1. Multiple Linear Regression Analysis
BSAD 30
Dave Novak
Fall 2018
Source: Ragsdale, 2018 Spreadsheet Modeling and Decision Analysis 8th edition © 2017 Cengage Learning

2. Overview
Last class we considered the relationship between one independent variable and one dependent variable
Referred to as “simple” linear regression
Today, we consider the relationship between more than one independent variable (X’s) and a single dependent variable (Y)
Referred to as “multiple” linear regression
Example

3. Multiple regression
When more than one independent variable can be used to explain variance in Y
𝑌 i = b0 + b1X1i + b2X2i + …. + bnXni
Where it is assumed all bi’s are independent

4. Multiple regression
Assume that you want to develop a model to predict the market value (or price) of houses in your town / city
We have access to the following data
Source: Ragsdale, 2018 Spreadsheet Modeling and Decision Analysis 8th

5. Multiple regression
We want to predict housing price (in thousands of $) using some combination of data we have on square footage, the garage size, and number of bedrooms (3 possible X values)
Even with three independent variables, we can create many different regression models
A model with the most X’s is often not the “best” model

6. Multiple regression Start by looking at scatter plots and correlation
Having access to many different independent variables does not necessarily mean that they all should be part of a regression model
Rule of thumb for linear regression: KEEP IT AS SIMPLE AS POSSIBLE
Start by looking at scatter plots and correlation

7. Multiple regression
Source: Ragsdale, 2018 Spreadsheet Modeling and Decision Analysis 8th

8. Multiple regression
Look at correlation coefficient (r) for each X / Y combination

9. Multiple regression
Given results from scatter and correlation, start with three separate simple linear regression models and compare results
𝑌 i = b0 + b1X1i (X1 = square footage)
𝑌 i = b0 + b2X2i (X2 = garage size)
𝑌 i = b0 + b3X3i (X3 = # bedrooms)

10. X1 – square footage

11. X2 – garage size

12. X3 – # bedrooms

13. Summary comparison X1 has highest R2 ,Adj R2, and lowest Std. error
Reasonable to start with X1 and build off that variable

14. Combinations of two variables
𝑌 i = b0 + b1X1i + b2X2i (square footage + garage)
𝑌 i = b0 + b1X1i + b3X3i (square footage + # bedrooms)

15. X1 + X2 (Sq ft + Garage)

16. X1 + X3 (Sq ft + Bedrooms)

17. Change in b0 and b1 𝑌 i = b0 + b1X1i (b0 = 109.5, b1 = 56.394)
Notice values of b0 and b1 have changed from the model where we just had X1
𝑌 i = b0 + b1X1i (b0 = 109.5, b1 = )
𝑌 i = b0 + b1X1i + b2X2i (b0 = , b1 = )
𝑌 i = b0 + b1X1i + b3X3i (b0 = , b1 = )

18. Summary comparison
Where X1 = sq. ft., X2 = garage, X3 = # bedrooms

19. Multicollinearity
Not surprising adding X3 (# of bedrooms) to regression model with X1 (total square footage) did not improve model
Both variables represent similar things – a measure of house size (sq ft)
These variables appear to be highly correlated

20. Combination of all three
𝑌 i = b0 + b1X1i + b2X2i + b3X3i (square footage + garage + # bedrooms)
As it is not a time consuming undertaking to test all three variables, we also want to examine the FULL (all independent variables) model

21. X1 + X2 + X3

22. Summary comparison

23. Best fit How do we choose?
The two variable model with X1 and X2 has highest adj R2 and lowest std. error of all models
Making the model more complex by adding all three variables doesn’t add anything to predictive power
We also know that X3 is highly correlated with X1 (X1 and X3 not necessarily independent)

24. Making predictions
Use our selected model to estimate average selling price of house with 2,100 sq ft and a 2-car garage
Y 𝑖 = X 1𝑖 X 2𝑖

25. Making predictions
95% prediction interval for the actual selling price:

26. Problem
In-class example problem