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%\VignetteIndexEntry{dexus: Manual for the R package}
%\VignetteDepends{dexus}
%\VignettePackage{dexus}
%\VignetteKeywords{RNASeq, RNA-Seq, RNA, Sequencing, NGS, differential expression,
%   transcripts, transcriptomics, negative binomial, dispersion,
%   overdispersion, maximum-a-posteriori, I/NI, INI, informative,
%   non-informative, replicates, unsupervised, clustering, unknown, conditions,
%   ENCODE, sample groups, case-control, treatment study, randomized controlled
%   study, RCS, non-randomized, cross-sectional, cross sectional, cohort}

\documentclass[article]{bioinf}

\usepackage[noae]{Sweave}
\usepackage{amsmath,amssymb}
\usepackage{hyperref}
\usepackage{float}
\usepackage[authoryear]{natbib}
\usepackage{bm}

\hypersetup{colorlinks=false,
   pdfborder=0 0 0,
   pdftitle={DEXUS -- Identifying Differential Expression in RNA-Seq Studies
   with Unknown Conditions},
   pdfauthor={G\"unter Klambauer},
   pdfsubject={RNA-Seq and Detection of Differential Expression},
   pdfkeywords={RNASeq, RNA-Seq, RNA, Sequencing, NGS, differential expression,
   transcripts, transcriptomics, negative binomial, dispersion,
   overdispersion, maximum-a-posteriori, I/NI, INI, informative,
   non-informative, replicates, unsupervised, clustering, unknown, conditions,
   ENCODE, sample groups, case-control, treatment study, randomized controlled
   study, RCS, non-randomized, cross-sectional, cross sectional, cohort}}

\title{DEXUS -- Identifying Differential Expression in RNA-Seq Studies
   with Unknown Conditions}
\author{G\"unter Klambauer and Thomas Unterthiner}
\affiliation{Institute of Bioinformatics, Johannes Kepler University
Linz\\Altenberger Str. 69, 4040 Linz, Austria\\
\email{klambauer@bioinf.jku.at}}


\newcommand{\DEXUS}{\texttt{DEXUS}}
\newcommand{\R}{R}
\newcommand{\Real}{\mathbb{R}}
\renewcommand{\vec}[1]{\mathbf{#1}}
\def\ga{\bm{\gamma}}
\def\ep{\bm{\epsilon}}
\def\al{\bm{\alpha}}
\def\Lambda{\lambda}
\def\pp{\bm{\Psi}}
\def\pps{\bm{\psi}}
\def\Si{\bm{\Sigma}}
\def\La{\bm{\Lambda}}
\def\m{\bm{m}}
\def\x{\bm{x}}
\def\r{\bm{r}}
\def\p{\bm{p}}
\def\y{\bm{y}}
\def\z{\bm{z}}
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\def\u{\bm{u}}
\def\1{\bm{1}}
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\def\le{{\bf \large 1}}
\def\lnull{{\bf \large 0}}
\def\I{\bm{I}}
\def\C{\bm{C}}
\def\c{\bm{c}}
\def\X{\bm{X}}
\def\Z{\bm{Z}}
\def\L{\bm{L}}
\def\muu{\bm{\mu}}
\def\RR{\mathbb{R}}
\def\NN{\mathbb{N}}
\def\sign{\mathrm{sign}}
\def\Xcal{\mathcal{X}}
\def\Fcal{\mathcal{F}}
\def\E{\mathrm{E}}
\def\D{\mathrm{D}}
\def\P{\mathrm{P}}
\def\ML{\mathrm{ML}}
\def\sgn{\mathop{\mathrm{sgn}\,}}
\def\AUCROC{$\mathrm{AUC}_{\mathrm{ROC}}$\ }
\def\AUCPR{$\mathrm{AUC}_{\mathrm{PR}}$\ }
\definecolor{grey}{rgb}{0.8,0.8,0.8}


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\newcommand{\pvalue}{$p$-value\ }
\newcommand{\pvalues}{$p$-values\ }
\newcommand{\pe}{$p$\hspace{2pt}=\hspace{2pt}}



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\SweaveOpts{eps=FALSE}

\begin{document}
<<echo=FALSE>>=
options(width=75)
set.seed(0)
library(dexus)
dexusVersion <- packageDescription("dexus")$Version
@
\newcommand{\dexusVersion}{\Sexpr{dexusVersion}}
\manualtitlepage[Version \dexusVersion, \today]

\vspace{1cm}

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\tableofcontents
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\section{Introduction}
\method{DEXUS} identifies differentially expressed transcripts in
RNA-Seq data under all possible study designs such as studies 
without replicates, without sample groups, and with unknown conditions.
\method{DEXUS} works also for known conditions, for example for
RNA-Seq data with two or multiple conditions. 
    
