---
author: "Belinda Phipson and Jovana Maksimovic"
title: "missMethyl: Analysing Illumina HumanMethylation BeadChip Data"
date: "`r BiocStyle::doc_date()`"
package: "`r BiocStyle::pkg_ver('missMethyl')`"
output: 
  BiocStyle::html_document
vignette: >
  %\VignetteEncoding{UTF-8}
  %\VignetteIndexEntry{missMethyl: Analysing Illumina HumanMethylation BeadChip Data}
  %\VignetteEngine{knitr::rmarkdown}
bibliography: bibliography.bib
editor_options: 
  chunk_output_type: console
---

# Introduction

The `r BiocStyle::Biocpkg("missMethyl")` package contains functions to analyse 
methylation data from Illumina's HumanMethylation450 and MethylationEPIC 
beadchip. These arrays are a cost-effective alternative to whole genome 
bisulphite sequencing, and as such are widely used to profile DNA methylation. 
Specifically, `r BiocStyle::Biocpkg("missMethyl")` contains functions to perform 
SWAN normalisation [@Maksimovic2012],perform differential methylation analysis 
using **RUVm** [@Maksimovic2015], differential variability analysis 
[@Phipson2014] and gene set enrichment analysis [@Phipson2016]. As our lab's 
research into specialised analyses of these arrays continues we anticipate that 
the package will be updated with new functions.

Raw data files are in IDAT format, which can be read into R using the
`r BiocStyle::Biocpkg("minfi")` package [@Aryee2014]. Statistical analyses are 
usually performed on M-values, and $\beta$ values are used for visualisation, 
both of which can be extracted from `MethylSet` data objects, which is a class 
of object created by `r BiocStyle::Biocpkg("minfi")`. For detecting 
differentially variable CpGs we recommend that the analysis is performed on 
M-values. All analyses described here are performed at the CpG site level.

## Analysis of Illumina MethylationEPIC v2.0 beadchip datasets

The `r BiocStyle::Biocpkg("missMethyl")` package has been updated to support 
Illumina's MethylationEPIC v2.0 beadchip. This array contains replicate probes 
which map to the same CpG dinucleotides. We recommend filtering these replicates
prior to analysis with the `r BiocStyle::Biocpkg("missMethyl")` package using 
the `r BiocStyle::Biocpkg("DMRcate")` package. An example of this is shown in 
this vignette.

# Reading data into R

We will use the data in the `r BiocStyle::Biocpkg("minfiData")` package to 
demonstrate the functions in `r BiocStyle::Biocpkg("missMethyl")`.
The example dataset has 6 samples across two slides. There are 3 cancer samples
and 3 normal sample. The sample information is in the targets file. An 
essential column in the targets file is the `Basename` column which tells us 
where the idat files to be read in are located. The R commands to read in the 
data are taken from the `r BiocStyle::Biocpkg("minfi")` User's Guide. For 
additional details on how to read the IDAT files into R, as well as information 
regarding quality control please refer to the `r BiocStyle::Biocpkg("minfi")` 
User's Guide.

```{r load-libs, message=FALSE}
library(missMethyl)
library(limma)
library(minfi)
library(minfiData)
```

```{r reading-data, message=FALSE}
baseDir <- system.file("extdata", package = "minfiData")
targets <- read.metharray.sheet(baseDir)
targets[,1:9]
targets[,10:12]
rgSet <- read.metharray.exp(targets = targets)
```

The data is now an `RGChannelSet` object and needs to be normalised and 
converted to a `MethylSet` object.

## Annotation for Illumina MethylationEPIC v2.0 beadchip datasets

When using MethylationEPIC v2.0 datasets, the array and annotation details in 
the `RGChannelSet` object must be set manually.

```{r epicv2-annotation, eval=FALSE}
### This code is not run within this vignette

# If using EPIC_V2, please run these lines:
annotation(rgSet)["array"] = "IlluminaHumanMethylationEPICv2"
annotation(rgSet)["annotation"] = "20a1.hg38"

### End of not run
```

# Subset-quantile within array normalization (SWAN)

SWAN (subset-quantile within array normalization) is a within-array
normalization method for Illumina 450k & EPIC BeadChips. Technical differencs
have been demonstrated to exist between the Infinium I and Infinium II
assays on a single Illumina HumanMethylation array 
[@Bibikova2011, @Dedeurwaerder2011]. Using the SWAN method
substantially reduces the technical variability between the assay
designs whilst maintaining important biological differences. The SWAN
method makes the assumption that the number of CpGs within the 50bp
probe sequence reflects the underlying biology of the region being
interrogated. Hence, the overall distribution of intensities of probes
with the same number of CpGs in the probe body should be the same
regardless of assay type. The method then uses a subset quantile
normalization approach to adjust the intensities of each array
[@Maksimovic2012].

`SWAN` can take a `MethylSet`, `RGChannelSet` or `MethyLumiSet` as input. It 
should be noted that, in order to create the normalization subset, `SWAN` 
randomly selects Infinium I and II probes that have one, two and three 
underlying CpGs; as such, we recommend using `set.seed` before to ensure that 
the normalized intensities will be
identical, if the normalization is repeated.

