---
title: "RNA-seq workflow: gene-level exploratory analysis and differential expression"
date: September 9, 2015
output: html_document
bibliography: bibliography.bib
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---

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<script type="text/javascript"
  src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML">
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Michael Love [1], Simon Anders [2,3], Vladislav Kim [3], Wolfgang Huber [3]

[1] Department of Biostatistics, Dana-Farber Cancer Institute and
Harvard School of Public Health, Boston, US; 

[2] Institute for Molecular Medicine Finland (FIMM), Helsinki, Finland;

[3] European Molecular Biology Laboratory (EMBL), Heidelberg, Germany.

```{r style, echo=FALSE, message=FALSE, warning=FALSE, results="asis"}
library("BiocStyle")
library("knitr")
library("rmarkdown")
options(width=100)
opts_chunk$set(message = FALSE, error = FALSE, warning = FALSE, fig.width=5, fig.height=5)
```

# Short links

* [Counting reads](#count) 
* [Building a DESeqDataSet](#construct)
* [Exploratory analysis and visualization](#eda)
* [Differential expression](#de)
* [Plotting results](#plots)
* [Annotating and exporting results](#annotate)
* [Accounting for unknown batches](#batch)
* [Time course experiments](#time)
* [Session information and references](#ref)

# Abstract

Here we walk through an end-to-end gene-level RNA-seq differential
expression workflow using Bioconductor packages.  We will start from
the FASTQ files, show how these were aligned to the reference genome,
and prepare a count matrix which tallies the number of RNA-seq
reads/fragments within each gene for each sample.  We will
perform exploratory data analysis (EDA) for quality assessment and to
explore the relationship between samples, perform differential gene
expression analysis, and visually explore the results.

# Introduction

Bioconductor has many packages which support analysis of
high-throughput sequence 
data, including RNA sequencing (RNA-seq). The packages which we
will use in this workflow include core packages maintained by the
Bioconductor core team for importing and processing raw sequencing
data and loading gene annotations. We will also use
contributed packages for statistical analysis and visualization
of sequencing data. Through scheduled releases every 6 months, the
Bioconductor project ensures that all the packages within a release
will work together in harmony (hence the "conductor" metaphor).
The packages used in this workflow are loaded with the
*library* function and can be installed by following the
[Bioconductor package installation instructions](http://bioconductor.org/install/#install-bioconductor-packages).

A published version of this workflow, including reviewer reports and comments
is available at [F1000Research](http://f1000research.com/articles/4-1070).

If you have questions about this workflow or any Bioconductor
software, please post these to the
[Bioconductor support site](https://support.bioconductor.org/).
If the questions concern a specific package, you can tag the post with
the name of the package, or for general questions about the workflow,
tag the post with `rnaseqgene`. Note the
[posting guide](http://www.bioconductor.org/help/support/posting-guide/)
for crafting an optimal question for the support site.

## Experimental data

The data used in this workflow is stored in the
`r Biocexptpkg("airway")` package that summarizes an RNA-seq experiment
wherein airway smooth muscle cells were treated with dexamethasone, a
synthetic glucocorticoid steroid with anti-inflammatory effects
[@Himes2014RNASeq].  Glucocorticoids are used, for example, 
by people with asthma to reduce inflammation of the airways. In the experiment,
four primary human airway smooth muscle cell lines were treated with 1
micromolar dexamethasone for 18 hours. For each of the four cell
lines, we have a treated and an untreated sample. For more description
of the experiment see the
[PubMed entry 24926665](http://www.ncbi.nlm.nih.gov/pubmed/24926665)
and for raw data see the
[GEO entry GSE52778](http://www.ncbi.nlm.nih.gov/geo/query/acc.cgi?acc=GSE52778).

<a id="count"></a>

# Preparing count matrices

As input, the count-based statistical methods, such as
`r Biocpkg("DESeq2")` [@Love2014Moderated],
`r Biocpkg("edgeR")` [@Robinson2009EdgeR],
`r Biocpkg("limma")` with the voom method [@Law2014Voom],
`r Biocpkg("DSS")` [@Wu2013New], 
`r Biocpkg("EBSeq")` [@Leng2013EBSeq] and
`r Biocpkg("BaySeq")` [@Hardcastle2010BaySeq],
expect input data as obtained, e.g., from RNA-seq or another high-throughput
sequencing experiment, in the form of a matrix of integer values.
The value in the *i*-th row and the *j*-th column of the matrix tells how
many reads (or fragments, for paired-end RNA-seq) have been
unambiguously assigned to gene *i* in sample *j*.  Analogously,
for other types of assays, the rows of the matrix might correspond
e.g., to binding regions (with ChIP-Seq), species of bacteria (with
metagenomic datasets), or peptide sequences (with
quantitative mass spectrometry).

The values in the matrix must be raw counts of sequencing
reads/fragments. This is important for *DESeq2*'s statistical model to
hold, as only the raw counts allow assessing the measurement precision
correctly. It is important to *never* provide counts that were
pre-normalized for sequencing depth/library size, as the
statistical model is most powerful when applied to raw counts, and is
designed to account for library size differences internally.

## Aligning reads to a reference genome

The computational analysis of an RNA-seq experiment begins earlier:
we first obtain a set of FASTQ files that
contain the nucleotide sequence of each read and a quality score at
each position.  These reads must first be aligned to a reference
genome or transcriptome.  It is important to know if the sequencing
experiment was single-end or paired-end, as the alignment software
will require the user to specify both FASTQ files for a paired-end
experiment. The output of this alignment step is commonly stored in a
file format called [SAM/BAM](http://samtools.github.io/hts-specs).

A number of software programs exist to align reads to a reference
genome, and the development is too rapid for this document to provide
an up-to-date list. We recommend consulting benchmarking papers that
discuss the advantages and disadvantages of each software, which
include accuracy, sensitivity in aligning reads over splice junctions, speed,
memory required, usability, and many other features.

The reads for this experiment were aligned to the Ensembl
release 75 [@Flicek2014Ensembl] human reference genome using the
[STAR read aligner](https://code.google.com/p/rna-star/)
[@Dobin2013STAR]. In this example, we have a file in the current
directory called `files` with each line containing an identifier for
each experiment, and we have all the FASTQ files in a subdirectory
`fastq`. If you have downloaded the FASTQ files from the
Sequence Read Archive, the identifiers would be SRA run IDs,
e.g. `SRR1039520`. You should have two files for a paired-end
experiment for each ID, `fastq/SRR1039520_1.fastq1` and
`fastq/SRR1039520_2.fastq`, which give the first and second read for
the paired-end fragments. If you have performed a single-end
experiment, you would only have one file per ID.
We have also created a subdirectory, `aligned`,
where STAR will output its alignment files.

```
for f in `cat files`; do STAR --genomeDir ../STAR/ENSEMBL.homo_sapiens.release-75 \
--readFilesIn fastq/$f\_1.fastq fastq/$f\_2.fastq \
--runThreadN 12 --outFileNamePrefix aligned/$f.; done
```

[SAMtools](http://www.htslib.org/doc/samtools.html) [@Li2009Sequence]
was used to generate BAM files. The `-@` flag can be used to allocate
additional threads.

```
for f in `cat files`; do samtools view -bS aligned/$f.Aligned.out.sam \
-o aligned/$f.bam; done
```

The BAM files for a number of sequencing runs can then be used to
generate count matrices, as described in the following section.

## Locating alignment files

Besides the count matrix that we will use later, the
`r Biocexptpkg("airway")` package also contains eight files 
with a small subset of the total number of reads in the experiment.
The reads were selected which aligned to a small region of
chromosome 1. We chose a subset of reads because the full alignment files are
large (a few gigabytes each), and because it takes between 10-30
minutes to count the fragments for each sample.
We will use these files to demonstrate how a count matrix can be constructed
from BAM files. Afterwards, we will load the full count matrix
corresponding to all samples and all data, which is already provided
in the same package, and will continue the analysis with that full
matrix.

We load the data package with the example data:

```{r}
library("airway")
```

The R function *system.file* can be used to find out where on your
computer the files from a package have been installed. Here we ask for
the full path to the `extdata` directory, where R packages store
external data, that is part of the `r Biocexptpkg("airway")` package.

```{r}
dir <- system.file("extdata", package="airway", mustWork=TRUE)
```

In this directory, we find the eight BAM files (and some other files):

```{r}
list.files(dir)
```

Typically, we have a table with detailed information for each of our
samples that links samples to the associated FASTQ and BAM files.
For your own project, you might create such a comma-separated
value (CSV) file using a text editor or spreadsheet software such as Excel.

