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@@ -1124,13 +1124,96 @@ literal "false"
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\end_layout
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\begin_layout Subsubsection
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-sva and ComBat for batch correction
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+ComBat and SVA for correction of known and unknown batch effects
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+\end_layout
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+
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+\begin_layout Standard
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+In addition to well-understood effects that can be easily normalized out,
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+ a data set often contains confounding biological effects that must be accounted
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+ for in the modeling step.
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+ For instance, in an experiment with pre-treatment and post-treatment samples
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+ of cells from several different donors, donor variability represents a
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+ known batch effect.
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+ The most straightforward correction for known batches is to estimate the
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+ mean for each batch independently and subtract out the differences, so
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+ that all batches have identical means for each feature.
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+ However, as with variance estimation, estimating the differences in batch
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+ means is not necessarily robust at the feature level, so the ComBat method
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+ adds empirical Bayes squeezing of the batch mean differences toward a common
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+ value, analogous to limma's empirical Bayes squeezing of feature variance
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+ estimates
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+\begin_inset CommandInset citation
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+LatexCommand cite
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+key "Johnson2007"
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+literal "false"
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+
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+\end_inset
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+
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+.
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+ Effectively, ComBat assumes that modest differences between batch means
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+ are real batch effects, but extreme differences between batch means are
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+ more likely to be the result of outlier observations that happen to line
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+ up with the batches rather than a genuine batch effect.
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+ The result is a batch correction that is more robust against outliers than
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+ simple subtraction of mean differences subtraction.
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+\end_layout
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+
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+\begin_layout Standard
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+In some data sets, unknown batch effects may be present due to inherent
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+ variability in in the data, either caused by technical or biological effects.
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+ Examples of unknown batch effects include variations in enrichment efficiency
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+ between ChIP-seq samples, variations in populations of different cell types,
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+ and the effects of uncontrolled environmental factors on gene expression
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+ in humans or live animals.
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+ In an ordinary linear model context, unknown batch effects cannot be inferred
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+ and must be treated as random noise.
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+ However, in high-throughput experiments, once again information can be
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+ shared across features to identify patterns of un-modeled variation that
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+ are repeated in many features.
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+ One attractive strategy would be to perform singular value decomposition
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+ (SVD) on the matrix os linear model residuals (which contain all the un-modeled
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+ variation in the data) and take the first few singular vectors as batch
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+ effects.
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+ While this can be effective, it makes the unreasonable assumption that
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+ all batch effects are uncorrelated with any of the effects being modeled.
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+ Surrogate variable analysis (SVA) starts with this approach, but takes
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+ some additional steps to identify batch effects in the full data that are
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+ both highly correlated with the singular vectors in the residuals and least
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+ correlated with the effects of interest
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+\begin_inset CommandInset citation
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+LatexCommand cite
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+key "Leek2007"
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+literal "false"
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+
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+\end_inset
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+
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+.
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+ Since the final batch effects are estimated from the full data, moderate
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+ correlations between the batch effects and effects of interest are allowed,
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+ which gives SVA much more freedom to estimate the true extent of the batch
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+ effects compared to simple residual SVD.
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+ Once the surrogate variables are estimated, they can be included as coefficient
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+s in the linear model in a similar fashion to known batch effects in order
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+ to subtract out their effects on each feature's abundance.
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\end_layout
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\begin_layout Subsubsection
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Factor analysis: PCA, MDS, MOFA
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\end_layout
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+\begin_layout Standard
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+\begin_inset Flex TODO Note (inline)
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+status open
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+
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+\begin_layout Plain Layout
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+Not sure if this merits a subsection here.
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+\end_layout
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+
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+\end_inset
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+
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+
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+\end_layout
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+
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\begin_layout Itemize
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Batch-corrected PCA is informative, but careful application is required
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to avoid bias
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