The technical differences between Infinium I and II assay designs can
result in aberrant $\beta$ value distributions (Figure \@ref(fig:betasByType), 
panel "Raw"). Using SWAN corrects for the technical differences between the
Infinium I and II assay designs and produces a smoother overall $\beta$
value distribution (Figure \@ref(fig:betasByType), panel "SWAN").

```{r ppraw}
mSet <- preprocessRaw(rgSet)
```

```{r swan}
mSetSw <- SWAN(mSet,verbose=TRUE)
```

```{r betasByType, fig.cap = "Beta value dustributions. Density distributions of beta values before and after using SWAN.", echo = TRUE, fig.width=10, fig.height=5}
par(mfrow=c(1,2), cex=1.25)
densityByProbeType(mSet[,1], main = "Raw")
densityByProbeType(mSetSw[,1], main = "SWAN")
```

# Filter out poor quality probes

Poor quality probes can be filtered out based on the detection p-value.
For this example, to retain a CpG for further analysis, we require that
the detection p-value is less than 0.01 in all samples.

```{r filtering}
detP <- detectionP(rgSet)
keep <- rowSums(detP < 0.01) == ncol(rgSet)
mSetSw <- mSetSw[keep,]
```

# Extracting $\beta$ and M-values

Now that the data has been `SWAN` normalised we can extract $\beta$ and
M-values from the object. We prefer to add an offset to the methylated
and unmethylated intensities when calculating M-values, hence we extract
the methylated and unmethylated channels separately and perform our own
calculation. For all subsequent analyses we use a random selection of
20000 CpGs to reduce computation time.

```{r extraction}
set.seed(10)
mset_reduced <- mSetSw[sample(1:nrow(mSetSw), 20000),]
meth <- getMeth(mset_reduced)
unmeth <- getUnmeth(mset_reduced)
Mval <- log2((meth + 100)/(unmeth + 100))
beta <- getBeta(mset_reduced)
dim(Mval)
```

```{r mdsplot, fig.cap = "MDS plot. A multi-dimensional scaling (MDS) plot of cancer and normal samples.", echo = TRUE, fig.small=TRUE}
par(mfrow=c(1,1))
plotMDS(Mval, labels=targets$Sample_Name, col=as.integer(factor(targets$status)))
legend("topleft",legend=c("Cancer","Normal"),pch=16,cex=1.2,col=1:2)
```

An MDS plot (Figure \@ref(fig:mdsplot)) is a good sanity check to make sure
samples cluster together according to the main factor of interest, in
this case, cancer and normal.

## Filtering replicate probes in Illumina MethylationEPIC v2.0 beadchip datasets

The MethylationEPIC v2.0 array contains replicate probes which map to the same 
CpG dinucleotides. We recommend filtering these replicate probes prior to continuing analysis. The `r BiocStyle::Biocpkg("DMRcate")` package contains [information](https://bioconductor.org/packages/release/bioc/vignettes/DMRcate/inst/doc/EPICv2.pdf) describing possible approaches for filtering replicate probes. We also suggest filtering out non-cg probes, as shown below.

```{r filter-replicate-probes_Mval, eval=FALSE}
### This code is not run within this vignette

# For EPIC_V2, please run these lines to remove replicate and non-cg probes:
# Remove non-cg probes
Mval <- Mval[grepl("^cg", rownames(Mval)),]

# Please also filter replicate probes using an appropriate strategy. See the
# DMRcate EPIC v2 vignette for details.

### End of not run
```

# Testing for differential methylation using limma

To test for differential methylation we use the `r BiocStyle::Biocpkg("limma")` 
package [@Smyth2005], which employs an empirical Bayes framework based on 
Guassian model theory. First we need to set up the design matrix. 
There are a number of ways to do this, the most straightforward is directly from 
the targets file. There are a number of variables, with the `status` column 
indicating **cancer/normal** samples. From the `person` column of the targets 
file, we see that the **cancer/normal** samples are matched, with 3 individuals 
each contributing both a **cancer** and **normal** sample. Since the 
`r BiocStyle::Biocpkg("limma")` model framework can handle any experimental 
design which can be summarised by a design matrix, we can take into account the 
paired nature of the data in the analysis. For more complicated experimental 
designs, please refer to the `r BiocStyle::Biocpkg("limma")` User's Guide.

```{r design}
group <- factor(targets$status,levels=c("normal","cancer"))
id <- factor(targets$person)
design <- model.matrix(~id + group)
design
```

Now we can test for differential methylation using the `lmFit` and `eBayes` 
functions from `r BiocStyle::Biocpkg("limma")`. As input data we use the matrix of 
M-values.

```{r diffmeth}
fit.reduced <- lmFit(Mval,design)
fit.reduced <- eBayes(fit.reduced, robust=TRUE)
```

The numbers of hyper-methylated (1) and hypo-methylated (-1) can be
displayed using the `decideTests` function in `r BiocStyle::Biocpkg("limma")` 
and the top 10 differentially methylated CpGs for *cancer* versus *normal* 
extracted using `topTable`.

```{r diffmeth-results}
summary(decideTests(fit.reduced))
top<-topTable(fit.reduced,coef=4)
top
```