We load such a CSV file with *read.csv*:

```{r}
csvfile <- file.path(dir,"sample_table.csv")
(sampleTable <- read.csv(csvfile,row.names=1))
```

**Note:** here and elsewhere in the workflow, the parentheses `()` around the
entire code of the last line above is an R trick to print the output of the function
in addition to saving it to `sampleTable`. This is equivalent to assigning
and then showing the object in two steps:

```{r eval=FALSE}
sampleTable <- read.csv(csvfile,row.names=1)
sampleTable
```

Once the reads have been aligned, there are a number of tools that
can be used to count the number of reads/fragments that can be
assigned to genomic features for each sample. These often take as
input SAM/BAM alignment files and a file specifying the genomic
features, e.g. a GFF3 or GTF file specifying the gene models.

## *tximport*: transcript abundance summarized to gene-level

An exception to the alignment-counting workflow
is the *tximport* workflow, which leverages transcript
quantification methods such as *Sailfish* [@Patro2014Sailfish],
*Salmon* [@Patro2015Salmon],
*kallisto* [@Bray2015Near],
and *RSEM* [@Li2011RSEM],
to estimate abundances without aligning reads (so skipping
the generation of BAM files). Advantages of using *tximport* to
produce gene-level count matrices and normalizing offsets, are:
this approach corrects for potential changes in gene
length across samples (e.g. from differential isoform usage)
[@Trapnell2013Differential]; 
some of these methods are substantially faster and require less memory
and disk usage compared to alignment-based methods;
and it is possible to avoid discarding those fragments that
can align to multiple genes with homologous sequence [@Robert2015Errors].
For more details,
see the manuscript describing this approach [@Soneson2015Differential].
The *tximport* package will be part of the release branch of Bioconductor
in April 2016.

## *DESeq2* import functions

The following tools can be used generate count matrices,
*summarizeOverlaps* [@Lawrence2013Software],
*featureCounts* [@Liao2014FeatureCounts],
*tximport* [@Soneson2015Differential],
or *htseq-count* [@Anders2015HTSeqa]:

function            | package                                              | framework      | output                 | *DESeq2* input function
--------------------|------------------------------------------------------|----------------|------------------------|-------------------------
*summarizeOverlaps* | `r Biocpkg("GenomicAlignments")`                     | R/Bioconductor | *SummarizedExperiment* | *DESeqDataSet*
*featureCounts*     | `r Biocpkg("Rsubread")`                              | R/Bioconductor | matrix                 | *DESeqDataSetFromMatrix*
*tximport*          | `r Biocpkg("tximport")`                              | R/Bioconductor | list of matrices       | *DESeqDataSetFromTximport*
*htseq-count*       | [HTSeq](http://www-huber.embl.de/users/anders/HTSeq) | Python         | files                  | *DESeqDataSetFromHTSeq* 

We now proceed using the *summarizeOverlaps* method of counting.
Using the `Run` column in the sample table, we construct the full
paths to the files we want to perform the counting operation on:

```{r}
filenames <- file.path(dir, paste0(sampleTable$Run, "_subset.bam"))
file.exists(filenames)
```

We indicate in Bioconductor that these files are BAM files using the
*BamFileList* function from the `r Biocpkg("Rsamtools")` package
that provides an R interface to BAM files.
Here we also specify details about how the BAM files should
be treated, e.g., only process 2 million reads at a time. See
`?BamFileList` for more information.

```{r}
library("Rsamtools")
bamfiles <- BamFileList(filenames, yieldSize=2000000)
```

**Note:** make sure that the chromosome names of the genomic features
in the annotation you use are consistent with the chromosome names of
the reference used for read alignment. Otherwise, the scripts might
fail to count any reads to features due to the mismatching names.
For example, a common mistake is when the alignment files contain
chromosome names in the style of `1` and the gene annotation in the
style of `chr1`, or the other way around. See the *seqlevelsStyle*
function in the `r Biocpkg("GenomeInfoDb")` package for solutions.
We can check the chromosome names (here called "seqnames")
in the alignment files like so:

```{r}
seqinfo(bamfiles[1])
```

## Defining gene models

Next, we need to read in the gene model that will be used for
counting reads/fragments. We will read the gene model from an Ensembl
[GTF file](http://www.ensembl.org/info/website/upload/gff.html) [@Flicek2014Ensembl],
using *makeTxDbFromGFF* from the `r Biocpkg("GenomicFeatures")` package.
GTF files can be downloaded from
[Ensembl's FTP site](http://www.ensembl.org/info/data/ftp/) or other gene model repositories.
A *TxDb* object is a database that can be used to
generate a variety of range-based objects, such as exons, transcripts,
and genes. We want to make a list of exons grouped by gene for
counting read/fragments.

There are other options for constructing a *TxDb*.
For the *known genes* track from the UCSC Genome Browser [@Kent2002Human],
one can use the pre-built Transcript DataBase:
`r Biocannopkg("TxDb.Hsapiens.UCSC.hg19.knownGene")`.
If the annotation file is accessible from
`r Biocpkg("AnnotationHub")` (as is the case for the Ensembl genes),
a pre-scanned GTF file can be imported using *makeTxDbFromGRanges*.
Finally, the *makeTxDbFromBiomart* function can be used to automatically
pull a gene model from Biomart using `r Biocpkg("biomaRt")` [@Durinck2009Mapping].

Here we will demonstrate loading from a GTF file:

```{r}
library("GenomicFeatures")
```

We indicate that none of our sequences (chromosomes) are circular
using a 0-length character vector.

```{r}
gtffile <- file.path(dir,"Homo_sapiens.GRCh37.75_subset.gtf")
(txdb <- makeTxDbFromGFF(gtffile, format="gtf", circ_seqs=character()))
```

The following line produces a *GRangesList* of all the exons grouped
by gene [@Lawrence2013Software]. Each element of the list is a
*GRanges* object of the exons for a gene.

```{r}
(ebg <- exonsBy(txdb, by="gene"))
```

## Read counting step

After these preparations, the actual counting is easy. The function
*summarizeOverlaps* from the `r Biocpkg("GenomicAlignments")`
package will do this. This produces a *SummarizedExperiment*
object that contains a variety of information about
the experiment, and will be described in more detail below.

**Note:** If it is desired to perform counting using multiple cores, one can use
the *register* and *MulticoreParam* or *SnowParam* functions from the
`r Biocpkg("BiocParallel")` package before the counting call below.
Expect that the `summarizeOverlaps` call will take at least 30 minutes
per file for a human RNA-seq file with 30 million aligned reads. By sending
the files to separate cores, one can speed up the entire counting process.

```{r}
library("GenomicAlignments")
library("BiocParallel")
```

Here we specify to use one core, not multiple cores. We could have
also skipped this line and the counting step would run in serial.

```{r}
register(SerialParam())
```

The following call creates the *SummarizedExperiment* object with counts:

```{r}
se <- summarizeOverlaps(features=ebg, reads=bamfiles,
                        mode="Union",
                        singleEnd=FALSE,
                        ignore.strand=TRUE,
                        fragments=TRUE )
```

We specify a number of arguments besides the `features` and the
`reads`. The `mode` argument describes what kind of read overlaps will
be counted. These modes are shown in Figure 1 of the
*Counting reads with summarizeOverlaps* vignette for the
`r Biocpkg("GenomicAlignments")` package.
Note that fragments will be counted only once to each gene, even if
they overlap multiple exons of a gene which may themselves be overlapping.
Setting `singleEnd` to `FALSE`
indicates that the experiment produced paired-end reads, and we want
to count a pair of reads (a fragment) only once toward the count
for a gene. 
The `fragments` argument can be used when `singleEnd=FALSE` to specify if unpaired
reads should be counted (yes if `fragments=TRUE`).

