When we study relationships between two variables, one of the first questions we ask is whether each variable is quantitative or categorical. For two quantitative variables, we use a scatterplot to examine the relationship, and we fit a line to the data if the relationship is approximately linear. If one of the variables is quantitative and the other is categorical, we can use the methods in Chapter 1 to describe the distribution of the quantitative variable for each value of the categorical variable. This leaves us with the situation where both variables are categorical. In this section, we discuss methods for studying these relationships.

Some variables—such as sex, race, and occupation—are inherently categorical. Other categorical variables are created by grouping values of a quantitative variable into classes. Published data are often reported in grouped form to save space. To describe categorical data, we use the counts (frequencies) or percents (relative frequencies) of individuals that fall into various categories.

The two-way table

A key idea in studying relationships between two variables is that both variables must be measured on the same individuals or cases. When both variables are categorical, the raw data are summarized in a two-way tabletwo-way table that gives counts of observations for each combination of values of the two categorical variables. Here is an example.

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For the calcium requirement example, we could view age as an explanatory variable and “met requirement” as a response variable. This is why we put age in the columns (like the x axis in a scatterplot) and “met requirement” in the rows (like the y axis in a scatterplot). We call “met requirement” the row variablerow variable because each horizontal row in the table describes whether or not the requirement was met. Age is the column variablecolumn variable because each vertical column describes one age group. Each combination of values for these two variables is called a cellcell. For example, the cell corresponding to children who are 5 to 10 years old and who have not met the requirement contains the number 194. This table is called a 2 × 2 table because there are two rows and two columns.

To describe relationships between two categorical variables, we compute different types of percents. Our job is easier if we expand the basic two-way table by adding various totals. We illustrate the idea with our calcium requirement example.

In this example, be sure that you understand how the table is obtained from the raw data. Think about a data file with one line per subject. There would be 2029 lines or records in this data set. In the two-way table, each individual is counted once and only once. As a result, the sum of the counts in the table is the total number of individuals in the data set. Most errors in the use of categorical-data methods come from a misunderstanding of how these tables are constructed.

Joint distribution

We are now ready to compute some proportions that help us understand the data in a two-way table. Suppose that we are interested in the children aged 5 to 10 years who do not meet the calcium requirement. The proportion of these is simply 194 divided by 2029, or 0.0956. We would estimate that 9.56% of children in the population from which this sample was drawn are 5- to 10-year-olds who do not meet the calcium requirement. For each cell, we can compute a proportion by dividing the cell entry by the total sample size. The collection of these proportions is the joint distributionjoint distribution of the two categorical variables.

How might we use the information in the joint distribution for this example? Suppose that we were to develop an outreach unit to increase the consumption of calcium. The distribution suggests that the older students should be targeted if we have to make a choice because of limited funds. Children who are 11 to 13 years old and do not meet the calcium requirement are 27.45% of the total; however, children who are 5 to 10 years old and do not meet the requirement are only 9.56% of the total. For other uses of these data, we may need to calculate different numerical summaries. Let’s look at the distribution of age.

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Marginal distributions

When we examine the distribution of a single variable in a two-way table, we are looking at a marginal distributionmarginal distribution. There are two marginal distributions, one for each categorical variable in the two-way table. They are very easy to compute.

Often, we prefer to use percents rather than proportions. Here is the marginal distribution of age described with percents:

Which form do you prefer?

The percent of children in each age group is approximately the same. This is interesting because the first category includes six ages (5, 6, 7, 8, 9, and 10); whereas the second includes only three ages (11, 12, and 13). Recall that the age categories were chosen in this way because the Institute of Medicine defined the calcium requirement differently for these age groups. In this study, the children were selected from grades 4, 5, and 6. The distribution of ages within these grades explains the marginal distribution of age for our sample.

The other marginal distribution for this example is the distribution of “met requirement.”

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Each marginal distribution from a two-way table is a distribution for a single categorical variable. We can use a bar graph or a pie chart to display such a distribution. For our two-way table, we will be content with numerical summaries: for example, 52% of the children are aged 5 to 10, and 37% of the children are not meeting their calcium requirement. When we have more rows or columns, the graphical displays are particularly useful.

