1. BPSChapter 61 Two-Way Tables

2. BPSChapter 62 To study associations between quantitative variables  correlation & regression (Ch 4 & Ch 5) To study associations between categorical variables  cross-tabulate frequencies & calculate conditional percents (this Chapter) Association

3. BPSChapter 63 Example: Age and Education Variables Marginal distributions “Age groups” is the categorical explanatory variable “Education level” is the categorical response variable

4. BPSChapter 64 Example: Marginal Totals Variables Marginal totals 37,786 81,435 56,008 27,858 58,077 44,465 44,828

5. BPSChapter 65 Marginal Distributions Marginal distributions are used as background information only. They do not address association

6. BPSChapter 66 Marginal Distribution, Row Variable % not completed HS =27,859 / 175,230 × 100% = 15.9% % graduated HS =58,077 / 175,230 × 100% = 33.1% % finished 1-3 yrs col. =44,465 / 175,230 × 100% = 25.4% % finished ≥4 yrs col. =44,828 / 175,230 × 100% = 25.6%

7. BPSChapter 67 Marginal Distribution, Column Variable % age 25–34 =37,786 / 175,230 × 100% = 21.6% % age 35–54 =81,435 / 175,230 × 100% = 46.5% % 55 and over =56,008 / 175,230 × 100% = 32.0%

8. BPSChapter 68 Association To determine associations, calculate conditional distributions (conditional percents) Two types of conditional distributions: Conditioned on row variable Conditioned on column variable

9. BPSChapter 69 Association If explanatory variable is in rows  calculate row percents  analyze row conditional distributions

10. BPSChapter 610 Association If explanatory variable is in columns  calculate column percents  analyze column conditional distribution

11. BPSChapter 611 Example: Column Percents Is AGE associated with EDUCATION? AGE is explanatory var.  use column percents

12. BPSChapter 612 Example: Association Percents completing college by age Age25-3435-5455+ % completed college 29.3%28.4%18.9% As age goes up, % completing college goes down NEGATIVE association between age and education

13. BPSChapter 613 No association: conditional percents nearly equal at all levels of explanatory variable Positive association: as explanatory variable rises  conditional percentages increase Negative associations: as explanatory variable rises  conditional percentages go down Association

14. BPSChapter 614 Statement of problem: Is ACCEPTANCE into a graduate program (response variable) predicted by GENDER (explanatory variable)? Example 2: Row Percent AcceptedNot accept.Total Male198162360 Female88112200 Total286274560 Explanatory variable (gender) is in rows  use row percents

15. BPSChapter 615 Example 2 AcceptedNot acceptTotal Male198162360 Female88112200 Total286274560 Explanatory variable in rows  use row percents Therefore: positive association with “maleness” Statement of problem: Is ACCEPTANCE associated with GENDER?

16. BPSChapter 616 Simpson’s Paradox In example 2, consider the lurking variable "major” –Business School (240 applicants) –Art School (320 applicants) Does this lurking variable explain the association? To address this potential problem, subdivide the data according to the lurking variable Lurking variables can change or even reverse the direction of an association

17. BPSChapter 617 Simpson’s Paradox Illustration Business School Applicants SuccessFailureTotal Male18102120 Female2496120 Total42198240 Male proportion = 18 / 120 = 0.15 Female prop. = 24 / 120 = 0.20 Negative association All Applicants SuccessFailureTotal Male198162360 Female88112200 Total286274560 Art School Applicants SuccessFailureTotal Male18060240 Female641680 Total24476320 Male proportion = 180 / 240 = 0.75 Female proportion = 64 / 80 = 0.80 Negative association

18. BPSChapter 618 Overall: higher proportion of men accepted than women Within majors  higher proportion of women accepted than men Reason  Men applied to easier majors  the initial association was an artifact of the lurking variable “MAJOR applied to” Simpson’s Paradox Illustration