In order to produce correct counts, it is important to know if the
RNA-seq experiment was strand-specific or not. This experiment was not
strand-specific so we set `ignore.strand` to `TRUE`.
However, certain strand-specific protocols could have the reads
align only to the opposite strand of the genes.
The user must check if the experiment was strand-specific and if so,
whether the reads should align to the forward or reverse strand of the genes.
For various counting/quantifying tools, one specifies counting on the
forward or reverse strand in different ways, although this task 
is currently easiest with *htseq-count*, *featureCounts*, or the
transcript abundance quantifiers mentioned previously.
It is always a good idea to check the column sums of the count matrix
(see below) to make sure these totals match the expected of the number
of reads or fragments aligning to genes.  Additionally, one can
visually check the read alignments using a genome visualization tool.

## SummarizedExperiment

```{r sumexp, echo=FALSE}
par(mar=c(0,0,0,0))
plot(1,1,xlim=c(0,100),ylim=c(0,100),bty="n",
     type="n",xlab="",ylab="",xaxt="n",yaxt="n")
polygon(c(45,90,90,45),c(5,5,70,70),col="pink",border=NA)
polygon(c(45,90,90,45),c(68,68,70,70),col="pink3",border=NA)
text(67.5,40,"assay")
text(67.5,35,'e.g. "counts"')
polygon(c(10,40,40,10),c(5,5,70,70),col="skyblue",border=NA)
polygon(c(10,40,40,10),c(68,68,70,70),col="skyblue3",border=NA)
text(25,40,"rowRanges")
polygon(c(45,90,90,45),c(75,75,95,95),col="palegreen",border=NA)
polygon(c(45,47,47,45),c(75,75,95,95),col="palegreen3",border=NA)
text(67.5,85,"colData")
```

**The component parts of a *SummarizedExperiment* object.** The `assay` (pink
block) contains the matrix of counts, the `rowRanges` (blue block) contains
information about the genomic ranges and the `colData` (green block)
contains information about the samples. The highlighted line in each
block represents the first row (note that the first row of `colData`
lines up with the first column of the `assay`). 

The *SummarizedExperiment* container is diagrammed in the Figure above
and discussed in the latest Bioconductor paper [@Huber2015Orchestrating].
In our case we have created a single matrix named "counts" that contains the fragment
counts for each gene and sample, which is stored in `assay`. It is
also possible to store multiple matrices, accessed with `assays`.
The `rowRanges` for our object is the *GRangesList* we used for
counting (one *GRanges* of exons for each row of the count matrix).
The component parts of the *SummarizedExperiment* are accessed with an
R function of the same name: `assay` (or `assays`), `rowRanges` and `colData`.

This example code above actually only counted a small subset of fragments
from the original experiment. Nevertheless, we
can still investigate the resulting *SummarizedExperiment* by looking at
the counts in the `assay` slot, the phenotypic data about the samples in
`colData` slot (in this case an empty *DataFrame*), and the data about the
genes in the `rowRanges` slot.

```{r}
se
dim(se)
assayNames(se)
head(assay(se), 3)
colSums(assay(se))
```

The `rowRanges`, when printed, only shows the first *GRanges*, and tells us
there are 19 more elements:

```{r}
rowRanges(se)
```

The `rowRanges` also contains metadata about the construction
of the gene model in the `metadata` slot. Here we use a helpful R
function, `str`, to display the metadata compactly:

```{r}
str(metadata(rowRanges(se)))
```

The `colData`:

```{r}
colData(se)
```

The `colData` slot, so far empty, should contain all the metadata.
Because we used a column of `sampleTable` to produce the `bamfiles`
vector, we know the columns of `se` are in the same order as the
rows of `sampleTable`. We can assign the `sampleTable` as the
`colData` of the summarized experiment, by converting
it into a *DataFrame* and using the assignment function:

```{r}
(colData(se) <- DataFrame(sampleTable))
```

## Branching point

At this point, we have counted the fragments which overlap the genes
in the gene model we specified. This is a branching point where we
could use a variety of Bioconductor packages for exploration and
differential expression of the count data, including
`r Biocpkg("edgeR")` [@Robinson2009EdgeR],
`r Biocpkg("limma")` with the voom method [@Law2014Voom],
`r Biocpkg("DSS")` [@Wu2013New],
`r Biocpkg("EBSeq")` [@Leng2013EBSeq] and
`r Biocpkg("BaySeq")` [@Hardcastle2010BaySeq].
@Schurch2016How compared performance of different statistical methods
for RNA-seq using a large number of biological replicates and can help
users to decide which tools make sense to use, and how many
biological replicates are necessary to obtain a certain sensitivity.
We will continue, using `r Biocpkg("DESeq2")` [@Love2014Moderated].
The *SummarizedExperiment* object is all we
need to start our analysis. In the following section we will show how
to use it to create the data object used by *DESeq2*.

<a id="construct"></a>

# The *DESeqDataSet*, sample information, and the design formula

Bioconductor software packages often define and use a custom class for
storing data that makes sure that all the needed data slots are
consistently provided and fulfill the requirements.  In addition,
Bioconductor has general data classes (such as the
*SummarizedExperiment*) that can be used to move data between
packages. Additionally, the core Bioconductor classes provide useful
functionality: for example, subsetting or reordering the rows or
columns of a *SummarizedExperiment* automatically subsets or reorders
the associated *rowRanges* and *colData*, which can help to prevent
accidental sample swaps that would otherwise lead to spurious
results. With *SummarizedExperiment* this is all taken care of behind
the scenes.

In *DESeq2*, the custom class is called *DESeqDataSet*. It is built on
top of the *SummarizedExperiment* class, and it is easy to convert
*SummarizedExperiment* objects into *DESeqDataSet* objects, which we
show below.  One of the two main differences is that the `assay` slot is
instead accessed using the *counts* accessor function, and the
*DESeqDataSet* class enforces that the values in this matrix are
non-negative integers.

A second difference is that the *DESeqDataSet* has an associated
*design formula*. The experimental design is specified at the
beginning of the analysis, as it will inform many of the *DESeq2*
functions how to treat the samples in the analysis (one exception is
the size factor estimation, i.e., the adjustment for differing library
sizes, which does not depend on the design formula).  The design
formula tells which columns in the sample information table (`colData`)
specify the experimental design and how these factors should be used
in the analysis.

The simplest design formula for differential expression would be
`~ condition`, where `condition` is a column in `colData(dds)` that
specifies which of two (or more groups) the samples belong to.  For
the airway experiment, we will specify `~ cell + dex`
meaning that we want to test for the effect of dexamethasone (`dex`)
controlling for the effect of different cell line (`cell`). We can see
each of the columns just using the `$` directly on the
*SummarizedExperiment* or *DESeqDataSet*:

```{r}
se$cell
se$dex
```

**Note:** it is prefered in R that the first level of a factor be the
reference level (e.g. control, or untreated samples), so we can
*relevel* the `dex` factor like so: 

```{r}
se$dex <- relevel(se$dex, "untrt")
se$dex
```

For running *DESeq2* models, you can use R's formula notation to
express any fixed-effects experimental design.
Note that *DESeq2* uses the same formula notation as, for instance, the *lm*
function of base R. If the research aim is to determine for which
genes the effect of treatment is different across groups, then
interaction terms can be included and tested using a design such as
`~ group + treatment + group:treatment`. See the manual page for
`?results` for more examples. We will show how to use an interaction
term to test for condition-specific changes over time in a 
time course example below.

In the following sections, we will demonstrate the construction of the
*DESeqDataSet* from two starting points:

* from a *SummarizedExperiment* object 
* from a count matrix and a sample information table

For a full example of using the *HTSeq* Python package for read
counting, please see the `r Biocexptpkg("pasilla")` vignette.  For an
example of generating the *DESeqDataSet* from files produced by
*htseq-count*, please see the `r Biocpkg("DESeq2")` vignette.

