%\VignetteIndexEntry{DEGseq}
%\VignetteKeywords{differentially expressed genes}
%\VignetteDepends{qvalue}
%\VignettePackage{DEGseq}

\documentclass[11pt]{article}
\usepackage{Sweave}
\usepackage{amsmath}
\usepackage{hyperref}
\usepackage[authoryear,round]{natbib}

\setlength{\textheight}{8.5in}
\setlength{\textwidth}{6in}
\setlength{\topmargin}{-0.25in}
\setlength{\oddsidemargin}{0.25in}
\setlength{\evensidemargin}{0.25in}
\newcommand{\Rpackage}[1]{{\textit{#1}}}

\begin{document}
\title{\bf How to use the DEGseq Package}
\author{Likun Wang$^1$$^,$$^2$, Xiaowo Wang$^1$ and Xuegong Zhang$^1$.}
\maketitle
\noindent
$^1$MOE Key Laboratory of Bioinformatics and Bioinformatics Division, TNLIST /Department
of Automation, Tsinghua University.\\
\noindent
$^2$Department of Biomedical Informatics, School of Basic Medical Sciences, Peking University Health Science Center.
\begin{center}
{\tt wanglk@pku.edu.cn; xwwang@tsinghua.edu.cn; zhangxg@tsinghua.edu.cn}
\end{center}
\tableofcontents

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
This document provides a discussion of the functions in the package \Rpackage{DEGseq}. \Rpackage{DEGseq} is
a free R package for identifying differentially expressed genes from RNA-seq data. The input of \Rpackage{DEGseq} is 
uniquely mapped reads from RNA-seq data with a gene annotation of the corresponding genome, 
or gene (or transcript isoform) expression values provided by other programs. 
The output of \Rpackage{DEGseq} includes a text file and an XHTML summary page. 
The text file contains the gene expression values for the samples, 
a \textit{P-value} and two kinds of \textit{Q-values} which are calculated by the methods described in \cite{Benjamini95} 
and \cite{Storey03} 
for each gene to denote its expression 
difference between libraries. 


We also provided a function {\tt samWrapper} using the method as described in SAM \citep{Tusher01} 
which can be applied to compare two sets of samples with multiple replicates or two groups of samples
from different individuals (e.g. disease samples, case vs. control).

The \Rpackage{DEGseq} package employs library \Rpackage{qvalue} and \Rpackage{samr}, 
which must be installed in advance.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Getting started}
To load the \Rpackage{DEGseq} package, type {\tt library(DEGseq)}. 
Total six methods are presented in this package. They are {\tt DEGexp}, 
{\tt DEGseq}, {\tt samWrapper}, {\tt getGeneExp} and {\tt readGeneExp}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Methods}

\subsection{MA-plot-based method with random sampling model}
Current observations suggest that typically RNA-seq experiments have low background noise 
and the Poisson model fits data well. In such cases, users could directly pool the technical replicates 
for each sample together to get higher sequencing depth and detect subtle gene expression changes.


\cite{Jiang09} modeled RNA sequencing as a random sampling process, in which 
each read is sampled independently and uniformly from every possible nucleotides 
in the sample. Based on this model, the number of reads 
coming from a gene (or transcript isoform) follows a binomial distribution 
(and could be approximated by a Poisson distribution). Using the statistical model, 
we proposed a novel method based on the MA-plot, 
which is a statistical analysis tool having been widely used to detect and visualize 
intensity-dependent ratio of microarray data \citep{Yang02}. 


Let $C_1$ and $C_2$ denote the counts of reads mapped to a specific gene obtained from two samples 
with $C_i \sim binomial(n_i, p_i), i = 1, 2$, where $n_i$ denotes the total number of mapped reads and $p_i$ 
denotes the probability of a read coming from that gene. We define $M = log_2^{C_1} - log_2^{C_2}$, 
and $A = (log_2^{C_1} + log_2^{C_2})/2$. 
We assume that $C_1$ and $C_2$ are independent. Let $X = log_2^{C_1}$ and $Y = log_2^{C_2}$, hence $M = X - Y$ 
and $A = (X + Y)/2$. 
We can prove that $X$ and $Y$ follow normal distributions approximately (when $n_i$ is large enough), denote

\begin{equation}
X \rightarrow  N(log_2(n_1p_1), (\frac{1-p_1}{n_1p_1})(log_2^e)^2) =  N(\mu_X, \sigma^2_X)
\end{equation} 
\begin{equation}
Y \rightarrow  N(log_2(n_2p_2), (\frac{1-p_2}{n_2p_2})(log_2^e)^2) =  N(\mu_Y, \sigma^2_Y)
\end{equation}

