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Multiple Regression I

Multiple Regression I. KNNL – Chapter 6. Models with Multiple Predictors. Most Practical Problems have more than one potential predictor variable Goal is to determine effects (if any) of each predictor, controlling for others Can include polynomial terms to allow for nonlinear relations

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Multiple Regression I

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  1. Multiple Regression I KNNL – Chapter 6

  2. Models with Multiple Predictors • Most Practical Problems have more than one potential predictor variable • Goal is to determine effects (if any) of each predictor, controlling for others • Can include polynomial terms to allow for nonlinear relations • Can include product terms to allow for interactions when effect of one variable depends on level of another variable • Can include “dummy” variables for categorical predictors

  3. First-Order Model with 2 Numeric Predictors

  4. Interpretation of Regression Coefficients • Additive: E{Y} = b0 + b1X1 + b2X2 ≡ Mean of Y @ X1, X2 • b0 ≡ Intercept, Mean of Y when X1=X2=0 • b1 ≡ Slope with Respect to X1 (effect of increasing X1 by 1 unit, while holding X2 constant) • b2 ≡ Slope with Respect to X2 (effect of increasing X2 by 1 unit, while holding X1 constant) • These can also be obtained by taking the partial derivatives of E{Y} with respect to X1 and X2, respectively • Interaction Model: E{Y} = b0 + b1X1 + b2X2 + b3X1X2 • When X2 = 0: Effect of increasing X1 by 1: b1(1)+b3(1)(0)= b1 • When X2 = 1: Effect of increasing X1 by 1: b1(1)+b3(1)(1)= b1+b3 • The effect of increasing X1 depends on level of X2, and vice versa

  5. General Linear Regression Model

  6. Special Types of Variables/Models - I • p-1 distinct numeric predictors (attributes) • Y = Sales, X1=Advertising, X2=Price • Categorical Predictors – Indicator (Dummy) variables, representing m-1 levels of a m level categorical variable • Y = Salary, X1=Experience, X2=1 if College Grad, 0 if Not • Polynomial Terms – Allow for bends in the Regression • Y=MPG, X1=Speed, X2=Speed2 • Transformed Variables – Transformed Y variable to achieve linearity Y’=ln(Y) Y’=1/Y

  7. Special Types of Variables/Models - II • Interaction Effects – Effect of one predictor depends on levels of other predictors • Y = Salary, X1=Experience, X2=1 if Coll Grad, 0 if Not, X3=X1X2 • E(Y) = b0 + b1X1 + b2X2 + b3X1X2 • Non-College Grads (X2 =0): • E(Y) = b0 + b1X1 + b2(0) + b3X1(0)= b0 + b1X1 • College Grads (X2 =1): • E(Y) = b0 + b1X1 + b2(1) + b3X1(1)= (b0 + b2)+(b1+ b3) X1 • Response Surface Models • E(Y) = b0 + b1X1 + b2X12 + b3X2 + b4X22 +b5X1X2 • Note: Although the Response Surface Model has polynomial terms, it is linear wrt Regression parameters

  8. Matrix Form of Regression Model

  9. Least Squares Estimation of Regression Coefficients

  10. Fitted Values and Residuals

  11. Analysis of Variance – Sums of Squares

  12. ANOVA Table, F-test, and R2

  13. Inferences Regarding Regression Parameters

  14. Estimating Mean Response at Specific X-levels

  15. Predicting New Response(s) at Specific X-levels

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