INTERPRETING REGRESSION

COEFFICIENTS

OUTLINE

1. Back to Basics

2. Form: The Regression Equation

3. Strength: PRE and r2

4. The Correlation Coefficient r

5. Significance: Looking Ahead

6. Example 1: Democracy in Latin America

7. Example 2: Wine Consumption and Heart Disease

BACK TO BASIC CONCEPTS

PRE = (E1 – E2)/E1 = 1 – E2/E1

E1 = Σ(Y – Y)2

Rule for “predicting” values of Y, given knowledge of X:

Yhati = a + bXi

E2 = Σ (Yi – 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.

STRENGTH OF ASSOCIATION

Symbol = r2 = PRE = (E1 – E2)/E1

= (total variance – unexplained variance)/total variance

Varies from 0 to 1

Some back-of-the-envelope thresholds:

0.10, 0.30, 0.50+

FOCUSING ON FORM

As given by equation Ŷi = a + bXi

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

Linear Regression Equation

THE CORRELATION COEFFICIENT

Symbol = r

Summary statement of form (from sign) and indirect statement of strength

r = square root of r2, varies from –1 to +1

subject to over-interpretation

useful for preliminary assessment of association

Symmetrical no matter which variable is X andwhich is Y (note: slope b is not symmetrical)

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

LOOKING AHEAD: MEASURING SIGNIFICANCE

1. Testing the null hypothesis:

F = r2(n-2)/(1-r2)

2. Standard errors and confidence intervals:

Dependent on desired significance level

Bands around the regression line

95% confidence interval ±1.96 x SE

Figure 1. Cycles of Political Change in Latin America, 1900-2000

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

1900 1905 1910 1915 1920 1925 1930 1935 1940 1945 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000

Year

Nu

mb

er Semi-Democracy

Oligarchy

Democracy

Coefficients for Regression of N Electoral Democracies (Y)on Change Over Time (X):

a = -1.427

b = +.126

r = + .883

r2 = .780, Adjusted r2 = .777

Standard error of slope = .0067

95% confidence interval for slope = (.0067)x1.96 = ± .0013setting confidence bands at .113 and .140

F for equation = 350.91, p < 0.000

Scatterplot: N Democracies by Year

• 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 democracyevery eight years

• and by 2000, total 11-12 democracies• PRE = .777

Interpreting the Equation

Example 2: Wine and Heart Disease

Data in Lectures 5-6

X = per capita annual consumption of alcohol from wine, in litersY = deaths from heart disease, per 100,000 people

Equation:

Ŷ = 260.6 - 22.97 X

r = - 0.843

What’s the interpretation?