What is the net effect of adding a new variable?
x
Did the new x variable add enough explanatory power to offset the loss of one degree of freedom?
Chap 1526
Shows the
proportion of variation in y explained
by
all x variables adjusted for the number of x
variables used
(where n = sample size, k = number of independent variables)
Penalize excessive use of unimportant independent
variables
Smaller than R
2
Useful in comparing among models
Adjusted R
2
(continued)





=
1
k
n
1
n
)
R
1
(
1
R
2
2
A
Chap 1527
Regression Statistics
Multiple R
0.72213
R Square
0.52148
Adjusted R Square
0.44172
Standard Error
47.46341
Observations
15
ANOVA
df
SS
MS
F
Significance F
Regression
2
29460.027
14730.013
6.53861
0.01201
Residual
12
27033.306
2252.776
Total
14
56493.333
.44172
R
2
A
=
44.2% of the variation in pie sales is
explained by the variation in price and
advertising, taking into account the sample
size and number of independent variables
Multiple Coefficient of
Determination
(continued)
Chap 1528
Is the Model Significant?
FTest for Overall Significance of the Model
Shows if there is a linear relationship between all
of the
x
variables considered together and
y
Use F test statistic
Hypotheses:
H
0
: β
1
= β
2
= … = β
k
= 0
(no linear relationship)
H
A
:
at least one
β
i
≠ 0
(at least one independent
variable affects y)
Chap 1529
FTest for Overall Significance
Test statistic:
where F has
(numerator)
D
1
= k
and
(denominator)
D
2
= (n – k – 1)
degrees of freedom
(continued)
MSE
MSR
1
k
n
SSE
k
SSR
F
=


=
Chap 1530
6.5386
2252.8
14730.0
MSE
MSR
F
=
=
=
Regression Statistics
Multiple R
0.72213
R Square
0.52148
Adjusted R Square
0.44172
Standard Error
47.46341
Observations
15
ANOVA
df
SS
MS
F
Significance F
Regression
2
29460.027
14730.013
6.53861
0.01201
Residual
12
27033.306
2252.776
Total
14
56493.333
(continued)
FTest for Overall Significance
With 2 and 12 degrees
of freedom
Pvalue for
the FTest
Chap 1531
H
0
: β
1
= β
2
= 0
H
A
: β
1
and β
2
not both zero
α
= .05
df
1
= 2
df
2
= 12
Test Statistic:
Decision:
Conclusion:
Reject H
0
at
α
= 0.05
The regression model does explain
a significant portion of the
variation in pie sales
(There is evidence that at least one
independent variable affects
y )
0
α
= .05
F
.05
=
3.885
Reject H
0
Do not
reject H
0
6.5386
MSE
MSR
F
=
=
Critical
Value:
F
α
=
3.885
FTest for Overall Significance
(continued)
F
Chap 1532
Are Individual Variables
Significant?
Use ttests of individual variable slopes
Shows if there is a linear relationship between the
variable
x
i
and
y
Hypotheses:
H
0
: β
i
= 0 (no linear relationship)
H
A
: β
i
≠ 0
(linear relationship does exist
between
x
i
and
y)
Chap 1533
Are Individual Variables
Significant?
H
0
: β
i
= 0 (no linear relationship)
H
A
: β
i
≠ 0
(linear relationship does exist
between
x
i
and
y )
Test Statistic:
(
df = n – k – 1)
i
b
i
s
0
b
t

=
(continued)
Chap 1534
Regression Statistics
Multiple R
0.72213
R Square
0.52148
Adjusted R Square
0.44172
Standard Error
47.46341
Observations
15
ANOVA
df
SS
MS
F
Significance F
Regression
2
29460.027
14730.013
6.53861
0.01201
Residual
12
27033.306
2252.776
Total
14
56493.333
tvalue for Price is
t = 2.306, with
pvalue .0398
tvalue for Advertising is t = 2.855,
with pvalue .0145
(continued)
Are Individual Variables
Significant?
Chap 1535
d.f. = 1521 = 12
α
= .05
t
α
/2
= 2.1788
Inferences about the Slope:
t
Test Example
H
0
: β
i
= 0
H
A
: β
i
≠
0
The test statistic for each variable falls
in the rejection region (pvalues < .05)
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 Spring '11
 BobSanders
 Regression Analysis, Model building