Based on the assumption that $C_1$ and $C_2$ are independent (so $X$ and $Y$ are independent), 
the distributions of $M$ and $A$ can be obtained:
\begin{equation}
M \sim N(\mu_X-\mu_Y, \sigma^2_X+\sigma^2_Y)=N(\mu_M, \sigma^2_M)
\end{equation}
\begin{equation}
A \sim N(\frac{1}{2}(\mu_X+\mu_Y), \frac{1}{4}(\sigma^2_X+\sigma^2_Y))=N(\mu_A, \sigma^2_A)
\end{equation}

Based on formulas (3) and (4), the conditional distribution of $M$ given 
that $A = a$ can be obtained: 
\begin{eqnarray*}
M|(A=a) \sim N(\mu_M+\rho\frac{\sigma_M}{\sigma_A}(a-\mu_A), \sigma^2_M(1-\rho^2)) , \\
\rho=\frac{Cov(M,A)}{\sigma_M\sigma_A}=\frac{\sigma^2_X-\sigma^2_Y}{\sigma^2_X+\sigma^2_Y} .
\end{eqnarray*}
Thus, 
\begin{eqnarray*}
E(M|A=a) & = & \mu_M+\rho\frac{\sigma_M}{\sigma_A}(a-\mu_A) \\
         & = & \mu_X-\mu_Y+2\frac{\sigma^2_X-\sigma^2_Y}{\sigma^2_X+\sigma^2_Y}(a-\frac{1}{2}(\mu_X+\mu_Y)) .
\end{eqnarray*}
and
\begin{eqnarray*}
Var(M|A=a) & = & \sigma^2_M(1-\rho^2) \\ 
           & = & 4\frac{\sigma^2_X\sigma^2_Y}{\sigma^2_X+\sigma^2_Y} .
\end{eqnarray*}

For gene $g$ with ($A=a$, $M=m$) on the MA-plot of two samples, 
we do the hypothesis test $H_0: p_1 = p_2 = p$ versus $H_1: p_1 \neq p_2$.
Based on above deduction,
        $$\mu_A=\frac{1}{2}(\mu_X+\mu_Y)=\frac{1}{2}log_2(n_1n_2p^2) .$$
Thus, 
        $$p=\sqrt{2^{2\mu_A}/(n_1n_2)} .$$
Use $a$ as an estimate of $\mu_A$ then
        $$\hat{p}=\sqrt{2^{2a}/(n_1n_2)} .$$
So the estimates of $E(M | A=a)$ and $Var(M | A=a)$ are
        $$\widehat{E}(M|A=a)=log_2(n_1)-log_2(n_2) ,$$
and
        $$\widehat{Var}(M|A=a)=\frac{4(1-\sqrt{2^{2a}/(n_1n_2)})(log_2^e)^2}{(n_1+n_2)\sqrt{2^{2a}/(n_1n_2)}} .$$       
Then use the two estimates to calculate a \textit{Z-score} for the gene $g$ with ($A=a$, $M=m$), 
and convert it to a two-sided \textit{P-value} which is used to indicate whether 
gene $g$ is differentially expressed or not.
        $$Z-score=\frac{|m-\widehat{E}(M|A=a)|}{\sqrt{\widehat{Var}(M|A=a)}} .$$

Given a \textit{Z-score} threshold, take four as an example, 
the two lines with the following equations are used to indicate the four-fold 
local standard deviation of $M$ according to the random sampling model:
\begin{eqnarray*}
m_1 & = & \widehat{E}(M|A=a)+4\cdot\sqrt{\widehat{Var}(M|A=a)} \\
    & = & log_2(n_1)-log_2(n_2)+4\cdot\sqrt{\frac{4(1-\sqrt{2^{2a}/(n_1n_2)})(log_2^e)^2}{(n_1+n_2)\sqrt{2^{2a}/(n_1n_2)}}}
\end{eqnarray*}
\begin{eqnarray*}
m_2 & = & \widehat{E}(M|A=a)-4\cdot\sqrt{\widehat{Var}(M|A=a)} \\
    & = & log_2(n_1)-log_2(n_2)-4\cdot\sqrt{\frac{4(1-\sqrt{2^{2a}/(n_1n_2)})(log_2^e)^2}{(n_1+n_2)\sqrt{2^{2a}/(n_1n_2)}}}
\end{eqnarray*}

We call the lines obtained by above equations \textit{theoretical} four-fold local standard deviations lines.

