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 (R 2 )

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 + b 1 X 1 + b 2 X 2  a = the Y intercept (Formula 17.6)  b 1 =the partial slope of X1 on Y (Formula 17.4)  b 2 =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 =  b 1 = 2.09  b 2 = -.43

Standardized Partial Slopes (beta-weights)  Partial slopes (b 1 and b 2 ) 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 X 1  Use Formula 17.8 to calculate the beta-weight for X 2

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 (R 2 )  The multiple correlation coefficient (R 2 ) shows the combined effects of all independent variables on the dependent variable.

Multiple Correlation (R2)  Formula allows X1 to explain as much of Y as it can and then adds in the effect of X2 after X1 is controlled.  Formula eliminates the overlap in the explained variance between X1 and X2.

Multiple Correlation (R 2 )  Zero order correlation between unemployment (X 1 ) and turnout. r =.95 X 1 explains 90% (r 2 =.90) of the variation in Y by itself.

Multiple Correlation (R 2 )  Zero order correlation between neg. advert. (X 2 ) and turnout. r = -.87 X 2 explains 76% (r 2 =.76) of the variation in Y by itself.

Multiple Correlation (R 2 )  Unemployment (X 1 ) explains 90% (r 2 =.90) of the variance by itself. R 2 =.98  To this, negative advertising (X 2 ) adds 8% for a total of 98%.