Understanding and visualizing a distance matrix

My PhD student has just received the data from a set of RNA samples analysed using a gene array. We discussed what should be done first to analyse these experiments. A good first step is to look at the distance between the samples and do unsupervised hierarchical clustering of all the samples to see how the biological replicates gather together. This represents an unbiased way to do some quality control for your experiments. Rather than just trying to identify different genes, we use all the data to determine if replicates gather together, if we have any outliers and to find patterns in the data. 

The first step is the generation of a distance matrix. A distance matrix tells us the difference between lists of numbers. If we put together a group of samples, then the distance matrix compares all of the samples. We discussed the purpose of the data matrix and visualising the matrix. I've written the following R script that tries to explain the concept. Mel helped me develop the script to visualise it using a script from this very informative Stack Overload post. 

Here is the a visualisation of a distance matrix using published data (Webber et al, 2014): 

Visualisation of distance matrix using data from Webber et al, 2014 

This distance matrix is used to do the hierarchical clustering that is plotted out here and shown here: 

Clustering samples using data from Webber et al, 2014.

# Trying to explain the concept of distance
# simple example 1: one digit different by 2 between two samples
samp1 <- c(0, 1, 2, 3, 4, 5, 6, 7, 8)
samp2 <- c(0, 1, 2, 3, 4, 5, 6, 7, 10)
dist(rbind(samp1, samp2))
# the dist() function compares each of the numbers in order along the two rows of the matrix. 
# answer is in this case is 2, as one number in the list is different by 2 to another. 

# simple example 2: one digit different by 72 between two samples
samp1 <- c(0, 1, 2, 3, 4, 5, 6, 7, 8)
samp2 <- c(0, 1, 2, 3, 4, 5, 6, 7, 80)
dist(rbind(samp1, samp2))
# answer is 72

# a little more complicated: two digits different
samp1 <- c(0, 1, 2, 3, 4, 5, 6, 7, 8)
samp2 <- c(0, 10, 2, 3, 4, 5, 6, 7, 80)
dist(rbind(samp1, samp2))
# now the answer is 72.56
# why? 
# well because of the equation used by the default method. 
# the default for the dist() function is euclidean 
# the equation for this is dist = sqrt(sum(x_i - y_i)^2)
# in this case, this calculates as sqrt(9^2 + 72^2)
# why the sqrt and the sum - in part so that the negatives and the positive differences don't cancel each other out.

# you can use other methods of calculating distance
help(dist)

# another is the "maximum" which just gives us the largest difference between the two arrays
# "Maximum distance between two components of x and y"
samp1 <- c(0, 1, 2, 3, 4, 5, 6, 7, 8)
samp2 <- c(0, 10, 2, 30, 4, 5, 6, 7, 80)
dist(rbind(samp1, samp2), method="maximum")
# in this case 72

# a third is the "manhattan" which sum of the absolute distances between the vectors
# "Absolute distance between the two vectors"
samp1 <- c(0, 1, 2, 3, 4, 5, 6, 7, 8)
samp2 <- c(0, 10, 2, 3, 4, 5, 6, 7, 80)
dist(rbind(samp1, samp2), method="manhattan")
# the answer now is 81

# now just envisage a more complicated situation when there are lots more numbers
# more than two samples and numbers that are higher and lower. 

# let's take a couple of examples from some data we analysed previously:

# http://www.mcponline.org/content/13/4/1050.full
# supplementary data is here: http://www.mcponline.org/content/suppl/2014/02/06/M113.032136.DC1/mcp.M113.032136-5.xlsx
# install if necessary:
# install.packages("readxl")
library(readxl)

# this is the link to the data
link <- "http://www.mcponline.org/content/suppl/2014/02/06/M113.032136.DC1/mcp.M113.032136-6.xlsx"

# the download.file() function downloads and saves the file with the name given
download.file(url=link,destfile="file.xlsx", mode="wb")

# then we can open the file and extract the data using the read_excel() function. 
data<- read_excel("file.xlsx")

View(data)

# the data has 762 observations. 

# we can only calculate distances in a matrix where all the values are the same mode - e.g numbers
# convert data frame (data) into a matrix 
# only want a subset of the data - the data from the samples. 
data.m <- as.matrix(data[2:7])
# transpose the data because a distance matrix works in rows
data.m.t <- t(data.m)

