Chapter 15

1. Chapter 15 Association Between Variables Measured at the Interval-Ratio Level

2. Chapter Outline • Interpreting the Correlation Coefficient: r 2 • The Correlation Matrix • Testing Pearson’s r for Significance • Interpreting Statistics: The Correlates of Crime

3. Scattergrams • Scattergrams have two dimensions: • The X (independent) variable is arrayed along the horizontal axis. • The Y (dependent) variable is arrayed along the vertical axis.

4. Scattergrams • Each dot on a scattergram is a case. • The dot is placed at the intersection of the case’s scores on X and Y.

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

6. 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.

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

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

9. 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.

10. 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.

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

12. 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

13. Regression Analysis • Before using the formula for the regression line, a and b must be calculated. • Compute b first, using Formula 15.3 (we won’t do any calculation for this chapter)

14. Regression Analysis • The Y intercept (a) is computed from Formula 15.4:

15. Regression Analysis • For the relationship between % college educated and turnout: • b (slope) = .42 • a (Y intercept)= 50.03 • Regression formula: Y = 50.03 + .42 X • 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.

16. 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? • This is a positive relationship so the value for Y increases as X increases: • For X =10, Y = 50.3 +.42(10) = 54.5 • For X =70, Y = 50.3 + .42(70) = 79.7

17. Pearson correlation coefficient • But of course, this is just an estimate of turnout based on % college educated, and many other factors also affect voter turnout. • How much of the variation in voter turnout depends on % college educated? The relevant statististic is the coefficient of determination (r squared), but first we need to learn about Pearson’s correlation coefficient (r).

18. Pearson’s r • Pearson’s r is a measure of association for I-R variables. • It varies from -1.0 to +1.0 • Relationship may be positive (as X increases, Y increases) or negative (as X increases, Y decreases) • For the relationship between % college educated and turnout, r =.32. • The relationship is positive: as level of education increases, turnout increases. • How strong is the relationship? For that we use R squared, but first, let’s look at the calculation process

19. Example of Computation • The computation and interpretation of a, b, and Pearson’s r will be illustrated using Problem 15.1. • The variables are: • Voter turnout (Y) • Average years of school (X) • The sample is 5 cities. • This is only to simplify computations, 5 is much too small a sample for serious research.

20. Example of Computation • The scores on each variable are displayed in table format: • Y = Turnout • X = Years of Education

21. Example of Computation • Sums are needed to compute b, a, and Pearson’s r.

22. Interpreting Pearson’s r • An r of 0.98 indicates an extremely strong relationship between average years of education and voter turnout for these five cities. • The coefficient of determination is r2 = .96. Knowing education level improves our prediction of voter turnout by 96%. This is a PRE measure (like lambda and gamma) • We could also say that education explains 96% of the variation in voter turnout.

23. Interpreting Pearson’s r • Our first example provides a more realistic value for r. • The r between turnout and % college educated for the 50 states was: • r = .32 • This is a weak to moderate, positive relationship. • The value of r2 is .10.Percent college educated explains 10% of the variation in turnout.