Chapter 17

Partial Correlation and Multiple Regression and Correlation

Chapter Outline

Introduction Partial Correlation Multiple Regression: Predicting the

Dependent Variable Multiple Regression: Assessing the

Effects of the Independent Variables

Chapter Outline

Multiple Correlation Interpreting Statistics: Another Look

at the Correlates of Crime The Limitations of Multiple Regression

and Correlation

In This Presentation

Multiple regression Using the multiple regression line to

predict Y Multiple correlation (R2)

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.

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)

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 X’s 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).

Formulas for Partial Slopes Formula 17.4

Formula 17.5

Formula for a Formula 17.6

Regression Coefficients for Problem 17.1

The Y intercept (a)

Partial slopes:

a = 70.25

b1 = 2.09 b2 = -.43

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.

Beta-weights Use Formula 17.7

to calculate the beta-weight for X1

Use Formula 17.8 to calculate the beta-weight for X2

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.

Multiple Correlation (R2)

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

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.

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.

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.

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%.