RNA-Seq read count data can be provided both by the S4 class 
{\tt CountDataSet} and by read count matrices.
Differentially expressed transcripts can be visualized by heatmaps,
in which unknown conditions, replicates, and samples groups are also indicated.
This software is fast as the core algorithm is written in C.
For very large data sets, a parallel version of \method{DEXUS}
is provided in this \Rpackage.
    
\method{DEXUS} is a statistical model that is selected in a Bayesian framework by
an EM algorithm. \method{DEXUS} does not need replicates to detect
differentially expressed transcript, since the replicates (or conditions) are 
estimated by the EM method for each transcript. This is an unsupervised machine
learning approach that does not require labeled data. The method provides an
informative/non-informative (I/NI) value to extract differentially expressed
transcripts at a desired significance level or power.
    
Detection of differential expression in RNA-Seq data is currently limited 
to studies in which two or more sample conditions are known {\em a priori}.
However, these biological conditions are typically unknown  in cohort,
cross-sectional, and non-randomized controlled studies such as the HapMap, the
ENCODE, or the 1000 Genomes project. \method{DEXUS} models read counts as
a finite mixture of negative binomial distributions.

See \url{http://www.bioinf.jku.at/software/dexus} for additional
information, data sets, and \R\ scripts. 


\section{Getting Started and Quick Guide}
To load the package, enter the following in the \R\ session:
<<>>=
library(dexus)
@
With the package {\tt dexus} we provide the ``Mice strains''
\citep{Bottomly2011}, ``Primate Liver'' \citep{Blekhman2010}, ``Maize leaves'' 
\citep{Li2010a}, ``European HapMap'' \citep{Montgomery2010}, and the ``Nigerian
HapMap'' \citep{Pickrell2010} data sets. The read counts are stored in the objects 
{\tt countsBottomly}, {\tt countsGilad}, {\tt countsLi}, {\tt countsMontgomery},
and {\tt countsPickrell}, respectively.

<<>>=
data(dexus)
ls()
@

\subsection{Unknown Conditions} 

One can simply run \method{DEXUS} by applying
the function {\tt dexus} to the count matrices. This is the mode in which the 
conditions are unknown, i.e. no labels that indicate the replicate groups have
to be provided.

<<eval=FALSE>>=
result <- dexus(countsBottomly[1:1000, ])
plot(result)
@


<<fig=FALSE,echo=FALSE,results=hide>>=
result <- dexus(countsBottomly[1:1000, ])
pdf("001.pdf")
plot(result,cexSamples=1)
dev.off()
@

\begin{figure}[H]
\begin{center}
\includegraphics[angle=0,width= 0.6\columnwidth]{001.pdf}
\caption[Result of \method{DEXUS} for data with {\em unknown} conditions.]{Result of
\method{DEXUS} for  data with {\em unknown} conditions. A heatmap of the log
read counts of the top ranked transcripts of the ``Mice strains'' \citep{Bottomly2011} data set is 
shown. Rows represent transcripts sorted  by their  I/NI
values. The trancript on the top has  the highest  I/NI value. Columns represent
different samples. The labels  ``D2'' and ``B6''  represent the two different
strains. Red crossed indicate the samples belonging to the second condition
that \method{DEXUS} has identified.
}
\end{center}
\end{figure} 

\subsection{Known Conditions} 

To test between two or more replicate
groups, \method{DEXUS} needs to be provided with the group labels:
<<eval=FALSE>>=
resultSupervised <- dexus(countsBottomly[1:1000, ],
		labels=substr(colnames(countsBottomly),1,2))
plot(resultSupervised)
@

