1. Multiple Regression and Model Building
Chapter 14
Multiple Regression and Model Building

2. Multiple Regression and Model Building
14.1 The Multiple Regression Model and the Least Squares Point Estimate 14.2 Model Assumptions and the Standard Error 14.3 R2 and Adjusted R The Overall F Test 14.5 Testing the Significance of an Independent Variable 14.6 Confidence and Prediction Intervals

3. Multiple Regression and Model Building Continued
14.7 The Sales Territory Performance Case: Evaluating Employee Performance 14.8 Using Dummy Variables to Model Qualitative Independent Variables 14.9 Using Squared and Interactive Terms Model Building and the Effects of Multicollinearity Residual Analysis in Multiple Regression Logistic Regression

4. LO 14-1: Explain the multiple regression model and the related least squares point estimates.
14.1 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

5. 14.2 Model Assumptions and the Standard Error
LO 14-2: Explain the assumptions behind multiple regression and calculate the standard
error.
14.2 Model Assumptions and the Standard Error
The model is y = β0 + β1x1 + β2x2 + … + βkxk + 
Assumptions for multiple regression are stated about the model error terms, ’s

6. LO 14-3: Calculate and interpret the multiple and adjusted multiple coefficients of determination.
14.3 R2 and Adjusted R2
Total variation is given by the formula Σ(yi - ȳ)2
Explained variation is given by the formula Σ(ŷi - ȳ)2
Unexplained variation is given by the formula Σ(yi - ŷi)2
Total variation is the sum of explained and unexplained variation
This section can be read anytime after reading Section 14.1

7. LO 14-4: Test the significance of a multiple regression model by using an F test.
14.4 The Overall F Test
To test H0: β1= β2 = …= βk = 0 versus Ha: At least one of β1, β2,…, βk ≠ 0
Reject H0 in favor of Ha if F(model) > F* or p-value < 
*F is based on k numerator and n-(k+1) denominator degrees of freedom

8. 14.5 Testing the Significance of an Independent Variable
LO 14-5: Test the significance of a single independent variable.
14.5 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 H0: βj = 0
Versus the alternative hypothesis Ha: βj ≠ 0

9. 14.6 Confidence and Prediction Intervals
LO 14-6: Find and interpret a confidence interval for a mean value and a prediction interval for an individual value.
14.6 Confidence and Prediction Intervals
The point on the regression line corresponding to a particular value of x1, x2,…, xk, of the independent variables is ŷ = b0 + b1x1 + b2x2 + … + bkxk
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 away the predicted value might be
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

10. 14.8 Using Dummy Variables to Model Qualitative Independent Variables
LO 14-7: Use dummy variables to model qualitative independent
variables.
14.8 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

11. 14.9 Using Squared and Interaction Variables
LO 14-8: Use squared and interaction variables.
14.9 Using Squared and Interaction Variables
Quadratic regression model is: y = β0 + β1x + β2x2 ε
where
β0 + β1x + β2x2 is μy
Β, β, and β2 are the regression parameters
ε is an error term

12. 14.10 Model Building and the Effects of Multicollinearity
LO 14-9: Describe multicollinearity and build a multiple regression model.
14.10 Model Building and the Effects of Multicollinearity
Multicollinearity: when “independent” variables are related to one another
Considered severe when the simple correlation exceeds 0.9
Even moderate multicollinearity can be a problem
Another measurement is variance inflation factors
Multicollinearity a problem when VIF>10
Moderate problem for VIF>5

13. 14.11 Residual Analysis in Multiple Regression
LO 14-10: Use residual analysis to check the assumptions of multiple
regression.
14.11 Residual Analysis in Multiple Regression
For an observed value of yi, the residual is ei = yi - ŷ = yi – (b0 + b1xi1 + … + bkxik)
If the regression assumptions hold, the residuals should look like a random sample from a normal distribution with mean 0 and variance σ2
Residual plots
Residuals versus each independent variable
Residuals versus predicted y’s
Residuals in time order (if the response is a time series)