## Starting from *SummarizedExperiment*

We now use R's *data* command to load a prepared
*SummarizedExperiment* that was generated from the publicly available
sequencing data files associated with @Himes2014RNASeq,
described above.  The steps we used to produce this object were
equivalent to those you worked through in the previous sections,
except that we used all the reads and all the genes. For more details
on the exact steps used to create this object, type
`vignette("airway")` into your R session. 

```{r}
data("airway")
se <- airway
```

Again, we want to specify that `untrt` is the reference level for the
dex variable:

```{r}
se$dex <- relevel(se$dex, "untrt")
se$dex
```

We can quickly check the millions of fragments that uniquely aligned
to the genes (the second argument of *round* tells how many decimal
points to keep).

```{r}
round( colSums(assay(se)) / 1e6, 1 )
```

Supposing we have constructed a *SummarizedExperiment* using
one of the methods described in the previous section, we now need to
make sure that the object contains all the necessary information about
the samples, i.e., a table with metadata on the count matrix's columns
stored in the `colData` slot:

```{r}
colData(se)
```

Here we see that this object already contains an informative
`colData` slot -- because we have already prepared it for you, as
described in the `r Biocexptpkg("airway")` vignette.
However, when you work with your own data, you will have to add the
pertinent sample / phenotypic information for the experiment at this stage.
We highly recommend keeping this information in a comma-separated
value (CSV) or tab-separated value (TSV) file, which can be exported
from an Excel spreadsheet, and the assign this to the `colData` slot,
making sure that the rows correspond to the columns of the
*SummarizedExperiment*. We made sure of this correspondence earlier by
specifying the BAM files using a column of the sample table.

Once we have our fully annotated *SummarizedExperiment* object,
we can construct a *DESeqDataSet* object from it that will then form
the starting point of the analysis.
We add an appropriate design for the analysis:

```{r}
library("DESeq2")
```

```{r}
dds <- DESeqDataSet(se, design = ~ cell + dex)
```

## Starting from count matrices

In this section, we will show how to build an *DESeqDataSet* supposing
we only have a count matrix and a table of sample information.

**Note:** if you have prepared a *SummarizedExperiment* you should skip this
section. While the previous section would be used to construct a
*DESeqDataSet* from a *SummarizedExperiment*, here we first extract
the individual object (count matrix and sample info) from the
*SummarizedExperiment* in order to build it back up into a new object
-- only for demonstration purposes.
In practice, the count matrix would either be read in from a file or
perhaps generated by an R function like *featureCounts* from the
`r Biocpkg("Rsubread")` package [@Liao2014FeatureCounts].

The information in a *SummarizedExperiment* object can be
accessed with accessor functions. For example, to see the actual data,
i.e., here, the fragment counts, we use the *assay* function. (The *head*
function restricts the output to the first few lines.)

```{r}
countdata <- assay(se)
head(countdata, 3)
```

In this count matrix, each row represents an Ensembl gene, each column
a sequenced RNA library, and the values give the raw numbers of
fragments that were uniquely assigned to the respective gene in each
library. We also have information on each of the samples (the columns of the
count matrix). If you've counted reads with some other software, it is
very important to check that the columns of the count matrix correspond to the rows
of the sample information table. 

```{r}
coldata <- colData(se)
```

We now have all the ingredients to prepare our data object in a form
that is suitable for analysis, namely:

* `countdata`: a table with the fragment counts
* `coldata`: a table with information about the samples

To now construct the *DESeqDataSet* object from the matrix of counts and the
sample information table, we use:

```{r}
(ddsMat <- DESeqDataSetFromMatrix(countData = countdata,
                                  colData = coldata,
                                  design = ~ cell + dex))
```

We will continue with the object generated from the
*SummarizedExperiment* section. 

<a id="eda"></a>

# Exploratory analysis and visualization

There are two separate paths in this workflow; the one we will
see first involves *transformations of the counts*
in order to visually explore sample relationships.
In the second part, we will go back to the original raw counts
for *statistical testing*. This is critical because
the statistical testing methods rely on original count data
(not scaled or transformed) for calculating the precision of measurements. 

## Pre-filtering the dataset

Our count matrix with our *DESeqDataSet* contains many rows with only zeros, and
additionally many rows with only a few fragments total. In
order to reduce the size of the object, and to increase the speed of
our functions, we can remove the rows that have no or nearly no
information about the amount of gene expression. Here we remove rows
of the *DESeqDataSet* that have no counts, or only a single count
across all samples:

```{r}
nrow(dds)
dds <- dds[ rowSums(counts(dds)) > 1, ]
nrow(dds)
```

## The rlog transformation

Many common statistical methods for exploratory analysis of
multidimensional data, for example clustering and *principal
components analysis* (PCA), work best for data that generally has the
same range of variance at different ranges of the mean values. When
the expected amount of variance is approximately the same across
different mean values, the data is said to be *homoskedastic*.  For
RNA-seq raw counts, however, the variance grows with the mean.  For
example, if one performs PCA directly on a matrix of
size-factor-normalized read counts, the result typically depends only
on the few most strongly expressed genes because they show the largest
absolute differences between samples.  A simple and often used
strategy to avoid this is to take the logarithm of the normalized
count values plus a small pseudocount; however, now the genes with the
very lowest counts will tend to dominate the results because, due to
the strong Poisson noise inherent to small count values, and the fact
that the logarithm amplifies differences for the smallest values,
these low count genes will show the strongest relative differences
between samples.

As a solution, *DESeq2* offers transformations for count data that
stabilize the variance across the mean. One such transformation is the 
*regularized-logarithm transformation* or *rlog* [@Love2014Moderated].
For genes with high counts, the rlog 
transformation will give similar result to the ordinary log2
transformation of normalized counts.
For genes with lower counts, however, the values are
shrunken towards the genes' averages across all samples. Using 
an empirical Bayesian prior on inter-sample differences in the form of
a *ridge penalty*, the rlog-transformed data
then becomes approximately *homoskedastic*, and can be used directly for
computing distances between samples and making PCA plots.
Another transformation,
the *variance stabilizing transformation* [@Anders2010Differential],
is discussed alongside the *rlog* in the *DESeq2* vignette. 

**Note:** the rlog transformation is provided for applications *other*
than differential testing. For differential testing we recommend the
*DESeq* function applied to raw counts, as described later
in this workflow, which also takes into account the dependence of the
variance of counts on the mean value during the dispersion estimation
step.

The function *rlog* returns a *SummarizedExperiment*
object that contains the rlog-transformed values in its *assay* slot.

```{r rlog}
rld <- rlog(dds, blind=FALSE)
head(assay(rld), 3)
```

We specify `blind=FALSE`, which means that differences between cell
lines and treatment should not add to the
variance-mean profile of the experiment.
However, the experimental design is not used directly in the
transformation, only in estimating the global amount of
variability in the counts. For a fully *unsupervised* transformation, one can set
`blind=TRUE` (which is the default).

**Note:** for large datasets (hundreds of samples), the
variance stabilizing transformation will be faster to compute.

To show the effect of the transformation, in the Figure below
we plot the first sample
against the second, first simply using the *log2* function (after adding
1, to avoid taking the log of zero), and then using the rlog-transformed
values. For the *log2* approach, we need to first estimate *size factors* to
account for sequencing depth, and then specify `normalized=TRUE`.
Sequencing depth correction is done automatically for the *rlog*
method (and for *varianceStabilizingTransformation*).

```{r rldplot, fig.width=7, fig.height=3.75}
par( mfrow = c( 1, 2 ) )
dds <- estimateSizeFactors(dds)
plot(log2(counts(dds, normalized=TRUE)[,1:2] + 1),
     pch=16, cex=0.3)
plot(assay(rld)[,1:2],
     pch=16, cex=0.3)
```

**Scatterplot of transformed counts from two samples**. Shown are
scatterplots using the log2
transform of normalized counts (left side) and using the rlog (right side).

We can see how genes with low counts (bottom left-hand corner)
seem to be excessively variable
on the ordinary logarithmic scale, while the rlog transform compresses
differences for the low count genes for which the data provide little
information about differential expression.

## Sample distances

A useful first step in an RNA-seq analysis is often to assess overall
similarity between samples: Which samples are similar to each other,
which are different? Does this fit to the expectation from the
experiment's design?