<<fig=FALSE,echo=FALSE,results=hide>>=
resultSupervised <- dexus(countsBottomly[1:1000, ],
		labels=substr(colnames(countsBottomly),1,2))
pdf("002.pdf")
plot(resultSupervised,cexSamples=1)
dev.off()
@

\begin{figure}[H]
\begin{center}
\includegraphics[angle=0,width= 0.6\columnwidth]{002.pdf}
\caption[Result of \method{DEXUS} for data with {\em known} conditions.]{Result of
\method{DEXUS} for data with  {\em known} conditions. 
A heatmap of the log read counts of the 
top ranked transcripts of the ``Mice strains'' \citep{Bottomly2011} data set is 
shown. Rows represent transcripts
sorted by their $p$-values. The trancript on the top
has the lowest $p$-value. Columns represent different samples. The labels 
``D2'' and ``B6'' represent the two different strains. Red crossed indicate the
samples belonging to the second condition, that was given by the labels}
\end{center}
\end{figure} 


\section{Input Data: Read Count Matrices or {\tt CountDataSets}}

\method{DEXUS} expects a table of counts per transcript, transcript, exon, or any
other region of interest as input in analogy to other RNA-Seq analysis methods
\citep{Anders2010, Robinson2010a, Hardcastle2010, Li2012, Wang2010, Li2011,
Tarazona2011, Wu2012}. The table should have the transcripts as rows and
samples as columns. An entry should correspond to the number of reads of 
the sample mapping to the transcript. Technical replicates of one sample should
be summed up so that each column corresponds to one sample.
There are various ways how to produce count matrices from BAM files:
\begin{itemize}
  \item A full guide on processing RNA-Seq data including the calculation of
  read count matrices is provided at
  \url{http://en.wikibooks.org/wiki/Next_Generation_Sequencing_(NGS)/RNA}.
  \item The function {\tt HTSeq-count} of {\tt HTSeq} Python package
        \url{http://www-huber.embl.de/users/anders/HTSeq/doc/count.html}.
  \item Ready-made count tables for a lot of studies are available at 
        \url{http://bowtie-bio.sourceforge.net/recount/} \citep{Frazee2011}.
  \item The function {\tt countOverlaps} of the Bioconductor package {\tt
  GenomicRanges} can also be utilized.
  \item The function {\tt getSegmentReadCountsFromBAM} of the Bioconductor
  package {\tt cn.mops} \citep{Klambauer2012} can also be utilized to extract
  read counts from BAM files efficiently.
\end{itemize}

\subsection{Read Count Matrices or Count Tables as Input for DEXUS}
A read count matrix or count table should look like the following:
<<>>=
data(dexus)
countsBottomly[1:10,1:5]
@


A numeric matrix of read counts can directly be used with \method{DEXUS}.
<<eval=FALSE>>=
result <- dexus(countsBottomly)
@

\subsection{{\tt CountDataSets} as input for DEXUS}
A {\tt CountDataSet}, such as the ones used in the package
\method{DESeq} \citep{Anders2010}, can also directly be used it with
\method{DEXUS}:

<<eval=FALSE>>=
library(DESeq)
cds <-  newCountDataSet(countData=countsBottomly, 
		conditions=substr(colnames(countsBottomly),1,2) )
result <- dexus(cds)
@



\section{General Study Designs: No Replicates, Unknown Sample Groups or
Conditions}

Examples of studies in which the groups are unknown, are the studies
of \citet{Montgomery2010} and \citet{Pickrell2010}. They sequenced the 
RNA of HapMap individuals to investigate eQTLs. \method{DEXUS} is able 
to identify differential expression in these data sets. The method estimates
the conditions for each transcript individually.

To run the method simply apply the function {\tt dexus} to the count table. In
the following example we run the algorithm only one the first 1000 transcripts.
<<>>=
resultMontgomery <- dexus(countsMontgomery[1:1000, ])
@

To show a summary of the result object, simply type the following.
<<>>=
resultMontgomery
@

The transcripts are in their original order; the displayed columns
give the whether a transcript is differentially
expressed ({\tt INICall}), 
the evidence for differential expression measured by the I/NI values 
({\tt INIValues}), and the means for each condition.

It is possible to sort the result object such that the transcripts with the
highest I/NI values are ranked highest.

<<eval=FALSE>>=
sort(resultMontgomery)
@

Transcripts with an I/NI value above $0.1$
are classfied as differentially expressed. The function {\tt INI} filters 
the result object for informative transcripts and sorts them by their I/NI calls.

