Title: Multiple%20Regression%20I
1Multiple Regression I
2Models 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
3First-Order Model with 2 Numeric Predictors
4Interpretation of Regression Coefficients
- Additive EY b0 b1X1 b2X2 Mean of Y _at_
X1, X2 - b0 Intercept, Mean of Y when X1X20
- 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 EY with respect to X1 and X2,
respectively - Interaction Model EY 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) b1b3 - The effect of increasing X1 depends on level of
X2, and vice versa
5General Linear Regression Model
6Special Types of Variables/Models - I
- p-1 distinct numeric predictors (attributes)
- Y Sales, X1Advertising, X2Price
- Categorical Predictors Indicator (Dummy)
variables, representing m-1 levels of a m level
categorical variable - Y Salary, X1Experience, X21 if College Grad,
0 if Not - Polynomial Terms Allow for bends in the
Regression - YMPG, X1Speed, X2Speed2
- Transformed Variables Transformed Y variable to
achieve linearity Yln(Y) Y1/Y
7Special Types of Variables/Models - II
- Interaction Effects Effect of one predictor
depends on levels of other predictors - Y Salary, X1Experience, X21 if Coll Grad, 0
if Not, X3X1X2 - 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
8Matrix Form of Regression Model
9Least Squares Estimation of Regression
Coefficients
10Fitted Values and Residuals
11Analysis of Variance Sums of Squares
12ANOVA Table, F-test, and R2
13Inferences Regarding Regression Parameters
14Estimating Mean Response at Specific X-levels
15Predicting New Response(s) at Specific X-levels