Note that since we performed our analysis on M-values, the `logFC` and
`AveExpr` columns are computed on the M-value scale. For interpretability
and visualisation we can look at the $\beta$ values. The $\beta$ values for
the top 4 differentially methylated CpGs shown in Figure \@ref(fig:top4).

```{r top4, fig.cap = "Top DM CpGs. The beta values for the top 4 differentially methylated CpGs.", echo = TRUE, fig.width=10,fig.height=9}
cpgs <- rownames(top)
par(mfrow=c(2,2))
for(i in 1:4){
stripchart(beta[rownames(beta)==cpgs[i],]~design[,4],method="jitter",
group.names=c("Normal","Cancer"),pch=16,cex=1.5,col=c(4,2),ylab="Beta values",
vertical=TRUE,cex.axis=1.5,cex.lab=1.5)
title(cpgs[i],cex.main=1.5)
}
```

# Removing unwanted variation when testing for differential methylation

Like other platforms, 450k array studies are subject to unwanted
technical variation such as batch effects and other, often unknown,
sources of variation. The adverse effects of unwanted variation have
been extensively documented in gene expression array studies and have
been shown to be able to both reduce power to detect true differences
and to increase the number of false discoveries. As such, when it is
apparent that data is significantly affected by unwanted variation, it
is advisable to perform an adjustment to mitigate its effects.

`r BiocStyle::Biocpkg("missMethyl")` provides a `r BiocStyle::Biocpkg("limma")`
inspired interface to functions from the CRAN package 
`r BiocStyle::CRANpkg("ruv")`, which enable the removal of unwanted variation 
when performing a differential methylation analysis [@Maksimovic2015].  

`RUVfit` uses the *RUV-inverse* method by default, as this does not require the
user to specify a $k$ parameter. The ridged version of *RUV-inverse* is also 
available by setting `method = rinv`. The *RUV-2* and *RUV-4* functions can also
be used by setting `method = ruv2` or `method = ruv4`, respectively, and 
specifying an appropriate value for *k* (number of components of unwanted 
variation to remove) where $0 \leq k < no. samples$.

All of the methods rely on negative control features
to accurately estimate the components of unwanted variation. Negative
control features are probes/genes/etc. that are known *a priori* to not
truly be associated with the biological factor of interest, but are
affected by unwanted variation. For example, in a microarray gene
expression study, these could be house-keeping genes or a set of
spike-in controls. Negative control features are extensively discussed
in Gagnon-Bartsch and Speed [-@Gagnon-Bartsch2012] and Gagnon-Bartsch et al.
[-@Gagnon-Bartsch2013]. Once the unwanted factors are accurately estimated from
the data, they are adjusted for in the linear model that describes the
differential analysis.

If the negative control features are not known *a priori*, they can be
identified empirically. This can be achieved via a 2-stage approach,
**RUVm**. Stage 1 involves performing a differential methylation analysis using 
*RUV-inverse* (by default) and the 613 Illumina negative controls (INCs) as 
negative control features. This will produce a list of CpGs ranked by p-value 
according to their level of association with the factor of interest. 
This list can then be used to identify a set of empirical control probes (ECPs), 
which will capture more of the unwanted variation than using the INCs alone. 
ECPs are selected by designating a proportion of the CpGs least associated with
the factor of interest as negative control features; this can be done
based on either an FDR cut-off or by taking a fixed percentage of probes
from the bottom of the ranked list. Stage 2 involves performing a second
differential methylation analysis on the original data using *RUV-inverse* 
(by default) and the ECPs. For simplicity, we are ignoring the paired
nature of the **cancer** and **normal** samples in this example.

```{r diffmeth2_part1}
# get M-values for ALL probes
meth <- getMeth(mSet)
unmeth <- getUnmeth(mSet)
M <- log2((meth + 100)/(unmeth + 100))
```

```{r filter-replicate-probes_M, eval=FALSE}
### This code is not run within this vignette

# For EPIC_V2, please run these lines to remove replicate and non-cg probes:
# remove non-cg probes
M <- M[grepl("^cg", rownames(M)),]

# Please also filter replicate probes using an appropriate strategy. See the
# DMRcate EPIC v2 vignette for details.

### End of not run
```

```{r diffmeth2_part2}
# setup the factor of interest
grp <- factor(targets$status, labels=c(0,1))
# extract Illumina negative control data
INCs <- getINCs(rgSet)
head(INCs)
# add negative control data to M-values
Mc <- rbind(M,INCs)
# create vector marking negative controls in data matrix
ctl1 <- rownames(Mc) %in% rownames(INCs)
table(ctl1)
rfit1 <- RUVfit(Y = Mc, X = grp, ctl = ctl1) # Stage 1 analysis
rfit2 <- RUVadj(Y = Mc, fit = rfit1)
```

Now that we have performed an initial differential methylation analysis
to rank the CpGs with respect to their association with the factor of
interest, we can designate the CpGs that are least associated with the
factor of interest based on FDR-adjusted p-value as ECPs.

```{r ruv1}
top1 <- topRUV(rfit2, num=Inf, p.BH = 1)
head(top1)
ctl2 <- rownames(M) %in% rownames(top1[top1$p.BH_X1.1 > 0.5,])
table(ctl2)
```

We can then use the ECPs to perform a second differential methylation
with *RUV-inverse*, which is adjusted for the unwanted variation
estimated from the data.

```{r ruv2}
# Perform RUV adjustment and fit
rfit3 <- RUVfit(Y = M, X = grp, ctl = ctl2) # Stage 2 analysis
rfit4 <- RUVadj(Y = M, fit = rfit3)
# Look at table of top results
topRUV(rfit4)
```