We use the R function *dist* to calculate the Euclidean distance
between samples. To ensure we have a roughly equal contribution from
all genes, we use it on the rlog-transformed data. We need to
transpose the matrix of values using *t*, because the *dist* function
expects the different samples to be rows of its argument, and
different dimensions (here, genes) to be columns.

```{r}
sampleDists <- dist( t( assay(rld) ) )
sampleDists
```

We visualize the distances in a heatmap in a Figure below, using the function
*pheatmap* from the `r CRANpkg("pheatmap")` package.

```{r}
library("pheatmap")
library("RColorBrewer")
```

In order to plot the sample distance matrix with the rows/columns
arranged by the distances in our distance matrix,
we manually provide `sampleDists` to the `clustering_distance`
argument of the *pheatmap* function.
Otherwise the *pheatmap* function would assume that the matrix contains
the data values themselves, and would calculate distances between the
rows/columns of the distance matrix, which is not desired.
We also manually specify a blue color palette using the
*colorRampPalette* function from the `r CRANpkg("RColorBrewer")` package.

```{r distheatmap, fig.width=6, fig.height=4.5}
sampleDistMatrix <- as.matrix( sampleDists )
rownames(sampleDistMatrix) <- paste( rld$dex, rld$cell, sep="-" )
colnames(sampleDistMatrix) <- NULL
colors <- colorRampPalette( rev(brewer.pal(9, "Blues")) )(255)
pheatmap(sampleDistMatrix,
         clustering_distance_rows=sampleDists,
         clustering_distance_cols=sampleDists,
         col=colors)
```

**Heatmap of sample-to-sample distances using the rlog-transformed values.**

Note that we have changed the row names of the distance matrix to
contain treatment type and patient number instead of sample ID, so
that we have all this information in view when looking at the heatmap.

Another option for calculating sample distances is to use the
*Poisson Distance* [@Witten2011Classification], implemented in the
`r CRANpkg("PoiClaClu")` package.
This measure of dissimilarity between counts
also takes the inherent variance
structure of counts into consideration when calculating the distances
between samples. The *PoissonDistance* function takes the original
count matrix (not normalized) with samples as rows instead of columns,
so we need to transpose the counts in `dds`.

```{r}
library("PoiClaClu")
poisd <- PoissonDistance(t(counts(dds)))
```

We plot the heatmap in a Figure below.

```{r poisdistheatmap, fig.width=6, fig.height=4.5}
samplePoisDistMatrix <- as.matrix( poisd$dd )
rownames(samplePoisDistMatrix) <- paste( rld$dex, rld$cell, sep="-" )
colnames(samplePoisDistMatrix) <- NULL
pheatmap(samplePoisDistMatrix,
         clustering_distance_rows=poisd$dd,
         clustering_distance_cols=poisd$dd,
         col=colors)
```

**Heatmap of sample-to-sample distances using the *Poisson Distance*.**

## PCA plot

Another way to visualize sample-to-sample distances is a
principal components analysis (PCA). In this ordination method, the
data points (here, the samples) are projected onto the 2D plane
such that they spread out in the two directions that explain most of
the differences (Figure below). The x-axis is the direction that separates the data
points the most. The values of the samples in this direction are
written *PC1*. The y-axis is a direction (it must be *orthogonal* to
the first direction) that separates the data the second most. The
values of the samples in this direction are written *PC2*.
The percent of the total variance that is contained in the direction
is printed in the axis label. Note that these percentages do not add to
100%, because there are more dimensions that contain the remaining
variance (although each of these remaining dimensions will explain
less than the two that we see). 

```{r plotpca, fig.width=6, fig.height=4.5}
plotPCA(rld, intgroup = c("dex", "cell"))
```

**PCA plot using the rlog-transformed values.** Each unique combination of
treatment and cell line is given its own color.

Here, we have used the function *plotPCA* that comes with *DESeq2*.
The two terms specified by `intgroup` are the interesting groups for
labeling the samples; they tell the function to use them to choose
colors. We can also build the PCA plot from scratch using the
`r CRANpkg("ggplot2")` package [@Wickham2009Ggplot2].
This is done by asking the *plotPCA* function
to return the data used for plotting rather than building the plot.
See the *ggplot2* [documentation](http://docs.ggplot2.org/current/)
for more details on using *ggplot*.

```{r}
(data <- plotPCA(rld, intgroup = c( "dex", "cell"), returnData=TRUE))
percentVar <- round(100 * attr(data, "percentVar"))
```

We can then use this data to build up a second plot in a Figure below, specifying that the
color of the points should reflect dexamethasone treatment and the
shape should reflect the cell line.

```{r}
library("ggplot2")
```

```{r ggplotpca, fig.width=6, fig.height=4.5}
ggplot(data, aes(PC1, PC2, color=dex, shape=cell)) + geom_point(size=3) +
  xlab(paste0("PC1: ",percentVar[1],"% variance")) +
  ylab(paste0("PC2: ",percentVar[2],"% variance")) +
  coord_fixed()
```

**PCA plot using the rlog-transformed values with custom *ggplot2* code.**
Here we specify cell line (plotting symbol) and dexamethasone treatment (color).

From the PCA plot, we see that the differences between cells (the
different plotting shapes) are considerable, though not stronger than the differences due to
treatment with dexamethasone (red vs blue color). This shows why it will be important to
account for this in differential testing by using a paired design
("paired", because each dex treated sample is paired with one
untreated sample from the *same* cell line). We are already set up for
this design by assigning the formula `~ cell + dex` earlier.

## MDS plot

Another plot, very similar to the PCA plot, can be made using the 
*multidimensional scaling* (MDS) function in base R. This is useful when we
don't have a matrix of data, but only a matrix of distances. Here we
compute the MDS for the distances calculated from the *rlog*
transformed counts and plot these in a Figure below.

```{r mdsrlog, fig.width=6, fig.height=4.5}
mdsData <- data.frame(cmdscale(sampleDistMatrix))
mds <- cbind(mdsData, as.data.frame(colData(rld)))
ggplot(mds, aes(X1,X2,color=dex,shape=cell)) + geom_point(size=3) +
  coord_fixed()
```

**MDS plot using rlog-transformed values.**

In a Figure below we show the same plot for the *PoissonDistance*:

```{r mdspois, fig.width=6, fig.height=4.5}
mdsPoisData <- data.frame(cmdscale(samplePoisDistMatrix))
mdsPois <- cbind(mdsPoisData, as.data.frame(colData(dds)))
ggplot(mdsPois, aes(X1,X2,color=dex,shape=cell)) + geom_point(size=3) +
  coord_fixed()
```

**MDS plot using the *Poisson Distance*.**

<a id="de"></a>

# Differential expression analysis

## Running the differential expression pipeline

As we have already specified an experimental design when we created
the *DESeqDataSet*, we can run the differential expression pipeline on
the raw counts with a single call to the function *DESeq*:

```{r airwayDE}
dds <- DESeq(dds)
```

This function will print out a message for the various steps it
performs. These are described in more detail in the manual page for
*DESeq*, which can be accessed by typing `?DESeq`. Briefly these are:
the estimation of size factors (controlling for differences in the
sequencing depth of the samples), the estimation of
dispersion values for each gene, and fitting a generalized linear model.

A *DESeqDataSet* is returned that contains all the fitted
parameters within it, and the following section describes how to
extract out results tables of interest from this object.

## Building the results table

Calling *results* without any arguments will extract the estimated
log2 fold changes and *p* values for the last variable in the design
formula. If there are more than 2 levels for this variable, *results*
will extract the results table for a comparison of the last level over
the first level. This comparison is printed at the top of the output:
`dex trt vs untrt`.

```{r}
(res <- results(dds))
```

As `res` is a *DataFrame* object, it carries metadata
with information on the meaning of the columns:

```{r}
mcols(res, use.names=TRUE)
```

The first column, `baseMean`, is a just the average of the normalized
count values, dividing by size factors, taken over all samples in the
*DESeqDataSet*.
The remaining four columns refer to a specific contrast, namely the
comparison of the `trt` level over the `untrt` level for the factor
variable `dex`. We will find out below how to obtain other contrasts.

The column `log2FoldChange` is the effect size estimate. It tells us
how much the gene's expression seems to have changed due to treatment
with dexamethasone in comparison to untreated samples.  This value is
reported on a logarithmic scale to base 2: for example, a log2 fold
change of 1.5 means that the gene's expression is increased by a
multiplicative factor of \(2^{1.5} \approx 2.82\).

Of course, this estimate has an uncertainty associated with it, which
is available in the column `lfcSE`, the standard error estimate for
the log2 fold change estimate.  We can also express the uncertainty of
a particular effect size estimate as the result of a statistical
test. The purpose of a test for differential expression is to test
whether the data provides sufficient evidence to conclude that this
value is really different from zero. *DESeq2* performs for each gene a
*hypothesis test* to see whether evidence is sufficient to decide
against the *null hypothesis* that there is zero effect of the treatment
on the gene and that the observed difference between treatment and
control was merely caused by experimental variability (i.e., the type
of variability that you can expect between different
samples in the same treatment group). As usual in statistics, the
result of this test is reported as a *p* value, and it is found in the
column `pvalue`. Remember that a *p* value indicates the probability
that a fold change as strong as the observed one, or even stronger,
would be seen under the situation described by the null hypothesis.