\subsection{MA-plot-based method with technical replicates}
To estimate the noise level of genes with different intensity, 
and identify gene expression difference in different sequencing libraries, 
we proposed another method which is also based on the MA-plot. 
Here $M$ is the $Y$-axis and represents the intensity ratio, 
and $A$ is the $X$-axis and represents the average intensity for each transcript. 
To estimate the random variation, we first draw a MA-plot using two technical replicates (e.g. two sequencing lanes) 
from the same library. A sliding window (each window includes 1\% points of the MA-plot) is applied to scan the MA-plot 
along the $A$-axis (average intensity). For the window which is centered at $A=a$, the local variation of $M$ conditioned 
on $A=a$ is estimated by all the $M$ values of the transcripts in the window. And a smoothed estimate of 
the intensity-dependent noise level (local variation of $M$) is estimated by lowess regression among the windows, 
and converted to the standard deviation, under the assumption of normal distribution. 
The local standard deviations $\sigma_a$ of $M$ conditioned on $A=a$ were then used to compare the observed difference between 
two different libraries. Next, we draw a second MA-plot for the data from two different libraries. 
For each transcript $g$ with ($A=a_g$, $M=m_g$) on the MA-plot, a \textit{Z-score} $=|m_g-\mu_g|/\sigma_g$ is calculated to evaluate whether 
this transcript is differentially expressed, where $\mu_g$ is the local mean of $M$ and $\sigma_g$ is the local standard deviation 
of $M$ conditioned on $A=a_g$ estimated by technical replicates. Finally, a \textit{P-value} is assigned to this gene according to 
the \textit{Z-score} under the assumption of normal distribution.

\section{Data}
The test RNA-seq data are from \cite{Marioni08}. 
In their research, the RNA samples from human liver and kidney were analyzed using the 
Illumina Genome Analyzer sequencing platform. 
Each sample was sequenced in seven lanes, split across two runs of the machine.
The raw data are available in the NCBI short 
read archive with accession number {\tt SRA000299}.
Please see \cite{Marioni08} for more details.

\section{Examples}
\subsection{Example for DEGexp}
If users already have the gene expression values (or the number of reads mapped to each gene),
this method can be used to identify differentially expressed genes between two samples
with or without multiple technical replicates directly.
In the package, there are test data for this method. The file GeneExpExample5000.txt includes the first 5000 lines 
in SupplementaryTable2.txt which is a supplementary file for \cite{Marioni08}. In this file, each line
includes the count of reads mapped to a gene for 14 lanes respectively (7 lanes for 
kidney and 7 lanes for liver). In the following examples, we only use the data sequenced at a concentration 
of 3 pM (five lanes for each sample). 

<<echo=TRUE,print=FALSE>>=
  library(DEGseq)
  geneExpFile <- system.file("extdata", "GeneExpExample5000.txt", package="DEGseq")
  geneExpMatrix1 <- readGeneExp(file=geneExpFile, geneCol=1, valCol=c(7,9,12,15,18))
  geneExpMatrix2 <- readGeneExp(file=geneExpFile, geneCol=1, valCol=c(8,10,11,13,16))
  write.table(geneExpMatrix1[30:31,],row.names=FALSE)
  write.table(geneExpMatrix2[30:31,],row.names=FALSE)
@
\noindent
To identify differentially expressed genes between the two samples (kidney and liver),
we first used the method {\tt MARS}: MA-plot-based method with Random Sampling model.
Five report graphs for the two samples will be shown following the example commands.
  
\begin{center}
<<echo=TRUE,print=FALSE,fig=TRUE>>=
layout(matrix(c(1,2,3,4,5,6), 3, 2, byrow=TRUE))
par(mar=c(2, 2, 2, 2))
DEGexp(geneExpMatrix1=geneExpMatrix1, geneCol1=1, expCol1=c(2,3,4,5,6), groupLabel1="kidney",
       geneExpMatrix2=geneExpMatrix2, geneCol2=1, expCol2=c(2,3,4,5,6), groupLabel2="liver",
       method="MARS")
@
\end{center}

\noindent
The red points in the last graph (MA-plot) are the genes identified as differentially expressed.
If the {\tt outputDir} is specified, 
a text file and an XHTML summary page will be generated. 
These files can be found in the output directory. 

