1. Education 793 Class Notes Multiple Regression 19 November 2003

2. Today’s Agenda Class and lab announcements –Draft of final paper to reviewer on the 15 th –Reviews and final version to instructors on the 19 th Your questions? Multiple Regression

3. Purpose: To help researcher predict some dependent variable from a set of predictors (X 1, X 2, X 3 …X n ) Based on prior research and theory, researchers are able to build comparative models

4. Review: Basic Equation for One Independent Variable Regression equation tells us that for every one unit increase in x, there is a b increase in y In analogous fashion, with k independent predictors: where and

5. Regression Coefficients a is the y intercept b’s are partial regression coefficients –A partial regression coefficient shows the relationship between the dependent variable and one independent variable controlling for the other independent variables in the model

6. Multiple Correlation Coefficient We can estimate the magnitude of the relationship between the dependent variable and the best linear combination of independent variables. R-multiple correlation coefficient ranges from 0 to 1. It is the correlation between the dependent variable and the predicted values from the regression equation

7. R2R2 The square of the multiple correlation coefficient is the proportion of variation in Y accounted for by the set of independent variables

8. Tests of Significance Ho: Is there a systematic relationship between the dependent variable and the set of predictors? The formal F-test compares the proportion of variance predictable by the X’s to the proportion that is unpredictable by the X’s. It is an omnibus test, it does not test the predictors individually.

9. Tests of Significance Besides the omnibus test, we have t- tests for each independent predictor individually. The t-test is the test of Ho:  =0 –If the p-value <.05, then the independent variable is a significant predictor of the dependent variable

10. Design Requirements There is one dependent variable and two or more independent variables that are correlated to the dependent Minimum sample size is approximately 50 and a general rule is that there should be at least 10 cases for every independent variable in the model

11. Assumptions Subjects are independent Dependent variable is normally distributed Constant variance across the range of predictor values The relationship between X and Y is linear

12. Example: Gender Effect on Predicting SAT Sample from 1998 University of Michigan CIRP Dependent Variable: SAT Verbal Independent Variables –HSGPA –Sex –Academic Rating Ability

13. Descriptive Statistics HSPGA is an 8 category ordinal variable that we treated as interval Sex is a 2 category variable 1=male, 2=female

14. Correlation Matrix What do we notice here?

15. ANOVA Table What do we notice here?

16. Test of the Coefficients Is Sex a significant predictor of SAT Verbal? Is HSGPA a significant predictor of SAT Verbal? What is the effect of HSGPA on SAT Verbal

17. Laptop Exercise Using the Cirp98, Run a multiple regression with a continuous dependent variable. –Chose 3-4 independent variables –Be ready to interpret your results

18. Next Week Enjoy Your Thanksgiving Break Please let the great bird live, choose a vegetarian alternative