APPLIED STATISTICS
Variable Selection
Dr Tao Zou
Research School of Finance, Actuarial Studies & Statistics
The Australian National University
Last Updated: Mon Sep 18 18:16:03 2017
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Overview
Motivation
Sequential Variable Selection
Variable Selection Among All Subsets
Cross Validation for Variable Selection Results
Multicollinearity
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References
1.
F.L. Ramsey and D.W. Schafer
(2012)
Chapter 12 of
The Statistical Sleuth
2.
The slides are made by
R Markdown
.
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Motivation
There are two prime reasons for variable selection:
1.
Simple models with less variables are preferable to complex models
with more variables.
2.
Including unnecessary variables in a model results in a loss of precision
⇒
overfitting.
Variable selection involves choosing a subset of explanatory variables to
construct the multiple linear regression model.
Because if the exlanatory variables selected in MLR are determined, then
the MLR model with those exlanatory variables is given.
Hence, sometimes we also call model selection.
Different subsets of explanatory variables determine different models. We
call those models candidate models.
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Motivation Example: Significance Depends on
Other Explanatory Variables in the Model (Con’d)
Suppose we are interested in predicting ANU students’ 2nd year GPA (
Y
)
given their 1st year GPA (
X
1
) and UAC score (
X
2
). The following regression
line is fit:
μ
{
Y

X
1
,
X
2
}
=
β
0
+
β
1
X
1
+
β
2
X
2
.
(1)
Based on the data, the
p
values for the
t
tests of whether
β
j
=
0 versus
β
j
=
0 for
j
=
1
,
2 are 0.15 and 0.20, respectively.
Does this mean that we
do not need to select both
X
1
and
X
2
in the
model? NO!
The test for
β
2
tells us whether
X
2
is needed in the model that already
contains
X
1
, i.e., does
X
2
offer any information about mean GPA over and
above that of
X
1
?
The meaning of the coefficient of an explanatory variable depends on what
other explanatory variables have been included in the regression.
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Motivation Example: Significance Depends on
Other Explanatory Variables in the Model (Con’d)
If we fit the following two models:
μ
{
Y

X
1
}
=
α
0
+
α
1
X
1
and
μ
{
Y

X
2
}
=
γ
0
+
γ
2
X
2
.
For both models, the
p
values for the
t
tests of
α
1
=
0 versus
α
1
=
0 and
γ
2
=
0 versus
γ
2
=
0 can be computed. Based on the data, the results of
the
p
values are 0.01 and 0.02, respectively.
Hence at least one of
X
1
and
X
2
is needed in the model.
In this example
X
1
and
X
2
are probably highly correlated so we might expect
this to be the case. The following
F
test of model (1) avoids this problem.
H
0
:
none of
X
1
and
X
2
is needed in the model
↔
H
a
:
at least one of
X
1
and
X
2
is needed in the model
.
However, the
F
test does not answer: which of
X
1
and
X
2
should be
selected in the model.
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Sequential Variable Selection  Backward
Elimination Steps
Backward Elimination Step for
j
Explanatory Variables
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Sequential Variable Selection  Forward Selection
Steps
Forward Selection Step for
j
Explanatory Variables
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Sequential Variable Selection
The idea behind sequential techniques is a sequential search through all
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