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Education 793 Class Notes Multiple Regression 19 November 2003 Today’s Agenda • Class and lab announcements – Draft of final paper to reviewer on the 15th – Reviews and final version to instructors on the 19th • Your questions? • Multiple Regression Multiple Regression • Purpose: To help researcher predict some dependent variable from a set of predictors (X1, X2, X3…Xn) • Based on prior research and theory, researchers are able to build comparative models Review: Basic Equation for One Independent Variable yˆ a bx where a y bx and b rxy sy sx 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: yˆ a b1 X 1 b2 X 2 ... bk X k Regression Coefficients yˆ a b1 X 1 b2 X 2 ... bk X k • 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 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 R2 • The square of the multiple correlation coefficient is the proportion of variation in Y accounted for by the set of independent variables 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. Tests of Significance • Besides the omnibus test, we have ttests for each independent predictor individually. • The t-test is the test of Ho: b=0 – If the p-value < .05, then the independent variable is a significant predictor of the dependent variable 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 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 Example: Gender Effect on Predicting SAT • Sample from 1998 University of Michigan CIRP • Dependent Variable: SAT Verbal • Independent Variables – HSGPA – Sex – Academic Rating Ability Descriptive Statistics HSPGA is an 8 category ordinal variable that we treated as interval Sex is a 2 category variable 1=male, 2=female Correlation Matrix • What do we notice here? ANOVA Table • What do we notice here? 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 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 Next Week • Enjoy Your Thanksgiving Break Please let the great bird live, choose a vegetarian alternative