1. BPS - 3rd Ed. Chapter 61 Two-Way Tables

2. BPS - 3rd Ed. Chapter 62 u In this chapter we will study the relationship between two categorical variables (variables whose values fall in groups or categories). u To analyze categorical data, use the counts or percents of individuals that fall into various categories. Categorical Variables

3. BPS - 3rd Ed. Chapter 63 u 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 u The number of observations falling into each combination of categories is entered into each cell of the table Two-Way Table

4. BPS - 3rd Ed. Chapter 64 u 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) Marginal Distributions

5. BPS - 3rd Ed. Chapter 65 u 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 u A bar graph could be used to graphically display marginal distributions for categorical variables Marginal Distributions

6. BPS - 3rd Ed. Chapter 66 Case Study 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

7. BPS - 3rd Ed. Chapter 67 Case Study Age and Education Variables Marginal distributions

8. BPS - 3rd Ed. Chapter 68 Case Study Age and Education Variables Marginal distributions 21.6% 46.5% 32.0% 15.9% 33.1% 25.4% 25.6%

9. BPS - 3rd Ed. Chapter 69 Case Study Age and Education Marginal Distribution for Education Level Not HS grad15.9% HS grad33.1% College 1-3 yrs25.4% College ≥4 yrs25.6%

10. BPS - 3rd Ed. Chapter 610 u 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 Conditional Distributions

11. BPS - 3rd Ed. Chapter 611 Case Study 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.

12. BPS - 3rd Ed. Chapter 612 Case Study 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.

13. BPS - 3rd Ed. Chapter 613 Case Study 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%

14. BPS - 3rd Ed. Chapter 614 u The conditional distribution of one variable can be calculated for each category of the other variable. u These can be displayed using bar graphs. u 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. u If there are significant differences in the conditional distributions for each category, then we say that there is an association between the two variables. Conditional Distributions

15. BPS - 3rd Ed. Chapter 615 Case Study Age and Education Conditional Distributions of Age for each level of Education:

16. BPS - 3rd Ed. Chapter 616 u 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. u 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. Simpson’s Paradox

17. BPS - 3rd Ed. Chapter 617 Consider the acceptance rates for the following group of men and women who applied to college. Discrimination? (Simpson’s Paradox) counts Accepted Not accepted Total Men198162360 Women88112200 Total286274560 percents Accepted Not accepted Men55%45% Women44%56% A higher percentage of men were accepted: Discrimination?

18. BPS - 3rd Ed. Chapter 618 Discrimination? (Simpson’s Paradox) 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 Lurking variable: Applications were split between the Business School (240) and the Art School (320). BUSINESS SCHOOL

19. BPS - 3rd Ed. Chapter 619 Discrimination? (Simpson’s Paradox) 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 Lurking variable: Applications were split between the Business School (240) and the Art School (320).

20. BPS - 3rd Ed. Chapter 620 u So within each school a higher percentage of women were accepted than men. There is not any discrimination against women!!! u 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. Discrimination? (Simpson’s Paradox)