produces one count file for each sample. \Rpackage{DESeq}
offers the function \Rfunction{newCountDataSetFromHTSeqCount},
described below, to get an analysis started from these files.

In this vignette, we will work with the gene level read counts from
the \Rpackage{pasilla} data package. This data set is from an
experiment on \emph{Drosophila melanogaster} cell cultures and
investigated the effect of RNAi knock-down of the splicing factor
\emph{pasilla}~\cite{Brooks2010}.  The detailed transcript of how we
produced the \Rpackage{pasilla} count table from second generation
sequencing (Illumina) FASTQ files is provided in the vignette of the
data package \Rpackage{pasilla}. The table is
supplied by the \Rpackage{pasilla} package as a text file of
tab-separated values. The function \Rfunction{system.file} tells
us where this file is stored on your computer.
%
<<systemFile>>=
datafile = system.file( "extdata/pasilla_gene_counts.tsv", package="pasilla" )
datafile
@
%
Have a look at the file with a text editor to see how it is formatted. To read
this file with R, we use the function \Rfunction{read.table}.
%
<<readTable>>=
pasillaCountTable = read.table( datafile, header=TRUE, row.names=1 )
head( pasillaCountTable )
@
%
Here, \texttt{header=TRUE} indicates that the first line contains column names
and \texttt{row.names=1} means that the first column should be used as
row names. This leaves us with a \Rclass{data.frame} containing integer
count values.

%--------------------------------------------------
\subsection{The metadata}
%--------------------------------------------------
The best data are useless without metadata. \Rpackage{DESeq} uses
familiar idioms in Bioconductor to manage the metadata that go with
the count table. The metadata can be divided into three groups:
information about the samples (table colums), about the features
(table rows), and about the overall experiment.

First, we need a description of the samples, which we keep in a
\Rclass{data.frame} whose columns correspond to different types of information, and whose
rows correspond to the \Sexpr{ncol(pasillaCountTable)} samples:
<<pasillaDesign>>=
pasillaDesign = data.frame(
   row.names = colnames( pasillaCountTable ),
   condition = c( "untreated", "untreated", "untreated",
      "untreated", "treated", "treated", "treated" ),
   libType = c( "single-end", "single-end", "paired-end",
      "paired-end", "single-end", "paired-end", "paired-end" ) )

pasillaDesign
@
%
Here, simply use a chunk of R code to set up this define. More often,
you will read this into R from a spreadsheet table, using the
R functions \Rfunction{read.table} or \Rfunction{read.csv}; or perhaps
even from a relational database table using R's database access
facilities.

To analyse these samples, we will have to account for the fact that we have
both single-end and paired-end method. To keep things simple at the start, we defer the
discussion of this to Section~\ref{sec:GLM} and first demonstrate a simple
analysis by using only the paired-end samples.
%
<<pairedSamples>>=
pairedSamples = pasillaDesign$libType == "paired-end"
countTable = pasillaCountTable[ , pairedSamples ]
condition = pasillaDesign$condition[ pairedSamples ]
@
%
Now, we have data input as follows.
%
<<>>=
head(countTable)
condition
@
%
For your own data, create such a factor simply with
<<condition,eval=FALSE>>=
#not run
condition = factor( c( "untreated", "untreated", "treated", "treated" ) )
@

<<conditionCheck,results=hide,echo=FALSE>>=
stopifnot( identical( condition, factor( c( "untreated", "untreated", "treated", "treated" ) ) ) )
@
%

We can now instantiate a \Rclass{CountDataSet}, which is the central
data structure in the \Rpackage{DESeq} package:
%
<<instantiate, results=hide>>=
library( "DESeq" )
cds = newCountDataSet( countTable, condition )
@

If you have used \emph{htseq-count} (see above) to obtain your read counts, you can
use the funtion \Rfunction{newCountDataSetFromHTSeqCount} to obtain a CountDataSe
directly from the count files produced by \emph{htseq-count}. To this end,
produce a data frame, similar to \emph{pasillaDesign} constructed above, with
the sample names in the first column, the file names in the scond column, and 
the condition in the third column (or, if you have more than one factor in your design matrix,
all these factors in the third and folling columns), and pass this data frame
to the function. See the help page (``\texttt{?newCountDataSetFromHTSeqCount}'')
for further details.

The \Rclass{CountDataSet} class is derived from \Rpackage{Biobase}'s
\Rclass{eSet} class and so shares all features of this standard
Bioconductor class. Furthermore, accessors are provided for its data
slots\footnote{In fact, the object \Robject{pasillaGenes}
  from the \Rpackage{pasilla} package is of class \Rclass{CountDataSet}, and we could do the subsequent
  analysis on the basis of this object.
  Here we re-create \Robject{cds} from elementary data types, a \Rclass{data.frame} and a \Rclass{factor},
  for pedagogic effect.}.  For example, the counts can be accessed with the
\Rfunction{counts} function, as we will see in the next section.

%------------------------------------------------------------
\subsection{Normalisation}
%------------------------------------------------------------
As a first processing step, we need to estimate the effective library
size. This step is sometimes also called \emph{normalisation}, even
though there is no relation to normality or a normal
distribution.  The effective library size information is called the
\emph{size factors} vector, since the package only needs to know the
relative library sizes.
If the counts of non-differentially
expressed genes in one sample are, on average, twice as high as in
another (because the library was sequenced twice as deeply), the size factor for the first sample should be twice that
of the other sample~\cite{Anders:2010:GB,Anders:2012:GR}. The function
\Rfunction{estimateSizeFactors} estimates the size factors from the
count data. (See the man page of
\Rfunction{estimateSizeFactorsForMatrix} for technical details on the
calculation.)
%
<<estimateSizeFactors>>=
cds = estimateSizeFactors( cds )
sizeFactors( cds )
@

If we divide each column of the count table by the size factor for this column, the
count values are brought to a common scale, making them comparable. When called
with \texttt{normalized=TRUE}, the \texttt{counts} accessor function performs this
calculation. This is useful, e.g., for visualization.
%
<<headcounts2>>=
head( counts( cds, normalized=TRUE ) )
@

%------------------------------------------------------------
\section{Variance estimation}
%------------------------------------------------------------

The inference in \Rpackage{DESeq} relies on an estimation of the typical
relationship between the data's variance and their mean, or, equivalently,
between the data's dispersion and their mean.

The \emph{dispersion} can be understood as the square of the coefficient
of biological variation. So, if a gene's expression typically differs
from replicate to replicate sample by 20\%, this gene's dispersion is $0.2^2=.04$.
Note that the variance seen between counts is the sum of two components: the
sample-to-sample variation just mentioned, and the uncertainty in measuring
a concentration by counting reads. The latter, known as shot noise or Poisson
noise, is the dominating noise source for lowly expressed genes.
The former dominates for highly expressed genes.
The sum of both, shot noise and dispersion, is considered in the differential expression inference.

Hence, the variance $v$ of count values is modelled as
\[ v = s \mu + \alpha s^2\mu^2, \]
where $\mu$ is the expected normalized count value (estimated by the average normalized count value),
$s$ is the size factor for the sample under consideration, and $\alpha$ is the dispersion
value for the gene under consideration.

