1. Prediction with multiple variables Statistics for the Social Sciences Psychology 340 Spring 2010

2. PSY 340 Statistics for the Social Sciences Outline Multiple regression –Comparing models, Delta r 2 –Using SPSS

3. PSY 340 Statistics for the Social Sciences Multiple Regression Typically researchers are interested in predicting with more than one explanatory variable In multiple regression, an additional predictor variable (or set of variables) is used to predict the residuals left over from the first predictor.

4. PSY 340 Statistics for the Social Sciences Multiple Regression Y = intercept + slope (X) + error Bi-variate regression prediction models

5. PSY 340 Statistics for the Social Sciences Multiple Regression Multiple regression prediction models “fit” “residual” Y = intercept + slope (X) + error Bi-variate regression prediction models

6. PSY 340 Statistics for the Social Sciences Multiple Regression Multiple regression prediction models First Explanatory Variable Second Explanatory Variable Fourth Explanatory Variable whatever variability is left over Third Explanatory Variable

7. PSY 340 Statistics for the Social Sciences Multiple Regression First Explanatory Variable Second Explanatory Variable Fourth Explanatory Variable whatever variability is left over Third Explanatory Variable Predict test performance based on: Study time Test time What you eat for breakfast Hours of sleep

8. PSY 340 Statistics for the Social Sciences Multiple Regression Predict test performance based on: Study time Test time What you eat for breakfast Hours of sleep Typically your analysis consists of testing multiple regression models to see which “fits” best (comparing r 2 s of the models) versus For example:

9. PSY 340 Statistics for the Social Sciences Multiple Regression Response variable Total variability it test performance Total study time r =.6 Model #1: Some co-variance between the two variables R 2 for Model =.36 64% variance unexplained If we know the total study time, we can predict 36% of the variance in test performance

10. PSY 340 Statistics for the Social Sciences Multiple Regression Response variable Total variability it test performance Test time r =.1 Model #2: Add test time to the model Total study time r =.6 R 2 for Model =.49 51% variance unexplained Little co-variance between these test performance and test time We can explain more the of variance in test performance

11. PSY 340 Statistics for the Social Sciences Multiple Regression Response variable Total variability it test performance breakfast r =.0 Model #3: No co-variance between these test performance and breakfast food Total study time r =.6 Test time r =.1 R 2 for Model =.49 51% variance unexplained Not related, so we can NOT explain more the of variance in test performance

12. PSY 340 Statistics for the Social Sciences Multiple Regression Response variable Total variability it test performance breakfast r =.0 We can explain more the of variance But notice what happens with the overlap (covariation between explanatory variables), can’t just add r’s or r 2 ’s Total study time r =.6 Test time r =.1 Hrs of sleep r =.45 R 2 for Model =.60 40% variance unexplained Model #4: Some co-variance between these test performance and hours of sleep

13. PSY 340 Statistics for the Social Sciences Multiple Regression in SPSS Setup as before: Variables (explanatory and response) are entered into columns A couple of different ways to use SPSS to compare different models

14. PSY 340 Statistics for the Social Sciences Regression in SPSS Analyze: Regression, Linear

15. PSY 340 Statistics for the Social Sciences Multiple Regression in SPSS Method 1: enter all the explanatory variables together –Enter: All of the predictor variables into the Independent Variable field Predicted (criterion) variable into Dependent Variable field

16. PSY 340 Statistics for the Social Sciences Multiple Regression in SPSS The variables in the model r for the entire model r 2 for the entire model Unstandardized coefficients Coefficient for var1 (var name) Coefficient for var2 (var name)

17. PSY 340 Statistics for the Social Sciences Multiple Regression in SPSS The variables in the model r for the entire model r 2 for the entire model Standardized coefficients Coefficient for var1 (var name)Coefficient for var2 (var name)

18. PSY 340 Statistics for the Social Sciences Multiple Regression –Which β to use, standardized or unstandardized? –Unstandardized β’s are easier to use if you want to predict a raw score based on raw scores (no z-scores needed). –Standardized β’s are nice to directly compare which variable is most “important” in the equation

19. PSY 340 Statistics for the Social Sciences Multiple Regression in SPSS Predicted (criterion) variable into Dependent Variable field First Predictor variable into the Independent Variable field Click the Next button Method 2: enter first model, then add another variable for second model, etc. –Enter:

