3.1 Contingency tables and bar plots

Table 3.1 summarizes two variables: application_type and homeownership. A table that summarizes data for two categorical variables in this way is called a contingency table. Each value in the table represents the number of times a particular combination of variable outcomes occurred.

For example, the value 3496 corresponds to the number of loans in the dataset where the borrower rents their home and the application type was by an individual. Row and column totals are also included. The row totals provide the total counts across each row and the column totals down each column. We can also create a table that shows only the overall percentages or proportions for each combination of categories, or we can create a table for a single variable, such as the one shown in Table 3.2 for the homeownership variable.

A bar plot is a common way to display a single categorical variable. The left panel of Figure 3.1 shows a bar plot for the homeownership variable. In the right panel, the counts are converted into proportions, showing the proportion of observations that are in each level.

3.2 Visualizing two categorical variables

We can display the distributions of two categorical variables on a bar plot concurrently. Such plots are generally useful for visualizing the relationship between two categorical variables. Figure 3.2 shows three such plots that visualize the relationship between homeownership and application_type variables. Plot A in Figure 3.2 is a stacked bar plot. This plot most clearly displays that loan applicants most commonly live in mortgaged homes. It is difficult to say, based on this plot alone, how different application types vary across the levels of homeownership. Plot B is a dodged bar plot. This plot most clearly displays that within each level of homeownership, individual applications are more common than joint applications. Finally, plot C is a standardized bar plot (also known as filled bar plot). This plot most clearly displays that joint applications are most common among loans for applicants who live in mortgaged homes, compared to renters and owners. This type of visualization is helpful in understanding the fraction of individual or joint loan applications for borrowers in each level of homeownership. Additionally, since the proportions of joint and individual loans vary across the groups, we can conclude that the two variables are associated for this sample.

3.3 Row and column proportions

In the previous sections we inspected visualizations of two categorical variables in bar plots and mosaic plots. However, we have not discussed how the values in the bar and mosaic plots that show proportions are calculated. In this section we will investigate fractional breakdown of one variable in another and we can modify our contingency table to provide such a view. Table 3.3 shows row proportions for Table 3.1, which are computed as the counts divided by their row totals. The value 3496 at the intersection of individual and rent is replaced by \(3496 / 8505 = 0.411,\) i.e., 3496 divided by its row total, 8505. So what does 0.411 represent? It corresponds to the proportion of individual applicants who rent.

A contingency table of the column proportions is computed in a similar way, where each is computed as the count divided by the corresponding column total. Table 3.4 shows such a table, and here the value 0.906 indicates that 90.6% of renters applied as individuals for the loan. This rate is higher compared to loans from people with mortgages (80.2%) or who own their home (86.5%). Because these rates vary between the three levels of homeownership (rent, mortgage, own), this provides evidence that app_type and homeownership variables may be associated.

Row and column proportions can also be thought of as conditional proportions as they tell us about the proportion of observations in a given level of a categorical variable conditional on the level of another categorical variable.

We could also have checked for an association between application_type and homeownership in Table 3.3 using row proportions. When comparing these row proportions, we would look down columns to see if the fraction of loans where the borrower rents, has a mortgage, or owns varied across the application types.

3.4 Pie charts

A pie chart is shown in Figure 3.3 alongside a bar plot representing the same information. Pie charts can be useful for giving a high-level overview to show how a set of cases break down. However, it is also difficult to decipher certain details in a pie chart. For example, it’s not immediately obvious that there are more loans where the borrower has a mortgage than rent when looking at the pie chart, while this detail is very obvious in the bar plot.

Pie charts can work well when the goal is to visualize a categorical variable with very few levels, and especially if each level represents a simple fraction (e.g., one-half, one-quarter, etc.). However they can be quite difficult to read when they are used to visualize a categorical variable with many levels. For example, the pie chart and the bar plot in Figure 3.4 both represent the distribution of loan grades (A through G). In this case, it is far easier to compare the counts of each loan grade using the bar plot than the pie chart.

3.5 Comparing numerical data across groups

Some of the more interesting investigations can be considered by examining numerical data across groups. In this section we will expand on a few methods we’ve already seen to make plots for numerical data from multiple groups on the same graph as well as introduce a few new methods for comparing numerical data across groups.

We will revisit the county dataset and compare the median household income for counties that gained population from 2010 to 2017 versus counties that had no gain. While we might like to make a causal connection between income and population growth, remember that these are observational data and so such an interpretation would be, at best, half-baked.

We have data on 3142 counties in the United States. We are missing 2017 population data from 3 of them, and of the remaining 3139 counties, in 1541 the population increased from 2010 to 2017 and in the remaining 1598 the population decreased. Table 3.6 shows a sample of 5 observations from each group.

Color can be used to split histograms for numerical variables by levels of a categorical variable. An example of this is shown in Plot A of Figure 3.5. The side-by-side box plot is another traditional tool for comparing across groups. An example is shown in Plot B of Figure 3.5, where there are two box plots, one for each group, placed into one plotting window and drawn on the same scale.

Another useful visualization for comparing numerical data across groups is a ridge plot, which combines density plots for various groups drawn on the same scale in a single plotting window. Figure 3.6 displays a ridge plot for the distribution of median household income in counties, split by whether there was a population gain or not.

One last visualization technique we’ll highlight for comparing numerical data across groups is faceting. In this technique we split (facet) the graphical display of the data across plotting windows based on groups. Plot A in Figure 3.7 displays the same information as Plot A in Figure 3.5, however here the distributions of median household income for counties with and without population gain are faceted across two plotting windows. We preserve the same scale on the x and y axes for easier comparison. An advantage of this approach is that it extends to splitting the data across levels of two categorical variables, which allows for displaying relationships between three variables. In Plot B in Figure 3.7 we have now split the data into four groups using the pop_change and metro variables:

• top left represents counties that are not in a metropolitan area with population gain,
• top right represents counties that are in a metropolitan area with population gain,
• bottom left represents counties that are not in a metropolitan area without population gain, and finally
• bottom right represents counties that are in a metropolitan area without population gain.

We can continue building upon this visualization to add one more variable, median_edu, which is the median education level in the county. In Figure 3.8, we represent median education level using color, where pink (solid line) represents counties where the median education level is high school diploma, yellow (dashed line) is some college degree, and red (dotted line) is Bachelor’s.

3.6 Chapter review

3.7 Exercises

Answers to odd numbered exercises can be found in Appendix A.3.