We can also summarize the results with the following line of code,
which reports some additional information, that will be covered in
later sections.

```{r}
summary(res)
```

Note that there are many genes with differential expression due to
dexamethasone treatment at the FDR level of 10%. This makes sense, as
the smooth muscle cells of the airway are known to react to
glucocorticoid steroids. However, there are two ways to be more strict
about which set of genes are considered significant:

* lower the false discovery rate threshold (the threshold on `padj` in
  the results table)
* raise the log2 fold change threshold from 0 using the `lfcThreshold`
  argument of *results*

If we lower the false discovery rate threshold, we should also
tell this value to `results()`, so that the function will use an
alternative threshold for the optimal independent filtering step:

```{r}
res.05 <- results(dds, alpha=.05)
table(res.05$padj < .05)
```

If we want to raise the log2 fold change threshold, so that we test
for genes that show more substantial changes due to treatment, we
simply supply a value on the log2 scale. For example, by specifying
`lfcThreshold=1`, we test for genes that show significant effects of
treatment on gene counts more than doubling or less than halving,
because \(2^1 = 2\).

```{r}
resLFC1 <- results(dds, lfcThreshold=1)
table(resLFC1$padj < 0.1)
```

Sometimes a subset of the *p* values in `res` will be `NA` ("not
available"). This is *DESeq*'s way of reporting that all counts for
this gene were zero, and hence no test was applied. In addition, *p*
values can be assigned `NA` if the gene was excluded from analysis
because it contained an extreme count outlier. For more information,
see the outlier detection section of the *DESeq2* vignette.

If you use the results from an R analysis package in published
research, you can find the proper citation for the software by typing
`citation("pkgName")`, where you would substitute the name of the
package for `pkgName`.  Citing methods papers helps to support and
reward the individuals who put time into open source software for
genomic data analysis.

## Other comparisons

In general, the results for a comparison of any two levels of a
variable can be extracted using the `contrast` argument to
*results*. The user should specify three values: the name of the
variable, the name of the level for the numerator, and the name of the
level for the denominator. Here we extract results for the log2 of the
fold change of one cell line over another:

```{r}
results(dds, contrast=c("cell", "N061011", "N61311"))
```

If results for an interaction term are desired, the `name`
argument of *results* should be used. Please see the 
help for the *results* function for more details.

## Multiple testing

In high-throughput biology, we are careful to not use the *p* values
directly as evidence against the null, but to correct for
*multiple testing*. What would happen if we were to simply threshold
the *p* values at a low value, say 0.05? There are
`r sum(res$pvalue < .05, na.rm=TRUE)` genes with a *p* value
below 0.05 among the `r sum(!is.na(res$pvalue))` genes, for which the
test succeeded in reporting a *p* value:

```{r}
sum(res$pvalue < 0.05, na.rm=TRUE)
sum(!is.na(res$pvalue))
```

Now, assume for a moment that the null hypothesis is true for all
genes, i.e., no gene is affected by the treatment with
dexamethasone. Then, by the definition of the *p* value, we expect up to
5% of the genes to have a *p* value below 0.05. This amounts to
`r round(sum(!is.na(res$pvalue)) * .05 )` genes.
If we just considered the list of genes with a *p* value below 0.05 as
differentially expressed, this list should therefore be expected to
contain up to
`r round(sum(!is.na(res$pvalue)) * .05)` /
`r sum(res$pvalue < .05, na.rm=TRUE)` =
`r round(sum(!is.na(res$pvalue))*.05 / sum(res$pvalue < .05, na.rm=TRUE) * 100)`%
 false positives.

*DESeq2* uses the Benjamini-Hochberg (BH) adjustment [@Benjamini1995Controlling] as implemented in
the base R *p.adjust* function; in brief, this method calculates for
each gene an adjusted *p* value that answers the following question:
if one called significant all genes with an adjusted *p* value less than or
equal to this gene's adjusted *p* value threshold, what would be the fraction
of false positives (the *false discovery rate*, FDR) among them, in
the sense of the calculation outlined above? These values, called the
BH-adjusted *p* values, are given in the column `padj` of the `res`
object.

The FDR is a useful statistic for many high-throughput
experiments, as we are often interested in reporting or focusing on a
set of interesting genes, and we would like to put an upper bound on the
percent of false positives in this set. 

Hence, if we consider a fraction of 10% false positives acceptable,
we can consider all genes with an adjusted *p* value below 10% = 0.1
as significant. How many such genes are there?

```{r}
sum(res$padj < 0.1, na.rm=TRUE)
```

We subset the results table to these genes and then sort it by the
log2 fold change estimate to get the significant genes with the
strongest down-regulation:

```{r}
resSig <- subset(res, padj < 0.1)
head(resSig[ order(resSig$log2FoldChange), ])
```

...and with the strongest up-regulation:

```{r}
head(resSig[ order(resSig$log2FoldChange, decreasing=TRUE), ])
```

<a id="plots"></a>

# Plotting results

A quick way to visualize the counts for a particular gene is to use
the *plotCounts* function that takes as arguments the
*DESeqDataSet*, a gene name, and the group over which to plot the
counts (Figure below).

```{r plotcounts}
topGene <- rownames(res)[which.min(res$padj)]
plotCounts(dds, gene=topGene, intgroup=c("dex"))
```

**Normalized counts for a single gene over treatment group.**

We can also make custom plots using the *ggplot* function from the
`r CRANpkg("ggplot2")` package (Figures below).

```{r ggplotcountsjitter, fig.width=6, fig.height=4.5}
data <- plotCounts(dds, gene=topGene, intgroup=c("dex","cell"), returnData=TRUE)
ggplot(data, aes(x=dex, y=count, color=cell)) +
  scale_y_log10() + 
  geom_point(position=position_jitter(width=.1,height=0), size=3)
```

**Normalized counts indicating cell line with color.**

```{r ggplotcountsdot, fig.width=6, fig.height=4.5}
ggplot(data, aes(x=dex, y=count, fill=dex)) +
  scale_y_log10() + 
  geom_dotplot(binaxis="y", stackdir="center")
```

**Normalized counts using a more structural arrangement.**
Here the color indicates treatment.

```{r ggplotcountsgroup, fig.width=6, fig.height=4.5}
ggplot(data, aes(x=dex, y=count, color=cell, group=cell)) +
  scale_y_log10() + geom_point(size=3) + geom_line()
```

**Normalized counts with lines connecting cell lines.**
Note that the *DESeq* test actually takes into account the cell line
effect, so this figure more closely depicts the difference being tested.

An *MA-plot* [@Dudoit2002Statistical] provides a useful overview for an experiment with a
two-group comparison (Figure below).

```{r plotma}
plotMA(res, ylim=c(-5,5))
```

**An MA-plot of changes induced by treatment.**
The log2 fold change for a particular
comparison is plotted on the y-axis and the average of the counts
normalized by size factor is shown on the x-axis ("M" for minus,
because a log ratio is equal to log minus log, and "A" for average).
Each gene is represented with a dot. Genes with an adjusted *p* value
below a threshold (here 0.1, the default) are shown in red.

The *DESeq2* package uses statistical techniques to moderate log2 fold
changes from genes with very low counts and highly variable counts, as
can be seen by the narrowing of the vertical spread of points on the
left side of the MA-plot. For a detailed explanation of the rationale
of moderated fold changes, please see the *DESeq2* paper
[@Love2014Moderated].  This plot demonstrates that only genes with a
large average normalized count contain sufficient information to yield
a significant call.

We can also make an MA-plot for the results table in which we raised
the log2 fold change threshold (Figure below).
We can label individual points on the MA-plot as well. Here we use the
*with* R function to plot a circle and text for a selected row of the
results object. Within the *with* function, only the `baseMean` and
`log2FoldChange` values for the selected rows of `res` are used.

```{r plotmalabel}
plotMA(resLFC1, ylim=c(-5,5))
topGene <- rownames(resLFC1)[which.min(resLFC1$padj)]
with(resLFC1[topGene, ], {
  points(baseMean, log2FoldChange, col="dodgerblue", cex=2, lwd=2)
  text(baseMean, log2FoldChange, topGene, pos=2, col="dodgerblue")
})
```

**An MA-plot of a test for large log2 fold changes.**
The red points indicate genes for which the log2 fold change was
significantly higher than 1 or less than -1 (treatment resulting in
more than doubling or less than halving of the normalized counts).
The point circled in blue indicates the gene with the lowest adjusted *p* value.