To estimate the dispersions, use this command.
<<estimateDispersions,cache=TRUE>>=
cds = estimateDispersions( cds )
@

We could now proceed straight to the testing for differetial
expression in Section~\ref{sec:DE}.  However, let us first spend a
little more time with looking at the results of
\Rfunction{estimateDispersions}, in order to better understand the
subsequent inference.  Also, verification of the dispersion plot
Figure~\ref{figFit}, explained below, is an aspect of data quality control: how well do
the data accord to the expectations of our analytic approach? Quality
assessment is a crucial step of the data analysis, and we will see
further quality diagnostics in Section~\ref{sec:quality}.

The function \Rfunction{estimateDispersions} performs three
steps. First, it estimates a dispersion value for each gene, then it
fits a curve through the estimates. Finally, it assigns to each gene a
dispersion value, using a choice between the per-gene estimate and the fitted
value.  To allow the user to inspect the intermediate steps, a
\Robject{fitInfo} object is stored, which contains the per-gene estimate, the fitted curve
and the values that will subsequently be used for inference.
%
<<str>>=
str( fitInfo(cds) )
@
%
\label{fitInfo}
It is useful to inspect the results of these steps visually.  We can
plot the per-gene estimates against the mean normalized counts
per gene and overlay the fitted curve by using the function
\Rfunction{plotDispEsts} (Fig.~\ref{figFit}).
%
\begin{figure}
\centering
\includegraphics[width=.6\textwidth]{DESeq-figFit}
\caption{Empirical (black dots) and fitted (red lines) dispersion values
plotted against the mean of the normalised counts.}
\label{figFit}
\end{figure}
%
<<figFit,fig=TRUE>>=
plotDispEsts( cds )
@

% check a claim made in the paragraph below.
<<echo=FALSE, results=hide>>=
all(table(conditions(cds))==2)
@

As we estimated the dispersions from a small number of replicates,
the estimates scatter with quite some sampling variance around their true values.
An initial assumption that one could make is that the regression line
shown in Figure~\ref{figFit} models the true underlying dispersions,
and that the variation of the point estimates around simply reflects
sampling variance. This is the assumption that we put forward in the
first paper on \Rpackage{DESeq}~\cite{Anders:2010:GB}.  However,
subsequent experience with larger data sets indicates that not all of
the variability of the points around the regression line seen in
Figure~\ref{figFit} is sampling variance: some of it reflects
differences of the true, underlying variance between different genes.
Hence, the default behaviour of \Rpackage{DESeq} now
uses a more prudent or conservative approach: if a per-gene estimates
lies below the regression line, we assume that this might indeed be
sampling variance, and shift the estimate upwards to the value
predicted by the regression line. If, however, the per-gene
estimate lies above the line, we do not shift it downwards to the
line, but rather keep it as is.

The option \texttt{sharingMode} of the function
\Rfunction{estimateDispersions} can be used to control this
behaviour. The value \texttt{sharingMode="maximum"} corresponds to the
default. If you have many replicates (where \emph{many} starts at
around 7), the choice \texttt{sharingMode="gene-est-only"} will
typically be more adequate.  If you would like to use the behaviour
described in~\cite{Anders:2010:GB}, this is achieved by specifying
\texttt{sharingMode="fit-only"}.

Another difference of the current \Rpackage{DESeq} version to the original method described in the
paper is the way how the mean-dispersion relation is fitted. By default,
\Rfunction{estimateDispersions} now performs a parametric fit: Using a gamma-family
GLM, two coefficients $\alpha_0, \alpha_1$ are found to
parametrize the fit as $\alpha = \alpha_0 + \alpha_1/\mu$. (The values of the two
coefficients can be found in the \texttt{fitInfo} object, as attribute \texttt{coefficients}
to \texttt{dispFunc}.) For some data sets, the parametric fit may give bad results, in
which case one should try a local fit (the method described in the paper), which
is available via the option \texttt{fitType="local"} to \Rfunction{estimateDispersions}.

In any case, the dispersion values which will be used by the subsequent testing are stored in
the feature data slot of \texttt{cds}:
%
<<head>>=
head( fData(cds) )
@
%
\fixme{Why is the value for FBgn00000014 Inf and not 0 (given that all counts are 0)?}
You can verify that these values are indeed the maxima from the two value vectors
in \Robject{fitInfo(cds)}, which we saw on page~\pageref{fitInfo}.
Advanced users who want to fiddle with the dispersion estimation can change the values
in \texttt{fData(cds)} prior to calling the testing function.

%------------------------------------------------------------
\section{Inference: Calling differential expression} \label{sec:DE}
%------------------------------------------------------------
\subsection{Standard comparison between two experimental conditions}

Having estimated the dispersion for each gene, it is straight-forward to look for
differentially expressed genes. To contrast two conditions, e.g., to see whether there is differential
expression between conditions ``untreated'' and ``treated'', we simply call the function \Rfunction{nbinomTest}.
It performs the tests as described in~\cite{Anders:2010:GB} and returns a data frame with the $p$ values and other
useful information.
%
<<nbt1,cache=TRUE>>=
res = nbinomTest( cds, "untreated", "treated" )
<<nbt2>>=
head(res)
@
%
<<checkClaims,echo=FALSE,results=hide>>=
stopifnot(identical(colnames(res), c("id", "baseMean", "baseMeanA", "baseMeanB", "foldChange",
                                     "log2FoldChange", "pval", "padj")))
@
The interpretation of the columns of \Rclass{\Sexpr{class(res)}} is as follows.

\noindent
\begin{tabular}{lp{0.8\textwidth}}
\texttt{id}&feature identifier\\
\texttt{baseMean}&mean normalised counts, averaged over all samples from both conditions\\
\texttt{baseMeanA}&mean normalised counts from condition A\\
\texttt{baseMeanB}&mean normalised counts from condition B\\
\texttt{foldChange}&fold change from condition A to B\\
\texttt{log2FoldChange}&the logarithm (to basis 2) of the fold change\\
\texttt{pval}&$p$ value for the statistical significance of this change\\
\texttt{padj}&$p$ value adjusted for multiple testing with the Benjamini-Hochberg procedure (see the R function
\Rfunction{p.adjust}), which controls false discovery rate (FDR)\\
\end{tabular}

\begin{figure}[ht]
\centering
\includegraphics[width=.6\textwidth]{DESeq-figDE}
\caption{Plot of normalised mean versus $log_2$ fold change
for the contrast \emph{untreated} versus \emph{treated}.}
\label{figDE}
\end{figure}

\vspace{1.5ex}
Let us first plot the $\log_2$ fold changes against the mean normalised counts, colouring in red those genes
that are significant at 10\% FDR (Figure~\ref{figDE}).
%
<<figDE,fig=TRUE>>=
plotMA(res)
@ %$
%
\begin{figure}[ht]
\centering
\includegraphics[width=.6\textwidth]{DESeq-histp}
\caption{Histogram of $p$-values from the call to nbinomTest.}
\label{fighistp}
\end{figure}

