1. 1 Chapter 5 Two-Way Tables Associations Between Categorical Variables

2. 2 Quantitative variables  correlation [Ch 3] & regression [Ch 4] categorical variables  two- way tables of frequency counts [Ch 5] Associations between variables

3. 3 Two-Way Table of Counts R-by-C tables Variables AGE variable = column variable (3 levels) EDUCATION variable = row variable (4 levels) This is a 4-by-3 table

4. 4 Marginal Distributions Variables Row variable marginal totals 37,786 81,435 56,008 27,858 58,077 44,465 44,828 Column variable marginal distribution

5. 5 Marginal Percents Relative frequencies (%s) for each variable separately notDescriptive purposes only; does not address association Illustrative Example (Distribution of education level) –Statement: Describe the distribution of education levels in the population –Plan: Calculate marginal percents for row variable “EDUCATION”

6. 6 Marginal Percents Example % not completing HS =27,859 / 175,230 × 100% = 15.9% % completing HS =58,077 / 175,230 × 100% = 33.1% % with 1-3 yrs college =44,465 / 175,230 × 100% = 25.4% % with 4+ yrs college =44,828 / 175,230 × 100% = 25.6% Step 3: “Solve” Row totals Table total

7. 7 Marginal Percents (Example) 16% did not complete high school 33% completed high school 25% completed 1 to 3 years of college 26% completed 4+ years of college Step 4: “Conclude” Merely descriptive statements

8. 8 Association If the row variable is the explanatory variable → compare conditional row proportions If the column variable is the explanatory variable → compare conditional column proportions Use conditional proportions to determine associations

9. 9 Example: Association between AGE & EDUCATION State : Is AGE associated with EDUCATION level? Plan : Since AGE is the explanatory variable  calculate conditional column proportions. We do not need to calculate every conditional proportion. (Be selective.) Let us calculate the proportion completing 4+ years of college by AGE

10. 10 Example: “Solve” & “Conclude” Conclude: As age goes up, % completing college goes down  Negative association between age and college completion

11. 11 No association: conditional percents nearly equal at all levels of the explanatory variable Positive association: as explanatory variable rises  conditional percentages increase Negative associations: as explanatory variable rises  conditional percentages go down Direction of association

12. 12 State : Is ACCEPTANCE into UC Berkeley graduate school (response variable) associated with GENDER (explanatory variable)? Example: Gender bias? AcceptedNot accept.Total Male198162360 Female88112200 Total286274560 Plan : Since GENDER is the explanatory variable  calculate row percents (acceptance “rates” by gender); compare % accepted by GENDER

13. 13 Example: “Gender bias?” AcceptedNot acceptTotal Male198162360 Female88112200 Total286274560 Conclude : positive association between “maleness” and acceptance Step 3: Solve

14. 14 Simpson’s Paradox Lurking variable  MAJOR applied to –Business school major (240 applicants) –Art school major (320 applicants) State : Does lurking variable explain association between maleness and acceptance? Plan : Subdivide (“stratify”) data into subgroups according to lurking variable MAJOR  then calculate acceptance rates by gender within subgroups Simpson’s Paradox ≡ lurking variable reverses direction of the association

15. 15 “Gender Bias” Data by MAJOR Business School Applicants SuccessFailureTotal Male18102120 Female2496120 Total42198240 All Applicants SuccessFailureTotal Male198162360 Female88112200 Total286274560 Art School Applicants SuccessFailureTotal Male18060240 Female641680 Total24476320

16. 16 Business School Applicants SuccessFailureTotal Male18102120 Female2496120 Total42198240 Conclude: Negative association with maleness

17. 17 Art School Applicants SuccessFailureTotal Male 18060240 Female 641680 Total 24476320 Conclude: Negative association with maleness

18. 18 Overall: higher acceptance rate for men Within Business school: higher acceptance rate for women Within Art school: higher acceptance rate for women Therefore, the lurking variable (MAJOR) reversed the direction of the association (Simpson’s Paradox) Acceptance to grad school at UC Berkeley favored women after “controlling for” MAJOR Gender Bias Example Conclusion

19. 19 HIV vaccine boost HIV vaccine boost (Exercise 5.6) State: Do data support that vaccine delivered by EP results in a higher proportion responding? Plan = ? Solution = ? Conclusion = ?

20. Kidney Stones Kidney Stones (Exercise 5.7) Small Stones Open Surgery Percutan eous Success 81234 Failure 636 Large Stones Open Surgery Percutan eous Success 19255 Failure 7125 (a) Find % of kidney stones, combining the data for small and large stones, that were successfully removed for each of the two procedures. Which procedure had the higher overall success rate? (b) What % of all small kidney stones were successfully removed? What % of all large kidney stones…? Which type of kidney stone is easier to treat?

21. 21 “Helicopter Evacuation” X Evacuation type Helicopter or Road Y Outcome Survived or Died ? Statement Statement: Does helicopter evacuation of motor vehicle accidents save lives?

22. 22 Helicopter Evacuation Simpson’s Paradox Helicopter Evacuation Lurking Variable /Simpson’s Paradox X Helicopter or Road Y Survived or Died Z Accident Severity

23. 23 Helicopter Evacuation /Simpson’s Serious Accidents DiedSurvivedTotal Helicopter 4852100 Road 6040100 Non-serious Accidents DiedSurvivedTotal Helicopter1684100 Road 2008001000 Find the percent dying combining the data for serious and non-serious accidents. Which evacuation method had the higher death rate.

24. 24 Helicopter Evacuation Plan= ? Solve = ? Conclusion = ? DiedSurvived Total Helicopter 64136200 Road 2608401100 What % of serious accident