Applied Statistics for High-throughput Biology: Session 4

Levi Waldron

Outline

• Exploratory data analysis (Chapter 2)
  • Quantile Quantile (QQ)-plots
  • Boxplots
  • Scatterplots
  • Volcano plots
  • p-value histograms
  • MA plots
  • heatmaps
• Batch effects (Chapter 10)

Exploratory data analysis

Introduction

“The greatest value of a picture is when it forces us to notice what we never expected to see.” - John W. Tukey

• Discover biases, systematic errors and unexpected variability in data
• Graphical approach to detecting these issues
• Represents a first step in data analysis and guides hypothesis testing
• Opportunities for discovery in the outliers

Quantile Quantile Plots

• Quantiles divide a distribution into equally sized bins
• Division into 100 bins gives percentiles
• Quantiles of a theoretical distribution are plotted against an experimental distribution
  • alternatively, quantiles of two experimental distributions
• Given a perfect fit, $x=y$
• Useful in determining data distribution (normal, t, etc.)

Example: Quantile Quantile Plots

Boxplots: About

• Provide a graph that is easy to interpret where data is not normally distributed
• Would be an appropriate choice to explore income data, as distribution is highly skewed
• Particularly informative in relation to outliers and range
• Possible to compare multiple distributions side by side

Boxplots: Example

Three different views of a continuous variable

Scatterplots And Correlation: About

• For two continuous variables, scatter plot and calculation of correlation is useful
• Provides a graphical and numeric estimation of relationships
• Quick and easy with plot() and cor()

Scatterplots And Correlation: Example

Exploratory data analysis in high dimensions

library(GSE5859Subset)
data(GSE5859Subset) ##this loads three tables
class(geneExpression)
#> [1] "matrix" "array"
dim(geneExpression)
#> [1] 8793   24
class(sampleInfo)
#> [1] "data.frame"
dim(sampleInfo)
#> [1] 24  4
head(sampleInfo)
#>     ethnicity       date         filename group
#> 107       ASN 2005-06-23 GSM136508.CEL.gz     1
#> 122       ASN 2005-06-27 GSM136530.CEL.gz     1
#> 113       ASN 2005-06-27 GSM136517.CEL.gz     1
#> 163       ASN 2005-10-28 GSM136576.CEL.gz     1
#> 153       ASN 2005-10-07 GSM136566.CEL.gz     1
#> 161       ASN 2005-10-07 GSM136574.CEL.gz     1

Volcano plots: Setup

T-test for every row (gene) of gene expression matrix:

library(genefilter)
g <- factor(sampleInfo$group)
system.time(results <- rowttests(geneExpression, g))
#>    user  system elapsed 
#>   0.004   0.000   0.005
pvals <- results$p.value

Note that these 8,793 tests are done in about 0.01s

Volcano plots: Example

Volcano plots: Summary

• Many small p-values with small effect size indicate low within-group variability
• Inspect for asymmetry
• Can color points by significance threshold

P-value histograms: Setup

• If all null hypotheses are true, expect a flat histogram of p-values:

m <- nrow(geneExpression)
n <- ncol(geneExpression)
set.seed(1)
randomData <- matrix(rnorm(n * m), m, n)
nullpvals <- rowttests(randomData, g)$p.value

P-value histograms: Example

P-value histograms: Example 2 (permutation)

Note that permuting these data doesn’t produce an ideal null p-value histogram due to batch effects:

permg <- sample(g)
permresults <- rowttests(geneExpression, permg)
hist(permresults$p.value)

P-value histograms: Summary

• Give a quick look at how many significant p-values there may be
• When using permuted labels, can expose non-independence among the samples
  • can be due to batch effects or family structure
• Most common approaches for correcting batch effects are:
  • ComBat: corrects for known batch effects by linear model), and
  • sva: creates surrogate variables for unknown batch effects, corrects the structure of permutation p-values
  • correction using control (housekeeping) genes
  • batchelor for single-cell analysis

ComBat and sva are available from the sva Bioconductor package

MA plot

• just a scatterplot rotated 45$^o$

rafalib::mypar(1, 2)
pseudo <- apply(geneExpression, 1, median)
plot(geneExpression[, 1], pseudo)
plot((geneExpression[, 1] + pseudo) / 2, (geneExpression[, 1] - pseudo))

MA plot: Summary

• useful for quality control of high-dimensional data
• plot all data values for a sample against another sample or a median “pseudosample”
• affyPLM::MAplots better MA plots
  • adds a smoothing line to highlight departures from horizontal line
  • plots a “cloud” rather than many data points

Heatmaps

• Detailed representation of high-dimensional dataset.
  • ComplexHeatmap package is the best as of 2024: large datasets, interactive heatmaps, simple defaults but many customizations possible