Another useful diagnostic plot is the histogram of the *p* values
(Figure below).
This plot is best formed by excluding genes with very small counts,
which otherwise generate spikes in the histogram.

```{r histpvalue2}
hist(res$pvalue[res$baseMean > 1], breaks=0:20/20, col="grey50", border="white")
```

**Histogram of *p* values for genes with mean normalized count larger than 1.**

## Gene clustering

In the sample distance heatmap made previously, the dendrogram at the
side shows us a hierarchical clustering of the samples. Such a
clustering can also be performed for the genes.  Since the clustering
is only relevant for genes that actually carry a signal, one usually
would only cluster a subset of the most highly variable genes. Here,
for demonstration, let us select the 20 genes with the highest
variance across samples. We will work with the *rlog* transformed
counts:

```{r}
library("genefilter")
topVarGenes <- head(order(rowVars(assay(rld)),decreasing=TRUE),20)
```

The heatmap becomes more interesting if we do not look at absolute
expression strength but rather at the amount by which each gene
deviates in a specific sample from the gene's average across all
samples. Hence, we center each genes' values across samples,
and plot a heatmap (Figure below). We provide a *data.frame* that instructs the
*pheatmap* function how to label the columns.

```{r genescluster}
mat <- assay(rld)[ topVarGenes, ]
mat <- mat - rowMeans(mat)
df <- as.data.frame(colData(rld)[,c("cell","dex")])
pheatmap(mat, annotation_col=df)
```

**Heatmap of relative rlog-transformed values across samples.**
Treatment status and cell line information are shown with colored bars
at the top of the heatmap.
Blocks of genes that covary across patients. Note that
a set of genes at the top of the heatmap are separating the N061011
cell line from the others. In the center of the heatmap, we see a set
of genes for which the dexamethasone treated samples have higher gene
expression. 

## Independent filtering

The MA plot highlights an important property of RNA-seq data.  For
weakly expressed genes, we have no chance of seeing differential
expression, because the low read counts suffer from such high Poisson
noise that any biological effect is drowned in the uncertainties from
the sampling at a low rate. We can also show this by examining the ratio of
small *p* values (say, less than 0.05) for genes binned by mean
normalized count. We will use the results table subjected to the threshold to show
what this looks like in a case when there are few tests with small *p*
value.

In the following code chunk, we create bins using the *quantile*
function, bin the genes by base mean using *cut*, rename the levels of
the bins using the middle point, calculate the ratio of *p* values
less than 0.05 for each bin, and finally plot these ratios (Figure below).

```{r sensitivityovermean, fig.width=6}
qs <- c(0, quantile(resLFC1$baseMean[resLFC1$baseMean > 0], 0:6/6))
bins <- cut(resLFC1$baseMean, qs)
levels(bins) <- paste0("~",round(signif(.5*qs[-1] + .5*qs[-length(qs)],2)))
ratios <- tapply(resLFC1$pvalue, bins, function(p) mean(p < .05, na.rm=TRUE))
barplot(ratios, xlab="mean normalized count", ylab="ratio of small p values")
```

**The ratio of small *p* values for genes binned by mean normalized count.**
The *p* values are from a test of log2 fold change greater than 1
or less than -1. This plot demonstrates that genes with very low mean count
are underpowered, and best excluded before multiple test correction.

At first sight, there may seem to be little benefit in filtering out
these genes. After all, the test found them to be non-significant
anyway. However, these genes have an influence on the multiple testing
adjustment, whose performance improves if such genes are removed. By
removing the low count genes from the input to the FDR
procedure, we can find more genes to be significant among those that
we keep, and so improved the power of our test. This approach is known
as *independent filtering*.

The *DESeq2* software automatically performs independent filtering
that maximizes the number of genes with adjusted *p* value
less than a critical value (by default, `alpha` is set to 0.1).  This
automatic independent filtering is performed by, and can be controlled
by, the *results* function.

The term *independent* highlights an important caveat. Such filtering
is permissible only if the statistic that we filter on (here the mean
of normalized counts across all samples) is independent of the
actual test statistic (the *p* value) under the null hypothesis.
Otherwise, the filtering would invalidate the
test and consequently the assumptions of the BH procedure.
The independent filtering software used inside
*DESeq2* comes from the `r Biocpkg("genefilter")` package, that
contains a reference to a paper describing the statistical foundation
for independent filtering [@Bourgon2010Independent].

<a id="annotate"></a>

# Annotating and exporting results

Our result table so far only contains information about Ensembl gene
IDs, but alternative gene names may be more informative for
collaborators.  Bioconductor's annotation packages help with mapping
various ID schemes to each other.
We load the `r Biocpkg("AnnotationDbi")` package and the annotation package
`r Biocannopkg("org.Hs.eg.db")`:

```{r}
library("AnnotationDbi")
library("org.Hs.eg.db")
```

This is the organism annotation package ("org") for
*Homo sapiens* ("Hs"), organized as an *AnnotationDbi*
database package ("db"), using Entrez Gene IDs ("eg") as primary key.
To get a list of all available key types, use:

```{r}
columns(org.Hs.eg.db)
```

We can use the *mapIds* function to add individual columns to our results
table. We provide the row names of our results table as a key, and
specify that `keytype=ENSEMBL`. The `column` argument tells the
*mapIds* function which information we want, and the `multiVals`
argument tells the function what to do if there are multiple possible
values for a single input value. Here we ask to just give us back the
first one that occurs in the database.
To add the gene symbol and Entrez ID, we call *mapIds* twice.

```{r}
res$symbol <- mapIds(org.Hs.eg.db,
                     keys=row.names(res),
                     column="SYMBOL",
                     keytype="ENSEMBL",
                     multiVals="first")
res$entrez <- mapIds(org.Hs.eg.db,
                     keys=row.names(res),
                     column="ENTREZID",
                     keytype="ENSEMBL",
                     multiVals="first")
```

Now the results have the desired external gene IDs:

```{r}
resOrdered <- res[order(res$padj),]
head(resOrdered)
```

## Exporting results

You can easily save the results table in a CSV file, that you can
then share or load with a spreadsheet program such as Excel. The call to
*as.data.frame* is necessary to convert the *DataFrame* object
(`r Biocpkg("IRanges")` package) to a *data.frame* object that can be
processed by *write.csv*. Here, we take just the top 100 genes for
demonstration.

```{r eval=FALSE}
resOrderedDF <- as.data.frame(resOrdered)[1:100,]
write.csv(resOrderedDF, file="results.csv")
```

Another more sophisticated package for exporting results from various
Bioconductor analysis packages is the `r Biocpkg("ReportingTools")`
package [@Huntley2013ReportingTools].
*ReportingTools* will automatically generate dynamic HTML documents,
including links to external databases using gene identifiers
and boxplots summarizing the normalized counts across groups.
See the *ReportingTools* vignettes for full details. The simplest
version of creating a dynamic *ReportingTools* report is performed
with the following code:

```{r eval=FALSE}
library("ReportingTools")
htmlRep <- HTMLReport(shortName="report", title="My report",
                      reportDirectory="./report")
publish(resOrderedDF, htmlRep)
url <- finish(htmlRep)
browseURL(url)
```

## Plotting fold changes in genomic space

If we have used the *summarizeOverlaps* function to count the reads,
then our *DESeqDataSet* object is built on top of ready-to-use
Bioconductor objects specifying the genomic ranges of the genes. We
can therefore easily plot our differential expression results in
genomic space. While the *results* function by default returns a
*DataFrame*, using the `format` argument, we can ask for *GRanges* or
*GRangesList* output.

```{r}
(resGR <- results(dds, lfcThreshold=1, format="GRanges"))
```

We need to add the symbol again for labeling the genes on the plot:

```{r}
resGR$symbol <- mapIds(org.Hs.eg.db, names(resGR), "SYMBOL", "ENSEMBL")
```

We will use the `r Biocpkg("Gviz")` package for plotting the GRanges
and associated metadata: the log fold changes due to dexamethasone treatment.

```{r}
library("Gviz")
```

