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Week 9: Chapter 15, 17 (and 16)

Week 9: Chapter 15, 17 (and 16). Association Between Variables Measured at the Interval-Ratio Level The Procedure in Steps. Step 1: Make Scattergrams and Regression Lines. Scattergrams have two dimensions: The X (independent) variable is arrayed along the horizontal axis.

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Week 9: Chapter 15, 17 (and 16)

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  1. Week 9: Chapter 15, 17 (and 16) Association Between Variables Measured at the Interval-Ratio Level The Procedure in Steps

  2. Step 1: Make Scattergrams and Regression Lines • Scattergrams have two dimensions: • The X (independent) variable is arrayed along the horizontal axis. • The Y (dependent) variable is arrayed along the vertical axis. • Each dot on a scattergram is a case. • The dot is placed at the intersection of the case’s scores on X and Y.

  3. Scattergrams • Shows the relationship between % College Educated (X) and Voter Turnout (Y) on election day for the 50 states.

  4. Scattergrams • Horizontal X axis - % of population of a state with a college education. • Scores range from 15.3% to 34.6% and increase from left to right.

  5. Scattergrams • Vertical (Y) axis is voter turnout. • Scores range from 44.1% to 70.4% and increase from bottom to top

  6. Scattergrams: Regression Line • A single straight line that comes as close as possible to all data points. • Indicates strength and direction of the relationship.

  7. Scattergrams:Strength of Regression Line • The greater the extent to which dots are clustered around the regression line, the stronger the relationship. • This relationship is weak to moderate in strength.

  8. Scattergrams: Direction of Regression Line • Positive: regression line rises left to right. • Negative: regression line falls left to right. • This a positive relationship: As % college educated increases, turnout increases.

  9. Scattergrams • Inspection of the scattergram should always be the first step in assessing the correlation between two I-R variables

  10. The Regression Line: Formula • This formula defines the regression line: • Y = a + bX • Where: • Y = score on the dependent variable • a = the Y intercept or the point where the regression line crosses the Y axis. • b = the slope of the regression line or the amount of change produced in Y by a unit change in X • X = score on the independent variable

  11. Regression Analysis • Before using the formula for the regression line, a and b must be calculated. • Compute b first, use Formula 15.3 (see Healey p. 401):

  12. Regression Analysis • The Y intercept (a) is computed from Formula 15.4 (see Healey p. 402):

  13. Regression Analysis • For the relationship between % college educated and turnout: • b (slope) = .42 • a (Y intercept)= 50.03 • A slope of .42 means that turnout increases by .42 (less than half a percent) for every unit increase of 1 in % college educated. • The Y intercept means that the regression line crosses the Y axis at Y = 50.03. • The regression line here is: Y = 50.03 + .42X

  14. Exercise: Predicting Y • What turnout would be expected in a state where only 10% of the population was college educated? • What turnout would be expected in a state where 70% of the population was college educated?

  15. Step 2: What is the pattern/direction of the association? • See results of step 1 • Focus on the b: a slope of .42 means that turnout increases by .42 (less than half a percent) for every unit increase of 1 in % college educated. • This a positive relationship: As % college educated increases, turnout increases.

  16. Step 3: How Strong is the Relationship? • See results of step 1 • The greater the extent to which dots are clustered around the regression line, the stronger the relationship • This relationship between education and voter turnout is weak to moderate in strength • Pearson’s r is a measure of association for I-R variables.

  17. Pearson’s r • Calculation Pearson’s r: Formula 15.5 and 15.6 (see Healey pp. 403-404). • For the relationship between % college educated and turnout, r =.32. • This relationship is positive and weak to moderate. • As level of education increases, turnout increases.

  18. Step 4: Is the Strength of the Association Significant? • Testing Pearson’s r for significance • See Chapter 15 of Healey (pp. 412-413) for five-step model for test to find out whether the strength of the association between the variables is significant or not

  19. Step 5: What is the r2? • The value of r2 is .10. • Interpretation • Percent college educated explains 10% of the variation in turnout • 10% of the variance in turnout is explained by education

  20. Step 6: Is there still an Association, if Control Variables are Added? • See Chapter 16 in Healey • See week 10 of this course

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