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Chapter 6

Chapter 6. Two-Way Tables . Association. 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). Variables.

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Chapter 6

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  1. Chapter 6 Two-Way Tables Chapter 6

  2. Association • 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) Chapter 6

  3. Variables Example: Age and Education “Age groups” is the categorical explanatory variable “Education level” is the categorical response variable Marginal distributions Chapter 6

  4. Variables 27,85858,07744,46544,828 37,786 81,435 56,008 Marginal totals Example: Marginal Totals Chapter 6

  5. Marginal Distributions Marginal distributions are used as background information only. They do not address association Chapter 6

  6. Marginal Distribution, Row Variable Chapter 6

  7. Marginal Distribution, Column Variable BPS Chapter 6 7

  8. Association To determine associations, calculate conditional distributions (conditional percents) Two types of conditional distributions: Conditioned on row variable Conditioned on column variable BPS Chapter 6 8

  9. Association • If explanatory variable is in rows • calculate row percents • analyze row conditional distributions Chapter 6

  10. Association • If explanatory variable is in columns • calculate column percents • analyze column conditional distribution BPS Chapter 6 10

  11. Example: Column Percents Is AGE associated with EDUCATION? AGE is explanatory var.  use column percents Chapter 6

  12. Example: Association As age goes up, % completing college goes down NEGATIVE association between age and education Chapter 6

  13. Association • 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 Chapter 6

  14. Statement of problem: Is ACCEPTANCE into a graduate program (response variable) predicted by GENDER (explanatory variable)? Example 2: Row Percent Explanatory variable (gender) is in rows  use row percents BPS Chapter 6 14

  15. Example 2 Statement of problem: Is ACCEPTANCE associated with GENDER? Explanatory variable in rows  use row percents Therefore: positive association with “maleness” BPS Chapter 6 15

  16. Simpson’s Paradox Lurking variables can change or even reverse the direction of an association • 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 Chapter 6

  17. Simpson’s Paradox Illustration Chapter 6

  18. Simpson’s Paradox Illustration • 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” Chapter 6

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