Chapter 8: Nonparametric Bootstrap Methods
4. Correlation and Regression
•
bootstrap approach to making inferences about the correlation coefficient
and the slope of a regression line
•
Bivariate Bootstrap Sampling
–
random sample (
X
i
, Y
i
)
i
= 1
, . . . , n
–
bivariate bootstrap sample:
n
n
possible bootstrap samples
–
bootstrap confidence interval for
ρ
(a) draw a speified number of vibarate bootstrap samples of size
n
from the data with replacement
(b) compute the bootstrap Pearson correlation coefficient
r
b
for each
bootstrap sample
(c) a confidence interval from the distribution of the
r
b
’s.
–
BCA method prefered / residual method has the undesirable property
–
limits developed for the vibariate normal distribution
Z
=
1
2
ln
parenleftbigg
1 +
r
1

r
parenrightbigg
∼
N
parenleftbigg
1
2
ln
parenleftbigg
1 +
ρ
1

ρ
parenrightbigg
,
1
n

3
parenrightbigg
100(1

α
)% confidence interval
(1 +
r
)
e

c

(1

r
)
(1 +
r
)
e

c
+ (1

r
)
< ρ <
(1 +
r
)
e
c

(1

r
)
(1 +
r
)
e
c
+ (1

r
)
where
c
= 2
z
α/
2
/
√
n

3
1
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•
Fixed
X
Bootstrap Sampling
–
regression model
Y
=
h
(
X
) +
ǫ
ǫ
∼
iid (0
, σ
2
)
–
when the
X
’s are regarded as fixed, sample the errors and then add
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 Spring '08
 Staff
 Statistics, Correlation, Normal Distribution, β1 Xi, − β1 Xi

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