AMS 315
Data Analysis
Chapter Twelve Study Guide
Multiple Regression and the General Linear Model
Fall 2010
Context
The statement of the theory of multiple linear regression in matrix terms is given
in Section 12.9. Understanding the results of this chapter is much easier if you invest in
learning how to use these tools. I will present the theory in matrix form in this chapter’s
study guide. We assume that
,
Z
X
Y
σ
β
+
=
where
Y
is an
1
×
n
vector of observations,
X
is
an
p
n
×
matrix of known constants,
is a
1
×
p
vector of unknown but constant
parameters,
,
0
and
Z
is an
1
×
n
vector of
)
1
,
0
(
NID
random variables. The sum of
squares function is given by
.
2
)
(
)
(
)
(
Xb
X
b
Y
X
b
Y
Y
Xb
Y
Xb
Y
b
SS
T
T
T
T
T
T
+

=


=
The OLS estimate
ˆ
is any solution of the system of normal equations:
Y
X
X
X
T
T
=
ˆ
)
(
.
If
1
)
(

X
X
T
exists, then the OLS estimate is unique and
Y
X
X
X
T
T
1
)
(
ˆ

=
.
The procedures in this chapter provide tools to deal with the association of one
continuous dependent variable and an arbitrary number of independent variables.
12.1. Introduction and Abstract of Research Study
This section contains a statement of the multiple linear regression model and key
definitions. It is an important section with key definitions.
12.2 The General Linear Model
This section contains the statement of the multiple linear regression model without using
matrix notation. You should work on Section 12.9.
12.3 Estimating Multiple Regression Coefficients
Ordinary Least Squares (OLS) estimates are the ones most commonly used. If
1
)
(

X
X
T
exists, then the OLS estimate is unique and
Y
X
X
X
T
T
1
)
(
ˆ

=
.
12.4 Inferences in Multiple Regression
Definition 12.2 is important. The overall F test statistic and the coefficient of
determination (multiple correlation coefficient squared)
2
.
1
k
x
x
y
R
are important statistics.
The supplemental material contains an example examination problem worked out in
detail.
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View Full Document12.5 Testing a Subset of Regression Coefficients
This is a common procedure. The most common example is to test the contribution of
variables sequentially.
12.6. Forecasting Using Multiple Regression
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 Fall '08
 Staff
 Regression Analysis, Caspi, Abstract of Research Study

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