1. Multiple Linear Regression
Probabilistic Multiple Regression Model Y = 0 + 1X1 + 2X2 + 3X kXk+ 
Y = the value of the dependent (response) variable
0 = the regression constant
1 = the regression coefficient of independent variable 1
2 = the regression coefficient of independent variable 2
k = the regression coefficient of independent variable k
k = the number of independent variables
 = the error of prediction

2. Multiple Variable Prediction Equation:

3. Model for Two Independent Variables
Population
Model
Estimated
Model

4. Model is Now a Plane in 3-D SpaceY 

5. General Multiple Regression Model
1) Test the Regression Model (k Indep Variables):
H0: Model is Not Significant ; β1=β2=β3=..βk=0
HA: Model is Significant ; At Least One βi≠0
R: F > Fα(k,n-k-1) F = MSR/MSE
ANOVA Table:
Source df SS MS
Regress k SSR MSR
Error n-k-1 SSE MSE
Total n-1 TSS
Multiple Coefficient of Determination: R2 = SSR/TSS
(% of Variation of Y Explained by all the X's)
Adjusted R Squared

6. 2) Test the Regression Model Coefficients:
H0: βi = 0 ; No Linear Dep i = 1,2,3,…,k
HA: βi ≠ 0 ; Linear Dep
R: t > tα/2,df=n-k-1 t = bi/Sbi
t < -tα/2,df=n-k-1
Interval Estimate for Coefficients:
bi e ≤ βi ≤ bi e
e = t•Sbi
Correlation Matrix - Test for Significant Correlation
H0: ρ = 0 ; No Sig Correlation
HA: ρ ≠ 0 ; Sig Correlation
R: t > tα/2,df=n-2
t < -tα/2,df=n-2

7. Example 1: Regress Sales on TV Ads and News Ads
SAS Program:
data;
input sales tv news;
datalines;
10 0 1
10 1 0
15 2 0
20 2 2
30 2 3
35 5 0
30 0 5
40 3 3
40 4 3
proc corr;
var sales tv news;
proc reg;
model sales=tv news;
run;

8. SAS Output:

9. Example 2: Housing Cost Model (p. 597)

10. Example 3: Regress Car Rating on Ride, Handling, & Comfort

11. Correlation Matrix:

12. Example 4: Regress Profit on Counter Sales and Drive-thru Sales

14. Wine Tasting with a PC

15. Mallows Coefficient Plot