<!-- Note, at present **RUVm** does not support contrasts, so only one factor of -->
<!-- interest can be interrogated at a time using a design matrix with an -->
<!-- intercept term. -->

## Alternative approach for RUVm stage 1

If the number of samples in your experiment is *greater* than the number of 
Illumina negative controls on the array platform used - 613 for 450k, 411 for 
EPIC - stage 1 of **RUVm** will not work. In such cases, we recommend performing
a standard `r BiocStyle::Biocpkg("limma")` analysis in stage 1. 

```{r limmaruv}
# setup design matrix
des <- model.matrix(~grp)
des
# limma differential methylation analysis
lfit1 <- lmFit(M, design=des)
lfit2 <- eBayes(lfit1) # Stage 1 analysis
# Look at table of top results
topTable(lfit2)
```
The results of this can then be used to define ECPs for stage 2, as in the 
previous example.

```{r limmaruv1}
topl1 <- topTable(lfit2, num=Inf)
head(topl1)
ctl3 <- rownames(M) %in% rownames(topl1[topl1$adj.P.Val > 0.5,])
table(ctl3)
```

We can then use the ECPs to perform a second differential methylation
with `RUV-inverse` as before.

```{r limmaruv2}
# Perform RUV adjustment and fit
rfit5 <- RUVfit(Y = M, X = grp, ctl = ctl3) # Stage 2 analysis
rfit6 <- RUVadj(Y = M, fit = rfit5)
# Look at table of top results
topRUV(rfit6)
```
## Visualising the effect of RUVm adjustment

To visualise the effect that the **RUVm** adjustment is having on the data, 
using an MDS plot for example, the `getAdj` function can be used to extract 
the adjusted values from the **RUVm** fit object produced by `RUVfit`. 
NOTE: The adjusted values should only be used for visualisations - it is NOT 
recommended that they are used in any downstream analysis.

```{r ruvadj}
Madj <- getAdj(M, rfit5) # get adjusted values
```

The MDS plots below show how the relationship between the samples changes with and 
without **RUVm** adjustment. **RUVm** reduces the distance between the samples in
each group by removing unwanted variation. It can be useful to examine this
type of plot when trying to decide on the best set of ECPs or to help select the 
optimal value of $k$, if using *RUV-4* or *RUV-2*. 

```{r mdsplotadj, fig.cap = "RUVm adjusted data. An MDS plot of cancer and normal data, before and after RUVm adjustment.", echo = TRUE, fig.width=10, fig.height=5}
par(mfrow=c(1,2))
plotMDS(M, labels=targets$Sample_Name, col=as.integer(factor(targets$status)),
        main="Unadjusted", gene.selection = "common")
legend("right",legend=c("Cancer","Normal"),pch=16,cex=1,col=1:2)
plotMDS(Madj, labels=targets$Sample_Name, col=as.integer(factor(targets$status)),
        main="Adjusted: RUV-inverse", gene.selection = "common")
legend("topright",legend=c("Cancer","Normal"),pch=16,cex=1,col=1:2)
```

To illustrate how the `getAdj` function can be used to help select an
appropriate value for $k$, we will run the second stage of the **RUVm** analysis 
using *RUV-4* with two different $k$ values.

```{r ruvadj1}
# Use RUV-4 in stage 2 of RUVm with k=1 and k=2
rfit7 <- RUVfit(Y = M, X = grp, ctl = ctl3,
                method = "ruv4", k=1) # Stage 2 with RUV-4, k=1
rfit9 <- RUVfit(Y = M, X = grp, ctl = ctl3,
                method = "ruv4", k=2) # Stage 2 with RUV-4, k=2
# get adjusted values
Madj1 <- getAdj(M, rfit7)
Madj2 <- getAdj(M, rfit9)
```

The following MDS plots show how the relationship between the samples changes
from the unadjusted data to data adjusted with *RUV-inverse* and *RUV-4* with 
two different $k$ values. For this small dataset, *RUV-inverse* appears to be
removing far too much variation as we can see the samples in each group are 
completely overlapping. Using *RUV-4* and choosing a smaller value for $k$
produces more sensible results.

```{r mdsplotadj1, fig.cap = "Effect of different adjustment methods and parameters. MDS plots of cancer and normal data before an after adjustment with RUV-inverse and RUV-4 with different k values.", echo = TRUE, fig.width=10, fig.height=9}
par(mfrow=c(2,2))
plotMDS(M, labels=targets$Sample_Name, col=as.integer(factor(targets$status)),
        main="Unadjusted", gene.selection = "common")
legend("top",legend=c("Cancer","Normal"),pch=16,cex=1,col=1:2)
plotMDS(Madj, labels=targets$Sample_Name, col=as.integer(factor(targets$status)),
        main="Adjusted: RUV-inverse", gene.selection = "common")
legend("topright",legend=c("Cancer","Normal"),pch=16,cex=1,col=1:2)
plotMDS(Madj1, labels=targets$Sample_Name, col=as.integer(factor(targets$status)),
        main="Adjusted: RUV-4, k=1", gene.selection = "common")
legend("bottom",legend=c("Cancer","Normal"),pch=16,cex=1,col=1:2)
plotMDS(Madj2, labels=targets$Sample_Name, col=as.integer(factor(targets$status)),
        main="Adjusted: RUV-4, k=2", gene.selection = "common")
legend("bottomright",legend=c("Cancer","Normal"),pch=16,cex=1,col=1:2)
```

More information about the various RUV methods can be found at 
[http://www-personal.umich.edu/~johanngb/ruv/](http://www-personal.umich.edu/~johanngb/ruv/),
including links to all relevant publications. Further examples of
RUV analyses, with code, can be found at [https://github.com/johanngb/ruv-useR2018](https://github.com/johanngb/ruv-useR2018). 
The tutorials demonstrate how the various plotting functions
available in the `r BiocStyle::CRANpkg("ruv")` package (which are not 
covered in this vignette) can be used to select sensible parameters and assess 
if the adjustment is "helping" your analysis.


