2. Distributions and Correlations

After the previous section 1, you should have loaded the following data:

Mice - mouse phenome dataset (790 mice)
miR & mRNA - microRNA and mRNA profiles of A375 cell line after stimulation with IFNg [Nazarov et al, Nucleic Acid Research, 2013]
Prot - protein-level results of MS/MS proteomics (from MaxQuant analysis) [Schuster et al, Nature Communication, ~2021]
mtcars & iris - standard R datasets (preloaded, no need to load)

If not, please load:

## load the script from Internet
source("http://r.modas.lu/readAMD.r")
source("http://r.modas.lu/readMaxQuant.r")

## Mouse 'phenome' data
Mice = read.table("http://edu.modas.lu/data/txt/mice.txt",
                  sep="\t",
                  header=TRUE,
                  stringsAsFactors = TRUE)

## miRNA data
miR = readAMD("http://edu.modas.lu/data/txt/mirna_ifng.amd.txt",stringsAsFactors = TRUE)

## mRNA data
mRNA = readAMD("http://edu.modas.lu/data/txt/mrna_ifng.amd.txt",
               stringsAsFactors = TRUE,
               index.column="GeneSymbol",
               sum.func="mean")

## Protein groups
Prot = readMaxQuant("http://edu.modas.lu/data/txt/proteingroups.txt")

2.1. Boxplot

Let’s use to check behavior of log-intensity in miRNA dataset. Each box includes 50% of data points (between 1st and 3rd quartiles). Median is shown by a thick line.

boxplot() - command to draw a boxplot

Try this:

boxplot(miR$X)

Hmm… not too nice! Let’s improve

col - color of the bars
las - axis text direction (2 for perpendicular)
cex.names,cex.axis - size of the names on the categorical axis and numerical axis
main - title of the figure. Alternatively title() can be called to add names after.
xlab,ylab - names of axes

col_mir = rainbow(10)[as.integer(miR$meta$time)]

boxplot(miR$X,
        las=2,
        col=col_mir,
        ylab="log intensity",
        main="Intensities in the samples")

Boxplot can be used also to summarize data over a factor using dependency operator ~. Let’s investigate weight of mice in different strains.

## define colors - individual for each strain
col_strain = rainbow(nlevels(Mice$Strain))

## build boxplots 
boxplot(Ending.weight ~ Strain,  
        data = Mice,
        las = 2,
        col = col_strain,
        cex.axis = 0.7,
        ylab="Weight, g",
        main="Mouse weight in Strains")

2.2. Histogram and density plots

In order to show distribution of the data you can use either histogram (precise, not beautiful) or estimation of probability density function (beautiful, approximate)

hist(miR$X,
     breaks=100,
     main="Histogram: log-intensities for miRNA",
     xlab="Values",
     col=4)

Density plots can shows a convolution of the data with smoothing Gaussian function. Parameter

plot(density(..))
width - width of the smoothing function (higher - more smoothed)

Let’s compare distributions of miRNA and mRNA:

## 2 plots per window
par(mfcol=c(1,2))

## plot density of miRNA data
plot(density(miR$X),main="Log-intensities for miRNA",lwd=2,col=4)
## optional: add line with different smoothing
lines(density(miR$X,width=0.1),lwd=1,col=2) ##  try with different smoothing

## plot density of miRNA data
plot(density(mRNA$X),main="Log-intensities for mRNA",lwd=2,col=4)

Additional functions:

lines() - adds a line to existing plot
points() - adds point-graph to the existing plot
legend() - adds legend to the existed plot
abline() - adds a line to plot (horizontal, vertical ot tilted)

I can propose you a custom function to visualize distribution in different samples. Here we will use proteomics data:

## load custom function plotDataPDF
source("http://r.modas.lu/plotDataPDF.r")

plotDataPDF(Prot$LX,ylim=c(0,0.4),lwd=2)

2.3 Violin plot

Violon plot combines boxplot and distribution into a single graph.

Please, first install packages vioplot and sm

#install.packages("vioplot")
#install.packages("sm")
source("http://r.modas.lu/violinplot.r")

## create a list to visualize
L = tapply(Mice$Ending.weight, Mice$Sex, c)
str(L)

## List of 2
##  $ f: num [1:396] 20.5 20.8 19.8 21 21.9 22.1 21.3 20.1 18.9 21.3 ...
##  $ m: num [1:394] 24.7 27.2 23.9 26.3 26 23.3 26.5 27.4 27.5 25.8 ...
##  - attr(*, "dim")= int 2
##  - attr(*, "dimnames")=List of 1
##   ..$ : chr [1:2] "f" "m"

violinplot(L,col="skyblue",main="Weight of male and female mice")

2.4. Scatter plot

A scatter plot is used to show potential dependencies between 2 variables (observations). For example, we could check how starting and ending weights of mice are connected. It can also show

plot(x = Mice$Starting.weight,
     y = Mice$Ending.weight,
     pch=19, col="#0000FF55",
     main="Scatter plot")

2.5. 2D distribution: `smoothScatter()`

Sometimes you need to show a scatter plot with many points. To keep information about their density - use smoothScatter()

par(mfcol=c(1,2))
smoothScatter(miR$X[,1],miR$X[,2], main="Scatter-plot (replicates)")
smoothScatter(miR$X[,1],miR$X[,15], main="Scatter-plot (T72 vs T00)")

2.6. Correlation

Correlation (Pearson correlation coefficient) shows the measure of the linear dependency between observations. Example:

Example of Pearson correlation (Wiki) Square of Pearson correlation (a.k.a. coefficient of determination, $R^{2}$) tells, which portion of the total variability in variable-1 can be explained by variable-2 and vice versa.

