11  Two continuous variables

In this chapter, we will look at techniques that explore the relationships between two continuous variables.

11.1 Scatterplot

11.1.1 Basics and implications

For the following example, we use data set SpeedSki.

library(GDAdata)
library(ggplot2)

ggplot(SpeedSki, aes(Year, Speed)) + 
  geom_point() +
  labs(x = "Birth year", y = "Speed achieved (km/hr)") +
  ggtitle("Skiers by birth year and speed achieved")

In our example, we simply use geom_point on variables Year and Speed to create the scatterplot. we try to capture if there is a relationship between the age of a player and the speed he/she can achieve. From the graph, it seems such relationship does not exist. Overall, scatterplots are very useful in understanding the correlation (or lack thereof) between variables. The scatterplot gives a good idea of whether that relationship is positive or negative and if there’s a correlation. However, don’t mistake correlation in a scatterplot for causation!

11.1.2 Overplotting

In some situations a scatter plot faces the problem of overplotting as there are so many points overlapping. Consider the following example from class. To save time, we randomly sample 20% of the data in advance.

library(dplyr)
library(ggplot2movies)

sample <- slice_sample(movies, prop = 0.2)

ggplot(sample,aes(x=votes,y=rating)) +
  geom_point() +
  ggtitle("Votes vs. rating") +
  theme_classic()

To create better visuals, we can use:

  • Alpha blending - alpha=...

  • Open circles - pch=21

  • smaller circles - size=... or shape="."

library(gridExtra)

f1 <- ggplot(sample,aes(x=votes,y=rating)) +
  geom_point(alpha=0.3) +
  theme_classic() +
  ggtitle("Alpha blending")

f2 <- ggplot(sample,aes(x=votes,y=rating)) +
  geom_point(pch = 21) +
  theme_classic() +
  ggtitle("Open circle")

f3 <- ggplot(sample,aes(x=votes,y=rating)) +
  geom_point(size=0.5) +
  theme_classic() +
  ggtitle("Smaller circle")

grid.arrange(f1, f2, f3,nrow = 3)

Other methods that directly deal with the data:

  • Randomly sample data - as shown in the first code chunk using sample_n

  • Subset - split data into bins using ntile(votes, 10)

  • Remove outliers

  • Transform to log scale

11.1.3 Interactive scatterplot

You can create an interactive scatterplot using plotly. In the following example, we take 1% of the movie data set to present a better visual. We plotted the votes vs. rating and grouped by the year they are released. In this graph:

  • You can hover on to the points to see the title of the movie

  • You can double click on the year legend to look at a certain year

  • You can zoom into a certain part of the graph to better understand the data points.

library(plotly)

sample2 <- slice_sample(movies,prop=0.01) |>
  filter(year > 2000)

plot_ly(sample2, x = ~votes, y = ~rating,
        color = ~as.factor(year), text= ~title,
        hoverinfo = 'text')

11.1.4 Modifications

11.1.4.1 Contour lines

Contour lines give a sense of the density of the data at a glance.

For these contour maps, we will use the SpeedSki dataset.

Contour lines can be added to the plot using geom_density_2d() and contour lines work best when combined with other layers

ggplot(SpeedSki, aes(Year, Speed)) + 
  geom_density_2d(bins=5) +
  geom_point() +
  ggtitle("Scatter plot with contour line")

You can use bins to control the number of contour bins.

11.1.4.2 Scatterplot matrices

If you want to compare multiple parameters to each other, consider using a scatterplot matrix. This will allow you to show many comparisons in a compact and efficient manner.

For these scatterplot matrices, we use the movies dataset from the ggplot2movies package.

As a default, the base R plot() function will create a scatterplot matrix when given multiple variables:

sample3 <- slice_sample(movies,prop=0.01) #sample data

splomvar <- sample3 |> 
  dplyr::select(length, budget, votes, rating, year)

plot(splomvar)

While this is quite useful for personal exploration of a dataset, it is not recommended for presentation purposes. Something called the Hermann grid illusion makes this plot very difficult to examine.

11.2 Heatmaps

11.2.1 Basics and implications

In the following example, we still use the SpeedSki data set.

ggplot(SpeedSki, aes(Year, Speed)) + 
  geom_bin2d()

To create a heatmap, simply substitute geom_point() with geom_bin2d(). Generally, heat maps are like a combination of scatterplots and histograms: they allow you to compare different parameters while also seeing their relative distributions.

11.2.2 Modifications

For the following section, we introduce some variations on heatmaps.

11.2.2.1 Change number of bins / binwidth

By default, geom_bin2d() use 30 bins. Similar to a histogram, we can change the number of bins or binwidth.

ggplot(SpeedSki, aes(Year, Speed)) + 
  geom_bin2d(binwidth = c(5,5)) +
  ggtitle("Changing binwidth")

Notice we are specifying the binwidth for both x and y axis.

11.2.2.2 Combine with a scatterplot

ggplot(SpeedSki, aes(Year, Speed)) +
  geom_bin2d(binwidth = c(10, 10), alpha = .4) + 
  geom_point(size = 2) +
  ggtitle("Combined with scatterplot")

11.2.2.3 Change color scale

You can change the continuous scale of color

ggplot(SpeedSki, aes(Year, Speed)) + 
  geom_bin2d() +
  ggtitle("Changing color scale") + 
  scale_fill_viridis_c()

11.2.2.4 Hex heatmap

One alternative is a hex heatmap. You can create the graph using geom_hex

ggplot(SpeedSki, aes(Year, Speed)) + 
  geom_hex(binswidth = c(10,10)) +
  ggtitle("Hex heatmap")

11.2.2.5 Alternative approach to color

If you look at all the previous examples, you might notice that lighter points correspond to more clustered points, which is somewhat counter-intuitive. The following example suggests an alternative approach in color scale.

ggplot(SpeedSki, aes(Year, Speed)) + 
  geom_hex(bins=12) +
  scale_fill_gradient(low = "grey", high = "purple") +  
  theme_classic(18) +
  ggtitle("Alternative approach to color")