It is also instructive to look at the histogram of $p$ values (Figure~\ref{fighistp}).
The enrichment of low $p$ values stems from the differentially expressed genes,
while those not differentially expressed are spread uniformly over the range
from zero to one (except for the $p$ values from genes with very low counts, which
take discrete values and so give rise to high counts for some bins at the right.)
%
<<histp,fig=TRUE>>=
hist(res$pval, breaks=100, col="skyblue", border="slateblue", main="")
@

We can filter for significant genes, according to some chosen threshold for the false dicovery rate (FDR),
<<ressig1>>=
resSig = res[ res$padj < 0.1, ]
@
and list, e.\,g., the most significantly differentially expressed genes:
<<ressig2>>=
head( resSig[ order(resSig$pval), ] )
@

We may also want to look at the most strongly down-regulated of the significant
genes,
<<ressig3>>=
head( resSig[ order( resSig$foldChange, -resSig$baseMean ), ] )
@
or at the most strongly up-regulated ones:
<<ressig4>>=
head( resSig[ order( -resSig$foldChange, -resSig$baseMean ), ] )
@

To save the output to a file, use the R functions \Rfunction{write.table} and
\Rfunction{write.csv}. (The latter is useful if you want to load the table in a
spreadsheet program such as Excel.)

<<writetable>>=
write.csv( res, file="My Pasilla Analysis Result Table.csv" )
@

\begin{figure}[ht]
\centering
\includegraphics[width=.6\textwidth]{DESeq-MArepl}
\caption{Plot of the log2 fold change between the untreated replicates
versus average expression strength.}
\label{figMArepl}
\end{figure}

Note in Fig.~\ref{figDE} how the power to detect significant differential expression depends on
the expression strength. For weakly expressed genes, stronger changes are
required for the gene to be called significantly expressed. To understand the reason for
this let, us compare the normalized counts between two replicate samples, here
taking the two untreated samples as an example:
%
<<ncu>>=
ncu = counts( cds, normalized=TRUE )[ , conditions(cds)=="untreated" ]
@
\Robject{ncu} is now a matrix with two columns.
%
<<MArepl,fig=TRUE>>=
plotMA(data.frame(baseMean = rowMeans(ncu),
                  log2FoldChange = log2( ncu[,2] / ncu[,1] )),
       col = "black")
@
%
As one can see in Figure~\ref{figMArepl}, the log fold changes between replicates
are stronger for lowly expressed genes than for highly expressed ones. We
ought to conclude that a gene's expression is influenced by the treatment only
if the change between treated and untreated samples is stronger than
what we see between replicates, and hence, the dividing line between red and black in
Figure~\ref{figDE} mimics the shape seen in Figure~\ref{figMArepl}.

%------------------------------------------------------------
\subsection{Working partially without replicates}
%------------------------------------------------------------

If you have replicates for one condition but not for the other, you can still proceed as
before. In such cases only the conditions with replicates will be used to estimate the dispersion.
Of course, this is only valid if you have good reason to believe that the unreplicated
condition does not have larger variation than the replicated one.

To demonstrate, we subset our data object to only three samples:

<<subset>>=
cdsUUT = cds[ , 1:3]
pData( cdsUUT )
@

\begin{figure}[ht]
\centering
\includegraphics[width=.6\textwidth]{DESeq-figDE_Tb}
\caption{MvA plot for the contrast ``treated'' vs.\ ``untreated'', using two treated and
only one untreated sample.}
\label{figDE_Tb}
\end{figure}

Now, we do the analysis as before.

<<est123,cache=TRUE>>=
cdsUUT = estimateSizeFactors( cdsUUT )
cdsUUT = estimateDispersions( cdsUUT )
resUUT = nbinomTest( cdsUUT, "untreated", "treated" )
@

We produce the analogous plot as before, again with
<<figDE_Tb,fig=TRUE>>=
plotMA(resUUT)
@ %$
Figure~\ref{figDE_Tb} shows the same symmetry in up- and down-regulation
as in Fig.~\ref{figDE}, but a certain asymmetry in the boundary line for significance. This
has an easy explanation: low counts suffer from proportionally stronger shot noise than high
counts, and since there is only one ``untreated'' sample versus two ``treated'' ones,
a stronger downward fold-change is required to be called significant than for the upward direction.

%--------------------------------------------------
\subsection{Working without any replicates}
%--------------------------------------------------

Proper replicates are essential to interpret a biological experiment. After all, if one compares two
conditions and finds a difference, how else can one know that this difference is due to the different conditions
and would not have arisen between replicates, as well, just due to experimental or biological noise? Hence, any attempt to work without
replicates will lead to conclusions of very limited reliability.

Nevertheless, such experiments are sometimes undertaken, and the \Rpackage{DESeq} package
can deal with them, even though the soundness of the results may depend much on the circumstances.

Our primary assumption is now that the mean is a good predictor for the dispersion.
Once we accept this assumption, we may argue as follows: Given two samples from
different conditions and a number of genes with comparable expression levels, of which we expect only a minority
to be influenced by the condition, we may take the dispersion estimated from comparing their counts \emph{across}
conditions as ersatz for a proper estimate of the variance across replicates. After all,
we assume most genes to behave the same within replicates as across conditions, and hence, the estimated variance
should not be affected too much by the influence of the hopefully few differentially expressed genes. Furthermore,
the differentially expressed genes will only cause the dispersion estimate to be too high, so that the test will err to the
side of being too conservative.

We shall now see how well this works for our example data.
We reduce our count data set to just two columns, one ``untreated'' and one ``treated'' sample:
<<subset2>>=
cds2 = cds[ ,c(  "untreated3", "treated3"   ) ]
@

Now, without any replicates at all, the \Rfunction{estimateDispersions} function will refuse to proceed unless we
instruct it to ignore the condition labels and estimate the variance by treating all
samples as if they were replicates of the same condition:
<<cds2,cache=TRUE>>=
cds2 = estimateDispersions( cds2, method="blind", sharingMode="fit-only" )
@
Note the option \texttt{sharingMode="fit-only"}. Remember that the default,
\texttt{sharingMode="maximum"}, takes care of outliers, i.e., genes with
dispersion much larger than the fitted values. Without replicates, we cannot
catch such outliers and so have to disable this functionality.

