• Batch effects are widespread in high-throughput biology. They are artifacts not related to the biological variation of scientific interests.

• For instance, two microarray experiments on the same technical replicates processed on two different days might present different results due to factors such as room temperature or the two technicians who did the two experiments.

• Batch effects can substantially confound the downstream analysis, especially meta-analysis across studies.

Two main approaches:

• Location-scale (LS)
• Matrix-factorization (MF)

• LS method assumes a model for the location (mean) and/or scale (variance) of the data within the batches.

• Adjusts the batches in order to agree with these models

• MF techniques assume that the variation in the data corresponding to batch effects is independent on the variation corresponding to the biological variable of interest

• Capture non-biological variability in a small set of factors

• Factors can be estimated through some matrix factorization methods

ComBat - Location-scale method

The core idea of ComBat was that the observed measurement \(Y_{ijg}\) for the expression value of gene \(g\) for sample \(j\) from batch \(i\) can be expressed as

\[Y_{ijg}=\alpha_g+X\beta_g+\gamma_{ig}+\delta_{ig}\epsilon_{ijg}\]

where \(X\) consists of covariates of scientific interests, while \(\gamma_{ig}\) and \(\delta_{ig}\) characterize the additive and multiplicative batch effects of batch \(i\) for gene \(g\).

https://www.bu.edu/jlab/wp-assets/ComBat/Abstract.html

After obtaining the estimators from the above linear regression, the raw data \(Y_{ijg}\) can be adjusted to \(Y_{ijg}^*\):

\[Y_{ijg}^*=\frac{Y_{ijg}-\hat{\alpha_g}-X\hat{\beta_g}-\hat{\gamma_{ig}}}{\hat{\delta_{ig}}}+\hat{\alpha_g}+X\hat{\beta_g}\]

For real application, an empirical Bayes method was applied for parameter estimation.

https://www.bu.edu/jlab/wp-assets/ComBat/Abstract.html

When batches were unknown, the surrogate variable analysis (SVA) was developed.

The main idea was to separate the effects caused by covariates of our primary interests from the artifacts not modeled.

Now the raw expression value \(Y_{jg}\) of gene \(g\) in sample \(j\) can be formulated as:

\[Y_{jg}=\alpha_g+X\beta_g+\sum_{k=1}^K{\lambda_{kg}\eta_{kj}}+\epsilon_{jg}\]

where \(\eta_{kj}\)s represent the unmodeled factors and are called as “surrogate variables”.

Once again, the basic idea was to estimate \(\eta_{kj}\)s and adjust them accordingly.

An iterative algorithm based on singular value decomposition (SVD) was derived to iterate between estimating the main effects \(\hat{\alpha_g}+X\hat{\beta_g}\) given the estimation of surrogate variables and estimating surrogate variables from the residuals \(r_{jg}=Y_{jg}-\hat{\alpha_g}-X\hat{\beta_g}\)

• Contains ComBat function for removing effects of known batches.
• Assume we have:
  • edata: a matrix for raw expression values
  • batch: a vector named for batch numbers.

modcombat = model.matrix(~1, data=as.factor(batch)) 

combat_edata = ComBat(dat=edata, batch=batch, mod=modcombat, par.prior=TRUE, prior.plot=FALSE)

https://bioconductor.org/packages/release/bioc/html/sva.html

For sequencing data, svaseq, the generalized version of SVA, suggested applying a moderated log transformation to the count data or fragments per kilobase of exon per million fragments mapped (FPKM) first to account for the nature of discrete distributions

Instead of a direct transformation on the raw counts or FPKM, remove unwanted variation (RUV) adopted a generalized linear model for \(Y_{jg}\)

A Bioconductor package with a GUI (shiny app).

https://github.com/mani2012/BatchQC

“ComBat, an Empirical Bayes method, outperformed the other five programs by most metrics”

Chen C et.al., "Removing Batch Effects in Analysis of Expression Microarray Data: An Evaluation of Six Batch Adjustment Methods" PLoS ONE 2011 http://journals.plos.org/plosone/article?id=10.1371/journal.pone.0017238

• Johnson, W. Evan, Cheng Li, and Ariel Rabinovic. “Adjusting Batch Effects in Microarray Expression Data Using Empirical Bayes Methods.” Biostatistics 2007 http://biostatistics.oxfordjournals.org/content/8/1/118.long

• Lazar, Cosmin, et.al. “Batch Effect Removal Methods for Microarray Gene Expression Data Integration: A Survey.” Briefings in Bioinformatics 2013 http://bib.oxfordjournals.org/content/14/4/469.long