Chapter 4
Simultaneous inferences and other topics in
Regression analysis
4.1 Bonferroni joint CIs
for
0
and
1
We will learn a procedure for constructing simultaneous CIs for
0
and
1
with a specified
family confidence coefficient
.
Note that the meaning of
family confidence coefficient
, say .95, is
that if repeated samples are selected and interval estimate for both
0
and
1
are calculated for each sample by specified procedure,
95% of the samples would lead to a family of estimates where both
CI are correct; for 5% of the samples, either one or both of the
interval estimates would be incorrect.
Consider a multiple hypotheses.
If each test in a family of
g
tests is
tested at level of significance
, the probability making one or more
Type I errors if all
g
hypotheses are true is
*
where:
g
)
1
(
1
*
(Bonferroni inequality)
Equality holds if the tests are mutually independent.
It is apparent that
1
)
1
(
1
*
g
g
That is, the probability is close to 1 that we falsely reject at least one
hypothesis even if they are all true.
If
is small then
g
g
)
1
(
1
*
.
Hence, if we perform each of
g
tests at the
g
/
level we get:
g
g
1
1
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This approach to multiple testing is referred to as the
Bonferroni
method
or
procedure
.
So, using the Bonferroni procedure, we perform each of
g
tests at the
g
/
level and are assured that the probability is
of making one
or more Type I errors if all
g
hypotheses are true.
For example, if we perform 4 tests and use the 0.05/4 = .0125 level
of significance for each test, the probability we falsely reject one or
more of the 4 corresponding hypotheses is less than or equal to 0.05.
In the simple linear regression model, simultaneous (Bonferroni)
confidence limits for estimating
0
and
1
are:
}
{
0
0
b
Bs
b
and
}
{
1
1
b
Bs
b
where
)
4
/
1
(
))
2
/(
1
(
2
2
n
n
t
g
t
B
Example: Toluca Company example
We are interested in finding Bonferroni joint CIs for
0
and
1
:
To estimate 90% CIs for
0
and
1
, we need
.
069
.
2
)
975
(.
)
4
/
1
.
1
(
)
4
/
1
(
23
2
25
2
t
t
t
B
n
From
The MEANS Procedure
Variable
Sum
ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ
cross
70690.00
xdev2
19800.00
The MEANS Procedure
Analysis Variable : lotsize
N
Mean
Std Dev
Minimum
Maximum
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '09
 Linear Regression, Normal Distribution, Regression Analysis, yh, Toluca Company

Click to edit the document details