Visualizing Cross Tabulations

Note on required packages: The following code requires the packages tidyverse, scales, stringr, and ggthemes.

If you have not already done so, download, install, and load the libraries with the following code:

Cross-tabulations or contingency tables are tables used to describe relationships between two categorical variables. This table is also used in the computation of the Chi-square test of independence, which you can read more about here.

1 Example Data Set

1.1 Download the Data

We will discuss cross tabulations in the context of an example data set on financial behaviors and financial literacy from The National Financial Capability Study. The study is performed by the FINRA Investor Education Foundation. The survey was administered in 2009, 2012, and 2015. The 2015 study includes responses by more than 27,000 individuals from every demographic group across the United States.

The data set findata.RData below includes a subset of the variables and observations from the full study. This subset includes 98 financial and demographic variables on 10,000 individuals across the country.

The code below downloads the data and opens it, creating a data frame object, df.

In this tutorial, we will focus on two categorical variables, FinDifficulty and LivingSituation.

1.2 Examining the Data: Financial Difficulty

FinDifficulty is an ordinal categorical variable that is a description of how difficult a person finds it to meet their expenses in a typical month.

In R, categorical variables are of class factor and ordinal variables are of class ordered factor. We can confirm the class for FinDifficulty with a call to class().

## [1] "ordered" "factor"

R refers to the possible categories a factor can take as levels. We can learn the levels for FinDifficulty with a call to the function levels()

## [1] "Very difficult"       "Somewhat difficult"   "Not at all difficult"

We can see that there are three categories corresponding to individuals finding it very difficult, somewhat difficult, or not at all difficult to meet their monthly expenses.

We can call the table() function to learn how many observations fall into each of these categories.

## 
##       Very difficult   Somewhat difficult Not at all difficult 
##                 1082                 3859                 4834

If is it more useful to see these in percentages of the full data set, we can save the table to an object (which we call findiff.tb below) and use that object to create a proportions table with the function, prop.table().

## 
##       Very difficult   Somewhat difficult Not at all difficult 
##            0.1106905            0.3947826            0.4945269

We can see here that approximately 11.1% of the sample finds it very difficult to meet their expenses in a typical month, approximately 39.5% find it somewhat difficult, and 49.4% of people find it not at all difficult.

Instead of looking at a table, we can visualize it with a bar graph. The code below uses the ggplot() function to plot a proportion table like the one we created above. (Note: This is not the only way, nor possibly even the most common way of creating a bar graph with counts or percentages for categorical variables, but it aligns with the strategy in the subsequent section of this tutorial for visualizing cross tabulations.)

In the first line we again compute the proportions for the table information, but this time save the result in findiff.ptb.

In the next line we call the function as.data.frame() to convert findiff.ptb to a data frame object and call this data frame findiff.df. We do this because we need to use an object of class data.frame in our call to ggplot(). An object of class table will not work.

In the next line we set the variable names of our data frame to FinDifficulty and Frequency.

Finally, we call ggplot() to create our plot. We set the data layer equal to our newly constructed findiff.df. We set the aesthetics layer by mapping the variable FinDifficulty to the x-axis and Frequency to the y-axis. We add the geometry layer that creates the columns with geom_col().

1.3 Improving the Data Visualization

The plot is useful visualization to the researcher, but it should be cleaned up some if you wanted to present this to an audience.

First, we should change the labels on the vertical axis so that it is clear these proportions are meant to communicate a percentage. We can change the y-scale with a call to scale_y_continuous() and change the labels by appropriately setting the label parameter to the function.

The parameter percent is actually a function that is included in the scales package. It converts fractions to a percent and presents the result with a percent sign.

Next we give our graph some color by setting some optional parameters in geom_col(). We set fill="darkgreen" to change the color of the columns, and we set alpha=0.5 to make the green bars partially transparent. With partially transparent bars, we can see through the bars to see the grid if necessary.

Next, we add some labels with a call to the labs() function. We set the title, a subtitle, and a caption to credit our source. We also remove the labels for the x-axis and y-axis because it is obvious from our other descriptions what these axes mean.