Now, we can attempt to find differential expression:
<<res2,cache=TRUE>>=
res2 = nbinomTest( cds2, "untreated", "treated" )
@

\begin{figure}[ht]
\centering
\includegraphics[width=.6\textwidth]{DESeq-figDE2}
\caption{MvA plot, from a test using no replicates.}
\label{figDE2}
\end{figure}

Unsurprisingly, we find much fewer hits, as can be seen from the plot (Fig.\ \ref{figDE2})
<<figDE2,fig=TRUE>>=
plotMA(res2)
@
and from this table, tallying the number of significant hits in our previous and our new, restricted analysis:
<<addmarg>>=
addmargins( table( res_sig = res$padj < .1, res2_sig = res2$padj < .1 ) )
@


%-------------------------------------------------------------------------------------------------
\section{Multi-factor designs} \label{sec:GLM}
%-------------------------------------------------------------------------------------------------
Let us return to the full pasilla data set. Remember that we started of with these data:
%
<<reminderFullData>>=
head( pasillaCountTable )
pasillaDesign
@

When creating a count data set with multiple factors, just pass a data frame
instead of the condition factor:

<<fct>>=
cdsFull = newCountDataSet( pasillaCountTable, pasillaDesign )
@

As before, we estimate the size factors and then the dispersions. Here, the default
method (\texttt{method=''pooled''} to \Rfunction{estimateDispersion}), means that,
as before, one dispersion is computed for each gene,
which is now an average over all cells (weighted by the number of samples for each cells), where
the term \emph{cell} denotes any of the four combinations of factor levels of
the design. An alternative is the option \texttt{method=''pooled-CR''}, which uses
a Cox-Reid-adjusted maximum likelihood estimator \cite{CR,edgeR_GLM}. The latter is
slower but useful when the cell concept is not applicable (e.g. in paired designs).

<<estsfdisp,cache=TRUE>>=
cdsFull = estimateSizeFactors( cdsFull )
cdsFull = estimateDispersions( cdsFull )
@

\begin{figure}[ht]
\centering
\includegraphics[width=.6\textwidth]{DESeq-figFitPooled}
\caption{Estimated (black) pooled dispersion values for all seven samples, with
regression curve (red).}
\label{figFitPooled}
\end{figure}


We check the fit (Fig.~\ref{figFitPooled}):

<<figFitPooled,fig=TRUE>>=
plotDispEsts( cdsFull )
@ %$

For inference, we now specify two \emph{models} by formulas. The \emph{full model}
regresses the genes' expression on both the library type and
the treatment condition, the \emph{reduced model} regresses them only on the library
type. For each gene, we fit generalized linear models (GLMs)
according to the two models, and then compare them in order to infer whether
the additional specification of the treatment improves the fit and hence, whether
the treatment has significant effect.

<<fit1,cache=TRUE,results=hide>>=
fit1 = fitNbinomGLMs( cdsFull, count ~ libType + condition )
fit0 = fitNbinomGLMs( cdsFull, count ~ libType  )
@
These commands take a while to execute. Also, they may produce a few warnings,
informing you that the GLM fit failed to converge (and the results from these
genes should be interpreted with care). The ``fit'' objects are
data frames with three columns:

<<fitstr>>=
str(fit1)
@

To perform the test, we call
<<pvalsGLM,cache=TRUE>>=
pvalsGLM = nbinomGLMTest( fit1, fit0 )
padjGLM = p.adjust( pvalsGLM, method="BH" )
@
%
\fixme{Can we add a paragraph on what to do if we only want to make specific comparisons, e.g.
  let \texttt{fac} be a factor with three levels A, B and C, and we want to test pairwise for differences
  between A and B (without C), between A and C (without B) etc.}

The function \Rfunction{nbinomTestGLM} returned simply a vector of $p$ values which
we handed to the standard R function \Rfunction{p.adjust} to adjust for multiple
testing using the Benjamini-Hochberg (BH) method.

\begin{figure}[ht]
\centering
\includegraphics[width=.5\textwidth]{DESeq-figDispScatter}
\caption{Comparison of per-gene estimates of the dispersion in the
  analysis using only the four paired-end samples ($x$-axis) and in
  the analysis using all seven samples ($y$-axis).  For visualisation,
  the values are on a logarithm-like scale, defined by the function
  \Rfunction{trsf}, which is of the form
  $f(x)=\log\left((x+\sqrt{x^2+1})/2\right)$. For $x>2$, the values of
  $f(x)$ and $\log(x)$ are approximately the same, whereas for values
of $x$ around 0, $f(x)\approx x$ is approximated by a straight line of
slope 1.}
\label{figDispScatter}
\end{figure}

Let's compare with the result from the four-samples test:
<<addmarg2>>=
tab1 = table( "paired-end only" = res$padj < .1, "all samples" = padjGLM < .1 )
addmargins( tab1 )
@
We see that the analyses find \Sexpr{tab1[2,2]} genes in common, while
\Sexpr{tab1[1,2]} were only found in the analysis using all samples and
\Sexpr{tab1[2,1]} were specific for the \emph{paired-end only} analysis.

In this case, the improvement in power that we gained from the additional
samples is rather modest. This might indicate that the single-end samples
(which are presumably older than the paired-end one) are of lower quality.
This is supported by the observation that the dispersion estimates in the full
data set tend to be larger than in the paired-end only one. So, despite the fact that
with 7 samples we gain power from having more degrees of freedom in the estimation of the treatment effect,
the additional samples also increase the dispersion (or equivalently, the variance), which loses power.
%
<<tablesignfitInfocdsperGeneDispEsts>>=
table(sign(fitInfo(cds)$perGeneDispEsts - fitInfo(cdsFull)$perGeneDispEsts))
@
See also Figure~\ref{figDispScatter}, which is produced by
%
<<figDispScatter,fig=TRUE,width=4.5, height=4.5>>=
trsf = function(x) log( (x + sqrt(x*x+1))/2 )
plot( trsf(fitInfo(cds)$perGeneDispEsts),
      trsf(fitInfo(cdsFull)$perGeneDispEsts), pch=16, cex=0.45, asp=1)
abline(a=0, b=1, col="red3")
@

Continuing with the analysis, we can now extract the significant genes
from the vector \Robject{padjGLM} as before.
To see the corresponding fold changes, we have a closer look at the object
\Robject{fit1}.
<<lookatfit1>>=
head(fit1)
@

The first three columns show the fitted coefficients, converted to a logarithm base 2 scale.
The log2 fold change due to the condition is shown in the third column. As indicated
by the column name, it is the effect of ``untreated'', i.e., the log ratio of
``untreated'' versus ``treated''. (This is unfortunately the other way round as
before, due to the peculiarities of contrast coding.) Note that the library type also
had noticeable influence on the expression, and hence would have increased the dispersion
estimates (and so reduced power) if we had not fitted an effect for it.

The column \emph{deviance}
is the deviance of the fit. (Comparing the deviances with a $\chi^2$ likelihood ratio test is
how \Rfunction{nbinomGLMTest} calculates the $p$ values.) The last column, \emph{converged},
indicates whether the calculation of coefficients and deviance has fully converged.
(If it is false too often, you can try to change the GLM control parameters, as
explained in the help page to \Rfunction{fitNbinomGLMs}.)
\fixme{This is a bit vague, 'too often' should be specified and ideally the remedy be automated.}

Finally, we show that taking the library type into account was important to have
good detection power by doing the analysis again using the standard workflow,
as outlined earlier, and without informing \Rpackage{DESeq} of the library types:

<<fullAnalysisSimple,cache=TRUE>>=
cdsFullB = newCountDataSet( pasillaCountTable, pasillaDesign$condition )
cdsFullB = estimateSizeFactors( cdsFullB )
cdsFullB = estimateDispersions( cdsFullB )
resFullB = nbinomTest( cdsFullB, "untreated", "treated" )
<<table>>=
tab2 = table(
   `all samples simple` = resFullB$padj < 0.1,
   `all samples GLM`    = padjGLM < 0.1 )
addmargins(tab2)
@ %$
%
We see that the two analyses find \Sexpr{tab2[2,2]} genes in common,
while in addition \Sexpr{tab2[1,2]} genes were found by the analysis
that took library type into account.  The small number, namely
\Sexpr{tab2[2,1]}, of genes that only made the FDR cutoff in the
simple analysis is likely due to sampling variation.

