1 Introduction

We guide you through the individual steps of the BEclear package in their own chapters. They will follow in the logical order of an example of correcting some batch affected DNA methylation data. This article should only give a small tutorial, more details about the individual methods can always be found in the help sections of the BEclear package, e.g. through typing calcBatchEffects in the R environment with the package loaded. To work with the methods contained in the BEclear package, a matrix or data.frame with genes as row-names and samples as column names as well as a samples data.frame with the first column named “sample_id” and the second column named “batch_id” is needed as input.

2 Installation

BEclear is available on Bioconductor. To install it you can therefore use the BiocManager:

if (!requireNamespace("BiocManager", quietly = TRUE)) {
  install.packages("BiocManager")
}
BiocManager::install("BEclear")

Otherwise you can also install BEclear from its Github repository by the following command:

if (!requireNamespace("devtools", quietly = TRUE)) {
  install.packages("devtools")
}
devtools::install_github("David-J-R/MoDentify")

We however recommend installing it through Bioconductor, as this takes care of installing the dependencies and furthermore you can refer to the release of Bioconductor, when using our package, which enables you to reproduce the exact conditions of your run.

During the compilation of the code, many parts of the software will be automatically tested for correct execution and reproduction of expected results. This is implemented in form of unit tests with the help of the testthat package.

When done with the installation you can simply load the package by typing:

library(BEclear)
#> Loading required package: BiocParallel

3 Data

The beta values stored in the ex.data matrix were obtained from level 3 BRCA data from the TCGA portal (Cancer Genome Atlas Research Network et al. 2013). Generally, beta values are calculated by dividing the methylated signal by the sum of the unmethylated and methylated signals from a DNA methylation microrarray. In the level 3 TCGA data, this calculation has already been done. The sample data used here contains averaged beta values of probes that belong to promoter regions of single genes. Another possibility would be to use beta values of single probes, whereby the probe names should then be used instead of the gene names as rownames of the matrix.

You can load our sample data via the following command:

data("BEclearData")

It contains one matrix with the beta values:

knitr::kable(ex.data[1:10,1:5], caption = 'Some entries from the example data-set')

And one data.frame containing the assignment of samples to batches:

knitr::kable(ex.samples[1:10,], caption = 'Some entries from the example sample annotation')

4 Detection of batch effects

For the detection of batch effects we calculate the median difference between the beta values of a gene in a batch and the values of this gene in all other batches. Furthermore we use a non-parametric Kolmogorov-Smirnov test (ks.test) to compare the distribution of the beta value for this gene in the batch and the other batches.

If one gene in a batch has a p-value determined by the ks.test of less or equal 0.01 and a median difference of greater or equal 0.05 it is considered batch effected.

batchEffect <- calcBatchEffects(
  data = ex.data, samples = ex.samples,
  adjusted = TRUE, method = "fdr"
)
#> INFO [2019-05-02 21:27:50] Transforming matrix to data.table
#> INFO [2019-05-02 21:27:50] Calculate the batch effects for 10 batches
#> INFO [2019-05-02 21:27:51] Adjusting p-values
mdifs <- batchEffect$med
pvals <- batchEffect$pval

summary <- calcSummary(medians = mdifs, pvalues = pvals)
#> INFO [2019-05-02 21:27:51] Generating a summary table
knitr::kable(head(summary), caption = 'Summary over the batch affected gene-sample combination of the example data set')

score <- calcScore(ex.data, ex.samples, summary, dir = getwd())
#> INFO [2019-05-02 21:27:51] Calculating the scores for 10 batches
knitr::kable(score, caption = 'Batch scores of the example data-set')

5 Imputation of missing values

For the imputation of missing values we use a slightly modified version of the stochastic gradient descent method described by Koren, Bell, and Volinsky (2009). In this section we will describe our implementation of this method and how to use it.

We assume that our complete data matrix \(D_{ij}\) can be described by the effects of a matrix \(L_i\), which represents the effect of the features (genes in our case) and a matrix \(R_j\) describing the effect of the samples in the following way:

The method can either be run on the complete data set or the data set can be divided into blocks on which the method is applied. This division into blocks allows for parallelisation of the method, which can be useful to speed up the process. We have found that a block-size of 60x60 works well(Akulenko, Merl, and Helms 2016).

We try to minimalise the following loss function through a gradient descent:

Where \(K\) is the set of tuples \((i,j)\) for which the value is present. \(\lambda\) is the penalty coefficient, which controls how restrictive the selection of variables should be. The default of \(\lambda\) is 1.

Another coefficient \(\gamma\) controls the size of the step by which the two matrices \(L_i\) and \(R_j\) are modified. It is initialized by default with 0.01 and its value changes during the iterations (epochs).

For the first iteration the matrices \(L_i\) and \(R_j\) are filled with random values generated by the rnorm function from the stats package and the initial loss and error matrix are calculated.

