Chapter 11
Multiple Regression
Chapter 11
1
Introduction
•
We have investigated the relationship between a
pair of quantitative variables with simple linear
regression (Chapter 2) and inference for
regression (Chapter 10).
•
Multiple regression
(Chapter 11) concerns the
use of more than one explanatory variable to
predict a response variable.
•
Many of the exploratory tools remain the same.
Chapter 11
2
Data for Multiple Regression
•
In simple linear regression, the data consist of
(
x
i
,
y
i
)
pairs for each individual, with the subscript
i
identifying the individuals
i
=
1
,
2
, . . . ,
n
•
In multiple regression we have more than one
explanatory variable, so the notation becomes
slightly more complicated.
•
It helps to think of a
data matrix
Chapter 11
3
Data Matrix
If we have
n
individuals and
p
explanatory variables:
Variables
Individual
y
x
1
x
2
. . .
x
p
1
y
1
x
11
x
12
. . .
x
1
p
2
y
2
x
21
x
22
. . .
x
2
p
.
.
.
.
.
.
n
y
n
x
n
1
x
n
2
. . .
x
np
•
Each explanatory variable has a different
subscript in addition to each individual having a
subscript.
Chapter 11
4
Simple Linear Regression
•
Recall the mathematical model for simple linear
regression:
μ
y
=
β
0
+
β
1
x
•
For any ﬁxed value of
x
, the response varies
according to a Normal distribution with mean
μ
y
and constant standard deviation
σ
.
•
An alternative formulation is
y
i
=
β
0
+
β
1
x
i
+
ε
i
ε
i
∼
N
(
0
,
σ
)
•
The parameters of the model are
β
0
,
β
1
and
σ
.
Chapter 11
5
Multiple Regression Model
•
The mathematical model for multiple regression is
μ
y
=
β
0
+
β
1
x
1
+
β
2
x
2
+
. . .
+
β
p
x
p
•
As in the simple case, the response varies
according to a Normal distribution with mean
μ
y
and constant standard deviation
σ
.
•
An alternative formulation is
y
i
=
β
0
+
β
1
x
i
1
+
β
2
x
i
2
+
. . .
+
β
p
x
ip
+
ε
i
ε
i
∼
N
(
0
,
σ
)
•
The parameters of the model are
β
0
,
β
1
,
β
2
, . . . ,
β
p
and
σ
.
Chapter 11
6
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View Full DocumentData Analysis Strategy
•
When we encounter several variables, our
analysis strategy consists of two exploratory
steps.
1. Examine the variables individually onebyone (shape, center,
spread, outliers)
2. Examine the relationship between pairs of variables with
scatterplots (form, direction, strength, outliers/inﬂuential points)
•
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 Spring '08
 ABBEY
 Linear Regression, Regression Analysis, Errors and residuals in statistics, µY

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