<<>>=
informativeTranscripts <- INI(resultMontgomery,threshold=0.2)
@


The result can be visualized by a heatmap using the {\tt plot} function.
<<eval=FALSE>>=
plot(informativeTranscripts)
@


<<fig=FALSE,echo=FALSE,results=hide>>=
pdf("004.pdf",width=12,height=6)
plot(informativeTranscripts)
dev.off()
@
\begin{figure}[H]
\begin{center}
\includegraphics[angle=0,width= \textwidth]{004.pdf}
\caption[Result of \method{DEXUS} in unsupervised mode with {\em
unknown conditions}.]{Result of \method{DEXUS} in unsupervised mode with {\em
unknown conditions}. A heatmap of the log read counts of the top ranked transcripts of
the ``European HapMap'' \citep{Montgomery2010} data set is shown. Rows represent
transcripts sorted  by their  I/NI values.
The trancript on the top has the highest I/NI. Columns represent
different samples. Red symbols indicate samples that belong to the minor
condition that was identified by \method{DEXUS}.
}
\end{center}
\end{figure} 


Information about a specific transcript can also be accessed from the result
object by subsetting it with the transcript name.

<<>>=
resultMontgomery["ENSG00000007038"]
@

Even more information can be obtained by using the {\tt as.data.frame} function.
<<eval=FALSE>>=
as.data.frame(resultMontgomery["ENSG00000007038"])
@
<<echo=FALSE>>=
as.data.frame(resultMontgomery["ENSG00000007038"])[,1:12]
@



To convert the full result object to a data frame that can be exported the 
function {\tt as.data.frame} can be used.
<<eval=FALSE>>=
as.data.frame(sort(resultMontgomery))
@
For more information on the result object, see Section \ref{s:ResultObject}.


\section{Case-Control Like Study Designs: Replicates, Known Sample Groups
or Conditions}
\subsection{Two Known Groups or Conditions}

In the study of \citet{Bottomly2011}, two strains of mice, C57BL/6J (B6) and
DBA/2J (D2), were compared using both RNA-Seq and microarrays.
The data set consists of 21 lanes from male mice (10 of the B6 strain and 11 of
D2 strain), produced using an Illumina GAIIx sequencing machine.
The data set was provided by the \method{ReCount} repository
\citep{Frazee2011} that is based on Ensembl 61 transcript definitions.
In this case of {\em two known conditions} we provide \method{DEXUS} with the
group labels, in order to detect transcripts that are differentially expressed
between the two mice strains. 


We apply the function {\tt dexus} to the count table of the first 1000
transcripts and provide the labels of the samples, and set the normalization
to ``Upper Quartile'' normalization.

<<>>=
resultSupervised <- dexus(countsBottomly[1:1000, ],
		labels=substr(colnames(countsBottomly),1,2),
		normalization="upperquartile")
@


To show a list of differentially expressed transcripts, simply type the name of the 
result object.
<<>>=
resultSupervised
@

To sort the transcripts in the result object by $p$-values use the {\tt sort}
method.

<<>>=
resultSupervised <- sort(resultSupervised)
@

To obtain a heatmap of the differentially expressed transcripts, type {\tt plot}.
<<eval=FALSE>>=
plot(resultSupervised)
@

To get the full list of transcripts together with additional information, such as
the I/NI values, conditions, dispersions, and means the function
{\tt as.data.frame} can be used.

<<eval=FALSE>>=
as.data.frame(resultSupervised)
@
For more information on the result object, see Section \ref{s:ResultObject}.


\subsection{Multiple Known Groups or Conditions}

\citet{Blekhman2010} investigated the differences in alternative splicing 
in liver tissue between humans, chimpanzees and rhesus macaques. 
For this purpose they  performed RNA-Seq on three male and three female liver
samples from each species. They focused on the expression values of exons that
had reliably determined orthologs in all species.  Read counts for exons were
provided by \citet{Blekhman2010}, who used transcript models from Ensemble (Release 50). 
In this case the three species are three distinct groups. The aim is to find
transcripts, that show large differences between these groups.

We run \method{DEXUS} on this data set and provide the method with the group
labels, i.e. the species.

