9. Venn diagram shows (R 2 ) the amount of variance in Y that is explained by X. Unexplained Variance in Y. (1-R 2 ) =.36, 36% R 2 =.64 (64%) Variance in Y that is explained by X

10. Multiple Regression Multiple regression is used to predict one Y (dependent) variable from two or more X (independent) variables. The advantage of multivariate or bivariate regression is – Provides lower standard error of estimate – Determines which variables contribute to the prediction and which do not.

11. Multiple Regression b 1, b 2, b 3, … b n are coefficients that give weight to the independent variables according to their relative contribution to the prediction of Y. X 1, X 2, X 3, … X n are the predictors (independent variables). C is a constant, similar to Y intercept. Body Fat = Abdominal + Tricep + Thigh

12. List the variables and order to enter into the equation 1.X 2 has biggest area (C), it comes in first. 2.X 1 comes in next area (A) is bigger than area (E). Both A and E are unique, not common to C. 3.X 3 comes in next, it uniquely adds area (E). 4.X 4 is not related to Y so it is NOT in the equation. 5.The first variable in is the variable with the highest correlation with Y. After that, it is the next variable with the highest Part Corr with Y.

13. Ideal Relationship Between Predictors and Y Each variable accounts for unique variance in Y Very little overlap of the predictors Order to enter? X 1, X 3, X 4, X 2, X 5

14. Guess the Order of Entry in Stepwise Advertising BudgetNo of Plays on RadioAttractiveness of the Band Record Sales.578.599.326 Advertising Budget.102.081 No of Plays on Radio.182 Step 1: Record Sales = No of Plays Step 2: Record Sales = No of Plays + Ad Budget Step 3: Record Sales = No of Plays + Ad Budget + Attractiveness

28. Multicollinearity

31. Multicollinearity was not violated if the following were met: (1)The largest VIF < 10 (2)the average VIF of predictors was not substantially > 1 (3)tolerance statistics > 0.2.

32. Record3.sav Outlier?

33. Is the Variance Equal?

34. Non Linear Regression Y = X 3 – 3X 2 + 4 Regress1.sav

38. Y = 1.515 (X) + 1.49 R 2 = 26.2%, SEE = 3.68

39. The Distribution is Not Normal

40. The Relationship between Xs and Y is Not Linear

41. Compute X Squared and X Cubed

42. Non Linear Regression Y = X 3 – 3X 2 + 4

43. Non-Linear Regression Output

44. Non-Linear Regression Output Y = X 3 – 3X 2 + 4 Y = X 3 – 3X 2 + 4 R 2 = 100%, SEE =.000000002

45. Assignment Assignment 1. Use the data set Regress3.sav to run a stepwise regression. Y = X1 + X2 + X3 + X4 After completing the linear regression compute the following non-linear terms: X3 2 and X4 3, then run stepwise regression with the following predictors: Y = X2 + X3 2 + X4 3 Assignment 2. Muscle Activation.sav The purpose of this experiment was to predict the time of muscle activation from fiber type and nerve conduction velocity. Fiber type was identified as slow, intermediate and fast twitch. Negative values for muscle activation indicate that the subject activated their muscles prior to ground contact and positive values indicated that they activated their muscles after ground contact. Use stepwise multiple regression. Check for outliers and remove any rows containing a standardized residual greater than ± 3. Check for, normality, homogeneity of variance and multicollinearity.