%--------------------------------------------------
\section{Independent filtering and multiple testing} \label{sec:indepfilt}
\subsection{Filtering by overall count} \label{sec:filtbycount}
%--------------------------------------------------
The analyses of the previous sections involve the application of
statistical tests, one by one, to each row of the data set, in order to
identify those genes that have evidence for differential expression.
The idea of \emph{independent filtering} is to filter out those tests
from the procedure that have no, or little chance of showing significant
evidence, without even looking at their test statistic. Typically, this results in
increased detection power at the same experiment-wide type I error.
Here, we  measure experiment-wide type I error in terms of the false discovery rate.
A good choice for a filtering criterion is one that
\begin{enumerate}
  \item\label{it:indp} is statistically independent from the test statistic
    under the null hypothesis,
  \item\label{it:corr} is correlated with the test statistic under the
    alternative, and
  \item\label{it:joint} does not notably change the dependence
    structure --if there is any-- between the
    tests that pass the filter, compared to the dependence structure
    between the tests before filtering.
\end{enumerate}
The benefit from filtering relies on property~\ref{it:corr}, and we
will explore it further in Section~\ref{sec:whyitworks}. Its
statistical validity relies on property \ref{it:indp} --which is
simple to formally prove for many combinations of filter criteria
with test statistics-- and \ref{it:joint}, which is less easy to theoretically
imply from first principles, but rarely a problem in practice.  We refer
to~\cite{Bourgon:2010:PNAS} for further discussion of this topic.
<<rs>>=
rs = rowSums ( counts ( cdsFull ))
theta = 0.4
use = (rs > quantile(rs, probs=theta))
table(use)
cdsFilt = cdsFull[ use, ]
@
<<check,echo=FALSE>>=
stopifnot(!any(is.na(use)))
@
%
Above, we consider as a filter criterion \Robject{rs}, the overall sum
of counts (irrespective of biological condition), and remove the genes
in the lowest \Sexpr{theta*100}\% quantile (as indicated by the
parameter \Robject{theta}).  We perform the testing as before in
Section~\ref{sec:GLM}.
%
<<fitFilt,cache=TRUE,results=hide>>=
fitFilt1  = fitNbinomGLMs( cdsFilt, count ~ libType + condition )
fitFilt0  = fitNbinomGLMs( cdsFilt, count ~ libType  )
pvalsFilt = nbinomGLMTest( fitFilt1, fitFilt0 )
padjFilt  = p.adjust(pvalsFilt, method="BH" )
@
<<doublecheck,echo=FALSE>>=
stopifnot(all.equal(pvalsFilt, pvalsGLM[use]))
@
Let us compare the number of genes found at an FDR of 0.1 by this
analysis with that from the previous one (\Robject{padjGLM}).
<<tab>>=
padjFiltForComparison = rep(+Inf, length(padjGLM))
padjFiltForComparison[use] = padjFilt
tab3 = table( `no filtering`   = padjGLM < .1,
             `with filtering` = padjFiltForComparison < .1 )
addmargins(tab3)
@
The analysis with filtering found an additional \Sexpr{tab3[1,2]}
genes, an increase in the detection rate by about
\Sexpr{round(100*tab3[1,2]/tab3[2,2])}\%,
while \Sexpr{tab3[2,1]} genes were only found by the previous analysis.

%--------------------------------------------------
\subsection{Why does it work?}\label{sec:whyitworks}
%--------------------------------------------------
First, consider Figure~\ref{figscatterindepfilt}, which shows that
among the 40--45\% of genes with lowest overall counts, \Robject{rs},
there are essentially none that achieved an (unadjusted) $p$ value less than
\Sexpr{signif(quantile(pvalsGLM[!use], 0.0001, na.rm=TRUE), 1)}
(this corresponds to about \Sexpr{signif(-log10(quantile(pvalsGLM[!use], 0.0001, na.rm=TRUE)), 2)} on the $-\log_{10}$-scale).
%
<<figscatterindepfilt,fig=TRUE>>=
plot(rank(rs)/length(rs), -log10(pvalsGLM), pch=16, cex=0.45)
@
\begin{figure}[ht]
\centering
\includegraphics[width=.5\textwidth]{DESeq-figscatterindepfilt}
\caption{Scatterplot of rank of filter criterion (overall sum of
  counts \Robject{rs}) versus the negative logarithm of the test statistic \Robject{pvalsGLM}.}
\label{figscatterindepfilt}
\end{figure}
This means that by dropping the 40\% genes with lowest \Robject{rs},
we do not loose anything substantial from our subsequent
results. Second, consider the $p$ value histogram in Figure~\ref{fighistindepfilt}.
It shows how the filtering ameliorates the multiple testing problem
-- and thus the severity of a multiple testing adjustment -- by
removing a background set of hypotheses whose $p$ values are distributed
more or less uniformly in $[0,1]$.
<<histindepfilt,width=7,height=5>>=
h1 = hist(pvalsGLM[!use], breaks=50, plot=FALSE)
h2 = hist(pvalsGLM[use], breaks=50, plot=FALSE)
colori = c(`do not pass`="khaki", `pass`="powderblue")
<<fighistindepfilt,fig=TRUE>>=
barplot(height = rbind(h1$counts, h2$counts), beside = FALSE, col = colori,
        space = 0, main = "", ylab="frequency")
text(x = c(0, length(h1$counts)), y = 0, label = paste(c(0,1)), adj = c(0.5,1.7), xpd=NA)
legend("topright", fill=rev(colori), legend=rev(names(colori)))
@
\begin{figure}[ht]
\centering
\includegraphics[width=.5\textwidth]{DESeq-fighistindepfilt}
\caption{Histogram of $p$ values for all tests (\Robject{pvalsGLM}).
  The area shaded in blue indicates the subset of those that pass the filtering,
  the area in khaki those that do not pass.}
\label{fighistindepfilt}
\end{figure}