Then for each iteration the following is done:

• \(L_i\) and \(R_j\) are modified proportional by \(\gamma\) through the following calculation:

  • \[\begin{equation} L_i = L_i + 2 \times \gamma \times (errorMatrix_{ij} \times R_j - \lambda \times L_i) \tag{4} \end{equation}\]
  • \[\begin{equation} R_j = R_j + 2 \times \gamma \times (errorMatrix_{ij} \times L_i - \lambda \times R_j) \tag{5} \end{equation}\]
• Then the new error matrix and loss are calculated.
• If the old loss is smaller than the new one:
  • \(\gamma = \gamma \div 2\)
• Else:
  • \(\gamma = \gamma \times 1.05\)

The \(L_i\) and \(R_j\) matrices at the end of the last iteration are then used to impute the missing data. The default number of iterations is 50.

cleared.data <- clearBEgenes(ex.data, ex.samples, summary)
#> INFO [2019-05-02 21:27:51] Removing values with batch effect:
#> INFO [2019-05-02 21:27:51] 510 values ( 5.1 % of the data) set to NA

corrected.data <- imputeMissingData(cleared.data,
  rowBlockSize = 60,
  colBlockSize = 60, epochs = 50,
  outputFormat = "", dir = getwd()
)
#> INFO [2019-05-02 21:27:51] Starting the imputation of missing values.
#> INFO [2019-05-02 21:27:51] This might take a while.
#> INFO [2019-05-02 21:27:51] BEclear imputation is started:
#> INFO [2019-05-02 21:27:51] block size: 60  x  60
#> INFO [2019-05-02 21:27:51] Impute missing data for block 1 of 4
#> INFO [2019-05-02 21:27:51] Impute missing data for block 2 of 4
#> INFO [2019-05-02 21:27:51] Impute missing data for block 3 of 4
#> INFO [2019-05-02 21:27:51] Impute missing data for block 4 of 4

corrected.data.valid<-replaceOutsideValues(corrected.data)
#> INFO [2019-05-02 21:27:51] Replacing values below 0 or above 1:
#> INFO [2019-05-02 21:27:51] 0 values replaced

6 Overall correction

Besides the individual methods BEclear also offers an overall method, which executes all the described previous steps in one call. It also applies some preprocessing to your data set if necessary.

result <- correctBatchEffect(data = ex.data, samples = ex.samples)
#> INFO [2019-05-02 21:27:51] Transforming matrix to data.table
#> INFO [2019-05-02 21:27:51] Calculate the batch effects for 10 batches
#> INFO [2019-05-02 21:27:52] Adjusting p-values
#> INFO [2019-05-02 21:27:52] Generating a summary table
#> INFO [2019-05-02 21:27:52] Calculating the scores for 10 batches
#> INFO [2019-05-02 21:27:52] Removing values with batch effect:
#> INFO [2019-05-02 21:27:52] 510 values ( 5.1 % of the data) set to NA
#> INFO [2019-05-02 21:27:52] Starting the imputation of missing values.
#> INFO [2019-05-02 21:27:52] This might take a while.
#> INFO [2019-05-02 21:27:52] BEclear imputation is started:
#> INFO [2019-05-02 21:27:52] block size: 60  x  60
#> INFO [2019-05-02 21:27:52] Impute missing data for block 1 of 4
#> INFO [2019-05-02 21:27:53] Impute missing data for block 2 of 4
#> INFO [2019-05-02 21:27:53] Impute missing data for block 3 of 4
#> INFO [2019-05-02 21:27:53] Impute missing data for block 4 of 4
#> INFO [2019-05-02 21:27:53] Replacing values below 0 or above 1:
#> INFO [2019-05-02 21:27:53] 0 values replaced

Returned is a list containing all results from the executed functions.

7 Parallelization

For parallelization we use the BiocParellel package. However by default all methods are executed in serial mode. The methods CalcBatchEffect, imputeMissingData and correctBatchEffect support parallelization through there parameter BPPARAM, which takes a BiocParallel::BiocParallelParam class as an argument.

Type the following to get an overview over the supported evaluation environments:

?BiocParallel::BiocParallelParam

8 Plotting

Additionally BEclear also includes a method for plotting the batch effects. Let us now use the makeBoxplot to compare the distributions of the values in the different samples before and after the batch effect correction:

makeBoxplot(ex.data, ex.samples, score,
  bySamples = TRUE,
  col = "standard", main = "Example data", xlab = "Batch",
  ylab = "Beta value", scoreCol = TRUE)

Figure 1: Distribution of the example beta values grouped by sample

makeBoxplot(corrected.data, ex.samples, score,
  bySamples = TRUE,
  col = "standard", main = "Corrected example data",
  xlab = "Batch", ylab = "Beta value", scoreCol = FALSE)

Figure 2: Distribution of the corrected beta values grouped by sample