INTERPRETING REGRESSION COEFFICIENTS

1
INTERPRETING REGRESSION COEFFICIENTS
2
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

3
BACK TO BASIC CONCEPTS PRE (E1 E2)/E1 1
E2/E1 E1 S(Y Y)2 Rule for predicting
values of Y, given knowledge of X Yhati a
bXi
4
E2 S (Yi Y)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 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
6

• FOCUSING ON FORM
• As given by equation Yi 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

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 r2,
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 ß 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,
ß r
10
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
11
(No Transcript)
12
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 .0013 setting confidence bands
at .113 and .140 F for equation 350.91, p lt
0.000
13
Scatterplot N Democracies by Year
14
(No Transcript)
15
Interpreting the Equation

• 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

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 Y
260.6 - 22.97 X r - 0.843 Whats the
interpretation?