%---------------------------------------------------
\subsection{Diagnostic plots for multiple testing}
%---------------------------------------------------
The Benjamini-Hochberg multiple testing adjustment
procedure~\cite{BH:1995} has a simple graphical illustration, which we
produce in the following code chunk. Its result is shown in the left
panel of Figure~\ref{figmulttest}.
%
<<sortP,cache=TRUE>>=
orderInPlot = order(pvalsFilt)
showInPlot = (pvalsFilt[orderInPlot] <= 0.08)
alpha = 0.1
<<sortedP,fig=TRUE, width=4.5, height=4.5>>=
plot(seq(along=which(showInPlot)), pvalsFilt[orderInPlot][showInPlot],
     pch=".", xlab = expression(rank(p[i])), ylab=expression(p[i]))
abline(a=0, b=alpha/length(pvalsFilt), col="red3", lwd=2)
@
<<doBH,echo=FALSE,results=hide>>=
whichBH = which(pvalsFilt[orderInPlot] <= alpha*seq(0, 1, length=length(pvalsFilt)))
## Test some assertions:
## - whichBH is a contiguous set of integers from 1 to length(whichBH)
## - the genes selected by this graphical method coincide with those
##   from p.adjust (i.e. padjFilt)
stopifnot(length(whichBH)>0,
          identical(whichBH, seq(along=whichBH)),
          padjFilt[orderInPlot][ whichBH] <= alpha,
          padjFilt[orderInPlot][-whichBH]  > alpha)
@
%
Schweder and Spj\o{}tvoll~\cite{SchwederSpjotvoll1982} suggested a diagnostic plot
of the observed $p$-values which permits estimation of the fraction of true null
hypotheses. For a series of hypothesis tests $H_1, \ldots, H_m$ with $p$-values
$p_i$, they suggested plotting
%
\begin{equation}
  \left( 1-p_i, N(p_i) \right) \mbox{ for } i \in 1, \ldots, m,
\end{equation}
%
where $N(p)$ is the number of $p$-values greater than $p$. An application of
this diagnostic plot to \Robject{pvalsFilt} is shown in the right panel of
Figure~\ref{figmulttest}.
When all null hypotheses are true, the $p$-values are each uniformly distributed
in $[0,1]$, Consequently, the cumulative distribution function of $(p_1, \ldots,
p_m)$ is expected to be close to the line $F(t)=t$. By symmetry, the same
applies to $(1 - p_1, \ldots, 1 - p_m)$.
When (without loss of generality) the first $m_0$ null hypotheses are true and
the other $m-m_0$ are false, the cumulative distribution function of $(1-p_1,
\ldots, 1-p_{m_0})$ is again expected to be close to the line $F_0(t)=t$. The
cumulative distribution function of $(1-p_{m_0+1}, \ldots, 1-p_{m})$, on the
other hand, is expected to be close to a function $F_1(t)$ which stays below
$F_0$ but shows a steep increase towards 1 as $t$ approaches $1$.
In practice, we do not know which of the null hypotheses are true, so we can
only observe a mixture whose cumulative distribution function is expected to be
close to
%
\begin{equation}
  F(t) = \frac{m_0}{m} F_0(t) + \frac{m-m_0}{m} F_1(t).
\end{equation}
%
Such a situation is shown in the right panel of
Figure~\ref{figmulttest}. If
$F_1(t)/F_0(t)$ is small for small $t$, then the mixture fraction
$\frac{m_0}{m}$ can be estimated by fitting a line to the left-hand portion of
the plot, and then noting its height on the right. Such a fit is shown by the
red line in the right panel of Figure~\ref{figmulttest}.
%
<<SchwSpjot,echo=FALSE,results=hide>>=
j  = round(length(pvalsFilt)*c(1, .66))
px = (1-pvalsFilt[orderInPlot[j]])
py = ((length(pvalsFilt)-1):0)[j]
slope = diff(py)/diff(px)
@
<<SchwederSpjotvoll,fig=TRUE, width=4.5, height=4.5>>=
plot(1-pvalsFilt[orderInPlot],
     (length(pvalsFilt)-1):0, pch=".",
     xlab=expression(1-p[i]), ylab=expression(N(p[i])))
abline(a=0, slope, col="red3", lwd=2)
@

\begin{figure}[ht]
\centering
\includegraphics[width=.49\textwidth]{DESeq-sortedP}
\includegraphics[width=.49\textwidth]{DESeq-SchwederSpjotvoll}
\caption{\emph{Left:} illustration of the Benjamini-Hochberg multiple testing
  adjustment procedure~\cite{BH:1995}.  The black line shows the
  $p$-values ($y$-axis) versus their rank ($x$-axis), starting with
  the smallest $p$-value from the left, then the second smallest, and
  so on. Only the first \Sexpr{sum(showInPlot)} $p$-values are shown.
  The red line is a straight line with slope $\alpha/n$, where
  $n=\Sexpr{length(pvalsFilt)}$ is the number of tests, and
  $\alpha=\Sexpr{alpha}$ is a target false discovery rate (FDR).  FDR
  is controlled at the value $\alpha$ if the genes are selected
  that lie to the left of the rightmost intersection between the red and black
  lines: here, this results in \Sexpr{length(whichBH)} genes.
  \emph{Right:} Schweder and Spj\o{}tvoll plot, as described in the text.
  For both of these plots, the $p$-values \Robject{pvalsFilt}
  from Section~\ref{sec:filtbycount} were used as a starting
  point. Analogously, one can produce these types of plots for any set of $p$-values,
  for instance those from the previous sections.}
\label{figmulttest}
\end{figure}

%---------------------------------------------------
\section{Variance stabilizing transformation}
%---------------------------------------------------
For some applications, it is useful to work with transformed versions
of the count data. Maybe the most obvious choice is logarithmic
transformation. Since count values for a gene can be zero in some
conditions (and non-zero in others), some people advocate the use of
\emph{pseudocounts}, i.\,e.\ transformations of the form
\begin{equation}\label{eq:shiftedlog}
  y = \log_2(n + 1)\quad\mbox{or more generally,}\quad y = \log_2(n + n_0),
\end{equation}
where $n$ represents the count values and $n_0$ is a somehow chosen
positive constant. In this section, we discuss a related, alternative
approach that offers more theoretical justification and a rational way
of chosen the parameter equivalent to $n_0$ above. It is based on
error modeling and the concept of variance stabilizing
transformations~\cite{Tibshirani1988,sagmb2003,Anders:2010:GB}. We
estimate an overall mean-dispersion relationship of the data using
\Rfunction{estimateDispersions} with the argument
\Robject{method="blind"} and call the function
\Rfunction{getVarianceStabilizedData}.
%
<<defvsd>>=
cdsBlind = estimateDispersions( cds, method="blind" )
vsd = varianceStabilizingTransformation( cdsBlind )
@
Here, we have used a parametric fit for the dispersion. In this case,
the a closed-form expression for the variance stabilizing transformation
is used by \Rfunction{getVarianceStabilizedData}, which is derived in
the file \texttt{vst.pdf}, that is distributed in the package
alongside this vignette. If a local fit is used (option \texttt{fittype="local"}
to \Rfunction{estimateDispersion}) a numerical integration is used instead.
%
The resulting transformation is shown in
Figure~\ref{figvsd1}.  The code that produces the figure is hidden from
this vignette for the sake of brevity, but can be seen in the
\texttt{.Rnw} or \texttt{.R} source file.
%
\begin{figure}[ht]
\centering
\includegraphics[width=.49\textwidth]{DESeq-vsd1}
\caption{Graphs of the variance stabilizing transformation for
  sample 1, in blue, and of the transformation $f(n)=log_2(n/s_1)$, in
  black. $n$ are the counts and $s_1$ is the size factor for the first sample.
}
\label{figvsd1}
\end{figure}
\begin{figure}[ht]
\centering
\includegraphics[width=\textwidth]{DESeq-vsd2}
\caption{Per-gene standard deviation (taken across samples), against
  the rank of the mean, for the shifted logarithm
  $\log_2(n+1)$ (left) and \Rpackage{DESeq}'s variance stabilising
  transformation (right).}
\label{figvsd2}
\end{figure}