# calculate the distances and put the calculations into an object called distances
distances <- dist(data.m.t)

# convert this distances object into a matrix. 
distances.m <- data.matrix(distances)

# you can look at this object.
View(distances.m)

# we can extract the size of the object and the titles
dim <- ncol(distances.m)
names <- row.names(distances.m)

# now to create the visualisation of the difference matrix. 
# first the coloured boxes
image(1:dim, 1:dim, distances.m, axes = FALSE, xlab = "", ylab = "")

# now label the axis
axis(3, 1:dim, names, cex.axis = 0.8, las=3)
axis(2, dim:1, names, cex.axis = 0.8, las=1)

# add the values of the differences
text(expand.grid(1:6, 6:1), sprintf("%0.1f", distances.m), cex=1)

# this example lacks subltety
# the exo samples are very close together and the cell samples are quite far apart.
# it explains why the cluster analysis is so dramatic. 

# export this image as a tiff file with width of 1000 seems to work well. 
# some of the other formats don't work as well. 

# to make the cluster dendrogram object using the hclust() function
hc <- hclust(distances) 
# plot the cluster diagram
# some interesting groups in the data
plot(hc, 
     xlab =expression(bold("All Samples")), 
     ylab = expression(bold("Distance")))
# replicates cluster together well. 



The visualisation was inspired by this:

• http://stackoverflow.com/questions/3081066/what-techniques-exists-in-r-to-visualize-a-distance-matrix

If you have feedback on this script, please leave a comment. 

Downloading and manipulating published proteomic data...

Update: 1 July 2025 - so a lot has happened in 10 years. This includes people moving jobs, promotions and the reorganisation of data on published website. 

Jason moved to Swansea University. 

Aled was promoted to Professor at Cardiff University 

Tim moved to the University of Westminister. 

----

There are many ways to get data into R. I want to illustrate a method of downloading published data within R, opening the data (an Excel file) and then doing visualisations and manipulations. 

I have chosen a paper from Molecular and Cellular Proteomics which uses aptamers to detect multiple proteins in exosomes and cells. 

The data was generated by colleagues that were working in the School of Medicine at Cardiff University including the first author - Dr Jason Webber, a Prostate Cancer UK funded Research Fellow and senior author, Professor Aled Clayton, a Senior Lecturer in Cancer & Genetics at Velindre Hospital. Dr Tim Stone was key to the data analysis. The protein detection method is from a company called SomaLogic. 

The first step was downloading the file from the Molecular and Cellular Proteomics website. I used the download.file() function. This saves the file into your current directory. This was opened using the readxl package (by Hadley Wickham) using the read_excel() function. 

I drew some graphs as I explored and manipulated the data. These are interspersed with the script below.  The visualisations included boxplots and a cluster diagram. 

I wanted to draw a volcano plot which expresses the fold change against the significant of the change (p-value). I couldn't do that with the data supplied so I had to reverse the transformation and calculate the fold change again. 

Here is the volcano plot:

Comparison of changes in exosomes compared cells. Proteins over-expressed in exosomes are on the right. Proteins over-expressed in cells are on the left. 

The plot indicates that there are more proteins over-expressed in cells (on the left) compared to exosomes (on the right). 

Update: 1 July 2025 - orignally, I was able to download the data directly from the Molecular and Cellular Proteomics website. However, at some point, they reorganised their site and put all the data into a zip file. This means that downloading it requires multiple steps outside of R. To make this analysis more stand alone, I have down loaded the data from this zip file and uploaded the spreadsheet onto my Github site. This means that the script will download and analyse the data. 