# Testing for differential variability (DiffVar)

## Methylation data

Rather than testing for differences in mean methylation, we may be
interested in testing for differences between group variances. For
example, it has been hypothesised that highly variable CpGs in cancer
are important for tumour progression [@Hansen2011]. Hence we may be
interested in CpG sites that are consistently methylated in the normal
samples, but variably methylated in the cancer samples.

In general we recommend at least 10 samples in each group for accurate
variance estimation, however for the purpose of this vignette we perform
the analysis on 3 vs 3. In this example, we are interested in testing
for differential variability in the cancer versus normal group. 

An important note on the `coef` parameter: please always explicitly state which 
columns of design matrix correspond to the groups that you are interested in 
testing for differential variability. The default setting for `coef` is to 
include all columns of the design matrix when calculating the Levene residuals, 
which is not suitable for when there are additional variables that need to be 
taken into account in the linear model. To avoid misspecification of the model, 
it is best to explicitly state the `coef` parameter. The additional nuisance or
confounding variables will still be taken into account in the linear 
modelling step, however we find that including them when calculating Levene
residuals often removes all the (possibly interesting) variation in the data.

When the design matrix includes an intercept term, the `coef` parameter must 
include both the intercept and groups of interest. Consider the design matrix 
that was used when performing the limma analysis:

```{r checkdesign}
design
```

The first column of the design matrix is the intercept term, and the fourth 
column tells us which samples are cancer and normal samples. The 2nd and 3rd
columns correspond to the ID parameter, which is not interesting in terms of 
finding differentially variable CpGs, but may be important to include in the 
linear model. Hence for this example we would specify `coef = c(1,4)` in the 
call to `varFit`.

For methylation data, the `varFit` function will take
either a matrix of M-values, $\beta$ values or a `MethylSet` object as input. If
$\beta$ values are supplied, a logit transformation is performed. Note
that as a default, `varFit` uses the robust setting in the 
`r BiocStyle::Biocpkg("limma")` framework, which requires the use of the 
`r BiocStyle::CRANpkg("statmod")` package.

```{r diffvar}
fitvar <- varFit(Mval, design = design, coef = c(1,4))
```

The numbers of hyper-variable (1) and hypo-variable (-1) genes in **cancer**
vs **normal** can be obtained using `decideTests`. In the cancer vs normal
context, we would expect to see more variability in methylation in cancer 
compared to normal (i.e. hyper-variable).

```{r diffvar-results}
summary(decideTests(fitvar))
topDV <- topVar(fitvar, coef=4)
topDV
```

An alternate parameterisation of the design matrix that does not include
an intercept term can also be used (i.e. a cell means model), and specific 
contrasts tested with `contrasts.varFit`.
Here we specify the design matrix such that the first two columns
correspond to the **normal** and **cancer** groups, respectively. Note that we 
now specify `coef=c(1,2)`.

```{r alternative}
design2 <- model.matrix(~0+group+id)
fitvar.contr <- varFit(Mval, design=design2, coef=c(1,2))
contr <- makeContrasts(groupcancer-groupnormal,levels=colnames(design2))
fitvar.contr <- contrasts.varFit(fitvar.contr,contrasts=contr)
```

The results are identical to before.

```{r altresults}
summary(decideTests(fitvar.contr))
topVar(fitvar.contr,coef=1)
```

The $\beta$ values for the top 4 differentially variable CpGs can be
seen in Figure \@ref(fig:top4DV).

```{r top4DV,fig.cap="Top DV CpGs. The beta values for the top 4 differentially variable CpGs.", fig.width=10, fig.height=9}
cpgsDV <- rownames(topDV)
par(mfrow=c(2,2))
for(i in 1:4){
stripchart(beta[rownames(beta)==cpgsDV[i],]~design[,4],method="jitter",
group.names=c("Normal","Cancer"),pch=16,cex=1.5,col=c(4,2),ylab="Beta values",
vertical=TRUE,cex.axis=1.5,cex.lab=1.5)
title(cpgsDV[i],cex.main=1.5)
}
```

## RNA-Seq expression data

Testing for differential variability in expression data is
straightforward if the technology is gene expression microarrays. The
matrix of expression values can be supplied directly to the `varFit` function.
For RNA-Seq data, the mean-variance relationship that occurs in count
data needs to be taken into account. In order to deal with this issue,
we apply a `voom` transformation [@Law2014] to obtain observation weights, which
are then used in the linear modelling step. For RNA-Seq data, the `varFit`
function will take a `DGElist` object as input.

To demonstrate this, we use data from the 
`r BiocStyle::Biocpkg("tweeDEseqCountData")` package. This data is part of the International HapMap project, consisting of 
RNA-Seq profiles from 69 unrelated Nigerian individuals [@Pickrell2010]. The 
only covariate is gender, so we can look at differentially variable expression 
between males and females. We follow the code from the 
`r BiocStyle::Biocpkg("limma")` vignette to read in and process the data before 
testing for differential variability.

First we load up the data and extract the relevant information.

```{r loadingdata}
library(tweeDEseqCountData)
data(pickrell1)
counts<-exprs(pickrell1.eset)
dim(counts)
gender <- pickrell1.eset$gender
table(gender)
rm(pickrell1.eset)
data(genderGenes)
data(annotEnsembl63)
annot <- annotEnsembl63[,c("Symbol","Chr")]
rm(annotEnsembl63)
```

We now have the counts, gender of each sample and annotation (gene
symbol and chromosome) for each Ensemble gene. We can form a `DGElist` object
using the `r BiocStyle::Biocpkg("edgeR")` package.

```{r dgelist}
library(edgeR)
y <- DGEList(counts=counts, genes=annot[rownames(counts),])
```

We filter out lowly expressed genes by keeping genes with at least 1
count per million reads in at least 20 samples, as well as genes that
have defined annotation. Finally we perform scaling normalisation.

```{r dgelist-filtering}
isexpr <- rowSums(cpm(y)>1) >= 20
hasannot <- rowSums(is.na(y$genes))==0
y <- y[isexpr & hasannot,,keep.lib.sizes=FALSE]
dim(y)
y <- calcNormFactors(y)
```

We set up the design matrix and test for differential variability. 

```{r testhapmap}
design.hapmap <- model.matrix(~gender)
fitvar.hapmap <- varFit(y, design = design.hapmap, coef=c(1,2))
fitvar.hapmap$genes <- y$genes
```

We can display the results of the test:

```{r resultshapmap}
summary(decideTests(fitvar.hapmap))
topDV.hapmap <- topVar(fitvar.hapmap,coef=ncol(design.hapmap))
topDV.hapmap
```

The log counts per million for the top 4 differentially variable genes
can be seen in Figure \@ref(fig:top4DVhapmap).

```{r top4DVhapmap,fig.cap="Top DV CpGs. The log counts per million for the top 4 differentially variably expressed genes.", fig.width=10, fig.height=9}
genesDV <- rownames(topDV.hapmap)
par(mfrow=c(2,2))
for(i in 1:4){
stripchart(cpm(y,log=TRUE)[rownames(y)==genesDV[i],]~design.hapmap[,ncol(design.hapmap)],method="jitter",
group.names=c("Female","Male"),pch=16,cex=1.5,col=c(4,2),ylab="Log counts per million",
vertical=TRUE,cex.axis=1.5,cex.lab=1.5)
title(genesDV[i],cex.main=1.5)
}
```

# Gene ontology analysis

Once a differential methylation or differential variability analysis has
been performed, it may be of interest to know which gene pathways are
targeted by the significant CpG sites. Geeleher et al.
[@Geeleher2013] showed there is a severe bias when performing gene
ontology analysis with methylation data. This is due to the fact that
there are differing numbers of probes per gene on several different
array technologies. For the Illumina Infinium HumanMethylation450 array
the number of probes per gene ranges from 1 to 1299, with a median of 15
probes per gene. For the EPIC array, the range is 1 to 1487, with a
median of 20 probes per gene. This means that when annotating CpG sites to
genes, a gene is more likely to be identified as differentially methylated if 
there are many CpG sites associated with the gene. We refer to this source of
bias as "probe number bias".

One way to take into account this selection bias is to model the
relationship between the number of probes per gene and the probability
of being differentially methylated. This can be performed by adapting the 
`r BiocStyle::Biocpkg("goseq")` method of Young et al. [@Young2010]. Each gene 
then has a prior probability associated with it, and a modified version of a 
hypergeometric test can be performed, testing for over-representation of the 
selected genes in each gene set. The `r BiocStyle::Biocpkg("BiasedUrn")` package
can be used to obtain p-values from the Wallenius' noncentral hypergeometric 
distribution, taking into account the odds of differential methylation for each 
gene set. Note that the `r BiocStyle::Biocpkg("BiasedUrn")` package can 
occassionally return p-values of 0. For the gene sets where a p-value of exactly 
zero is outputted we perform a hypergeometric test, which ensures non-zero 
p-values and hence false discovery rates.

We have recently uncovered a new source of bias in gene set testing for 
methylation array data [@Maksimovic2020.08.24.265702] that we refer to as 
"multi-gene bias". This second source of bias arises due to the fact that
around 10% of gene-annotated CpGs are annotated to more than one gene, violating 
assumptions of independently measured genes. This can lead to some GO categories
being identified as significantly enriched as they contain genes with 
methylation measurements from shared CpGs. This can occur for large gene 
families such as the protocadherin gamma gene cluster which happen to be 
over-represented in the GO category "GO:0007156: homophilic cell adhesion via 
plasma membrane adhesion molecules". This is now taken into account in the 
gene set testing functions in `r BiocStyle::Biocpkg("missMethyl")`.

We have developed methods for both CpG and region level analyses. The `gometh` 
function performs gene set testing on GO categories or KEGG pathways for 
significant CpG sites. The `gsameth` function is a more generalised gene set 
testing function which can take as input a list of user specified gene sets.
Note that for `gsameth`, the format for the gene ids for each gene in the gene
set needs to be **Entrez Gene IDs**. For example, the entire curated gene
set list (C2) from the Broad's Molecular Signatures Database can be
specified as input. The R version of these lists can be downloaded from
[http://bioinf.wehi.edu.au/software/MSigDB/index.html](here). Both functions
take a vector of significant CpG probe names as input. For a region level 
analysis, using the `r BiocStyle::Biocpkg("DMRcate")` package, for example, 
`goregion` and `gsaregion` can perform gene set enrichment analysis using the 
ranged object as input.

NOTE ON UPDATES IN SEPTEMBER 2020: We have added new functionality to the 
gene set testing functions to allow the user to limit the input CpGs to specific
genomic features of interest using the `genomic.features` argument. This is 
based on the annotation information in the manifest files for the 450K and EPIC
arrays. Possible values are ""ALL", "TSS200", "TSS1500", "Body", "1stExon", 
"3'UTR", "5'UTR" and "ExonBnd" (only for EPIC), and any combination can be 
specified. In order to include the significant genes that overlap with the gene
set of interest, the `sig.genes` parameter can be set to `TRUE`. This adds an 
additional column to the results dataframe.

## CpG level analysis 

To illustrate how to use `gometh`, consider the results from the differential
methylation analysis with **RUVm**.

```{r gometh1}
top <- topRUV(rfit4, number = Inf, p.BH = 1)
table(top$p.BH_X1.1 < 0.01)
```

At a 1% false discovery rate cut-off, there are more than 60,000 CpG sites 
differentially methylated. These CpGs are annotated to almost 10,000 genes, 
which means that a gene ontology analysis is unlikely to be relevant or 
reveal anything biologically interesting. One option for selecting CpGs in this 
context is to apply not only a false discovery rate cut-off, but also a 
$\Delta\beta$ cut-off. However, for this dataset, taking a relatively large
$\Delta\beta$ cut-off of 0.25 still leaves more than 30000 CpGs
differentially methylated, which can be annotated to more than 6000 genes.

```{r gometh2}
beta <- getBeta(mSet)
# make sure that order of beta values matches orer after analysis
beta <- beta[match(rownames(top),rownames(beta)),]
beta_norm <- rowMeans(beta[,grp==0])
beta_can <- rowMeans(beta[,grp==1])
Delta_beta <- beta_can - beta_norm
sigDM <- top$p.BH_X1.1 < 0.01 & abs(Delta_beta) > 0.25
table(sigDM)
```

Instead, we take the top 10000 CpG sites as input to `gometh` which can be 
annotated to around 2500 genes.

```{r gometh3}
topCpGs<-topRUV(rfit4,number=10000)
sigCpGs <- rownames(topCpGs)
sigCpGs[1:10]

# Check number of genes that significant CpGs are annotated to
check <- getMappedEntrezIDs(sig.cpg = sigCpGs)
length(check$sig.eg)
```

The`gometh` function takes as input a character vector of CpG names, and 
optionally, a character vector of all CpG sites tested. This is important to 
include if filtering of the CpGs has been performed prior to differential
methylation analysis. If the `all.cpg` argument is omitted, all the CpGs on the 
array are used as background. To change the array type, the `array.type` 
argument can be specified as either "450K", "EPIC" or "EPIC_V2". The default is 
"450K".

If the `plot.bias` argument is `TRUE`, a figure showing the relationship
between the probability of differential methylation and the number of probes per
gene will be displayed.

For testing of GO terms, the `collection` argument takes the value
"GO", which is the default setting. For KEGG pathway analysis, set
`collection` to "KEGG". The function `topGSA` shows the top enriched GO
categories. The `gsameth` function is called for GO and KEGG pathway analysis 
with the appropriate inputs.

For GO testing on our example dataset:

```{r gometh4, fig.cap="Probe number bias in the cancer dataset.", fig.width=6, fig.height=5}
library(IlluminaHumanMethylation450kanno.ilmn12.hg19)
gst <- gometh(sig.cpg=sigCpGs, all.cpg=rownames(top), collection="GO",
              plot.bias=TRUE)
topGSA(gst, n=10)
```

Testing all GO categories (>20,000) can be a little slow. To demonstrate 
the `genomic.features` parameter, let us rather focus on KEGG pathways 
(~330 pathways).

```{r gometh5}
gst.kegg <- gometh(sig.cpg=sigCpGs, all.cpg=rownames(top), collection="KEGG")
topGSA(gst.kegg, n=10)
```

We can limit the input CpGs to those in the promoter regions of genes:

```{r gometh6}
gst.kegg.prom <- gometh(sig.cpg=sigCpGs, all.cpg=rownames(top), 
                        collection="KEGG", genomic.features = c("TSS200",
                                                                "TSS1500",
                                                                "1stExon"))
topGSA(gst.kegg.prom, n=10)
```

We can see if the results are different if we only include CpGs in gene bodies:

```{r gometh7}
gst.kegg.body <- gometh(sig.cpg=sigCpGs, all.cpg=rownames(top), 
                        collection="KEGG", genomic.features = c("Body"))
topGSA(gst.kegg.body, n=10)
```

The KEGG pathways are quite different when limiting CpGs to those in gene bodies 
versus CpGs in promoters. To include the significant genes that overlap with 
each gene set, set the `sig.genes` parameter to TRUE.

```{r gometh8}
gst.kegg.body <- gometh(sig.cpg=sigCpGs, all.cpg=rownames(top), 
                        collection="KEGG", genomic.features = c("Body"), 
                        sig.genes = TRUE)
topGSA(gst.kegg.body, n=5)
```


For a more generalised version of gene set testing in methylation data
where the user can specify the gene set to be tested, the function `gsameth` can
be used. To display the top 20 pathways, `topGSA` can be called. `gsameth` can 
take a single gene set, or a list of gene sets. The gene identifiers in the
gene set must be **Entrez Gene IDs**. To demonstrate 
`gsameth`, we download and use the Hallmark gene set.

```{r gsameth}
hallmark <- readRDS(url("http://bioinf.wehi.edu.au/MSigDB/v7.1/Hs.h.all.v7.1.entrez.rds"))
gsa <- gsameth(sig.cpg=sigCpGs, all.cpg=rownames(top), collection=hallmark)
topGSA(gsa, n=10)
```

Note that if it is of interest to obtain the **Entrez Gene IDs** that the
significant CpGs are mapped to, the `getMappedEntrezIDs` function can be called.


## Region level analysis

We have extended `gometh` and `gsameth` to perform gene set testing following
a region analysis. The region gene set testing function analogues are `goregion`
and `gsaregion`. Instead of inputting significant CpGs, a ranged object as 
typically outputted from region finding software is used. The CpGs overlapping 
the significant differentially methylated regions (DMRs) are extracted and the 
same statistical framework in `gometh` and `gsameth` is applied to test for 
enrichment of gene sets. We find that genes that have more CpGs measuring 
methylation are more likely to be called as differentially methylated regions, 
and hence it is important to take into account this bias when performing gene 
set testing.

To demonstrate `goregion` and `gsaregion` we will perform a region analysis 
on the cancer dataset using the `r BiocStyle::Biocpkg("DMRcate")` package.

```{r dmrcate1}
library(DMRcate)
```

First, the matrix of M-values is annotated with the relevant annotation 
information about the probes such as their genomic position, gene annotation, 
etc. By default, this is done using the ilmn12.hg19 annotation. The limma 
pipeline is then used for differential methylation analysis to calculate 
moderated t-statistics.

```{r dmrcate2}
myAnnotation <- cpg.annotate(object = M, datatype = "array", what = "M", 
                             arraytype = c("450K"), 
                             analysis.type = "differential", design = design, 
                             coef = 4)
```

Next we can use the `dmrcate` function to combine the CpGs to identify
differentially methylated regions. We can use the `extractRanges` function to 
extract a `GRanges` object with the genomic location and the relevant 
statistics associated with each DMR. This object can then be used as input
to `goregion` and `gsaregion`.

```{r dmrcate3}
DMRs <- dmrcate(myAnnotation, lambda=1000, C=2)
results.ranges <- extractRanges(DMRs)
results.ranges
```

We can visualise the top DMR using the `DMR.plot` function:

```{r dmrcatetopDMR,fig.cap="Top DMR from DMRcate.", fig.width=10, fig.height=9}
cols <- c(2,4)[group]
names(cols) <-group
beta <- getBeta(mSet)
par(mfrow=c(1,1))
DMR.plot(ranges=results.ranges, dmr=2, CpGs=beta, phen.col=cols, 
         what="Beta", arraytype="450K", genome="hg19")
```

Now that we have performed our region analysis, we can use `goregion` and 
`gsaregion` to perform gene set testing. Setting `plot.bias` to TRUE, we can
see strong probe number bias in the data. This can be interpretted that a
gene is more likely to have a DMR called if it has more CpGs measuring 
methylation.

```{r goregion1, fig.cap="Probe number bias for DMRs in the cancer dataset.", fig.width=6, fig.height=5}
gst.region <- goregion(results.ranges, all.cpg=rownames(M), 
                       collection="GO", array.type="450K", plot.bias=TRUE)
```

```{r goregion2}
topGSA(gst.region, n=10)
```

We can also test for enrichment of KEGG pathways:

```{r goregion3}
gst.region.kegg <- goregion(results.ranges, all.cpg=rownames(M), 
                       collection="KEGG", array.type="450K")
topGSA(gst.region.kegg, n=10)
```

What is interesting is that although very similar pathways are highly 
ranked compared to the CpG level analysis, gene set testing on the regions 
is more powerful i.e. the p-values are more significant. In experiments 
where there are tens of thousands of significant CpGs from a CpG level analysis, 
we recommend that a good quality region analysis can be a more powerful approach 
for gene set enrichment analysis.

We can also perform gene set testing on the Hallmark gene sets using 
`gsaregion`:

```{r gsaregion}
gsa.region <- gsaregion(results.ranges, all.cpg=rownames(M), 
                        collection=hallmark)
topGSA(gsa.region, n=10)
```

Note that the DMR analysis can be further refined by imposing a 
$\Delta\beta$ value cut-off and changing various parameters. Please refer to the 
`r BiocStyle::Biocpkg("DMRcate")` package vignette for more details on how to 
do this.

# Session information

```{r sessionInfo, eval=TRUE, results='asis'}
sessionInfo()
```

# References