<<vsd1, fig=TRUE, echo=FALSE, width=4.5, height=4.5>>=
##par(mai=ifelse(1:4 <= 2, par("mai"), 0))
px     = counts(cds)[,1] / sizeFactors(cds)[1]
ord    = order(px)
ord    = ord[px[ord] < 150]
ord    = ord[seq(1, length(ord), length=50)]
last   = ord[length(ord)]
vstcol = c("blue", "black")
matplot(px[ord],
        cbind(exprs(vsd)[, 1], log2(px))[ord, ],
        type="l", lty=1, col=vstcol, xlab="n", ylab="f(n)")
legend("bottomright",
       legend = c(
        expression("variance stabilizing transformation"),
        expression(log[2](n/s[1]))),
       fill=vstcol)
@
%
Figure~\ref{figvsd2} plots the standard deviation of the transformed
data, across samples, against the mean, first using the shifted
logarithm transformation~(\ref{eq:shiftedlog}), then using
\Rpackage{DESeq}'s variance stabilising transformation. While for the
latter, the standard deviation is roughly constant along the whole
dynamic range, the shifted logarithm results in a highly elevated
standard deviation in the lower count range.
%
<<vsd2, fig=TRUE, width=8.5, height=4.5, results=hide>>=
library("vsn")
par(mfrow=c(1,2))
notAllZero = (rowSums(counts(cds))>0)
meanSdPlot(log2(counts(cds)[notAllZero, ] + 1), ylim = c(0,2.5))
meanSdPlot(vsd[notAllZero, ], ylim = c(0,2.5))
@

%---------------------------------------------------------------
\subsection{Application to moderated fold change estimates}
%---------------------------------------------------------------
In the beginning of Section~\ref{sec:DE}, we have seen in the
\Rclass{data.frame} \Robject{res} the (logarithm base 2) fold change estimate
computed from the size-factor adjusted counts.  When the involved
counts are small, these (logarithmic) fold-change estimates can be
highly variable, and can even be infinite. For some purposes, such as
the clustering of samples or genes according to their expression
profiles, or for visualisation of the data, this high variability from
ratios between low counts tends to drown informative, systematic signal
in other parts of the data.  The variance stabilizing transformation
offers one way to \emph{moderate} the fold change estimates, so that
they are more amenable to plotting or clustering.
%
<<modlr>>=
mod_lfc = (rowMeans( exprs(vsd)[, conditions(cds)=="treated", drop=FALSE] ) -
           rowMeans( exprs(vsd)[, conditions(cds)=="untreated", drop=FALSE] ))
@
%
Let us compare these to the original ($\log_2$) fold changes. First we
find that many of the latter are infinite (resulting from division of
a finite value by 0) or \emph{not a number} (NaN, resulting from
division of 0 by 0).
%
<<dah>>=
lfc = res$log2FoldChange
table(lfc[!is.finite(lfc)], useNA="always")
@
%
For plotting, let us bin genes by their $\log_{10}$ mean to colour
them according to expression strength,
%
<<colourramp>>=
logdecade = 1 + round( log10( 1+rowMeans(counts(cdsBlind, normalized=TRUE)) ) )
lfccol = colorRampPalette( c( "gray", "blue" ) )(6)[logdecade]
@
%
and then compare their ordinary log-ratios (\Robject{lfc}) against the
\emph{moderated} log-ratios (\Robject{mod\_lfc}) in a scatterplot.
%
<<figmodlr, fig=TRUE, width=4.5, height=4.5>>=
ymax = 4.5
plot( pmax(-ymax, pmin(ymax, lfc)), mod_lfc,
      xlab = "ordinary log-ratio", ylab = "moderated log-ratio",
      cex=0.45, asp=1, col = lfccol,
      pch = ifelse(lfc<(-ymax), 60, ifelse(lfc>ymax, 62, 16)))
abline( a=0, b=1, col="red3")
@
The result is shown in Figure~\ref{figmodlr}.
\begin{figure}[htb]
\centering
\includegraphics[width=.5\textwidth]{DESeq-figmodlr}
\caption{Scatterplot of ordinary (\Robject{lfc}) versus
  \emph{moderated} log-ratios (\Robject{mod\_lfc}).  The points are
  coloured in a scale from grey to blue, representing weakly to
  strongly expressed genes. For the highly expressed genes (blue),
  \Robject{lfc} and \Robject{mod\_lfc} agree.  Differences arise
  when the involved counts are small (grey points): in this case, the
  moderated log-ratio is shrunk towards 0, compared to the ordinary
  log-ratio.  The $>$ and $<$ symbols correspond to values of
  \Robject{lfc} whose absolute value was larger than \Sexpr{ymax}
  (including those that were infinite).  All values of
  \Robject{mod\_lfc} vary in a finite range.}
\label{figmodlr}
\end{figure}

%---------------------------------------------------------------
\section{Data quality assessment by sample clustering and visualisation}\label{sec:quality}
%---------------------------------------------------------------
Data quality assessment and quality control (i.\,e.\ the removal of
insufficiently good data) are essential steps of any data
analysis. Even though we present these steps towards the end of this vignette,
they should typically be performed very early in the analysis of a new data set,
preceding or in parallel to the differential expression testing.

We define the term \emph{quality} as \emph{fitness for
  purpose}\footnote{\url{http://en.wikipedia.org/wiki/Quality_\%28business\%29}}.
Our purpose is the detection of differentially expressed genes, and we
are looking in particular for samples whose experimental treatment
suffered from an anormality that renders the data points obtained from
these particular samples detrimental to our purpose.