Here is the script with some other plots along the way:

START

# pull down a file from the internet, do some analysis and draw a graph...

# choose Jason Webber's MCP Paper...

# http://www.mcponline.org/content/13/4/1050.full

# the data can be downloaded using this link with give zip file. 

# It isn't necessary to do that for this script. 

# install if necessary:

# install.packages(c("ggplot2", "readxl")) 

library(ggplot2)

library(readxl)

# this is the link to the data

link <- "https://github.com/brennanpincardiff/RforBiochemists/raw/master/R_for_Biochemists_101/data/mcp.M113.032136-6.xlsx"

# the download.file() function downloads and saves the file with the name given

download.file(url=link,destfile="file.xlsx", mode="wb")

# then we can open the file and extract the data using the read_excel() function. 

data<- read_excel("file.xlsx")

View(data)

# plot the data - always an important first step!

boxplot(data[2:7], 

        las =2, # las = 2 turns the text around to show sample names

        ylab = expression(bold("expression")),

        main="Boxplot of expression data")  

Two types of sample - exosomes (n=3) and cells (n=3).

# do a cluster analysis to quality control the different groups

# convert to matrix first

data.m <- as.matrix(data[2:7])

dim(data.m)   # gives the dimensions of the matrix

# ans: 762   6

# calculate the distances using the dist() function. 

# various methods are possible - default is Euclidean. 

distances <- dist(data.m)

summary(distances) # have a look at the object

# make the cluster dendrogram object using the hclust() function

hc <- hclust(distances) 

# plot the cluster diagram

# some interesting groups in the data

plot(hc, 

     xlab =expression(bold("All Samples")), 

     ylab = expression(bold("Distance")))

# ah, not what I intended. 

# it clustered the proteins NOT the samples. 

# transpose the data and try again...

data.m.t <- t(data.m)

dim(data.m.t)

# ans = 6  762 - so that has worked. 

# repeat cluter analysis

# calculate the distances using the dist() function. 

# various methods are possible - default is Euclidean. 

distances <- dist(data.m.t)

summary(distances)

# make the cluster dendrogram object using the hclust() function

hc <- hclust(distances) 

# plot the cluster diagram

# some interesting groups in the data

plot(hc, 

     xlab =expression(bold("All Samples")), 

     ylab = expression(bold("Distance")))

# replicates cluster together well. 

# the adjusted P value are in a column entiteld: BH - P.value

# this is a little awkward so rename this and Fold Change column:

colnames(data)[9] <- "P.Value"

colnames(data)[10] <- "Fold.Change"

plot(data$P.Value ~ data$Fold.Change)

# this works but we would like to turn it into a volcano plot 

# with log2 at the bottom and -p-value. 

# we can't log fold changes as half of them are negative numbers 

# we need to re-calculate the raw data

# reverse the log2 transformation. 

# mean the values & calculate fold change in decimal format (no +/-)

data$exo1 <- 2^data$exoRFU1

data$exo2 <- 2^data$exoRFU2

data$exo3 <- 2^data$exoRFU3

data$cell1 <- 2^data$cellRFU1

data$cell2 <- 2^data$cellRFU2

data$cell3 <- 2^data$cellRFU3

# calculate means of the replicates

data$exoMean <- rowMeans(data[,13:15])

data$cellMean <- rowMeans(data[,16:18])

# always good to visualise the data:

plot(log2(data$exoMean)~log2(data$cellMean))

# calculate fold change exo/cell

data$FoldChange <- data$exoMean/data$cellMean

plot(log2(data$FoldChange)) # transform for plotting

# make column in data.frame with transformed data

data$Log2.Fold.Change <- log2(data$FoldChange)

## Identify the genes that have a p-value < 0.05

data$threshold = as.factor(data$P.Value < 0.05)

## Construct the volcano plot object using ggplot

g <- ggplot(data=data, 

            aes(x=Log2.Fold.Change, y =-log10(P.Value), 

                colour=threshold)) +

  geom_point(alpha=0.4, size=1.75) +

  xlim(c(-6, 6)) +

  xlab("log2 fold change") + ylab("-log10 p-value") +

  theme_bw() +

  theme(legend.position="none") + 

  ggtitle("Volcano Plot comparing protein expression in exosomes vs cells ")  # add a title

  

g # show the plot 

END of script

So I think that the threshold of p<0.05 is too low for this volcano plot. It's relatively easy to change and would make a good exercise. Perhaps a threshold of p<0.00001 would be better. 

Useful resources (updated 1 July 2025)

• Here is a script about making multiple volcano plots
• More information about volcano plots on Wikipedia. 
• This Enhanced Volcano package looks interesting but I haven't played with it myself.