Handout 11
Multiple regression
For the Electric bill data we have
Y
= annual electric bill,
X
1
= monthly household income,
X
2
=number of persons,
X
3
=living area.
The model is
Y
=
β
0
+
β
1
X
1
+
β
2
X
2
+
β
3
X
3
+
ε
.
The fitted regression is
ˆ
Y
=

358
.
4 +
.
0751
X
1
+ 55
.
09
X
2
+
.
2811
X
3
, SSE
(
X
1
, X
2
, X
3
) = 550163
, SSTO
= 3701668
.
The correlation matrix is:
Bill
Income
Persons
Area
Bill
1
.837
.494
.905
Income
1
.143
.961
Person
1
.366
Area
1
Three testing problems.
I. An overall test for the regression:
H
0
:
β
1
=
β
2
=
β
3
= 0 against
H
1
: at least one of
β
1
, β
2
, β
3
is nonzero.
Note that ”
H
0
is true” is equivalent to ”knowing
X
1
, X
2
, X
3
does not help in predicting
Y
”.
II. Can a predictor variable, say
X
1
, can be dropped from the model? This is equivalent to asking ”does
adding variable
X
1
to the model
Y
=
β
0
+
β
2
X
2
+
β
3
X
3
+
ε
improve prediction of
Y
”.
In order to settle the question ”can we can drop variable
X
1
from the full model
Y
=
β
0
+
β
1
X
1
+
β
2
X
2
+
β
3
X
3
+
ε
” we need to test
H
0
:
β
1
= 0 against
H
1
:
β
1
6
= 0.
There are two (equivalent) tests for this: a) ttest, b) (partial) Ftest.
III. Can we drop more than one variable, say
X
1
and
X
3
, from the model
Y
=
β
0
+
β
1
X
1
+
β
2
X
2
+
β
3
X
3
+
ε
?
An equivalent question is:
does the addition of variables
X
1
and
X
3
to the model
Y
=
β
0
+
β
2
X
2
+
ε
significantly improve the prediction of
Y
?
Statistically this is equivalent to testing
H
0
:
β
1
=
β
3
= 0 against
H
1
: not both of
β
1
and
β
3
are zero.
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 Summer '09
 Jiang
 Regression Analysis, 1%, X1, 1 M

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