Heatmaps: Summary

• Clustering becomes slow and memory-intensive for thousands of rows
  • probably too detailed for thousands of rows
• can show co-expressed genes, groups of samples

Colors

• Types of color pallettes:
  • sequential: shows a gradient
  • diverging: goes in two directions from a center
  • qualitative: for categorical variables
• Keep color blindness in mind (10% of all men)

Colors (cont’d)

Combination of RColorBrewer package and colorRampPalette() can create anything you want

Plots To Avoid

“Pie charts are a very bad way of displaying information.” - R Help

• Avoid pie charts
• Avoid doughnut charts too
• Avoid pseudo 3D
• Use color judiciously

Batch effects

Pervasiveness of batch Effects

• pervasive in genomics (e.g. Leek et al. Nat Rev Genet. 2010 Oct;11(10):733-9.)
• affect DNA and RNA sequencing, proteomics, imaging, microarray…
• have caused high-profile problems and retractions
  • you can’t get rid of them
  • but you can make sure they are not confounded with your experiment

Batch Effects - an example

• Nat Genet. 2007 Feb;39(2):226-31. Epub 2007 Jan 7.
• Title: Common genetic variants account for differences in gene expression among ethnic groups.
  • “The results show that specific genetic variation among populations contributes appreciably to differences in gene expression phenotypes.”

library(Biobase)
library(genefilter)
library(GSE5859)
data(GSE5859)
geneExpression = exprs(e)
sampleInfo = pData(e)

• Note: the ExpressionSet object is obsolete, we use SummarizedExperiment now

Date of processing as a proxy for batch

• Sample metadata included date of processing:

head(table(sampleInfo$date))
#> 
#> 2002-10-31 2002-11-15 2002-11-19 2002-11-21 2002-11-22 2002-11-27 
#>          2          2          2          5          4          2

year <-  as.integer(format(sampleInfo$date, "%y"))
year <- year - min(year)
month = as.integer(format(sampleInfo$date, "%m")) + 12 * year
table(year, sampleInfo$ethnicity)
#>     
#> year ASN CEU HAN
#>    0   0  32   0
#>    1   0  54   0
#>    2   0  13   0
#>    3  80   3   0
#>    4   2   0  24

Visualizing batch effects by PCA

pc <- prcomp(t(geneExpression), scale. = TRUE)

Visualizing batch effects by MDS

A starting point for a color palette:

RColorBrewer::display.brewer.all(n = 3, colorblindFriendly = TRUE)

Interpolate one color per month on a quantitative palette:

col3 <- c(RColorBrewer::brewer.pal(n = 3, "Greys")[2:3], "black")
MYcols <- colorRampPalette(col3, space = "Lab")(length(unique(month)))

Visualizing batch effects by MDS

d <- as.dist(1 - cor(geneExpression))
mds <- cmdscale(d)
plot(mds, col = MYcols[as.integer(factor(month))],
     main = "MDS shaded by month")

The batch effects impact clustering

hcclass <- cutree(hclust(d), k = 5)
table(hcclass, year)
#>        year
#> hcclass  0  1  2  3  4
#>       1  0  2  8  1  0
#>       2 25 43  0  2  0
#>       3  6  0  0  0  0
#>       4  1  7  0  0  0
#>       5  0  2  5 80 26

Approaches to correcting for batch effects

• No correction
  • in my experience, the best choice for machine learning applications
• Simple rescaling
  • Rescale observations (cells) to the same mean, variance in each batch
  • maintains sparsity (ie zeros remain zeros)
  • batchelor::rescaleBatches()
• Linear modeling to regress out batch effects
  • use when batches are known. Fits a linear model and use residuals
  • assumes the same composition of cells across batches
  • limma::removeBatchEffect(), sva::comBat(), batchelor::regressBatches()
• Linear modeling to achieve a flat p-value histogram when permuting labels
  • can be used when batches are unknown
  • “Surrogate Variables Analysis” implemented by the sva package
• Mutual Nearest Neighbors
  • developed specifically for single-cell RNA-seq
  • no assumption of the same composition of cells across batches, but still assumes no meaningful biological differences exist between batches
  • batchelor::fastMNN()

Batch Effects - summary

• batches can be corrected for only if not overlapping with conditions of interest
  • if confounded with treatment or outcome of interest, nothing can help you
  • randomization of samples to batches in study design protects against this
• during experimental design:
  • keep track of anything that might cause a batch effect for post-hoc analysis
  • include control samples in each batch
• tend to affect many or all measurements by a little bit

Exercises

• OSCA Multi-sample Chapter 1: Correcting batch effects

Links

• A built html version of this lecture is available.
• The source R Markdown is available from Github.