<<>>=
resultMultipleGroups <- dexus(countsGilad[1:1000, ],
		labels=substr(colnames(countsGilad),1,2))
@

To show a list of differentially expressed transcripts, simply type the name of the 
result object.
<<>>=
resultMultipleGroups
@

To sort the transcripts in the result object by $p$-values use the {\tt sort}
method.

<<>>=
resultMultipleGroups <- sort(resultMultipleGroups)
@


To obtain a heatmap of the top-ranked transcripts use the {\tt plot} function.

<<eval=FALSE>>=
plot(resultMultipleGroups)
@

<<fig=FALSE,echo=FALSE,results=hide>>=
pdf("003.pdf")
plot(resultMultipleGroups,cexSamples=1)
dev.off()
@

\begin{figure}[H]
\begin{center}
\includegraphics[angle=0,width= 0.6\columnwidth]{003.pdf}
\caption[Result of \method{DEXUS} in supervised mode with {\em multiple 
known groups}.]{Result of \method{DEXUS} in supervised mode with {\em multiple 
known groups}. A heatmap of the log read counts of the 
top ranked transcripts of the ``Primate Liver'' \citep{Blekhman2010} data set is 
shown. Rows represent transcripts sorted  by their  $p$-values. 
The trancript on the top has the lowest $p$-value. Columns represent
different samples. The labels  ``HS'',``PT'' and ``MM''  represent the three
different species. Red symbols indicate the different species.
}
\end{center}
\end{figure} 

To get the full list of transcripts together with their $p$-values 
the function {\tt getResult} can be used.
<<eval=FALSE>>=
as.data.frame(resultMultipleGroups)
@
For more information on the result object, see Section \ref{s:ResultObject}.



\section{Calling Differential Expression, Visualization and the Result Object}
\label{s:ResultObject}

\subsection{Calling Differential Expression by the Informative/Non-Informative
Call}

In a setting in which the conditions or sample groups are unknown, or in which
there are no replicates, the I/NI value measures the evidence for differential
expression. At different thresholds \method{DEXUS} has different detection
powers (sensitivity) and significance levels (specificity).  On 2,400 simulated
data sets, I/NI value thresholds of 0.025, 0.05, and 0.1 yielded average
specificities of 92\%, 97\%, and 99\% at sensitivities of 76\%, 61\%, and 38\%
respectively. The threshold for the I/NI values is set by the function
{\tt INIThreshold}. The function {\tt INI} filters out non-informative
transcripts.

<<echo=TRUE>>=
informativeTranscripts2 <- INI(resultMontgomery,threshold=0.25)
@

The object {\tt informativeTranscripts2} contains only the informative, i.e. the
differentially expressed transcripts.

\subsection{Visualization}

There is a generic plotting function that can be applied to the result object
of \method{DEXUS}. The log read counts are visualized as a heatmap, in which we
also indicate the identified sample condition.
We can select how many transcripts we want to plot by using the parameter {\tt n}.

<<eval=FALSE>>=
plot(informativeTranscripts2)
@

<<fig=FALSE,echo=FALSE,results=hide>>=
pdf("005.pdf",width=12,height=6)
plot(informativeTranscripts2)
dev.off()
@

\begin{figure}[H]
\begin{center}
\includegraphics[angle=0,width= \textwidth]{005.pdf}
\caption[Result of \method{DEXUS} in unsupervised mode with {\em
unknown conditions}.]{Result of \method{DEXUS} in unsupervised mode with {\em
unknown conditions}. A heatmap of the log read counts of the top ranked transcripts of
the ``European HapMap'' \citep{Montgomery2010} data set is shown. Rows represent
transcripts sorted  by their  I/NI values.
The trancript on the top has the highest I/NI. Columns represent
different samples. Red symbols indicate samples that belong to the minor
condition that was identified by \method{DEXUS}.
}
\end{center}
\end{figure} 