We next call some theme functions. In the code below, we use a black-and-white theme with a call to theme_bw() to get rid of the gray background. If this visual is to be used in a PowerPoint presentation, we will want to make the text larger. We do this by calling the theme() function and configuring the title and axis text. Finally, we remove the vertical grid lines in a call to the theme() function by setting panel.grid.major.x to element_blank().

The code below recreates the same graph with a number of other measures to enhance it for effective visual communication.

Finally, let us add the frequencies to the top of the bars with a call to geom_text(). In the geom_text() call, the parameters x and y determine where on the plot the text will be placed, the parameter label determines what text will be written, the size parameter scales the size of the text (does not correspond perfectly to font sizes), and nudge_y tells geom_text() to move the text up just a little but above the top of the bar.

1.4 Examining the Data: Living Situation

The variable LivingSituation refers to who, if anyone, that the individual answering the survey lives with. It is a nominal categorical variable. We can confirm the class and learn the levels of the variable with calls to class() and levels().

## [1] "factor"
## [1] "Alone"               "With Parnter/Spouse" "With Parents"       
## [4] "With Roommates"

We can see that living situation includes living alone, living with a roommate, living with one’s parents, and living with a spouse.

In the code that follows we discover what proportion of our sample is in each living situation.

## 
##               Alone With Parnter/Spouse        With Parents 
##              0.2430              0.5698              0.0955 
##      With Roommates 
##              0.0917

We can make a bar plot for living situation like we did above.

We can configure this plot to make our communication more effective and visually appealing following the same strategies above.

2 Contingency Table

2.1 Computing the Contingency Table

A contingency table or cross-tabulation breaks both variables into its categories and reports how many observations in the data set fall into each paired subgroup. We can also use the function table() to create a contingency table. We create the contingency table and save the output to an object we call tb.

##                      
##                       Very difficult Somewhat difficult
##   Alone                          344                988
##   With Parnter/Spouse            463               2016
##   With Parents                   131                421
##   With Roommates                 144                434
##                      
##                       Not at all difficult
##   Alone                               1056
##   With Parnter/Spouse                 3140
##   With Parents                         322
##   With Roommates                       316

The table reveals that of the 10,000 individuals in the sample, there are 316 that are living alone and report meeting monthly expenses is typically very difficult, 991 individuals are living along and report meeting monthly expenses is somewhat difficult, etc.

It is usually more convenient to view a contingency tables in terms of row percentages or column percentages. Whether to choose the row depends on what makes more sense in your application and how you ordered your variables in the call to table().

We will take row percentages, that is, for each living situation, determine what percentage of those in that living situation finds meeting monthly expenses very difficult, somewhat difficult, or not at all difficult. In the code below, we call the prop.table() function, pass in our contingency table, and instruct the function to compute proportions long dimension 1. We use dimension 1 because LivingSituation is the first variable in the table() call. We save the output to an object we call tb.prob.

##                      
##                       Very difficult Somewhat difficult
##   Alone                    0.1440536          0.4137353
##   With Parnter/Spouse      0.0823990          0.3587827
##   With Parents             0.1498856          0.4816934
##   With Roommates           0.1610738          0.4854586
##                      
##                       Not at all difficult
##   Alone                          0.4422111
##   With Parnter/Spouse            0.5588183
##   With Parents                   0.3684211
##   With Roommates                 0.3534676

If we look along the first row, we see that of those that live alone, approximately 13.4% find it very difficult to meet monthly expenses, 42.1% of people find it somewhat difficult, and 44.5% of people do not find it difficult at all.

If you look in the next row, you can see the story is somewhat different for individuals living with a partner or spouse. Of those, only 8% find it very difficult to meet monthly expenses, 35.7% find it somewhat difficult, and a majority, 56.2%, do not find it difficult at all.

2.2 Testing for Independence: The Chi-Square Test of Independence

In this tutorial we describe the Chi-Square Test of Independence for determining statistical evidence for whether two categorical variables are related to one another.

Our casual examination of the contingency table in the previous subsection suggested that a person’s degree of financial difficulty (measured by FinDifficulty) may be related to one’s living situation (measured by LivingSituation).

We can formally test this with a call to chisq.test(). In the call below, we save the output to an object we call cht which we will use later in the tutorial.

