F-2011 STAT 410/510
1
Chapter 6.
Multiple Regression I
6.1 Multiple regression models:
First-order model with two predictor variables:
i
i
i
i
X
X
Y
2
2
1
1
0
Assuming that
,
0
}
{
i
E
2
2
1
1
0
}
{
X
X
Y
E
Then, the regression function is a plane.
See Figure 6.1.
The parameter
0
indicates the
Y
intercept of the regression plane.
If the scope of the model includes
,
0
,
0
2
1
X
X
then
0
indicates
the mean response
}
{
Y
E
at
.
0
,
0
2
1
X
X
Otherwise, it does not
have any particular meaning.
The parameter
1
indicates the change in the mean response
}
{
Y
E
per unit increase in
1
X
when
2
X
is held constant.
Likewise,
2
indicates the change in the mean response per unit increase in
2
X
when
1
X
is held constant. (
additive effects
).
The parameters
1
and
2
are sometimes called
partial regression
coefficients
because they reflect the partial effect of one predictor
variable when the other predictor variable is included in the model
and is held constant.

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F-2011 STAT 410/510
2
Qualitative predictor variables:
Consider a regression analysis to predict the length of hospital stay
(
Y
) based on the age (
1
X
) and gender (
2
X
) of the patient.
Then, we define
2
X
as follows:
patient
male
if
0
patient
female
if
1
2
X
Then, the first-order regression model is
i
i
i
i
X
X
Y
2
2
1
1
0
.
For male patient
,
0
2
X
and response function becomes
1
1
0
}
{
X
Y
E
and
for female patient
,
,
1
2
X
1
1
2
0
)
(
}
{
X
Y
E
These two response functions represent
parallel
straight lines with
different intercepts.
In general, we represent a qualitative variable with
c
classes by means
of
1
c
indicator variables.
Polynomial regression:
i
i
i
i
X
X
Y
2
2
1
0
Despite the curvilinear nature of the response function, it is a special
case of general linear regression model.
If we let
i
i
X
X
1
and
,
2
2
i
i
X
X
i
i
i
i
X
X
Y
2
2
1
1
0
Transformed variable:
i
i
i
i
i
X
X
X
Y
3
3
2
2
1
1
0
)
log(
If we let
)
log(
i
i
Y
Y
,
i
i
i
i
i
X
X
X
Y
3
3
2
2
1
1
0
Another example would be
i
i
i
i
X
X
Y
2
2
1
1
0
1
By letting
,
/
1
i
i
Y
Y
i
i
i
i
X
X
Y
2
2
1
1
0
Interaction effects: