# The DSS User’s Guide

Hao Wu, hao.wu@emory.edu

#### 29 October 2019

#### Abstract

This vignette introduces the use of the Bioconductor package DSS (Dispersion Shrinkage for Sequencing data), which is designed for differential analysis on high-throughput sequencing data. It provide functionalities for RNA-seq differential expression, and bisulfite sequencing (BS-seq) differential methylation. The core of DSS is a procedure based on Bayesian hierarchical model to estimate and shrink gene- or CpG site-specific dispersions, then conduct Wald tests for detecting differential expression/methylation.

#### Package

DSS 2.34.0

# 1 Introduction

## 1.1 Background

Recent advances in various high-throughput sequencing technologies have revolutionized genomics research. Among them, RNA-seq is designed to measure the the abundance of RNA products, and Bisulfite sequencing (BS-seq) is for measuring DNA methylation. A fundamental question in functional genomics research is whether gene expression or DNA methylation vary under different biological contexts. Thus, identifying differential expression genes (DEGs) or differential methylation loci/regions (DML/DMRs) are key tasks in RNA-seq or BS-seq data analyses.

The differential expression (DE) or differential methylation (DM) analyses are often based on gene- or CpG-specific statistical test. A key limitation in RNA- or BS-seq experiments is that the number of biological replicates is usually limited due to cost constraints. This can lead to unstable estimation of within group variance, and subsequently undesirable results from hypothesis testing. Variance shrinkage methods have been widely applied in DE analyses in microarray data to improve the estimation of gene-specific within group variances. These methods are typically based on a Bayesian hierarchical model, with a prior imposed on the gene-specific variances to provide a basis for information sharing across all genes.

A distinct feature of RNA-seq or BS-seq data is that the measurements are in the form of counts and have to be modeld by discrete distributions. Unlike continuous distributions such as Gaussian, the variances depend on means in these distributions. This implies that the sample variances do not account for biological variation, and shrinkage cannot be applied on variances directly. In DSS, we assume that the count data are from the negative-binomial (for RNA-seq) or beta-binomial (for BS-seq) distribution. We then parameterize the distributions by a mean and a dispersion parameters. The dispersion parameters, which represent the biological variation for replicates within a treatment group, play a central role in the differential analyses.

DSS implements a series of DE/DM detection algorithms based on the dispersion shrinkage method followed by Wald statistical test to test each gene/CpG site for differential expression/methylation. It provides functions for RNA-seq DE analysis for both two group comparision and multi-factor design, BS-seq DM analysis for two group comparision, multi-factor design, and data without biological replicate.

For more details of the data model, the shrinkage method, and test procedures, please read (Wu, Wang, and Wu 2012) for differential expression from RNA-seq, (Feng, Conneely, and Wu 2014) for differential methylation for two-group comparison from BS-seq, (Wu et al. 2015) for differential methylation for data without biological replicate, and (Park and Wu 2016) for differential methylation for general experimental design.

## 1.2 Citation

- For differential expression in RNA-seq, cite Wu, Wang, and Wu (2012).
- For general differential methylation in BS-seq, cite Feng, Conneely, and Wu (2014).
- For differential methylation in BS-seq when there’s no biological replicate, cite Wu et al. (2015).
- For differential methylation in BS-seq under general experimental design, cite Park and Wu (2016).

# 2 Using DSS for RNA-seq differential expression analysis

## 2.1 Input data preparation

DSS requires a count table (a matrix of integers) for gene expression values (rows are for genes and columns are for samples). This is different from the isoform expression based analysis such as in cufflink/cuffdiff, where the gene expressions are represented as non-integers values.

There are a number of ways to obtain the count table from raw sequencing data (fastq file). Aligners such as STAR automatically output a count table. Several Bioconductor packages serve this purpose, for example, Rsubread, QuasR, and easyRNASeq. Other tools like Salmon, Kallisto, or RSEM can also be used. Please refer to the package manuals for more details.

## 2.2 Single factor experiment

In single factor RNA-seq experiment, DSS requires a vector representing experimental designs. The length of the design vector must match the number of columns of the count table. Optionally, normalization factors or additional annotation for genes can be supplied.

The basic data container in the package is `SeqCountSet` class, which is directly inherited from `ExpressionSet` class defined in `Biobase`. An object of the class contains all necessary information for a DE analysis: gene expression values, experimental designs, and additional annotations.

A typical DE analysis contains the following simple steps. - Create a `SeqCountSet` object using `newSeqCountSet`. - Estimate normalization factor using `estNormFactors`. - Estimate and shrink gene-wise dispersion using `estDispersion`. - Two-group comparison using `waldTest`.

The usage of DSS is demonstrated in the simple simulation below.

- First load in the library, and make a `SeqCountSet` object from simulated counts for 2000 genes and 6 samples.

    library(DSS)
    counts1 = matrix(rnbinom(300, mu=10, size=10), ncol=3)
    counts2 = matrix(rnbinom(300, mu=50, size=10), ncol=3)
    X1 = cbind(counts1, counts2)
    X2 = matrix(rnbinom(11400, mu=10, size=10), ncol=6)
    X = rbind(X1,X2)  ## these are 100 DE genes
    designs = c(0,0,0,1,1,1)
    seqData = newSeqCountSet(X, designs)
    seqData

    ## SeqCountSet (storageMode: lockedEnvironment)
    ## assayData: 2000 features, 6 samples
    ##   element names: exprs
    ## protocolData: none
    ## phenoData
    ##   sampleNames: 1 2 ... 6 (6 total)
    ##   varLabels: designs
    ##   varMetadata: labelDescription
    ## featureData: none
    ## experimentData: use 'experimentData(object)'
    ## Annotation:

- Estimate normalization factor.

    seqData = estNormFactors(seqData)

- Estimate and shrink gene-wise dispersions

    seqData = estDispersion(seqData)

- With the normalization factors and dispersions ready, the two-group comparison can be conducted via a Wald test:

    result=waldTest(seqData, 0, 1)
    head(result,5)

    ##    geneIndex      muA      muB       lfc   difExpr     stats         pval
    ## 26        26 4.333333 63.84762 -2.588764 -59.51429 -5.989386 2.106344e-09
    ## 66        66 5.666667 70.77778 -2.447426 -65.11111 -5.944167 2.778660e-09
    ## 31        31 7.000000 72.22381 -2.271766 -65.22381 -5.866847 4.441595e-09
    ## 62        62 5.333333 60.39365 -2.345540 -55.06032 -5.830946 5.511388e-09
    ## 77        77 7.666667 71.02381 -2.169970 -63.35714 -5.787195 7.157156e-09
    ##       local.fdr          fdr
    ## 26 1.643259e-05 1.643259e-05
    ## 66 1.678304e-05 1.643259e-05
    ## 31 2.119235e-05 1.877446e-05
    ## 62 2.369321e-05 2.162590e-05
    ## 77 2.686691e-05 2.162590e-05

A higher level wrapper function `DSS.DE` is provided for simple RNA-seq DE analysis in a two-group comparison. User only needs to provide a count matrix and a vector of 0’s and 1’s representing the design, and get DE test results in one line. A simple example is listed below:

    counts = matrix(rpois(600, 10), ncol=6)
    designs = c(0,0,0,1,1,1)
    result = DSS.DE(counts, designs)

    ## Warning in locfdr(normstat, plot = 0): CM estimation failed, middle of
    ## histogram non-normal

    head(result)

    ##    geneIndex       muA       muB        lfc   difExpr     stats        pval
    ## 40        40 13.666667  6.000000  0.7790896  7.666667  2.837117 0.004552288
    ## 86        86  4.666667 10.000000 -0.7091475 -5.333333 -2.317292 0.020487850
    ## 66        66  8.000000 14.333333 -0.5568107 -6.333333 -2.192056 0.028375428
    ## 12        12 13.666667  7.666667  0.5508310  6.000000  2.130100 0.033163327
    ## 45        45 13.666667  7.666667  0.5508310  6.000000  2.130100 0.033163327
    ## 3          3 10.666667  5.666667  0.5937747  5.000000  2.052657 0.040105825
    ##    local.fdr       fdr
    ## 40 0.2534392 0.2534392
    ## 86 0.2210275 0.3437303
    ## 66 0.3432557 0.3685097
    ## 12 0.5684478 0.3587111
    ## 45 0.5684478 0.3587111
    ## 3  0.6610655 0.3840798

## 2.3 Multifactor experiment

`DSS` provides functionalities for dispersion shrinkage for multifactor experimental designs. Downstream model fitting (through genearlized linear model) and hypothesis testing can be performed using other packages such as `edgeR`, with the dispersions estimated from DSS.

Below is an example, based a simple simulation, to illustrate the DE analysis of a crossed design.

- First simulate data for a 2x2 crossed experiments. Note the counts are randomly generated.

    library(DSS)
    library(edgeR)
    counts = matrix(rpois(800, 10), ncol=8)
    design = data.frame(gender=c(rep("M",4), rep("F",4)),
                        strain=rep(c("WT", "Mutant"),4))
    X = model.matrix(~gender+strain, data=design)

- make SeqCountSet, then estimate size factors and dispersion

    seqData = newSeqCountSet(counts, as.data.frame(X))
    seqData = estNormFactors(seqData)
    seqData = estDispersion(seqData)

- Using edgeR’s function to do glm model fitting, but plugging in the estimated dispersion from DSS.

    fit.edgeR = glmFit(counts, X, dispersion=dispersion(seqData))

- Using edgeR’s function to do hypothesis testing on the second parameter of the model (gender).

    lrt.edgeR = glmLRT(glmfit=fit.edgeR, coef=2)
    head(lrt.edgeR$table)

    ##         logFC   logCPM         LR    PValue
    ## 1  0.09043562 13.50417 0.06266792 0.8023278
    ## 2  0.06852230 13.53391 0.03588961 0.8497436
    ## 3  0.09232706 13.47360 0.06237494 0.8027809
    ## 4  0.46944115 13.30414 1.04076962 0.3076425
    ## 5  0.15069765 13.59340 0.18745782 0.6650409
    ## 6 -0.53979714 13.66475 2.52977665 0.1117160

# 3 Using DSS for BS-seq differential methylation analysis

## 3.1 Overview

To detect differential methylation, statistical tests are conducted at each CpG site, and then the differential methylation loci (DML) or differential methylation regions (DMR) are called based on user specified threshold. A rigorous statistical tests should account for biological variations among replicates and the sequencing depth. Most existing methods for DM analysis are based on ad hoc methods. For example, using Fisher’s exact ignores the biological variations, using t-test on estimated methylation levels ignores the sequencing depth. Sometimes arbitrary filtering are implemented: loci with depth lower than an arbitrary threshold are filtered out, which results in information loss

The DM detection procedure implemented in DSS is based on a rigorous Wald test for beta-binomial distributions. The test statistics depend on the biological variations (characterized by dispersion parameter) as well as the sequencing depth. An important part of the algorithm is the estimation of dispersion parameter, which is achieved through a shrinkage estimator based on a Bayesian hierarchical model (Feng, Conneely, and Wu 2014). An advantage of DSS is that the test can be performed even when there is no biological replicates. That’s because by smoothing, the neighboring CpG sites can be viewed as pseudo-replicates, and the dispersion can still be estimated with reasonable precision.

DSS also works for general experimental design, based on a beta-binomial regression model with arcsine link function. Model fitting is performed on transformed data with generalized least square method, which achieves much improved computational performance compared with methods based on generalized linear model.

DSS depends on bsseq Bioconductor package, which has neat definition of data structures and many useful utility functions. In order to use the DM detection functionalities, `bsseq` needs to be pre-installed.

## 3.2 Input data preparation

DSS requires data from each BS-seq experiment to be summarized into following information for each CG position: chromosome number, genomic coordinate, total number of reads, and number of reads showing methylation. For a sample, this information are saved in a simple text file, with each row representing a CpG site. Below shows an example of a small part of such a file:

    chr     pos     N       X
    chr18   3014904 26      2
    chr18   3031032 33      12
    chr18   3031044 33      13
    chr18   3031065 48      24

One can follow below steps to obtain such data from raw sequence file (fastq file), using bismark (version 0.10.0, commands for newer versions could be different) for BS-seq alignment and count extraction. These steps require installation of bowtie or bowtie2, `bismark`, and the fasta file for reference genome.

1. Prepare Bisulfite reference genome. This can be done using the `bismark_genome_preparation` function (details in bismark manual). Example command is:

    bismark_genome_preparation --path_to_bowtie /usr/local/bowtie/ \
      --verbose /path/to/refgenomes/

2. BS-seq alignment. Example command is:

    bismark -q -n 1 -l 50  --path_to_bowtie \
      /path/bowtie/ BS-refGenome reads.fastq

This step will produce two text files `reads.fastq_bismark.sam` and `reads.fastq_bismark_SE_report.txt`.

3. Extract methylation counts using `bismark_methylation_extractor` function:

    bismark_methylation_extractor -s --bedGraph reads.fastq_bismark.sam

This will create multiple txt files to summarize methylation call and cytosine context, a bedGraph file to display methylation percentage, and a coverage file containing counts information. The count file contain following columns: `chr, start, end, methylation%, count methylated, count unmethylated`. This file can be modified to make the input file for DSS.

A typical DML detection contains two simple steps. First one conduct DM test at each CpG site, then DML/DMR are called based on the test result and user specified threshold.

## 3.3 DML/DMR detection from two-group comparison

Step 1. Load in library. Read in text files and create an object of `BSseq` class, which is defined in `bsseq` Bioconductor package.

    library(DSS)
    require(bsseq)
    path = file.path(system.file(package="DSS"), "extdata")
    dat1.1 = read.table(file.path(path, "cond1_1.txt"), header=TRUE)
    dat1.2 = read.table(file.path(path, "cond1_2.txt"), header=TRUE)
    dat2.1 = read.table(file.path(path, "cond2_1.txt"), header=TRUE)
    dat2.2 = read.table(file.path(path, "cond2_2.txt"), header=TRUE)
    BSobj = makeBSseqData( list(dat1.1, dat1.2, dat2.1, dat2.2),
         c("C1","C2", "N1", "N2") )[1:1000,]
    BSobj

    ## An object of type 'BSseq' with
    ##   1000 methylation loci
    ##   4 samples
    ## has not been smoothed
    ## All assays are in-memory

Step 2. Perform statistical test for DML by calling `DMLtest` function. This function basically performs following steps: (1) estimate mean methylation levels for all CpG site; (2) estimate dispersions at each CpG sites; (3) conduct Wald test. For the first step, there’s an option for smoothing or not. Because the methylation levels show strong spatial correlations, smoothing can help obtain better estimates of mean methylation when the CpG sites are dense in the data (such as from the whole-genome BS-seq). However for data with sparse CpG, such as from RRBS or hydroxyl-methylation, smoothing is not recommended.

To perform DML test without smoothing, do:

    dmlTest = DMLtest(BSobj, group1=c("C1", "C2"), group2=c("N1", "N2"))
    head(dmlTest)

To perform statistical test for DML with smoothing, do:

    dmlTest.sm = DMLtest(BSobj, group1=c("C1", "C2"), group2=c("N1", "N2"),
                         smoothing=TRUE)

User has the option to smooth the methylation levels or not. For WGBS data, smoothing is recommended so that information from nearby CpG sites can be combined to improve the estimation of methylation levels. A simple moving average algorithm is implemented for smoothing. In RRBS since the CpG coverage is sparse, smoothing might not alter the results much. If smoothing is requested, smoothing span is an important parameter which has non-trivial impact on DMR calling. We use 500 bp as default, and think that it performs well in real data tests.

Step 3. With the test results, one can call DML by using `callDML` function. The results DMLs are sorted by the significance.

      dmls = callDML(dmlTest, p.threshold=0.001)
      head(dmls)

    ##       chr     pos        mu1       mu2       diff    diff.se      stat
    ## 450 chr18 3976129 0.01027497 0.9390339 -0.9287590 0.06544340 -14.19179
    ## 451 chr18 3976138 0.01027497 0.9390339 -0.9287590 0.06544340 -14.19179
    ## 638 chr18 4431501 0.01331553 0.9430566 -0.9297411 0.09273779 -10.02548
    ## 639 chr18 4431511 0.01327049 0.9430566 -0.9297862 0.09270080 -10.02997
    ## 710 chr18 4564237 0.91454619 0.0119300  0.9026162 0.05260037  17.15988
    ## 782 chr18 4657576 0.98257334 0.0678355  0.9147378 0.06815000  13.42242
    ##            phi1       phi2         pval          fdr postprob.overThreshold
    ## 450 0.052591567 0.02428826 1.029974e-45 2.499403e-43                      1
    ## 451 0.052591567 0.02428826 1.029974e-45 2.499403e-43                      1
    ## 638 0.053172411 0.07746835 1.177826e-23 1.429096e-21                      1
    ## 639 0.053121697 0.07746835 1.125518e-23 1.429096e-21                      1
    ## 710 0.009528898 0.04942849 5.302004e-66 3.859859e-63                      1
    ## 782 0.010424723 0.06755651 4.468885e-41 8.133371e-39                      1

By default, the test is based on the null hypothesis that the difference in methylation levels is 0. Alternatively, users can specify a threshold for difference. For example, to detect loci with difference greater than 0.1, do:

      dmls2 = callDML(dmlTest, delta=0.1, p.threshold=0.001)
      head(dmls2)

    ##       chr     pos        mu1       mu2       diff    diff.se      stat
    ## 450 chr18 3976129 0.01027497 0.9390339 -0.9287590 0.06544340 -14.19179
    ## 451 chr18 3976138 0.01027497 0.9390339 -0.9287590 0.06544340 -14.19179
    ## 638 chr18 4431501 0.01331553 0.9430566 -0.9297411 0.09273779 -10.02548
    ## 639 chr18 4431511 0.01327049 0.9430566 -0.9297862 0.09270080 -10.02997
    ## 710 chr18 4564237 0.91454619 0.0119300  0.9026162 0.05260037  17.15988
    ## 782 chr18 4657576 0.98257334 0.0678355  0.9147378 0.06815000  13.42242
    ##            phi1       phi2         pval          fdr postprob.overThreshold
    ## 450 0.052591567 0.02428826 1.029974e-45 2.499403e-43                      1
    ## 451 0.052591567 0.02428826 1.029974e-45 2.499403e-43                      1
    ## 638 0.053172411 0.07746835 1.177826e-23 1.429096e-21                      1
    ## 639 0.053121697 0.07746835 1.125518e-23 1.429096e-21                      1
    ## 710 0.009528898 0.04942849 5.302004e-66 3.859859e-63                      1
    ## 782 0.010424723 0.06755651 4.468885e-41 8.133371e-39                      1

When delta is specified, the function will compute the posterior probability that the difference of the means is greater than delta. So technically speaking, the threshold for p-value here actually refers to the threshold for 1-posterior probability, or the local FDR. Here we use the same parameter name for the sake of the consistence of function syntax.

Step 4. DMR detection is also Based on the DML test results, by calling `callDMR` function. Regions with many statistically significant CpG sites are identified as DMRs. Some restrictions are provided by users, including the minimum length, minimum number of CpG sites, percentage of CpG site being significant in the region, etc. There are some post hoc procedures to merge nearby DMRs into longer ones.

    dmrs = callDMR(dmlTest, p.threshold=0.01)
    head(dmrs)

    ##      chr   start     end length nCG meanMethy1 meanMethy2 diff.Methy
    ## 27 chr18 4657576 4657639     64   4   0.506453   0.318348   0.188105
    ##    areaStat
    ## 27 14.34236

Here the DMRs are sorted by `areaStat`, which is defined in `bsseq` as the sum of the test statistics of all CpG sites within the DMR.

Similarly, users can specify a threshold for difference. For example, to detect regions with difference greater than 0.1, do:

      dmrs2 = callDMR(dmlTest, delta=0.1, p.threshold=0.05)
      head(dmrs2)

    ##      chr   start     end length nCG meanMethy1 meanMethy2 diff.Methy
    ## 31 chr18 4657576 4657639     64   4  0.5064530  0.3183480   0.188105
    ## 19 chr18 4222533 4222608     76   4  0.7880276  0.3614195   0.426608
    ##    areaStat
    ## 31 14.34236
    ## 19 12.91667

Note that the distribution of test statistics (and p-values) depends on the differences in methylation levels and biological variations, as well as technical factors such as coverage depth. It is very difficulty to select a natural and rigorous threshold for defining DMRs. We recommend users try different thresholds in order to obtain satisfactory results.

The DMRs can be visualized using `showOneDMR` function, This function provides more information than the `plotRegion` function in `bsseq`. It plots the methylation percentages as well as the coverage depths at each CpG sites, instead of just the smoothed curve. So the coverage depth information will be available in the figure.

To use the function, do

      showOneDMR(dmrs[1,], BSobj)

The result figure looks like the following. Note that the figure below is not generated from the above example. The example data are from RRBS experiment so the DMRs are much shorter.

Figure 1: An example DMR

### 3.3.1 Parallel computing for DML/DMR detection from two-group comparison

We now implement parallel computing for two-group DML test to speed up the computation. The parallelism is achieved by specifying the `BPPARAM` parameter in `DMLtest` function,
through the functionalities provided in BiocParallel package. Please see the help for `DMLtest` function for more description and example codes. To know more about the BiocParallel, please read its documentations.

We did some simple tests. The figure below shows the time spent in `DMLtest` function, for different numbers of CpG sites in the data and using different number of cores. It shows significant improvements using multiple cores.

Figure 2: Parallelization in DML calling

## 3.4 DML/DMR detection from general experimental design

In DSS, BS-seq data from a general experimental design (such as crossed experiment, or experiment with covariates) is modeled through a generalized linear model framework. We use arcsine link function instead of the typical logit link for it better deals with data at boundaries (methylation levels close to 0 or 1). Linear model fitting is done through ordinary least square on transformed methylation levels. Variance/covariance matrices for the estimates are derived with consideration of count data distribution and transformation.

### 3.4.1 Hypothesis testing in general experimental design

In a general design, the data are modeled through a multiple regression framework, thus there are several regression coefficients. In contrast, there is only one parameter in two-group comparison which is the difference between two groups. Under this type of design, hypothesis testing can be performed for one, multiple, or any linear combination of the parameters.

DSS provides flexible functionalities for hypothesis testing. User can test one parameter in the model through a Wald test, or any linear combination of the parameters through an F-test.

The `DMLtest.multiFactor` function provide interfaces for testing one parameter (through `coef` parameter), one term in the model (through `term` parameter), or linear combinations of the parameters (through `Contrast` parameter). We illustrate the usage of these parameters through a simple example below. Assume we have an experiment from three strains (A, B, C) and two sexes (M and F), each has 2 biological replicates (so there are 12 datasets in total).

    Strain = rep(c("A", "B", "C"), 4)
    Sex = rep(c("M", "F"), each=6)
    design = data.frame(Strain,Sex)
    design

    ##    Strain Sex
    ## 1       A   M
    ## 2       B   M
    ## 3       C   M
    ## 4       A   M
    ## 5       B   M
    ## 6       C   M
    ## 7       A   F
    ## 8       B   F
    ## 9       C   F
    ## 10      A   F
    ## 11      B   F
    ## 12      C   F

To test the additive effect of Strain and Sex, a design formula is `~Strain+Sex`, and the corresponding design matrix for the linear model is:

    X = model.matrix(~Strain+ Sex, design)
    X

    ##    (Intercept) StrainB StrainC SexM
    ## 1            1       0       0    1
    ## 2            1       1       0    1
    ## 3            1       0       1    1
    ## 4            1       0       0    1
    ## 5            1       1       0    1
    ## 6            1       0       1    1
    ## 7            1       0       0    0
    ## 8            1       1       0    0
    ## 9            1       0       1    0
    ## 10           1       0       0    0
    ## 11           1       1       0    0
    ## 12           1       0       1    0
    ## attr(,"assign")
    ## [1] 0 1 1 2
    ## attr(,"contrasts")
    ## attr(,"contrasts")$Strain
    ## [1] "contr.treatment"
    ##
    ## attr(,"contrasts")$Sex
    ## [1] "contr.treatment"

Under this design, we can do different tests using the `DMLtest.multiFactor` function:

- If we want to test the sex effect, we can either specify `coef=4` (because the 4th column in the design matrix corresponds to sex), `coef="SexM"`, or `term="Sex"`. It is important to note that when using character for coef, the character must match the column name of the design matrix, i.e., one cannot do `coef="Sex"`. It is also important to note that using `term="Sex"` only tests a single paramter in the model because sex only has two levels.

- If we want to test the effect of Strain B versus Strain A (this is also testing a single parameter), we do `coef=2` or `coef="StrainB"`.

- If we want to test the whole Strain effect, it becomes a compound test because Strain has three levels. We do `term="Strain"`, which tests `StrainB` and `StrainC` simultaneously. We can also make a Contrast matrix L as following. It’s clear that testing \(L^T \beta = 0\) is equivalent to testing StrainB=0 and StrainC=0.

    L = cbind(c(0,1,0,0),c(0,0,1,0))
    L

    ##      [,1] [,2]
    ## [1,]    0    0
    ## [2,]    1    0
    ## [3,]    0    1
    ## [4,]    0    0

- One can perform more general test, for example, to test StrainB=StrainC, or that strains B and C has no difference (but they could be different from Strain A). In this case, we need to make following contrast matrix:

    matrix(c(0,1,-1,0), ncol=1)

    ##      [,1]
    ## [1,]    0
    ## [2,]    1
    ## [3,]   -1
    ## [4,]    0

### 3.4.2 Example analysis for data from general experimental design

Step 1. Load in data distributed with `DSS`. This is a small portion of a set of RRBS experiments. There are 5000 CpG sites and 16 samples. The experiment is a \(2\times2\) design (2 cases and 2 cell types). There are 4 replicates in each case-by-cell combination.

    data(RRBS)
    RRBS

    ## An object of type 'BSseq' with
    ##   5000 methylation loci
    ##   16 samples
    ## has not been smoothed
    ## All assays are in-memory

    design

    ##    case cell
    ## 1    HC   rN
    ## 2    HC   rN
    ## 3    HC   rN
    ## 4   SLE   aN
    ## 5   SLE   rN
    ## 6   SLE   aN
    ## 7   SLE   rN
    ## 8   SLE   aN
    ## 9   SLE   rN
    ## 10  SLE   aN
    ## 11  SLE   rN
    ## 12   HC   aN
    ## 13   HC   aN
    ## 14   HC   aN
    ## 15   HC   aN
    ## 16   HC   rN

Stepp 2. Fit a linear model using `DMLfit.multiFactor` function, include case, cell, and case by cell interaction. Similar to in a multiple regression, the model only needs to be fit once, and then the parameters can be tested based on the model fitting results.

    DMLfit = DMLfit.multiFactor(RRBS, design=design, formula=~case+cell+case:cell)

    ## Fitting DML model for CpG site:

Step 3. Use `DMLtest.multiFactor` function to test the cell effect. It is important to note that the `coef` parameter is the index of the coefficient to be tested for being 0. Because the model (as specified by `formula` in `DMLfit.multiFactor`) include intercept, the cell effect is the 3rd column in the design matrix, so we use `coef=3` here.

    DMLtest.cell = DMLtest.multiFactor(DMLfit, coef=3)

Alternatively, one can specify the name of the parameter to be tested. In this case, the input `coef` is a character, and it must match one of the column names in the design matrix. The column names of the design matrix can be viewed by

    colnames(DMLfit$X)

    ## [1] "(Intercept)"    "caseSLE"        "cellrN"         "caseSLE:cellrN"

The following line also works. Specifying `coef="cellrN"` is the same as specifying {coef=3}.

    DMLtest.cell = DMLtest.multiFactor(DMLfit, coef="cellrN")

Result from this step is a data frame with chromosome number, CpG site position, test statistics, p-values (from normal distribution), and FDR. Rows are sorted by chromosome/position of the CpG sites. To obtain top ranked CpG sites, one can sort the data frame using following codes:

    ix=sort(DMLtest.cell[,"pvals"], index.return=TRUE)$ix
    head(DMLtest.cell[ix,])

    ##       chr     pos     stat        pvals         fdrs
    ## 1273 chr1 2930315 5.280301 1.289720e-07 0.0006448599
    ## 4706 chr1 3321251 5.037839 4.708164e-07 0.0011770409
    ## 3276 chr1 3143987 4.910412 9.088510e-07 0.0015147517
    ## 2547 chr1 3069876 4.754812 1.986316e-06 0.0024828953
    ## 3061 chr1 3121473 4.675736 2.929010e-06 0.0029290097
    ## 527  chr1 2817715 4.441198 8.945925e-06 0.0065858325

**Step 4*. DMRs for multifactor design can be called using {callDMR} function:

    callDMR(DMLtest.cell, p.threshold=0.05)

    ##      chr   start     end length nCG   areaStat
    ## 33  chr1 2793724 2793907    184   5  12.619968
    ## 413 chr1 3309867 3310133    267   7 -12.093850
    ## 250 chr1 3094266 3094486    221   4  11.691413
    ## 262 chr1 3110129 3110394    266   5  11.682579
    ## 180 chr1 2999977 3000206    230   4  11.508302
    ## 121 chr1 2919111 2919273    163   4   9.421873
    ## 298 chr1 3146978 3147236    259   5   8.003469
    ## 248 chr1 3090627 3091585    959   5  -7.963547
    ## 346 chr1 3200758 3201006    249   4  -4.451691
    ## 213 chr1 3042371 3042459     89   5   4.115296
    ## 169 chr1 2995532 2996558   1027   4  -2.988665

Note that for results from for multifactor design, {delta} is NOT supported. This is because in multifactor design, the estimated coefficients in the regression are based on a GLM framework (loosely speaking), thus they don’t have clear meaning of methylation level differences. So when the input DMLresult is from {DMLtest.multiFactor}, {delta} cannot be specified.

### 3.4.3 More flexible way to construct a hypothesis test

Following 4 tests should produce the same results, since ‘case’ only has two levels. However the p-values from F-tests (using term or Contrast) are slightly different, due to normal approximation in Wald test.

    ## fit a model with additive effect only
    DMLfit = DMLfit.multiFactor(RRBS, design, ~case+cell)

    ## Fitting DML model for CpG site:

    ## test case effect
    test1 = DMLtest.multiFactor(DMLfit, coef=2)
    test2 = DMLtest.multiFactor(DMLfit, coef="caseSLE")
    test3 = DMLtest.multiFactor(DMLfit, term="case")
    Contrast = matrix(c(0,1,0), ncol=1)
    test4 = DMLtest.multiFactor(DMLfit, Contrast=Contrast)
    cor(cbind(test1$pval, test2$pval, test3$pval, test4$pval))

    ##      [,1] [,2] [,3] [,4]
    ## [1,]    1    1    1    1
    ## [2,]    1    1    1    1
    ## [3,]    1    1    1    1
    ## [4,]    1    1    1    1

The model fitting and hypothesis test procedures are computationally very efficient. For a typical RRBS dataset with 4 million CpG sites, it usually takes less than half hour. In comparison, other similar software such as RADMeth or BiSeq takes at least 10 times longer.

# 4 Frequently Asked Questions

## 4.1 For BS-seq data analysis

Q: Do I need to filter out CpG sites with low or no coverage?

A: No. DSS will take care of the coverage depth information. The depth will be factored into the statistical computation.

Q: Can I use estimated methylation levels (beta values) with DSS?

A: No. DSS only takes count data. I don’t recommend compute methylation levels out of the counts and then use the point estimates as input, since that will complete ignore the sequencing depth information. For example, 1 out 2 and 100 out of 200 both give 50% methylation level, but the precision of the point estimates are very different.

Q: Should I always do smoothing, even for RRBS data?

A: Smoothing is always recommended, even for RRBS. In RRBS, CpG sites are likely to be clustered locally within small genomic regions, so smoothing will still help to improve the methylation estimation.

Q: Why doesn’t `DMLtest.multiFactor` function return the relative methylation levels?

A: There are several reasons. First, in functions for general design, we use arcsin link function with a multiple regression setting, thus the estimated regression coefficients are not relative methylation levels. Returning these values will not be very meaningful. Secondly, the relative methylation levels in a multiple regression setting is a quantity to difficult to comprehend for user without formal statistical training.
It is similar to a multiple regression, where the interpretation of the coefficient is something like “the change of Y with 1 unit increase of X, adjusting for other covariates”. Considering X can be anything (even continuous), we don’t feel there is a clear definition of relative methylation level, unlike in two-group comparison setting. Due to these reasons, we decided not to compute and return those values.

Q: Why doesn’t callDMR function return FDR for identified DMRs?

A: In DMR calling, the statistical test is perform for each CpG, and then the significant CpG are merged into regions. It is very difficult to estimate FDR at the region-level, based on the site level test results (p-values or FDR). This is a difficult question for all types of genome-wide assays such as the ChIP-seq. So I rather not to report a DMR-level FDR if it’s not accurate.

Q: What’s the meaning of areaStat parameter?

A: We adapt that from the `bsseq` package. It is the sum of the test statistics of all CpG sites within a DMR. It doesn’t have a direct biological meaning. One can imagine when we try to rank the DMRs, we don’t know whether the height or the width is more important? AreaStat is a combination of the two. This is an ad hoc way to rank DMRs, but larger AreaStat is more likely to be a DMR

Q: How can I get genome annotation (like the nearby genes) for the DMRs?

A: DSS doesn’t provide this functionality. There are many other tools can do this, for example, HOMER.

Q: What’s the memory requirement for DSS?

A: It is really difficult to tell. According to my experience, a whole genome BS-seq dataset with 20+ million CpG sites and 4 samples can run on my laptop with 16G RAM. I once used DelayedArray in DSS to reduce the memory usage. But it makes the computation significantly slower so I changed back.

Q: How to consider the batch effect with DSS?

A: I think the best strategy will be using the multifactor functions with batch as a covariate in the model.

Q: Does DSS work for longitudinal or paired data?

A. No. But these are on our future development plan.

# 5 Session Info

    sessionInfo()

    ## R version 3.6.1 (2019-07-05)
    ## Platform: x86_64-w64-mingw32/x64 (64-bit)
    ## Running under: Windows Server 2012 R2 x64 (build 9600)
    ##
    ## Matrix products: default
    ##
    ## locale:
    ## [1] LC_COLLATE=C
    ## [2] LC_CTYPE=English_United States.1252
    ## [3] LC_MONETARY=English_United States.1252
    ## [4] LC_NUMERIC=C
    ## [5] LC_TIME=English_United States.1252
    ##
    ## attached base packages:
    ##  [1] splines   stats4    parallel  stats     graphics  grDevices utils
    ##  [8] datasets  methods   base
    ##
    ## other attached packages:
    ##  [1] edgeR_3.28.0                limma_3.42.0
    ##  [3] DSS_2.34.0                  bsseq_1.22.0
    ##  [5] SummarizedExperiment_1.16.0 DelayedArray_0.12.0
    ##  [7] BiocParallel_1.20.0         matrixStats_0.55.0
    ##  [9] GenomicRanges_1.38.0        GenomeInfoDb_1.22.0
    ## [11] IRanges_2.20.0              S4Vectors_0.24.0
    ## [13] Biobase_2.46.0              BiocGenerics_0.32.0
    ## [15] BiocStyle_2.14.0
    ##
    ## loaded via a namespace (and not attached):
    ##  [1] gtools_3.8.1             locfit_1.5-9.1
    ##  [3] xfun_0.10                HDF5Array_1.14.0
    ##  [5] lattice_0.20-38          rhdf5_2.30.0
    ##  [7] colorspace_1.4-1         snow_0.4-3
    ##  [9] htmltools_0.4.0          rtracklayer_1.46.0
    ## [11] yaml_2.2.0               XML_3.98-1.20
    ## [13] rlang_0.4.1              R.oo_1.22.0
    ## [15] R.utils_2.9.0            GenomeInfoDbData_1.2.2
    ## [17] stringr_1.4.0            zlibbioc_1.32.0
    ## [19] Biostrings_2.54.0        munsell_0.5.0
    ## [21] R.methodsS3_1.7.1        evaluate_0.14
    ## [23] knitr_1.25               permute_0.9-5
    ## [25] highr_0.8                Rcpp_1.0.2
    ## [27] scales_1.0.0             BSgenome_1.54.0
    ## [29] BiocManager_1.30.9       XVector_0.26.0
    ## [31] Rsamtools_2.2.0          digest_0.6.22
    ## [33] stringi_1.4.3            bookdown_0.14
    ## [35] grid_3.6.1               tools_3.6.1
    ## [37] bitops_1.0-6             magrittr_1.5
    ## [39] RCurl_1.95-4.12          Matrix_1.2-17
    ## [41] data.table_1.12.6        DelayedMatrixStats_1.8.0
    ## [43] rmarkdown_1.16           Rhdf5lib_1.8.0
    ## [45] GenomicAlignments_1.22.0 compiler_3.6.1

# References

Feng, Hao, Karen N Conneely, and Hao Wu. 2014. “A Bayesian Hierarchical Model to Detect Differentially Methylated Loci from Single Nucleotide Resolution Sequencing Data.” Nucleic Acids Research 42 (8). Oxford University Press:e69–e69.

Park, Yongseok, and Hao Wu. 2016. “Differential Methylation Analysis for Bs-Seq Data Under General Experimental Design.” Bioinformatics 32 (10). Oxford University Press:1446–53.

Wu, Hao, Chi Wang, and Zhijin Wu. 2012. “A New Shrinkage Estimator for Dispersion Improves Differential Expression Detection in Rna-Seq Data.” Biostatistics 14 (2). Oxford University Press:232–43.

Wu, Hao, Tianlei Xu, Hao Feng, Li Chen, Ben Li, Bing Yao, Zhaohui Qin, Peng Jin, and Karen N Conneely. 2015. “Detection of Differentially Methylated Regions from Whole-Genome Bisulfite Sequencing Data Without Replicates.” Nucleic Acids Research 43 (21). Oxford University Press:e141–e141.