\subsection{Heatmap of the count table}\label{sec:hmc}
To explore a count table, it is often instructive to look at it as a
heatmap.  Below we show how to produce such a heatmap from the
variance stabilisation transformed data for all \Sexpr{ncol(cdsFull)}
samples.
%
<<cdsFullBlind,cache=TRUE>>=
cdsFullBlind = estimateDispersions( cdsFull, method = "blind" )
vsdFull = varianceStabilizingTransformation( cdsFullBlind )
<<heatmap>>=
library("RColorBrewer")
library("gplots")
select = order(rowMeans(counts(cdsFull)), decreasing=TRUE)[1:30]
hmcol = colorRampPalette(brewer.pal(9, "GnBu"))(100)
<<figHeatmap2a,fig=TRUE,width=7,height=10>>=
heatmap.2(exprs(vsdFull)[select,], col = hmcol, trace="none", margin=c(10, 6))
@
%
\begin{figure}
\centering
\includegraphics[width=.49\textwidth]{DESeq-figHeatmap2a}
\includegraphics[width=.49\textwidth]{DESeq-figHeatmap2b}
\caption{Heatmaps showing the expression data of the
  \Sexpr{length(select)} most highly expressed genes. Left, the
  variance stabilisation transformed data (\Robject{vsdFull}) are
  shown, right, the original count data (\Robject{cdsFull}). In the
  left panel, the sample clustering aligns with the experimental
  factor (\emph{treated} / \emph{untreated}).  The clustering and the
  colour scale in the right plot is dominated by a small number of data
  points with large values.}
\label{figHeatmap2}
\end{figure}
%
The left panel of Figure~\ref{figHeatmap2} shows the resulting heatmap
for the \Sexpr{length(select)} most highly expressed genes.  For
comparison, let us also do the same with the untransformed counts.
%
<<figHeatmap2b,fig=TRUE,width=7,height=10>>=
heatmap.2(counts(cdsFull)[select,], col = hmcol, trace="none", margin=c(10,6))
@
\begin{figure}
\centering
\includegraphics[width=.6\textwidth]{DESeq-figHeatmapSamples}
\caption{Heatmap showing the Euclidean distances between the samples
  as calculated from the variance stabilising transformation of the
  count data.}
\label{figHeatmapSamples}
\end{figure}

\subsection{Heatmap of the sample-to-sample distances}\label{sec:dists}
Another use of variance stabilized data is sample clustering. Here, we apply the
\Rfunction{dist} function to the transpose of the transformed count matrix to get
sample-to-sample distances.
%
<<sampleClust>>=
dists = dist( t( exprs(vsdFull) ) )
@
%
A heatmap of this distance matrix gives us an overview over similarities
and dissimilarities between samples (Figure~\ref{figHeatmapSamples}):

<<figHeatmapSamples,fig=TRUE,width=7,height=7>>=
mat = as.matrix( dists )
rownames(mat) = colnames(mat) = with(pData(cdsFullBlind), paste(condition, libType, sep=" : "))
heatmap.2(mat, trace="none", col = rev(hmcol), margin=c(13, 13))
@
%
The clustering correctly reflects our experimental design, i.e.,
samples are more similar when they have the same treatment or the same
library type.  (To avoid potential circularities in this conclusion,
it was important to re-estimate the dispersions with
\Robject{method="blind"} in the calculation for \Robject{cdsFullBlind}
above, as only then, the variance stabilizing transformation is not
informed about the design, and we can be sure that it is not biased
towards a result supporting the design.)

\subsection{Principal component plot of the samples}
Related to the distance matrix of Section~\ref{sec:dists} is the PCA
plot of the samples, which we obtain as follows (Figure~\ref{figPCA}).
%
<<figPCA,fig=TRUE,width=5,height=5>>=
print(plotPCA(vsdFull, intgroup=c("condition", "libType")))
@
\begin{figure}
\centering
\includegraphics[width=.6\textwidth]{DESeq-figPCA}
\caption{PCA plot. The \Sexpr{ncol(vsdFull)} samples shown in the 2D
  plane spanned by their first two principal components. This type of
  plot is useful for visualizing the overall effect of experimental
  covariates and batch effects.  For this data set, no batch effects
  besides the known effects of \emph{condition} and \emph{libType} are
  discernible.}
\label{figPCA}
\end{figure}

\subsection{arrayQualityMetrics}
Heatmaps and PCA plots similar to Figures~\ref{figHeatmapSamples} and
\ref{figPCA} as well as some further diagnostics are also provided by
the package \Rpackage{arrayQualityMetrics}. The package may be
particularly useful for larger data set with dozens of
samples. You can call its main function, which is also called
\Rfunction{arrayQualityMetrics}, with an \Rclass{ExpressionSet} such
as \Robject{vsdFull}, which you will have produced from your
\Rclass{CountDataSet} with calls to \Rfunction{estimateDispersions}
and \Rpackage{varianceStabilizingTransformation} as in the beginning
of Section~\ref{sec:hmc}.

\clearpage
%------------------------------------------------------------
\section{Further reading}\label{sec:further}
%------------------------------------------------------------

For more information on the statistical method, see our paper~\cite{Anders:2010:GB}. For
more information on how to customize the \Rpackage{DESeq} work flow, see the package
help pages, especially the help page for \Rfunction{estimateDispersions}.

%------------------------------------------------------------
\section{Changes since publication of the paper}\label{sec:changes}
%------------------------------------------------------------
Since our paper on \Rpackage{DESeq} was published in Genome Biology in
Oct 2010, we have made a number of changes to algorithm and
implementation, which are listed here.

\begin{itemize}

\item \Rfunction{nbinomTest} calculates a $p$ value by summing up the probabilities
of all per-group count sums $a$ and $b$ that sum up to the observed count
sum $k_{iS}$ and are more extreme than the observed count sums $k_{iA}$ and $k_{iB}$.
Equation~(11) of the paper defined \emph{more extreme} as those pairs of values $(a,b)$
that had smaller probability than the observed pair. This caused problems in cases
where the dispersion exceeded 1. Hence, we now sum instead the probabilities of all
values pairs that are \emph{further out} in the sense that they cause a more extreme
fold change $(a/s_A)/(b/s_B)$, where $s_A$ and $s_B$ are the sums of the size factors of
the samples in conditions $A$ and $B$, respectively. We do this in a one-tailed manner and
double the result. Furthermore, we no longer approximate the sum, but always
calculate it exactly.

\item We added the possibility to fit GLMs of the negative binomial
family with log link.
This new functionality is described in this vignette. $p$ values are calculated by a
$\chi^2$ likelihood ratio test. The logarithms of the size factors
are provided as offsets to the GLM fitting function.

\item The option \texttt{sharingMode='maximum'} was added to \Rfunction{estimateDispersion}
and made default. This change makes DESeq robust against variance outliers and was not yet
discussed in the paper.

\item By default, DESeq now estimates one pooled dispersion estimate across all
(replicated) conditions. In the original version, we estimated a separate
dispersion-mean relation for each condition. The ``maximum'' sharing mode achieves
its goal of making DESeq robust against outliers only with pooled dispersion
estimate, and hence, this is now the default. The option \texttt{method='per-condition'} to
\Rfunction{estimateDispersions} allows the user to go back to the old method.

\item In the paper, the mean-dispersion relation is fitted by local regression. Now,
DESeq also offers a parametric regression, as described in this vignette. The option
\texttt{fitType} to \Rfunction{estimateDispersions} allows the user to choose between these.
If a parametric regression is used, the variance stabilizing transformation is calculated
using the closed-form expression given in the vignette supplement file \texttt{vst.pdf}.

\item Finally, instead of the term \emph{raw squared coefficient of variance}
used in the paper we now prefer the more standard term \emph{dispersion}.

\end{itemize}

\section{Session Info}
<<sessi>>=
sessionInfo()
@

\bibliographystyle{unsrt}
\bibliography{DESeq}


\end{document}
