Copyright © 2011 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin Multiple Regression Chapter 14

14-2 Multiple Regression 14.1The Multiple Regression Model and the Least Squares Point Estimate 14.2Model Assumptions and the Standard Error 14.3R 2 and Adjusted R The Overall F Test 14.5Testing the Significance of an Independent Variable

14-3 Multiple Regression Continued 14.6Confidence and Prediction Intervals 14.7The Sales Territory Performance Case 14.8Using Dummy Variables to Model Qualitative Independent Variables 14.9The Partial F Test: Testing the Significance of a Portion of a Regression Model 14.10Residual Analysis in Multiple Regression

The Multiple Regression Model and the Least Squares Point Estimate Simple linear regression used one independent variable to explain the dependent variable Some relationships are too complex to be described using a single independent variable Multiple regression uses two or more independent variables to describe the dependent variable This allows multiple regression models to handle more complex situations There is no limit to the number of independent variables a model can use Multiple regression has only one dependent variable LO 1: Explain the multiple regression model and the related least squares point estimates.

14-5 The Multiple Regression Model The linear regression model relating y to x1, x2,…, xk is y = β0 + β1x1 + β2x2 +…+ βkxk +  µy = β0 + β1x1 + β2x2 +…+ βkxk is the mean value of the dependent variable y when the values of the independent variables are x1, x2,…, xk β0, β1, β2,… βk are unknown the regression parameters relating the mean value of y to x1, x2,…, xk  is an error term that describes the effects on y of all factors other than the independent variables x1, x2,…, xk LO1

Model Assumptions and the Standard Error The model is y = β 0 + β 1 x 1 + β 2 x 2 + … + β k x k +  Assumptions for multiple regression are stated about the model error terms,  ’s LO 2: Explain the assumptions behind multiple regression and calculate the standard error.

14-7 The Regression Model Assumptions Continued 1. Mean of Zero Assumption The mean of the error terms is equal to 0 2. Constant Variance Assumption The variance of the error terms σ 2 is, the same for every combination values of x 1, x 2,…, x k 3. Normality Assumption The error terms follow a normal distribution for every combination values of x 1, x 2,…, x k 4. Independence Assumption The values of the error terms are statistically independent of each other LO2

R 2 and Adjusted R 2 1. Total variation is given by the formula Σ(y i - y ̄) 2 2. Explained variation is given by the formula Σ(y ̂ i - y ̄) 2 3. Unexplained variation is given by the formula Σ(y i - y ̂ i ) 2 4. Total variation is the sum of explained and unexplained variation This section can be read anytime after reading Section 14.1 LO 3: Calculate and interpret the multiple and adjusted multiple coefficients of determination.

The Overall F Test To test H 0 : β 1 = β 2 = …= β k = 0 versus H a : At least one of β 1, β 2,…, β k ≠ 0 The test statistic is Reject H 0 in favor of H a if F(model) > F  * or p-value <  * F  is based on k numerator and n-(k+1) denominator degrees of freedom LO 4: Test the significance of a multiple regression model by using an F test.

Testing the Significance of an Independent Variable A variable in a multiple regression model is not likely to be useful unless there is a significant relationship between it and y To test significance, we use the null hypothesis H 0 : β j = 0 Versus the alternative hypothesis H a : β j ≠ 0 LO 5: Test the significance of a single independent variable.

Confidence and Prediction Intervals The point on the regression line corresponding to a particular value of x 01, x 02,…, x 0k, of the independent variables is y ̂ = b 0 + b 1 x 01 + b 2 x 02 + … + b k x 0k It is unlikely that this value will equal the mean value of y for these x values Therefore, we need to place bounds on how far the predicted value might be from the actual value We can do this by calculating a confidence interval for the mean value of y and a prediction interval for an individual value of y LO 6: Find and interpret a confidence interval for a mean value and a prediction interval for an individual value.

Using Dummy Variables to Model Qualitative Independent Variables So far, we have only looked at including quantitative data in a regression model However, we may wish to include descriptive qualitative data as well For example, might want to include the gender of respondents We can model the effects of different levels of a qualitative variable by using what are called dummy variables Also known as indicator variables LO 7: Use dummy variables to model qualitative independent variables.

The Partial F Test: Testing the Significance of a Portion of a Regression Model So far, we have looked at testing single slope coefficients using t test We have also looked at testing all the coefficients at once using F test The partial F test allows us to test the significance of any set of independent variables in a regression model LO 8: Test the significance of a portion of a regression model by using an F test.

Residual Analysis in Multiple Regression For an observed value of y i, the residual is e i = y i - y ̂ = y i – (b 0 + b 1 x i 1 + … + b k x ik ) If the regression assumptions hold, the residuals should look like a random sample from a normal distribution with mean 0 and variance σ 2 LO 9: Use residual analysis to check the assumptions of multiple regression.