In R correlation between variables is calculated by function cor(). Parameters:

use - how to deal with missing values. Use “complete.obs” or “pairwise.complete.obs” instead of “everything”.
method - method used for correlation: pearson - standard parametric Pearson correlation (linear dependency), spearman - non-parametric Spearman correlation (rank correlation), kendall - Kendall rank test (ordinal associations).

Pearson correlation usually converges faster, but is sensitive to influential outliers. Spearman correlation is less powerful, but is consider each pair of points equally important. Kendall behaves like Spearman, but is very slow.

If you would like to check statistical significance of correlation, use cor.test().

Here is the script demonstrating differences in correlation (no need to run!).

x1 = 1:30 + 10*rnorm(30) # create x1 variable
x2 = 1:30 + 10*rnorm(30) # create x2 variable
x2[25] = 200 # add an outlier
y1 = x1 + 10*rnorm(length(x1))  # create correlated y1 variable
y2 = 10*rnorm(length(x2)); y2[25] = 150   # create correlated y1 variable
mod1 = lm(y1~x1) # linear regression 1
mod2 = lm(y2~x2) # linear regression 2

res1p= unlist(cor.test(x1,y1,method="pearson")[c("estimate","p.value")])
res1s= unlist(cor.test(x1,y1,method="spearman")[c("estimate","p.value")])
res2p= unlist(cor.test(x2,y2,method="pearson")[c("estimate","p.value")])
res2s= unlist(cor.test(x2,y2,method="spearman")[c("estimate","p.value")])

par(mfcol=c(1,2))
plot(x1,y1,pch=19,main="`Normal` data")
abline(a=mod1$coef[1],b=mod1$coef[2],col=2)
legend("topleft",sprintf("Pearson: r=%.2f, pv=%.1e",res1p[1],res1p[2]))
legend("bottomright",sprintf("Spearman: r=%.2f, pv=%.1e",res1s[1],res1s[2]))

plot(x2,y2,pch=19,main="Data with outliers")
abline(a=mod2$coef[1],b=mod2$coef[2],col=2)
legend("topleft",sprintf("Pearson: r=%.2f, pv=%.1e",res2p[1],res2p[2]))
legend("bottomright",sprintf("Spearman: r=%.2f, pv=%.1e",res2s[1],res2s[2]))

You can test significance of correlation using cor.test(). The default null hypothesis is r=0.

Note: Number of observations is important!

Let’s see how correlation behaves between two random vectors of different sizes.

Correlation between random

Now let’s try to calculate and visualized correlations for all variables in the dataset.

One example is to use PerformanceAnalytics (please install)

#install.packages("PerformanceAnalytics")

## require() is similar to library()
require(PerformanceAnalytics) 

## let's look into mtcars 
str(mtcars)

## 'data.frame':    32 obs. of  11 variables:
##  $ mpg : num  21 21 22.8 21.4 18.7 18.1 14.3 24.4 22.8 19.2 ...
##  $ cyl : num  6 6 4 6 8 6 8 4 4 6 ...
##  $ disp: num  160 160 108 258 360 ...
##  $ hp  : num  110 110 93 110 175 105 245 62 95 123 ...
##  $ drat: num  3.9 3.9 3.85 3.08 3.15 2.76 3.21 3.69 3.92 3.92 ...
##  $ wt  : num  2.62 2.88 2.32 3.21 3.44 ...
##  $ qsec: num  16.5 17 18.6 19.4 17 ...
##  $ vs  : num  0 0 1 1 0 1 0 1 1 1 ...
##  $ am  : num  1 1 1 0 0 0 0 0 0 0 ...
##  $ gear: num  4 4 4 3 3 3 3 4 4 4 ...
##  $ carb: num  4 4 1 1 2 1 4 2 2 4 ...

?mtcars

chart.Correlation(mtcars)

## Another example - miRNA data 
chart.Correlation(miR$X)

Another way to represent correlations in the dataset - via corrplot or heatmap

Corrplot:

#install.packages("corrplot")
library(corrplot)

## calculate correlations
r = cor(mtcars)

## different visualization
par(mfrow=c(2,2))
corrplot(r, method="circle")
corrplot(r, method="pie", type="upper")
corrplot(r, method="number")
corrplot(r, method="color")

Heatmap:

library(pheatmap)

r = cor(Mice[,-c(1:3)],use = "pairwise.complete.obs")

pheatmap(r)

Exercises

Build boxplot for proteomics data (data are stored in Prot$LX).

boxplot()

Buils a boxplot of IRF1 expression vs IFNg-treatment time (data: mRNA). You can access expression by mRNA$X["IRF1",] and time by mRNA$meta$time .

boxplot(values ~ factor)

Build distribution of bleeding time for Mice dataset

plot(), density(...,na.rm=TRUE)

Analyse correlations within protein-group data. Plot and visually compare correlations in original data Prot$X0 and in log-transformed data Prot$LX

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By Petr Nazarov

2. Distributions and Correlations

Petr V. Nazarov, LIH

2022-10-10

2.1. Boxplot

2.2. Histogram and density plots

2.3 Violin plot

2.4. Scatter plot

2.5. 2D distribution: `smoothScatter()`

2.6. Correlation

Exercises

2. Distributions and Correlations

Petr V. Nazarov, LIH

2022-10-10

2.1. Boxplot

2.2. Histogram and density plots

2.3 Violin plot

2.4. Scatter plot

2.5. 2D distribution: smoothScatter()

2.6. Correlation

Exercises

2.5. 2D distribution: `smoothScatter()`