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Chapter 5

Chapter 5. Two-Way Tables Associations Between Categorical Variables. Associations between variables. Quantitative variables  correlation [Ch 3] & regression [Ch 4] categorical variables  two-way tables of frequency counts [Ch 5]. Variables.

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Chapter 5

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

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

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

  4. Variables Column variablemarginal distribution Marginal Distributions 27,85858,07744,46544,828 37,786 81,435 56,008 Row variablemarginal totals

  5. Relative frequencies (%s) for each variable separately Descriptive 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” Marginal Percents

  6. Marginal Percents Example Step 3: “Solve” Row totals Table total

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

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

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

  11. Direction of association • 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

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

  13. Example: “Gender bias?” Step 3: Solve Conclude: positive association between “maleness” and acceptance 13

  14. Simpson’s Paradox Simpson’s Paradox ≡ lurking variable reverses direction of the association • 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

  15. “Gender Bias” Data by MAJOR

  16. Business School Applicants Conclude: Negative association with maleness Conclude: Negative association with maleness 16

  17. Art School Applicants Conclude: Negative association with maleness 17

  18. Gender Bias Example Conclusion • 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

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

  20. Kidney Stones(Exercise 5.7) (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. Helicopter EvacuationLurking Variable /Simpson’s Paradox X Helicopter or Road Y Survived or Died Z Accident Severity 22

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