## 
##  Pearson's Chi-squared test
## 
## data:  df$LivingSituation and df$FinDifficulty
## X-squared = 277.5, df = 6, p-value < 2.2e-16

The p-value for the test is less than \(2.2 \times 10^{-16}\) (which is machine-zero), which implies we have statistical evidence that there is a relationship between financial difficulty and living situation.

What remains now is to construct some visualizations that can effectively communicate what that relationship is.

3 Visualizing the Cross Tabulation Proportions

3.1 Converting Contingency Table to Data Frame

The ggplot() function requires a data frame for creating a plot, so we must first convert our contingency table from a class table to a class data.frame. In the code that follows, we convert the proportions table to a data frame with the function as.data.frame() and examine it with the function glimpse() from the tidyverse.

## Observations: 12
## Variables: 3
## $ Var1 <fct> Alone, With Parnter/Spouse, With Parents, With Roommates, A…
## $ Var2 <fct> Very difficult, Very difficult, Very difficult, Very diffic…
## $ Freq <dbl> 0.1440536, 0.0823990, 0.1498856, 0.1610738, 0.4137353, 0.35…

We can see that we have three variables, named Var1, Var2, and Freq. From the first few values, we can see that Var1 is the living situation, Var2 is the financial difficulty variable, and Freq is the proportion in our contingency table.

Let us rename these variables appropriately.

3.2 Bar Charts

We can make a very ugly stacked-bar chart with our new data frame. This kind of visualization is very common, but not very effective. Still, it makes a good start.

The first parameter to ggplot() is the data frame where are contingency table proportions are stored, tb.df. The second parameter calls aes() to set up the aesthetics layer. We map the x-axis to the variable Living Situation to give us the living situation categories along the bottom. We map the y-axis to Frequency so that the height of the bars are the proportions. The last aesthetic we set is fill, which is the color the bars are filled with. We map fill to the variable FinDifficulty so each category for financial difficulty is shown with a different color.

A stacked-bar chart stacks the bars on top of one another. This is often convenient when it makes intuitive sense to add up the responses within each x-axis category. In this case, it does makes sense that within each living situation, the proportions add up to 100% (or 1.00).

This stacked bar chart is convenient for comparing the bottom (blue) bars between living situation categories, or even the top (red) bars. It is more difficult to compare the lengths of the middle (green) bars, and it is extremely to eye the difference in the length of the different color bars within a category.

Instead of a stacked-bar chart, we can make a multiple-bar chart. We do this by setting the optional parameter, position in the geom_col() function. By setting position="dodge", we tell ggplot() to not let the columns stack on top of each other (dodge each other).

This makes it easier to compare heights of bars within a living situation. For example, we can see among people with partners or spouses, many more of them have no difficulty meeting monthly expenses (blue bar) than finding it somewhat difficult (green bar). That situation is reversed for people living with parents.

3.3 Line Plots

Generally, line plot are reserved for time-series variables, where time moves forward as you move to the right along the x-axis, and the y-axis shows the value the a variable takes over time.

However, time plots are often useful for comparing across categories, for multiple categorical variables. The code below creates another plot with many of the same aesthetics as above.

Instead of a fill aesthetic, we have a color aesthetic for the color of the line. We map FinDifficulty to color. We also have a group aesthetic that we also map to FinDifficulty. The group aesthetic is used to draw different lines, instead of having all the points connect is a single line. We want a separate line for each FinDifficulty category, so we set group=FinDifficulty.

Finally, instead of calling geom_col() to create columns, we call geom_line() to create lines.

This line plot is useful for a researcher, and with some work, can be useful to communicate to an audience. The red line on the bottom illustrates the proportions of people that find it very difficult to meet their monthly finances. You can see that these have lower proportions than the other two categories (somewhat and not-at-all). You can also follow it with ease to see with what living situations it is highest (living with roommates) and where it is lowest (living with a partner/spouse).

The blue line illustrates those who have no difficulty, you can see that with two living situations, the blue line is highest (alone and with partner/spouse), and with two situations is is the second most common (with parents and with roommates).