\subsection{The Structure of the Result Object}

\method{DEXUS} returns an instance of ``DEXUSResult'' that contains the
following slots:
\begin{itemize}
   \item {\tt transcriptNames}: The names of the transcripts, genes, exons, or
   regions of interest.
   \item {\tt sampleNames}: The sample names, as they were given in the 
   input matrix.
   \item {\tt inputData}: The original read count matrix.
   \item {\tt normalizedData}: The normalized read count matrix.
   \item {\tt sizeFactors}: The size factors that were calculated for the
      normalization. This is that factor that scales each column or sample.
   \item {\tt INIValues}: An informative/non-informative (I/NI) value for  each
   sample that measures the evidence for differential expression.
   \item {\tt INIThreshold}: The threshold for the I/NI values. Transcript with
   I/NI values above the threshold will be considered as differentially
   expressed.
   \item {\tt INICalls}: A binary value for each transcript indicating whether
   it is differentially expressed.
   \item {\tt pvals:} In case of {\em two known conditions} or {\em multiple 
   known conditions} it is possible to calculate a $p$-value for each
   transcript. This value is given in this slot.
   \item {\tt responsibilites:} A matrix of the size of the input matrix. It
   indicates the condition for each sample and transcript. The condition named
   ``1'' is the major condition. All other conditions are minor conditions. In
   case of supervised ({\em two known conditions} or {\em multiple 
   known conditions}) analyses this clustering matrix will be the same for all
   transcripts.
   \item {\tt posteriorProbs}: An array of the dimension of transcripts times
   samples times conditions. It gives the probability that a certain read count 
   $x$ was generated under a condition.
   \item {\tt logFC}: The log foldchanges between the conditions. The reference
   is always condition ``1''.
   \item {\tt conditionSizes}: The ratio of samples belonging to that condition.
   These are the $\alpha_i$ values of the model.
   \item {\tt sizeParameters}: The size parameter estimates for each condition.
   These are the $r_i$ values of the model.
   \item {\tt means}: The mean of each condition. The $\mu _i$ values of the
   model. 
   \item {\tt dispersions}: The dispersion estimates for each condition. The
   inverse size parameters.
   \item {\tt params}: The input parameters of the \method{DEXUS} algorithm.
\end{itemize}
        


\section{Parameter Settings of DEXUS}
The input parameters of the \method{DEXUS} algorithm are the following:
\begin{itemize}
  \item {\tt X}: The read count matrix. If the reads are already normalized,
  then set {\tt normalization} to ``{\tt none}''.
  \item {\tt nclasses}: The number of conditions that \method{DEXUS} should
  model.
  The number should be much smaller than the number of samples. In
  supervised mode ({\em two known conditions} or {\em multiple 
   known conditions}) the algorithm uses the number of different
   labels as {\tt nclasses}. Needs not be specified in supervised mode.
  \item {\tt alphaInit}: The initialization of the $\alpha_i$ values of the
  model.
  A vector with length of the number of conditions. The algorithm internally
  scales this vector to sum 1. Needs not be specified in supervised mode.
  \item {\tt G}: The weight of the Dirichlet prior. An important parameter that
  guides the EM algorithm. The higher this value the more transcripts will be
  explained by one condition and will therefore be classified as not
  differentially expressed. The lower the value of {\tt G} the more transcripts
  will be found to be differentially expressed. Needs not be specified in
  supervised mode.
  \item {\tt cyc}: The number of cycles of the EM algorithm per transcript.
  Needs not be specified in supervised mode.
  \item {\tt labels}: If the conditions, groups, or classes are known, then they
  can be passed to the algorithm through this parameter.
  \item {\tt normalization}: The normalization method to be used. Choices are
  ``RLE'', ``upperquartile'', and ``none''.
  \item {\tt kmeansIter}: For the initialization of the algorithm a k-means
  clustering is run. This is the number of iterations of the clustering.
  \item {\tt ignoreIfAllCountsSmaller}: A transcript is considered as ``not
  expressed'', if counts of all samples are below this value. The algorithm is
  not applied to these transcripts.
  \item {\tt theta}: The weight of the exponential prior on the size parameter
  of the negative binomial distributions. The higher this parameter, the lower the
  estimates of the size parameters, and consequently the higher the estimates of
  the overdispersions. 
  \item {\tt minMu}: The minimal value for the mean parameter of the negative
  binomial distribution.
  \item {\tt rmax}: An upper bound for the size parameter and thereby a lower
  bound for the overdispersion. The value is set to the value that \method{DESeq} uses for
  this purpose.
  \item {\tt initializiation}: How the initial estimates of the conditions are
  determined. Possible choices are ``kmeans'' and ``quantiles''. Needs not be
  specified in supervised mode.
  \item {\tt multiClassPhiPoolingFunction}: In case of multiple known conditions
  it is possible to calculate one overdispersion value per transcript. This can be
  calculated over all conditions or as mean, maximum or minimum over the
  specified conditions. Usually the option ``NULL'' (calculation of the 
  overdispersion across all conditions) performs best.
\end{itemize}