Next, we performed the function {\tt DEGexp} with the method {\tt MATR}: 
MA-plot-based method with technical replicates.

\begin{center}
<<echo=TRUE,print=FALSE,fig=TRUE,results=hide>>=
layout(matrix(c(1,2,3,4,5,6), 3, 2, byrow=TRUE))
par(mar=c(2, 2, 2, 2))
DEGexp(geneExpMatrix1=geneExpMatrix1, expCol1=2, groupLabel1="kidneyR1L1", 
       geneExpMatrix2=geneExpMatrix2, expCol2=2, groupLabel2="liverR1L2",
       replicateExpMatrix1=geneExpMatrix1, expColR1=3, replicateLabel1="kidneyR1L3", 
       replicateExpMatrix2=geneExpMatrix1, expColR2=4, replicateLabel2="kidneyR1L7",
       method="MATR")
@
\end{center}

\noindent
The red points in the last graph (MA-plot) are the genes identified as differentially expressed.
The blue points are from the replicates (kidneyR1L3 and kidneyR1L7), 
and the blue lines show the four-fold local standard deviation of $M$ estimated by the two technical replicates.

\subsection{Example for DEGseq}
The method {\tt DEGseq} takes uniquely mapped reads from RNA-seq data with a gene annotation as input. 
This function
first counts the number of reads mapped to each gene for samples with or without multiple technical replicates.
And then it invokes {\tt DEGexp} to identify significant genes.

<<echo=TRUE,print=FALSE>>=
  kidneyR1L1 <- system.file("extdata", "kidneyChr21.bed.txt", package="DEGseq")
  liverR1L2  <- system.file("extdata", "liverChr21.bed.txt", package="DEGseq")
  refFlat    <- system.file("extdata", "refFlatChr21.txt", package="DEGseq")
  mapResultBatch1 <- c(kidneyR1L1)  ## only use the data from kidneyR1L1 and liverR1L2
  mapResultBatch2 <- c(liverR1L2)
  outputDir <- file.path(tempdir(), "DEGseqExample")
  DEGseq(mapResultBatch1, mapResultBatch2, fileFormat="bed", refFlat=refFlat,
         outputDir=outputDir, method="LRT")
@

\subsection{Example for samWrapper}
To compare two sets of samples with multiple replicates or two groups of samples from different individuals 
(e.g. disease samples vs. control samples), we provided a method which employs the methods in package \Rpackage{samr}. 
The strategy used in \Rpackage{samr} was first described in \cite{Tusher01}, 
and is used for significance analysis of microarrays. Note: This function was removed from version 1.34.1.

\subsection{Example for getGeneExp}
This method is used to count the number of reads mapped to each gene for one sample.
The sample can have multiple technical replicates.
The input of this method is the uniquely mapped reads with a gene annotation. And 
the output is a text file containing gene expression values for the sample. For example,
<<echo=TRUE,print=FALSE>>=
  kidneyR1L1 <- system.file("extdata", "kidneyChr21.bed.txt", package="DEGseq")
  refFlat    <- system.file("extdata", "refFlatChr21.txt", package="DEGseq")
  mapResultBatch <- c(kidneyR1L1)
  output <- file.path(tempdir(), "kidneyChr21.bed.exp")
  exp <- getGeneExp(mapResultBatch, refFlat=refFlat, output=output)
  write.table(exp[30:32,], row.names=FALSE)
@
The gene annotation file must follow the UCSC refFlat format.
See \url{http://genome.ucsc.edu/goldenPath/gbdDescriptionsOld.html#RefFlat}. 

\subsection{Example for readGeneExp}
This function is used to read gene expression values from a file to a matrix in R workspace. 
For example,
<<echo=TRUE,print=FALSE>>=
  geneExpFile <- system.file("extdata", "GeneExpExample1000.txt", package="DEGseq")
  exp <- readGeneExp(file=geneExpFile, geneCol=1, valCol=c(7,9,12,15,18,8,10,11,13,16))
  write.table(exp[30:32,], row.names=FALSE)
@

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\newpage
\bibliographystyle{apalike}
\bibliography{DEGseq}
\end{document}