That the lines cross are not all horizontal, and that they sometimes cross, are visual clues that there is a relationship between living situation and financial difficulty. If there were no relationship, we would expect the lines to be of possibly different heights, but to not change height with each living situation category. If there were no relationship, then we should see approximately the same percentage of people having financial difficulty regardless of their living situation.

3.4 Improving the Data Visualization

Some Basics: Thicker Lines, Titles, and Labels

Let us make the lines thicker, make the y-axis labels percentages, and give some descriptive titles and labels:

Include the Chi-Square Test Results

In Section 2.2, we computed the Chi-Square Test of Independence and found out that there is statistically significance evidence of a relationship between financial difficulty and living situation. We saved the results in an object we called cht. In the code below, we remind ourselves of this object, and view the statistic object and p.value object with it.

## 
##  Pearson's Chi-squared test
## 
## data:  df$LivingSituation and df$FinDifficulty
## X-squared = 277.5, df = 6, p-value < 2.2e-16
## X-squared 
##  277.4957
## [1] 5.398428e-57

Let us report this statistical evidence in a subtitle of our graph. In the code below, we use the sprintf() function to create a formatted string. We save the string in an object that we call, sbtitle.

The first parameter to sprintf() is our text, which includes the format codes, %.1f and %.3f. The ‘%’ indicates what follows is going to be a format code, the ‘f’ in these codes means to print a floating point number (i.e. a number with one or more decimal places), the number after the decimal point indicates how many decimal places to show.

The numbers that are shown are the parameters that follow. The first number that will be shown rounded to one decimal point will be the result of cht.statistic. The second number that will be shown rounded to three decimal points will be the result of cht.p.value.

Let us look at the string we created.

## [1] "Chi-Square Test of Independence: Test Statistic = 277.5, P-value = 0.000"

Let us make this string our subtitle. We recreate the ggplot() code above, this time adding a parameter for subtitle in the labs() function.

Themes: Background and text-size

We can alter the theme so that the background is white instead of gray, and that the text is larger, which can be useful if the graph is to be used for presentation slides.

Avoid Legends if Possible

The legend takes up a lot of space that could be used to illustrate the data. It also takes time for the audience to move back and forth between a legend and a graph. This is especially problematic in oral presentations, but less problematic in written communication. While legends reveal important information, there may be more effective ways of communicating the same information.

Let us try to remove the legend, and embed the labels directly on the graph next to the lines with geom_text(). In the code below, the call to theme(legend.position="none") removes the legend, and geom_text() places this information on the graph.

That is not quite what we wanted. We have labels next to the lines, but more labels than we wanted. Let us look at the call to geom_text() in the last line and see if we understand what happened.

We set the aesthetics in the call to aes() within geom_text(). The parameters x=LivingSituation and y=Frequency tells geom_text() the location to place the labels. In this case, we place the labels at every point on the graph. The label we placed is given in the variable FinDifficulty.

This is almost what we want, but we only want each label to appear once. How about we put labels only near the end of the line, that is only near the “With Roommates” category.

Let us create a new data frame that is only for our labels. We will take our existing data frame tb.df and create from that a new data frame that only includes the rows for “With Roommates”.

##   LivingSituation        FinDifficulty Frequency
## 1  With Roommates       Very difficult 0.1610738
## 2  With Roommates   Somewhat difficult 0.4854586
## 3  With Roommates Not at all difficult 0.3534676

You can see that our new data frame includes only three rows, the rows where Living Situation="With Roomates".

Now, let us repeat the code to plot above, put this time, use only the data frame labels.df in the geom_text() call.

That looks closer to what we wanted. Let us make the text a little bigger, and nudge the text down a little bit (with nudge_y) and nudge the text to left a little bit (with nudge_x)

Change Colors

Finally, we can change the color scale if we are interested in doing so. It is best to avoid both red and green in the same visual for those who are colorblind. With a call to scale_color_manual(), we can manually set some colors. I obtained the color codes below using the colorpicker tool,

http://tristen.ca/hcl-picker/#/hlc/3/1/D67368/5997B3.

James M. Murray, Ph.D.
University of Wisconsin - La Crosse
R Tutorials for Applied Statistics