Adding additional independent variables to the regression equation
will always increase the value of R²

Even though not all independent variables are worth
including in the regression model
o
Showing It In Action
▪
Add 3
rd
variable that affects car price, seller age
▪
Compute multiple regression analysis in PHStat2

Same procedure as in earlier examples
o
Results
▪
Regression Statistics – R Square and Adjusted R Square

Adjusted = with third variable included

R² increases and adjusted R² decreases
▪
Why R² increases

Adding 3
rd
IV increases SSR

Adding new IV to model will ALWAYS increase R²
▪
MSR decreases

Harder to reject null hypothesis when test for significance
o
Evaluation
▪
If added IV causes reduction in adjusted R²  evidence that the new
variable might not be worth keeping in model

Benefit of increasing SSR offset by having cost of
additional DF
▪
IN our example – Adjusted R² decreases

Shouldn’t include this 3
rd
variable in the regression analysis
o
Other Notes
▪
Adjusted R² will always be ≤ R²
▪
If R² is very small and large # of IV

Adjusted R² COULD be negative

→ report value as zero
➢
15.3 – INFERENCES ABOUT THE INDEPENDENT VARIABLES
▪
Examine e/independent variable separately to make sure that it
belongs in the final regression model
▪
Confidence intervals for the regression coefficients
●
A Significance Test for the Regression Coefficients
o
The Basics
▪
Obtain regression coefficients for all 3 IVs

PHStat2 procedures

Coefficients Column under last table
o
Step 1 – Hypothesis Statement
▪
H0: M = 0

No relationship exists
▪
H1: M ≠ 0

A relationship does exist
▪
Rejecting the null hypothesis

Support of a significant relationship b/w mileage / price

Want to reject null hypothesis b/c included IVs in model
that we think belong there
▪
Set ∞ = 0.05
o
Step 2 – Test Statistic
T = (b – B) ÷ Sb

Sb = standard error of the regression coefficient

b = regression coefficient

B = population regression coefficient
▪
Standard error of the regression coefficient
(Sb) – measures the
variation in the regression coefficient, bi, among several samples

Larger variation → larger value of Sb

In our example, Sb = 0.0415 coefficient for mileage
▪
In our example (mileage variable)

b = 0.1434

B = 0 b/c null hypothesis assumes no relationship (= 0)

Sb = 0.0415

→ T = 3.46
o
Step 3 – Critical TScore
▪
Follows Tdistribution
▪
DF = (n – k – 1)

In our example, 20 – 3 – 1 = 16
▪
Twotail test at ∞ = 0.05 → ±2.120
▪
FUNCTION in Excel (twotail)

= TINV (∞, DF)

for onetail, double value of ∞
o
Step 4 – Comparing T and TScore
▪
3.46 < 2.120 → REJECT
o
Step 5 – Conclusions
▪
Reject null hypothesis
▪
Conclude that mileage population coefficient is NOT equal to zero
▪
Relationship b/w mileage and asking price is significant
o
3
rd
Variable – Seller Age
▪
Pvalue (0.778) > ∞ = 0.05 → DO NOT REJECT

No relationship b/w the variables

The seller’s age is not a good predictor of car price
▪
Should remove this independent variable from regression model
o
When to Use F and TTest
▪
FTest

Det. if overall regression model (all IVs included) is
statistically signficiant
▪
TTest

Examine difference of individual independent variables
●
Confidence Intervals for the Regression Coefficients
▪
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 Fall '12
 Donnelly
 Normal Distribution, Null hypothesis, Hypothesis testing, Statistical hypothesis testing