1. Chapter 6 Two-Way Tables BPS - 5th Ed.Chapter 61

2. Categorical Variables In this chapter we will study the relationship between two categorical variables (variables whose values fall in groups or categories). To analyze categorical data, use the counts or percents of individuals that fall into various categories. BPS - 5th Ed.Chapter 62

3. Two-Way Table When there are two categorical variables, the data are summarized in a two-way table ◦ each row in the table represents a value of the row variable ◦ each column of the table represents a value of the column variable The number of observations falling into each combination of categories is entered into each cell of the table BPS - 5th Ed.Chapter 63

4. Two-Way Table Data from Wissahickon High School and their AP course involvement What are the two categorical data pieces? ◦ # of AP courses ◦ Year in school BPS - 5th Ed.Chapter 64

5. Marginal Distributions A distribution for a categorical variable tells how often each outcome occurred ◦ totaling the values in each row of the table gives the marginal distribution of the row variable (totals are written in the right margin) ◦ totaling the values in each column of the table gives the marginal distribution of the column variable (totals are written in the bottom margin) BPS - 5th Ed.Chapter 65

6. Marginal Distributions It is usually more informative to display each marginal distribution in terms of percents rather than counts ◦ each marginal total is divided by the table total to give the percents A bar graph could be used to graphically display marginal distributions for categorical variables BPS - 5th Ed.Chapter 66

7. Case Study BPS - 5th Ed.Chapter 67 Data from the U.S. Census Bureau for the year 2000 on the level of education reached by Americans of different ages. (Statistical Abstract of the United States, 2001) Age and Education

8. Case Study BPS - 5th Ed.Chapter 68 Age and Education Variables Marginal distributions

9. Case Study BPS - 5th Ed.Chapter 69 Age and Education Variables Marginal distributions 21.6% 46.5% 32.0% 15.9% 33.1% 25.4% 25.6%

10. Case Study BPS - 5th Ed.Chapter 610 Age and Education Marginal Distribution for Education Level Not HS grad15.9% HS grad33.1% College 1-3 yrs25.4% College ≥4 yrs25.6%

11. Conditional Distributions Relationships between categorical variables are described by calculating appropriate percents from the counts given in the table ◦ prevents misleading comparisons due to unequal sample sizes for different groups BPS - 5th Ed.Chapter 611

12. Case Study BPS - 5th Ed.Chapter 612 Age and Education Compare the 25-34 age group to the 35- 54 age group in terms of success in completing at least 4 years of college: Data are in thousands, so we have that 11,071,000 persons in the 25-34 age group have completed at least 4 years of college, compared to 23,160,000 persons in the 35-54 age group. The groups appear greatly different, but look at the group totals.

13. Case Study BPS - 5th Ed.Chapter 613 Age and Education Compare the 25-34 age group to the 35-54 age group in terms of success in completing at least 4 years of college: Change the counts to percents: Now, with a fairer comparison using percents, the groups appear very similar.

14. Case Study BPS - 5th Ed.Chapter 614 Age and Education If we compute the percent completing at least four years of college for all of the age groups, this would give us the conditional distribution of age, given that the education level is “completed at least 4 years of college”: Age:25-3435-5455 and over Percent with ≥ 4 yrs college: 29.3%28.4%18.9%

15. Conditional Distributions The conditional distribution of one variable can be calculated for each category of the other variable. These can be displayed using bar graphs. If the conditional distributions of the second variable are nearly the same for each category of the first variable, then we say that there is not an association between the two variables. If there are significant differences in the conditional distributions for each category, then we say that there is an association between the two variables. BPS - 5th Ed.Chapter 615

16. Case Study BPS - 5th Ed.Chapter 616 Age and Education Conditional Distributions of Age for each level of Education:

17. Simpson’s Paradox When studying the relationship between two variables, there may exist a lurking variable that creates a reversal in the direction of the relationship when the lurking variable is ignored as opposed to the direction of the relationship when the lurking variable is considered. The lurking variable creates subgroups, and failure to take these subgroups into consideration can lead to misleading conclusions regarding the association between the two variables. BPS - 5th Ed.Chapter 617

18. Discrimination? (Simpson’s Paradox) Consider the acceptance rates for the following group of men and women who applied to college. BPS - 5th Ed.Chapter 618 counts Accepted Not accepted Total Men198162360 Women88112200 Total286274560 percents Accepted Not accepted Men55%45% Women44%56% A higher percentage of men were accepted: Discrimination?

19. Discrimination? (Simpson’s Paradox) Lurking variable: Applications were split between the Business School (240) and the Art School (320). BPS - 5th Ed.Chapter 619 counts Accepted Not accepted Total Men18102120 Women2496120 Total42198240 percents Accepted Not accepted Men15%85% Women20%80% A higher percentage of women were accepted in Business BUSINESS SCHOOL

20. Discrimination? (Simpson’s Paradox) Lurking variable: Applications were split between the Business School (240) and the Art School (320). BPS - 5th Ed.Chapter 620 counts Accepted Not accepted Total Men18060240 Women641680 Total24476320 percents Accepted Not accepted Men75%25% Women80%20% ART SCHOOL A higher percentage of women were also accepted in Art

21. Discrimination? (Simpson’s Paradox) So within each school a higher percentage of women were accepted than men. There is not any discrimination against women!!! This is an example of Simpson’s Paradox. When the lurking variable (School applied to: Business or Art) is ignored the data seem to suggest discrimination against women. However, when the School is considered the association is reversed and suggests discrimination against men. BPS - 5th Ed.Chapter 621