Bivariate regression

1. Bivariate regression The slope, explained variance, residuals

3. What is the formula for a slope? • A. e = mc2 • B. Yi = a + bxi + ei • C. ŷ = a + bx • D. y ≥ x ≥ a

4. What information does the slope provide? • A. whether the relationship is statistically significant • B. whether a case is a severe outlier, like Buchanan’s share of the vote • C. on average, what is the predicted value of y, given various values of x • D. which baseball batter is likely to hit best in the next game

5. What is a? • A. the y intercept • B. the value of y when x = 0 • C. where the slope crosses the y a axis • D. all of the above

6. Bivariate Relationships Plotting a Line

7. Review: Covariance • When it tends to be the case that x is greater than the mean when y is greater than the mean AND x is lower than the mean when y is lower than the mean, then there is a positive covariation

8. Plot showing positive covariance

9. Expected value • But we may want to know more specific knowledge than that – we may want to know the expected value of y for each increased value of x • I may know the mean of everyone’s height in class • But if I know gender, then I can generate two expected values • If you remember, we are always trying to do better than the mean

10. Substantive effect • For every 10K dollars given in humanitarian aid, there is an increase in 3K spent on weapons • For every 10K dollars given in humanitarian aid, there is a .5K increase spent on weapons • For every 10K dollars given in humanitarian aid, there is a 8K increase spent on weapons • Unit of analysis?

11. Regression equation • y = a + bx + e • ŷ = a + bx • ŷ is also known as yhat • y is the dependent variable value • yhat is the predicted value • a is the intercept

12. X and Y • Y X • 2 1 • 2 • 4 3 • 3 4 • 6 5 • 5 6

13. X and Y • Y X • 2 1 • 2 • 4 3 • 3 4 • 6 5 • 5 6

14. Theory Living in an urban area allows better access to prenatal care.

15. Output Source SS df MS Number of obs = 41 F( 1, 39) = 9.09 Model 860.523694 1 860.523694 Prob > F = 0.0045 Residual 3693.55683 39 94.7065855 R-squared = 0.1890 Adj R-squared = 0.1682 Total 4554.08053 40 113.852013 Root MSE = 9.7317 prenatalcarepctCoef. Std. Err. t P>t [95% Conf. Interval] urbanpctoftotal.2517241 .083509 3.01 0.005 .0828111 .4206371 _cons 76.35186 4.367962 17.48 0.000 67.51682 85.18689

16. Linear Equation

17. ŷ= a + bx • b is slope – rise over run • a is the y intercept; constant • Standard error is the average error from the actual points to the slope • T is the ratio of the slope divided by the standard error • Beta = Pearson r in bivariate analysis

18. Other examples

19. Occupational Prestige and Education Dependent Variable: perceived prestige of occupation

20. Policy Liberalism and Public Opinion (Erikson, Wright & McIver, 1987) Q: Where does the diagonal line come from?

21. Extending Interval Correlations • Regression and correlation are intimately related concepts. • You’ve probably all tried to map lines in Cartesian spaces before…

23. Formula to find the slope of a line

27. Slopes

28. Policy Liberalism and Public Opinion (Erikson, Wright & McIver, 1987) Q: Where does the diagonal line come from?

29. Revisiting Erikson (1972)

31. 1. Compute the Variable Means

32. Mean of Votes Mean of Seats

33. 2a. Compute the Deviations - Votes

34. 2b. Compute the Deviations - Seats

35. The Regression Line Seats = -80.32 + 2.53*Votes

36. Estimating the magnitude of the effect from the slope • Find the range of x • (x may vary from 0-4, 1-5, 0-100000) • Multiply the range by the slope to find the effect of change in y, going from the lowest value of x to the highest value of x • If x ranges from 0-4, the range is 4 • If the slope is .25, the magnitude of the effect is 1