Advanced High School Statistics: Third Edition

Table 2.4.1 summarizes two variables: app_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 data set 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 (e.g. \(3496+3839+1170=8505\)), and column totals are total counts 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 2.4.2 for the homeownership variable.

A bar chart (also called bar plot or bar graph) is a common way to display a single categorical variable. The left panel of Figure 2.4.3 shows a bar chart for the homeownership variable. In the right panel, the counts are converted into proportions, showing the proportion of observations that are in each level (e.g. \(3858/10000=0.3858\) for rent).

Sometimes it is useful to understand the fractional breakdown of one variable in another, and we can modify our contingency table to provide such a view. Table 2.4.4 shows the row proportions for Table 2.4.1, which are computed as the counts divided by their row totals. The value 3496 at the intersection of individual and rent is replaced by \(3496/8505=0.411\text{,}\) 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 column proportion is computed as the count divided by the corresponding column total. Table 2.4.5 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 (85.1%). Because these rates vary between the three levels of homeownership (rent, mortgage, own), this provides evidence that the app_type and homeownership variables are associated.

We could also have checked for an association between app_type and homeownership in Table 2.4.4 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 individual to joint application types.

Example 2.4.8 points out that row and column proportions are not equivalent. Before settling on one form for a table, it is important to consider each to ensure that the most useful table is constructed. However, sometimes it simply isn’t clear which, if either, is more useful.

Contingency tables using row or column proportions are especially useful for examining how two categorical variables are related. Segmented bar charts provide a way to visualize the information in these tables.

A segmented bar chart, or stacked bar chart, is a graphical display of contingency table information. For example, a segmented bar chart representing Table 2.4.5 is shown in Figure 2.4.(a), where we have first created a bar chart using the homeownership variable and then divided each group by the levels of app_type .

One related visualization to the segmented bar chart is the side-by-side bar chart, where an example is shown in Figure 2.4.(b).

For the last type of bar chart we introduce, the column proportions for the app_type and homeownership contingency table have been translated into a standardized segmented bar chart in Figure 2.4.(c). 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 vary across the groups, we can conclude that the two variables are associated.

A mosaic plot is a visualization technique suitable for contingency tables that resembles a standardized segmented bar chart with the benefit that we still see the relative group sizes of the primary variable as well.

To get started in creating our first mosaic plot, we’ll break a square into columns for each category of the homeownership variable, with the result shown in Figure 2.4.13.(a). Each column represents a level of homeownership, and the column widths correspond to the proportion of loans in each of those categories. For instance, there are fewer loans where the borrower is an owner than where the borrower has a mortgage. In general, mosaic plots use box areas to represent the number of cases in each category.

To create a completed mosaic plot, the single-variable mosaic plot is further divided into pieces in Figure 2.4.13.(b) using the app_type variable. Each column is split proportional to the number of loans from individual and joint borrowers. For example, the second column represents loans where the borrower has a mortgage, and it was divided into individual loans (upper) and joint loans (lower). As another example, the bottom segment of the third column represents loans where the borrower owns their home and applied jointly, while the upper segment of this column represents borrowers who are homeowners and filed individually. We can again use this plot to see that the homeownership and app_type variables are associated, since some columns are divided in different vertical locations than others, which was the same technique used for checking an association in the standardized segmented bar chart.

In Figure 2.4.14, we chose to first split by the homeowner status of the borrower. However, we could have instead first split by the application type, as in Figure 2.4.14. Like with the bar charts, it’s common to use the explanatory variable to represent the first split in a mosaic plot, and then for the response to break up each level of the explanatory variable, if these labels are reasonable to attach to the variables under consideration.

A pie chart is shown in Figure 2.4.15 alongside a bar chart 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 details in a pie chart. For example, it takes a couple seconds longer to recognize 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 chart. While pie charts can be useful, we prefer bar charts for their ease in comparing groups.