• Drawing scatterplots of pairs of continuous variables
  • Using transparency in your visualisations

You will need to install the following packages for today’s workshop:

  • scales for the alpha function for transparency in plots

Custom packages are exceptionally useful in R, as they provide more specialised functionality beyond what is included in the base version of R. We will typically make use of custom packages to access specialised visualisation functions or particular data sets that are included in the package.

There are many ways to install a new package. Here are a few of the easiest:

  • In RStudio, open the Tools menu and select Install Packages.... Type the name of the package, and click install.
  • In the console, type install.packages("pkgname") where you replace pkgname with the name of the package you want to install.
install.packages("scales")

1 Drawing Scatterplots

Scatterplots are another rather simple plot, but incredibly effective at showing structure in the data. By simply plotting points at the coordinates of two numerical variables, we can easily detect any patterns that may appear.

The plot function produces a scatterplot of its two arguments. For illustration, let us use the mtcars data set again, containing information on the characteristics of 23 cars. We can plot miles per gallon against weight with the command

data(mtcars)
plot(x=mtcars$wt, y=mtcars$mpg)

Unsurprisingly, heavier cars do fewer miles per gallon and are less efficient. The relationship here, while clear and negative, is far from exact with a lot of noise and variation.

If the argument labels x and y are not supplied to plot, R will assume the first argument is x and the second is y. If only one vector of data is supplied, this will be taken as the \(y\) value and will be plotted against the integers 1:length(y), i.e. in the sequence in which they appear in the data.

1.1 Customising the plot type

Another useful optional argument is type, which can substantially change how plot draws the data. The type argument can take a number of different values to produce different types of plot:

  • type="p" - draws a standard scatterplot with a point for every \((x,y)\) pair
  • type="l" - connects adjacent \((x,y)\) pairs with straight lines, does not draw points. Note this is a lowercase L, not a number 1.
  • type="b" - draws both points and connecting line segments
  • type="s" - connects points with ‘steps’ rather than straight lines

1.2 Plot symbols

The symbols used for points in scatter plots can be changed by specifying a value for the argument pch {#pch} (which stands for plot character). Specifying values for pch works in the same way as col, though pch only accepts integers between 1 and 20 to represent different point types. The default is pch=1 which is a hollow circle. The possible values of pch are shown in the plot below:

  • Experiment with using the pch argument to adjust the plot points used in the scatterplot above. Which plot symbols do you find most effective?

1.3 Overplotting and transparency

To deal with issues of overplotting - where dense areas of points are drawn ontop of each other - we can use transparency to make the plot symbols. For example, here are 500 points randomly generated from a 2-D normal distribution. Notice how the middle of the plot is a solid lump of black

plot(x=rnorm(5000),y=rnorm(5000),pch=16)

To ‘fix’ this, we can specify a transparent colour in the col argument by using the alpha function from the scales package:

library(scales)
plot(x=rnorm(5000),y=rnorm(5000),pch=16,col=alpha('black',0.2))

The alpha function takes two arguments - a colour first, and then the alpha level itself. This should be a number in \([0,1]\) with smaller values being more transparent. Finding a good value for alpha is usually a case of trial-and-error, but in general it will be smaller than you might first expect!

Now with the transparency we can see a bit more structure in the data, and the darker areas now highlight regions of high data density.

1.4 Data Set 3: Engine pollutants

Download data: engine

This rather simple data set contains three numerical variables, each representing different amounts of pollutants emitted by 46 light-duty engines. The pollutants recorded are Carbon monoxide (CO), Hydrocarbons (HC), and Nitrogen oxide (NOX), all recorded as grammes emitted per mile.

  • Construct three scatterplots to investigate the relationships between every pair of pollutants:
    • CO versus HC
    • CO versus NOX
    • HC versus NOX
  • What are the relationships between the amounts of different pollutants emitted by the various engines? In particular, which pollutants are positively associated and which are negatively associated?
  • The strength of a linear relationship can be assessed through the correlation, which is a number between 0 (no linear relationship) and 1 (perfect linear relationship). Have a guess at the correlation between these variables, and then check the numerical values with the cor function.
  • Do any of the scatterplots have any outliers?
  • If so, can you identify which data points correspond to these outliers. Look at the other data values for these outlying cases - are they extreme on other variables or just one?
  • Can you visually indicate the outliers on your scatterplot, e.g with colour?

