Partial Correlation and Multiple Regression and Correlation

1
Chapter 17

• Partial Correlation and Multiple Regression and
  Correlation

2
Chapter Outline

• Introduction
• Partial Correlation
• Multiple Regression Predicting the Dependent
  Variable
• Multiple Regression Assessing the Effects of the
  Independent Variables

3
Chapter Outline

• Multiple Correlation
• Interpreting Statistics Another Look at the
  Correlates of Crime
• The Limitations of Multiple Regression and
  Correlation

4
In This Presentation

• Multiple regression
• Using the multiple regression line to predict Y
• Multiple correlation (R2)

5
Introduction

• Multiple Regression and Correlation allow us to
• Disentangle and examine the separate effects of
  the independent variables.
• Use all of the independent variables to predict
  Y.
• Assess the combined effects of the independent
  variables on Y.

6
Multiple Regression

• Y a b1X1 b2X2
• a the Y intercept (Formula 17.6)
• b1 the partial slope of X1 on Y (Formula 17.4)
• b2 the partial slope of X2 on Y (Formula 17.5)

7
Partial Slopes

• The partial slopes the effect of each
  independent variable on Y while controlling for
  the effect of the other independent variable(s).
• Show the effects of the Xs in their original
  units.
• These values can be used to predict scores on Y.
• Partial slopes must be computed before computing
  a (the Y intercept).

8
Formulas for Partial Slopes

• Formula 17.4
• Formula 17.5

9
Formula for a

• Formula 17.6

10
Regression Coefficients for Problem 17.1

• The Y intercept (a)
• Partial slopes

• a 70.25

• b1 2.09
• b2 -.43

11
Standardized Partial Slopes(beta-weights)

• Partial slopes (b1 and b2) are in the original
  units of the independent variables.
• To compare the relative effects of the
  independent variables, compute beta-weights (b).
• Beta-weights show the amount of change in the
  standardized scores of Y for a one-unit change in
  the standardized scores of each independent
  variable while controlling for the effects of all
  other independent variables.

12
Beta-weights

• Use Formula 17.7 to calculate the beta-weight for
  X1
• Use Formula 17.8 to calculate the beta-weight for
  X2

13
Beta-weights for Problem 17.1

• The Beta-weights show that the independent
  variables have roughly similar but opposite
  effects.
• Turnout increases with unemployment and decreases
  with negative advertising.

14
Multiple Correlation (R2)

• The multiple correlation coefficient (R2) shows
  the combined effects of all independent variables
  on the dependent variable.

15
Multiple Correlation (R2)

• Formula 17.11 allows X1 to explain as much of Y
  as it can and then adds in the effect of X2 after
  X1 is controlled.
• Formula 17.11 eliminates the overlap in the
  explained variance between X1 and X2.

16
Multiple Correlation (R2)

• Zero order correlation between unemployment (X1)
  and turnout.
• r .95
• X1 explains 90 (r2 .90) of the variation in Y
  by itself.

17
Multiple Correlation (R2)

• Zero order correlation between neg. advert. (X2)
  and turnout.
• r - .87
• X2 explains 76 (r2 .76) of the variation in Y
  by itself.

18
Multiple Correlation (R2)

• Unemployment (X1) explains 90(r2 .90) of the
  variance by itself.
• R2 .98
• To this, negative advertising (X2) adds 8 for a
  total of 98.