Multiple Linear Regression Probabilistic Multiple Regression Model Y = 0 + 1X1 + 2X2 + 3X3 + . . . + 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

Multiple Variable Prediction Equation:

Model for Two Independent Variables Population Model Estimated Model

Model is Now a Plane in 3-D Space Y 

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

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

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;

SAS Output:

Example 2: Housing Cost Model (p. 597)

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

Correlation Matrix:

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

Wine Tasting with a PC

Mallows Coefficient Plot