1. INTERPRETING REGRESSION COEFFICIENTS

2. OUTLINE 1.Back to Basics 2.Form: The Regression Equation 3.Strength: PRE and r 2 4.The Correlation Coefficient r 5.Significance: Looking Ahead 6.Example 1: Democracy in Latin America 7.Example 2: Wine Consumption and Heart Disease

3. BACK TO BASIC CONCEPTS PRE = (E 1 – E 2 )/E 1 = 1 – E 2 /E 1 E 1 = Σ(Y – Y) 2 Rule for “predicting” values of Y, given knowledge of X: Yhat i = a + bX i

4. E 2 = Σ (Y i – Ŷ) 2 that is, sum of squared differences between observed values of Y and predicted values of Y (values of Y as “predicted” by the regression equation) Thus the elements of PRE.

5. STRENGTH OF ASSOCIATION Symbol = r 2 = PRE = (E 1 – E 2 )/E 1 = (total variance – unexplained variance)/total variance Varies from 0 to 1 Some back-of-the-envelope thresholds: 0.10, 0.30, 0.50+

6. FOCUSING ON FORM As given by equation Ŷ i = a + bX i Constant a = intercept = predicted value of Y when X = 0 Coefficient b = slope = average change in Y for change in X Magnitude (large or small) Sign (positive or negative) Key to much interpretation

7. Linear Regression Equation

8. THE CORRELATION COEFFICIENT Symbol = r Summary statement of form (from sign) and indirect statement of strength r = square root of r 2, varies from –1 to +1 subject to over-interpretation useful for preliminary assessment of association Symmetrical no matter which variable is X and which is Y (note: slope b is not symmetrical)

9. ON THE CORRELATION COEFFICIENT r Analogous to slope b (with removal of intercept a) The “standardized regression coefficient,” or beta weight: β= b (stand.dev. X/stand.dev. Y) employs slope, values, and dispersion of variables thus a “standardized” slope Question: How much action on Y do you get from X? In bivariate (or “simple”) regression, β = r

10. LOOKING AHEAD: MEASURING SIGNIFICANCE 1. Testing the null hypothesis: F = r 2 (n-2)/(1-r 2 ) 2. Standard errors and confidence intervals: Dependent on desired significance level Bands around the regression line 95% confidence interval ±1.96 x SE

12. Coefficients for Regression of N Electoral Democracies (Y) on Change Over Time (X): a = -1.427 b = +.126 r = +.883 r 2 =.780, Adjusted r 2 =.777 Standard error of slope =.0067 95% confidence interval for slope = (.0067)x1.96 = ±.0013 setting confidence bands at.113 and.140 F for equation = 350.91, p < 0.000

13. Scatterplot: N Democracies by Year

15. N democracies = - 1.427 +.126 year intercept = nonsense, but allows calculation of year that predicted value of Y would be zero, in this case 1910 slope = +.126 so, one additional democracy every eight years and by 2000, total 11-12 democracies PRE =.777 Interpreting the Equation

16. Example 2: Wine and Heart Disease Data in Lectures 5-6 X = per capita annual consumption of alcohol from wine, in liters Y = deaths from heart disease, per 100,000 people Equation: Ŷ = 260.6 - 22.97 X r = - 0.843 What’s the interpretation?