\section{The Method}
\method{DEXUS} models read counts as
a finite mixture of negative binomial distributions
in which each mixture component
corresponds to a condition.
\method{DEXUS} classifies a transcript as differentially expressed
if modeling of its read counts requires more than one condition.
To account for the high overdispersion
observed in RNA-Seq data,
\method{DEXUS} assumes that under each condition the read counts are
drawn from a 
negative binomial distribution.
Read count $x$ is explained by a mixture of $n$ negative binomial
distributions:
\begin{align}
\label{eq:DEXUS}
p(x) \ = \ \sum_{i=1}^{n}\alpha_{i} \ \mathrm{NB}(x \ ; \ \mu_{i},r_{i}) \ ,
\end{align}
where $\alpha_{i}$ is the probability of being in 
condition $i$ out of $n$ possible conditions. 
In condition $i$, read counts are drawn from 
a negative binomial distribution with mean $\mu_i$ and size $r_i$,
where the size parameter $r_{i}$ is the inverse of the overdispersion $\phi_i$.
An expectation maximization (EM) algorithm is used to estimate
mean and overdispersion parameters of the negative binomials
as well as the condition under which a particular read count was generated.
\method{DEXUS}
decomposes read count variation into variation due to noise and
variation due to differential expression.
The evidence for differential expression is 
measured by an informative/non-informative (I/NI) value.
\method{DEXUS} applies a threshold to the I/NI value
to extract differentially expressed transcripts 
with a desired specificity (significance level)
or sensitivity (power).

\method{DEXUS} performs excellently in identifying differentially
expressed transcripts on data with unknown conditions. 
\method{DEXUS} was tested on 2,400 simulated data sets.
For I/NI value thresholds of 0.025, 0.05, and 0.1, it
yielded average specificities of 92\%, 97\%, and 99\% at 
sensitivities of 76\%, 61\%, and 38\%, respectively.
Subsequently, \method{DEXUS} was tested on real-world data sets,
in which it identified differentially
expressed transcripts between subgroups defined by sex, species,
or tissue although information about these subgroups was withheld.
On HapMap individuals, \method{DEXUS} detected several differentially
expressed transcripts, the vast majority of which are related to sex,
eQTLs, or copy number variable regions.
However, we were unable to interpret the conditions for some differentially
expressed transcripts 
which hints at the existence of another cause of differential expression.

\section{A MAP Estimate for the Size Parameter and the
Overdispersion of a Negative Binomial}

We provide the function {\tt getSizeNB} that gives an estimate for the size
parameter of a negative binomial distribution from given data. In this function
the maximum-likelihood estimate is used, if the argument {\tt eta} is set to
0, and if {\tt eta} is set to a value greater than 0, a maximum-a-posteriori
estimator for the size parameter is calculated. In that case an exponential 
prior is used. The argument {\tt eta} determines the weight of this prior.

The maximum-likelihood estimator overestimates the size parameter and, thus,
underestimates the overdispersion parameter \cite{Piegorsch1990}. The
maximum-a-posteriori estimator can correct for this bias and decreases the
variance of the estimator, as we show in the following example. Another
problem is that, if the mean of the given data exceeds the variance, the
maximum-likelihood-estimator tends to infinity \cite{Anscombe1950}. By setting
the argument {\tt rmax} to a positive value, one can infer an upper bound on
the size parameter and, thereby, a lower bound on the overdispersion.


<<eval=FALSE>>=
trueSizeParameter <- 2
x <- rnbinom(n=5, size=trueSizeParameter, mu=40)
(sizeML <- getSizeNB(x,eta=0))
@
<<echo=FALSE>>=
trueSizeParameter <- 2
x <- c(38,40,30,55,36)
(sizeML <- getSizeNB(x,eta=0))
@
<<>>=
(sizeMAP <- getSizeNB(x,eta=1))
@
<<>>=
(trueDispersion <- 1/trueSizeParameter)
@
<<>>=
(dispersionML <- 1/getSizeNB(x,eta=0))
@
<<>>=
(dispersionMAP <- 1/getSizeNB(x,eta=1))
@



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\end{document}