Describing relations in two-way tables

The table in Example 2.34 contains much more information than the two marginal distributions of age alone and “met requirement” alone. We need to do a little more work to examine the relationship. Relationships among categorical variables are described by calculating appropriate percents from the counts given. What percents do you think we should use to describe the relationship between age and meeting the calcium requirement?

Conditional distributions

In Example 2.38, we looked at the children aged 5 to 10 alone and examined the distribution of the other categorical variable, “met requirement.” Another way to say this is that we conditioned on the value of age, 5 to 10 years old. Similarly, we can condition on the value of age being 11 to 13 years old. When we condition on the value of one variable and calculate the distribution of the other variable, we obtain a conditional distributionconditional distribution. Note that in Example 2.38, we calculated only the percent for children aged 5 to 10 years. The complete conditional distribution gives the proportions or percents for all possible values of the conditioning variable.

Comparing the conditional distributions (Example 2.39 and Exercise 2.118) reveals the nature of the association between age and meeting the calcium requirement. In this set of data, the older children are more likely to fail to meet the calcium requirement.

Bar graphs can help us to see relationships between two categorical variables. No single graph (such as a scatterplot) portrays the form of the relationship between categorical variables, and no single numerical measure (such as the correlation) summarizes the strength of an association. Bar graphs are flexible enough to be helpful, but you must think about what comparisons you want to display. For numerical measures, we must rely on well-chosen percents or on more advanced statistical methods.²⁵

A two-way table contains a great deal of information in compact form. Making that information clear almost always requires finding percents. You must decide which percents you need. Of course, we prefer to use software to compute the joint, marginal, and conditional distributions.

Most software packages order the row and column labels numerically or alphabetically. In general, it is better to use words rather than numbers for the column labels. This sometimes involves some additional work, but it avoids the kind of confusion that can result when you forget the real values associated with each numerical value. You should verify that the entries in Figure 2.28 correspond to the calculations that we performed in Examples 2.34 through 2.39. In addition, verify the calculations for the conditional distributions of age for each value of “met requirement.”

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The JMP output in Figure 2.28 includes a graphical display of the data called a mosaic plotmosaic plot. The sizes of the four boxes display the joint distribution. The narrow bar to the right shows the marginal distribution of Met and the widths of the vertical bars show the marginal distribution of age. The conditional distribution of Met for each Age is represented in each of these vertical bars by the heights of the blue and red sections. Notice that they always add to one.

Simpson’s paradox

As is the case with quantitative variables, the effects of lurking variables can strongly influence relationships between two categorical variables. Here is an example that demonstrates the surprises that can await the unsuspecting consumer of data.

Let’s look at the data in a little more detail. The data summarized come from two different weeks in the year.

These results can be explained by a lurking variable, Week. The first week was during a period when the product had been in use for several months. Most of the calls to the customer service center concerned problems that had been encountered before. The representatives were trained to answer these questions and usually had no trouble in meeting the goal of resolving the problems quickly. On the other hand, the second week occurred shortly after the release of a new version of the product. Most of the calls during this week concerned new problems that the representatives had not yet encountered. Many more of these questions took longer than the 10-minute goal to resolve.

Look at the totals in the bottom row of the detailed table. During the first week, when calls were easy to resolve, Alexis handled 180 calls and Peyton handled 20. The situation was exactly the opposite during the second week, when the calls were difficult to resolve. There were 20 calls for Alexis and 180 for Peyton.

The original two-way table, which did not take account of week, was misleading. This example illustrates Simpson’s paradox.

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The lurking variables in our Simpson’s paradox example, Week and problem difficulty, are categorical. That is, they break the observations into groups by workweek. Simpson’s paradox is an extreme form of the fact that observed associations can be misleading when there are lurking variables.

The data in Example 2.42 are given in a three-way tablethree-way table that reports counts for each combination of three categorical variables: week, representative, and whether or not the goal was met. In our example, we constructed the three-way table by constructing two two-way tables for representative by goal, one for each week. The original table in Example 2.41 can be obtained by adding the corresponding counts for these two tables. This process is called aggregatingaggregation the data. When we aggregated data in Example 2.41, we ignored the variable week, which then became a lurking variable. Conclusions that seem obvious when we look only at aggregated data can become quite different when the data are examined in more detail.