The following code chunk specifies a window of 1 million base pairs
upstream and downstream from the gene with the smallest *p* value.
We create a subset of our full results, for genes within the window
We add the gene symbol as a name, if the symbol exists or is not duplicated in
our subset.

```{r}
window <- resGR[topGene] + 1e6
strand(window) <- "*"
resGRsub <- resGR[resGR %over% window]
naOrDup <- is.na(resGRsub$symbol) | duplicated(resGRsub$symbol)
resGRsub$group <- ifelse(naOrDup, names(resGRsub), resGRsub$symbol)
```

We create a vector specifying if the genes in this subset had a low
false discovery rate. 

```{r}
sig <- factor(ifelse(resGRsub$padj < .1 & !is.na(resGRsub$padj),"sig","notsig"))
```

We can then plot the results using `r Biocpkg("Gviz")` functions
(Figure below). We
create an axis track specifying our location in the genome, a track
that will show the genes and their names, colored by significance,
and a data track that will draw vertical bars showing the moderated
log fold change produced by *DESeq2*, which we know are only large
when the effect is well supported by the information in the counts.

```{r gvizplot}
options(ucscChromosomeNames=FALSE)
g <- GenomeAxisTrack()
a <- AnnotationTrack(resGRsub, name="gene ranges", feature=sig)
d <- DataTrack(resGRsub, data="log2FoldChange", baseline=0,
               type="h", name="log2 fold change", strand="+")
plotTracks(list(g,d,a), groupAnnotation="group", notsig="grey", sig="hotpink")
```

**log2 fold changes in genomic region surrounding the gene with smallest
  adjusted *p* value.** Genes highlighted in pink have adjusted *p*
  value less than 0.1.

<a id="batch"></a>

# Removing hidden batch effects

Suppose we did not know that there were different cell lines involved
in the experiment, only that there was treatment with
dexamethasone. The cell line effect on the counts then would represent
some hidden and unwanted variation that might be affecting
many or all of the genes in the dataset. We can use statistical
methods designed for RNA-seq from the `r Biocpkg("sva")` package [@Leek2014Svaseq] to
detect such groupings of the samples, and then we can add these to the
*DESeqDataSet* design, in order to account for them. The *SVA*
package uses the term *surrogate variables* for the estimated
variables that we want to account for in our analysis. Another
package for detecting hidden batches is the `r Biocpkg("RUVSeq")`
package [@Risso2014Normalization], with the acronym "Remove Unwanted Variation".

```{r}
library("sva")
```

Below we obtain a matrix of normalized counts for which the average count across
samples is larger than 1. As we described above, we are trying to
recover any hidden batch effects, supposing that we do not know the
cell line information. So we use a full model matrix with the
*dex* variable, and a reduced, or null, model matrix with only
an intercept term. Finally we specify that we want to estimate 2
surrogate variables. For more information read the manual page for the
*svaseq* function by typing `?svaseq`.

```{r}
dat <- counts(dds, normalized=TRUE)
idx <- rowMeans(dat) > 1
dat <- dat[idx,]
mod <- model.matrix(~ dex, colData(dds))
mod0 <- model.matrix(~ 1, colData(dds))
svseq <- svaseq(dat, mod, mod0, n.sv=2)
svseq$sv
```

Because we actually do know the cell lines, we can see how well the
SVA method did at recovering these variables (Figure below).

```{r svaplot}
par(mfrow=c(2,1),mar=c(3,5,3,1))
stripchart(svseq$sv[,1] ~ dds$cell,vertical=TRUE,main="SV1")
abline(h=0)
stripchart(svseq$sv[,2] ~ dds$cell,vertical=TRUE,main="SV2")
abline(h=0)
```

**Surrogate variables 1 and 2 plotted over cell line.**
Here, we know the hidden source of variation (cell line), and
therefore can see how the SVA procedure is able to identify a source
of variation which is correlated with cell line.

Finally, in order to use SVA to remove any effect on the counts from
our surrogate variables, we simply add these two surrogate variables
as columns to the *DESeqDataSet* and then add them to the design:

```{r}
ddssva <- dds
ddssva$SV1 <- svseq$sv[,1]
ddssva$SV2 <- svseq$sv[,2]
design(ddssva) <- ~ SV1 + SV2 + dex
```

We could then produce results controlling for surrogate variables by
running *DESeq* with the new design: 

```{r svaDE, eval=FALSE}
ddssva <- DESeq(ddssva)
```

<a id="time"></a>

# Time course experiments

*DESeq2* can be used to analyze time course experiments, for example
to find those genes that react in a condition-specific manner over
time, compared to a set of baseline samples.
Here we demonstrate a basic time course analysis with the
 `r Biocexptpkg("fission")` data package,
that contains gene counts for an RNA-seq time course of fission
yeast [@Leong2014Global]. The yeast were exposed to oxidative stress, and half of the
samples contain a deletion of the gene *atf21*.
We use a design formula that models the strain difference at time 0,
the difference over time, and any strain-specific differences over
time (the interaction term `strain:minute`).

```{r}
library("fission")
data("fission")
ddsTC <- DESeqDataSet(fission, ~ strain + minute + strain:minute)
```

The following chunk of code performs a likelihood ratio test, where we remove
the strain-specific differences over time. Genes with small *p* values
from this test are those which at one or more time points after time
0 showed a strain-specific effect. Note therefore that this will not
give small *p* values to genes that moved up or down over time in the
same way in both strains.

```{r fissionDE}
ddsTC <- DESeq(ddsTC, test="LRT", reduced = ~ strain + minute)
resTC <- results(ddsTC)
resTC$symbol <- mcols(ddsTC)$symbol
head(resTC[order(resTC$padj),],4)
```

This is just one of the tests that can be applied to time
series data. Another option would be to model the counts as a
smooth function of time, and to include an interaction term of the
condition with the smooth function. It is possible to build such a
model using spline basis functions within R.

We can plot the counts for the groups over time using
`r CRANpkg("ggplot2")`, for the gene with the smallest adjusted *p* value,
testing for condition-dependent time profile and accounting for
differences at time 0 (Figure below). Keep in mind that the interaction terms are the
*difference* between the two groups at a given time after accounting for
the difference at time 0.

```{r fissioncounts, fig.width=6, fig.height=4.5}
data <- plotCounts(ddsTC, which.min(resTC$padj), 
                   intgroup=c("minute","strain"), returnData=TRUE)
ggplot(data, aes(x=minute, y=count, color=strain, group=strain)) + 
  geom_point() + stat_smooth(se=FALSE,method="loess") +  scale_y_log10()
```

**Normalized counts for a gene with condition-specific changes over time.**

Wald tests for the log2 fold changes at individual time points can be
investigated using the `test` argument to *results*:

```{r}
resultsNames(ddsTC)
res30 <- results(ddsTC, name="strainmut.minute30", test="Wald")
res30[which.min(resTC$padj),]
```

We can furthermore cluster significant genes by their profiles. We
extract a matrix of the shrunken log2 fold changes using the *coef* function:

```{r}
betas <- coef(ddsTC)
colnames(betas)
```

We can now plot the log2 fold changes in a heatmap (Figure below). 

```{r fissionheatmap}
library("pheatmap")
topGenes <- head(order(resTC$padj),20)
mat <- betas[topGenes, -c(1,2)]
thr <- 3 
mat[mat < -thr] <- -thr
mat[mat > thr] <- thr
pheatmap(mat, breaks=seq(from=-thr, to=thr, length=101),
         cluster_col=FALSE)
```

**Heatmap of log2 fold changes for genes with smallest adjusted *p* value.**
The bottom set of genes show strong induction of expression for the
baseline samples in minutes 15-60 (red boxes in the bottom left
corner), but then have slight differences for the mutant strain
(shown in the boxes in the bottom right corner).

<a id="ref"></a>

# Session information

As the last part of this document, we call the function *sessionInfo*,
which reports the version numbers of R and all the packages used in
this session. It is good practice to always keep such a record of this
as it will help to track down what has happened in case an R script
ceases to work or gives different results because the functions have
been changed in a newer version of one of your packages. By including
it at the bottom of a script, your reports will become more reproducible.

The session information should also *always*
be included in any emails to the
[Bioconductor support site](https://support.bioconductor.org) along
with all code used in the analysis.

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

# References

