1. Fundamental Statistics for the Behavioral Sciences, 5th edition David C. Howell Chapter 11 Multiple Regression © 2003 Brooks/Cole Publishing Company/ITP

2. 2Chapter 11 Multiple Regression Major Points The problemThe problem An exampleAn example Multiple correlationMultiple correlation Regression equationRegression equation PredictionsPredictions Cont.

3. 3Chapter 11 Multiple Regression Major Points--cont. ResidualsResiduals Hypothesis testingHypothesis testing Review questionsReview questions

4. 4Chapter 11 Multiple Regression The Problem Using several predictors to predict the dependent variableUsing several predictors to predict the dependent variable Finding a measure of overall fitFinding a measure of overall fit Weighting each predictorWeighting each predictor

5. 5Chapter 11 Multiple Regression An Example Study by Kliewer et al. (1998) on effect of violence on internalizing behaviorStudy by Kliewer et al. (1998) on effect of violence on internalizing behavior XDefine internalizing behavior PredictorsPredictors XDegree of witnessing violence XMeasure of life stress XMeasure of social support

6. 6Chapter 11 Multiple Regression Violence and Internalizing Subjects are children 8-12 yearsSubjects are children 8-12 years XLived in high-violence areas XHypothesis: violence and stress lead to internalizing behavior. Data available atData available at Xwww.uvm.edu/~dhowell/StatPages/ More_Stuff/Kliewer.dat www.uvm.edu/~dhowell/StatPages/ More_Stuff/Kliewer.datwww.uvm.edu/~dhowell/StatPages/ More_Stuff/Kliewer.dat

7. 7Chapter 11 Multiple Regression Intercorrelation Matrix

8. 8Chapter 11 Multiple Regression Preliminary Stuff Note that both Stress and Witnessing Violence are significantly correlated with Internalizing.Note that both Stress and Witnessing Violence are significantly correlated with Internalizing. Note that predictors are largely independent of each other.Note that predictors are largely independent of each other.

9. 9Chapter 11 Multiple Regression Multiple Correlation Directly analogous to simple rDirectly analogous to simple r Always capitalized (e.g. R)Always capitalized (e.g. R) Always positiveAlways positive XCorrelation of with observed Y where is computed from regression equationwhere is computed from regression equation XOften reported as R 2 instead of R

10. 10Chapter 11 Multiple Regression R 2R 2R 2R 2

11. 11Chapter 11 Multiple Regression Regression Coefficients Slopes and an intercept.Slopes and an intercept. Each variable adjusted for all others in the model.Each variable adjusted for all others in the model. Just an extension of slope and intercept in simple regressionJust an extension of slope and intercept in simple regression SPSS output on next slideSPSS output on next slide

12. 12Chapter 11 Multiple Regression Slopes and Intercept

13. 13Chapter 11 Multiple Regression Regression Equation A separate coefficient for each variableA separate coefficient for each variable XThese are slopes An intercept (here called b 0 instead of a)An intercept (here called b 0 instead of a)

14. 14Chapter 11 Multiple RegressionInterpretation Note slope for Witness and Stress are positive, but slope for Social Support is negative.Note slope for Witness and Stress are positive, but slope for Social Support is negative. XDoes this make sense? If you had two subjects with identical Stress and SocSupp, a one unit increase in Witness would produce 0.038 unit increase in Internal.If you had two subjects with identical Stress and SocSupp, a one unit increase in Witness would produce 0.038 unit increase in Internal. Cont.

15. 15Chapter 11 Multiple RegressionInterpretation--cont. The same holds true for other predictors.The same holds true for other predictors. t test on two slopes are significantt test on two slopes are significant XSocSupp not significant. XElaborate R 2 has same interpretation as r 2.R 2 has same interpretation as r 2. X13.5% of variability in Internal accounted for by variability in Witness, Stress, and SocSupp.

16. 16Chapter 11 Multiple RegressionInterpretation--cont. Intercept usually not meaningful.Intercept usually not meaningful. XPrediction when all predictors are 0.0

17. 17Chapter 11 Multiple RegressionPredictions Assume Witness = 20, Stress = 5, and SocSupp = 35.Assume Witness = 20, Stress = 5, and SocSupp = 35.

18. 18Chapter 11 Multiple Regression Hypothesis Testing Test on R 2 given in Analysis of Variance tableTest on R 2 given in Analysis of Variance table Cont.

19. 19Chapter 11 Multiple RegressionTesting--cont. Tests on regression coefficients given along with the coefficients.Tests on regression coefficients given along with the coefficients. See next slideSee next slide Note tests on each coefficient.Note tests on each coefficient.

20. 20Chapter 11 Multiple Regression Testing Slopes and Intercept

21. 21Chapter 11 Multiple Regression Review Questions How does multiple regression differ from simple regression?How does multiple regression differ from simple regression? Can R 2 decrease as you add predictors?Can R 2 decrease as you add predictors? What do we mean by “controlling for?”What do we mean by “controlling for?” How do we calculate our prediction?How do we calculate our prediction? Is prediction even an important issue in the violence example?Is prediction even an important issue in the violence example? Cont.

22. 22Chapter 11 Multiple Regression Review Questions--cont. Is it likely that a slope would be significant if the overall R is not?Is it likely that a slope would be significant if the overall R is not? Give an example where multiple regression might help you to understand behavior.Give an example where multiple regression might help you to understand behavior.