20. PSY 340 Statistics for the Social Sciences Multiple Regression in SPSS Method 2 cont: –Enter: Second Predictor variable into the Independent Variable field Click Statistics

21. PSY 340 Statistics for the Social Sciences Multiple Regression in SPSS –Click the ‘R squared change’ box

22. PSY 340 Statistics for the Social Sciences Multiple Regression in SPSS The variables in the first model (math SAT) Shows the results of two models The variables in the second model (math and verbal SAT)

23. PSY 340 Statistics for the Social Sciences Multiple Regression in SPSS The variables in the first model (math SAT) r 2 for the first model Coefficients for var1 (var name) Shows the results of two models The variables in the second model (math and verbal SAT) Model 1

24. PSY 340 Statistics for the Social Sciences Multiple Regression in SPSS The variables in the first model (math SAT) Coefficients for var1 (var name) Coefficients for var2 (var name) Shows the results of two models r 2 for the second model The variables in the second model (math and verbal SAT) Model 2

25. PSY 340 Statistics for the Social Sciences Multiple Regression in SPSS The variables in the first model (math SAT) Shows the results of two models The variables in the second model (math and verbal SAT) Change statistics: is the change in r 2 from Model 1 to Model 2 statistically significant?

26. PSY 340 Statistics for the Social Sciences Hypothesis testing with Regression Multiple Regression “residual” “fit” –We can test hypotheses about the overall model

27. PSY 340 Statistics for the Social Sciences Multiple Regression in SPSS Null Hypotheses H 0 : University GPA is not predicted by SAT verbal or SAT Math scores p < 0.05, so reject H 0, SAT math and verbal predict University GPA

28. PSY 340 Statistics for the Social Sciences Hypothesis testing with Regression First Explanatory Variable Second Explanatory Variable Fourth Explanatory Variable Third Explanatory Variable Multiple Regression –We can test hypotheses about each of these explanatory hypotheses within a regression model So it’ll tell us whether that variable is explaining a “significant”amount of the variance in the response variable –We can test hypotheses about the overall model

29. PSY 340 Statistics for the Social Sciences Multiple Regression in SPSS Null Hypotheses H 0 : Coefficient for var1 = 0 p < 0.05, so reject H 0, var1 is a significant predictor H 0 : Coefficient for var2 = 0 p > 0.05, so fail to reject H 0, var2 is a not a significant predictor

30. PSY 340 Statistics for the Social Sciences Hypothesis testing with Regression Multiple Regression –We can test hypotheses about each of these explanatory hypotheses within a regression model So it’ll tell us whether that variable is explaining a “significant”amount of the variance in the response variable –We can also use hypothesis testing to examine if the change in r 2 is statistically significant –We can test hypotheses about the overall model

31. PSY 340 Statistics for the Social Sciences Hypothesis testing with Regression The variables in the first model (math SAT) r 2 for the first model Coefficients for var1 (var name) Shows the results of two models The variables in the second model (math and verbal SAT) Model 1

32. PSY 340 Statistics for the Social Sciences Hypothesis testing with Regression The variables in the first model (math SAT) Coefficients for var1 (var name) Coefficients for var2 (var name) Shows the results of two models r 2 for the second model The variables in the second model (math and verbal SAT) Model 2

33. PSY 340 Statistics for the Social Sciences Hypothesis testing with Regression The variables in the first model (math SAT) Shows the results of two models The variables in the second model (math and verbal SAT) Change statistics: is the change in r 2 from Model 1 to Model 2 statistically significant? The 0.002 change in r 2 is not statistically significant (p = 0.46) The 0.002 change in r 2 is not statistically significant (p = 0.46)

34. PSY 340 Statistics for the Social Sciences Regression in Research Articles Bivariate prediction models rarely reported Multiple regression results commonly reported

35. PSY 340 Statistics for the Social Sciences Cautions in Multiple Regression We can use as many predictors as we wish but we should be careful not to use more predictors than is warranted. –Simpler models are more likely to generalize to other samples. –If you use as many predictors as you have participants in your study, you can predict 100% of the variance. Although this may seem like a good thing, it is unlikely that your results would generalize to any other sample and thus they are not valid. –You probably should have at least 10 participants per predictor variable (and probably should aim for about 30).