Reading 3.3 Case C->C

Learning Objectives

Produce a two-way table, and interpret the information stored in it about the association between two categorical variables by comparing conditional percentages.

Case C→C: Two Categorical Variables

Recall the role-type classification table for framing our discussion about the relationship between two variables:

		Response
		Categorical	Quantitative
Explanatory	Categorical	C→C	√C→Q
Explanatory	Quantitative	Q→C	Q→Q

We are done with case C→Q, and will now move on to case C→C, where we examine the relationship between two categorical variables.

Earlier in the course, (when we discussed the distribution of a single categorical variable) we examined the data obtained when a random sample of 1,200 U.S. college students were asked about their body image (underweight, overweight, or about right.) We are now returning to this example, to address the following question:

If we had separated our sample of 1,200 U.S. college students by gender and looked at males and females separately, would we have found a similar distribution across body-image categories? More specifically, are men and women just as likely to think their weight is about right? Among those students who do not think their weight is about right, is there a difference between the genders in feelings about body image?

Answering these questions requires us to examine the relationship between two categorical variables, gender and body image. Because the question of interest is whether there is a gender effect on body image,

the explanatory variable is gender, and
the response variable is body image.

Here is what the raw data look like when we include the gender of each student:

A table of the data. There are three columns, "Student", "Gender", and "Body Image". "Gender" is the Explanatory variable, and "Body Image" is the Response variable. Some example data: ... (abbreviated) ... student 25, M, overweight; student 26, M, about right; student 27, F, underweight; student 28, F, about right; student 29, M, about right; ... (abbreviated) ...

Once again the raw data is a long list of 1,200 genders and responses, and thus not very useful in that form. To start our exploration of how body image is related to gender, we need an informative display that summarizes the data. In order to summarize the relationship between two categorical variables, we create a display called a two-way table.

Here is the two-way table for our example:

		Body Image
		About right	Overweight	Underweight	Total
Gender	Female	560	163	37	760
	Male	295	72	73	440
	Total	855	235	110	1200

The table has the possible genders in the rows, and the possible responses regarding body image in the columns. At each intersection between row and column, we put the counts for how many times that combination of gender and body image occurred in the data. We sum across the rows to fill in the Total column, and we sum across the columns to fill in the Total row.

Make sure you can identify all of the following in the table:

The number of females in our sample who feel that they are underweight.
The number of males in our sample who feel that they are about right.
The total number of females in our sample.
The total number of students who feel they are overweight.

Comments

Note that from the way the two-way table is constructed, the Total row or column is a summary of one of the two categorical variables, ignoring the other. In our example:

The Total row gives the summary of the categorical variable body image:

Body Image

About right Overweight Underweight Total

Gender Female 560 163 37 760

Male 295 72 73 440

Total 855 235 110 1200

(These are the same counts we got earlier in the course when we looked at the single categorical variable body image, and did not consider gender.)
The Total column gives the summary of the categorical variable gender:

Body Image

About right Overweight Underweight Total

Gender Female 560 163 37 760

Male 295 72 73 440

Total 855 235 110 1200

Analyzing the Relationship

So far, we have organized the raw data in a much more informative display—the two-way table:

		Body Image
		About right	Overweight	Underweight	Total
Gender	Female	560	163	37	760
	Male	295	72	73	440
	Total	855	235	110	1200

Remember, though, that our primary goal is to explore how body image is related to gender. Exploring the relationship between two categorical variables (in this case, body image and gender) amounts to comparing the distributions of the response variable (in this case body image) across the different values of the explanatory variable (in this case, males and females). So we'll compare the distribution in the female row with the distribution in the male row:

		Body Image
		About right	Overweight	Underweight	Total
Gender	Female	560	163	37	760
	Male	295	72	73	440
	Total	855	235	110	1200

Note that it doesn't make sense to compare raw counts, because there are more females than males overall. So, for example, it is not very informative to say, "There are 560 females who responded 'about right' compared to only 295 males," since the 560 females are out of a total of 760, and the 295 males are out of a total of only 440.

We need to supplement our display, the two-way table, with some numerical summaries that will allow us to compare the distributions. These numerical summaries are found by simply converting the counts to percentages within (or restricted to) each value of the explanatory variable separately.

In our example, we look at each gender separately and convert the counts to percentages within that gender. Let's start with females:

		Body Image
		About right	Overweight	Underweight	Total
Gender	Female	560/760= 73.7%	163/760= 21.5%	37/760= 4.9%	760/760= 100%
Gender	Male	%	%	%	%

Note that each count is converted to percents by dividing by the total number of females, 760. These numerical summaries are called conditional percentages, since we find them by "conditioning" on one of the genders.

Let's try finishing the table together:

Questions

Fill in the percentages in the table for the distribution of men's body image.

		Body Image
		About right	Overweight	Underweight	Total
Gender	Female	560/760= 73.7%	163/760= 21.5%	37/760= 4.9%	760/760= 100%
Gender	Male	%	%	%	%

Answers

		Body Image
		About right	Overweight	Underweight	Total
Gender	Female	560/760= 73.7%	163/760= 21.5%	37/760= 4.9%	760/760= 100%
Gender	Male	295/440= 67%	72/440= 16.4%	73/440= 16.6%	440/440= 100%

Comments

In our example, we chose to organize the data with the explanatory variable gender in rows and the response variable body image in columns, and thus our conditional percentages were row percentages, calculated within each row separately. Similarly, if the explanatory variable happens to sit in columns and the response variable in rows, our conditional percentages will becolumn percentages, calculated within each column separately. For an example, see the "Did I Get This?" exercises below.
Another way to visualize the conditional percentages, instead of in a table, is to use a double bar chart. This display is quite common in newspapers.

Now that we have summarized the relationship between the categorical variables gender and body image, here's how we could interpret the results in the context of the question we posed:

The results suggest that the proportion of males who are happy with their body weight is slightly less than among female students. Females who are not happy with their body weight more often feel they are overweight. Males who are not happy with their body weight feel they are about as often as they feel they are underweight.

Summary

The relationship between two categorical variables is summarized using
- Data display: Two-way table, supplemented by
- Numerical summaries: Conditional percentages.
Conditional percentages are calculated for each value of the explanatory variable separately. They can be row percentages if the explanatory variable "sits" in the rows, or column percentages if the explanatory variable "sits" in the columns.
When we try to understand the relationship between two categorical variables, we compare the distributions of the response variable for values of the explanatory variable. In particular, we look at how the pattern of conditional percentages differs between the values of the explanatory variable.

When you finish reading any text that's detailed and dense, like math text usually is, it makes sense to write down some notes about what you learned! Even if you've been taking notes as you read, now's a good time to jot down a summary at the end of what the most important points in this reading were.

About This Text

This text is from the Probability + Statistics text produced by Carnegie-Mellon University's Open Learning Initiative, with some modifications by Clare Heimer. It's shared under a Creative Commons Attribution-NonCommercial-ShareAlike license.