1.5 Data Set 4: The evils of drink?

Download data: DrinksWages

Karl Pearson was another prominent early statistician, responsible for number of major contributions: correlation, p-values, chi-square tests, and the simple histogram to name a few.

In 1910, hen weighed in on the debate, fostered by the temperance movement, on the evils done by alcohol not only to drinkers, but to their families. The report “A first study of the influence of parental alcholism on the physique and ability of their offspring” was an ambitious attempt to the new methods of statistics to bear on an important question of social policy, to see if the hypothesis that children were damaged by parental alcoholism would stand up to statistical scrutiny.

Working with his assistant, Ethel M. Elderton, Pearson collected voluminous data in Edinburgh and Manchester on many aspects of health, stature, intelligence, etc. of children classified according to the drinking habits of their parents. His conclusions where almost invariably negative: the tendency of parents to drink appeared unrelated to any thing he had measured. The publication of their report caused substantial controversy at the time, and the data set DrinksWages is just one of Pearson’s many tables, that he published in a letter to The Times, August 10, 1910.

The data contain 70 observations on the following 6 variables, giving the number of non-drinkers (sober) and drinkers (drinks) in various occupational categories (trade).

  • class - wage class: a factor with levels A B C
  • trade - a factor with levels: baker barman billposter … wellsinker wireworker
  • sober - the number of non-drinkers, a numeric vector
  • drinks - the number of drinkers, a numeric vector
  • wage - weekly wage (in shillings), a numeric vector
  • n - total number, a numeric vector
  • Use the drinks and n columns to compute the proportion of drinkers for each row of the data set.
  • Plot the proportion of drinkers against the weekly wage.
  • Is there any evidence of a relationship between these variables?
  • Do you see anything unusual happening at either end of the scale of the proportion of drinkers?

It is perhaps a little surprising that some trades have 100% drinkers and some 100% non-drinkers. A look at the distribution of numbers in the study by trade (or just a summary table) will help explain why:

  • Make a histogram of n, the counts in each row. What do you find?
  • Of all the different trades, how many only have 1 or 2 member in this survey?
  • Can you find the biggest group with 100% drinkers? What about non-drinkers?
With very small samples such as these, our proportions are somewhat unreliable. Excluding the smaller trades should give more reliable results:
  • Create a new dataset which contains only those trades where n is 5 or more.
  • Redraw your scatterplot.
  • How has the plot changed? Has your interpretation of the relationship between these variables changed?

2 Variations on Standard Plots

2.1 A hexbin

When overplotting of data in a scatterplot is an issue, we saw above that we can use transparency to ‘see through’ our data points and reveal areas of high density. A hexbin can be though of as a two-dimensional histogram, where we plot the counts of the data which fall in small hexagonal regions (squares would work, but hexagons just look cooler). The shades of the bins then take the place of the heights of the bars. This technique is computed in the hexbin package.

# you'll need to install `hexbin` if you don't have it
# install.packages("hexbin")
library(hexbin)
library(grid)

# Create some dense random normal values
x <- rnorm(mean=1.5, 5000)
y <- rnorm(mean=1.6, 5000)
# Bin the data 
bin <-hexbin(x, y, xbins=40)
# Make the plot
plot(bin, main="" , colramp = colorRampPalette(c("darkorchid4","darkblue","green","yellow", "red")))

3 More Practice

3.1 Data Set 5: Exam marks in mathematics

Download data: marks

This data set contains the exam marks of 88 students taking an exam on five different topics in mathematics. The data contain the separate marks for each topic on a scale of 0-100:

  • MECH - mechanics
  • VECT - vectors
  • ALG - algebra
  • ANL - analysis
  • STAT - statistics
  • Explore the data!
  • If that’s a little too vague:
    • Start with some plots of the individual variables. Look for commonality and obvious differences.
    • Think about what you might expect to see here, and investigate those hypotheses.
    • Look at the scatterplots between the variables - maybe focus on STATs for now, as the most familiar subject.
    • Which subjects have a strong relationship between their exam performance? Which have a weak relationship?
    • If we take this as a proxy for the similarity between the subjects, what would you infer?