Biology Reference
In-Depth Information
The merged set,
M
, would have nine elements:
M
5
f
C
1
;
;
;
;
;
;
;
;
D
4
g
C
2
C
3
C
4
C
5
D
1
D
2
D
3
(8A.16)
To draw two bootstrap sets out of
M
, we would form a list of five random integers
(because there are five elements in
C
), and the elements in
M
corresponding to this list
would be the elements in the bootstrap version of
C
:
L
1
5
f
75185
g
(8A.17)
C
Bootstrap
5
f
D
2
;
C
5
;
C
1
;
D
3
;
C
5
g
(8A.18)
Note that two elements in
C
Bootstrap
come from
D
. A second list of four integers is used
to form a bootstrap version of
D
:
L
2
5
f
2499
g
(8A.19)
D
Bootstrap
5
f
C
2
;
C
4
;
D
4
;
D
4
g
(8A.20)
The approach we used to produce the bootstrap versions of
C
and
D
reflects the null
hypothesis that
C
and
D
come from a common underlying distribution. The elements of
C
and
are thus interchangeable.
The difference between means of the bootstrapped versions of
D
can be deter-
mined by many repetitions, developing a bootstrap estimate of the distribution of the differ-
ences between means produced by the null hypothesis (given the data). When we carry out
this bootstrap t-test on our numerical example, sets
C
and
D
, we find that 268 of 1000 boot-
strap sets (26.8%) have a difference between means as large or larger than that between the
means of
X
and
Y
X
Y
. Thus, we cannot reject the null hypothesis that these samples were drawn
from populations with equal means, the difference between them being due solely to
chance. Using a t-test based on the normal distribution, we would have rejected that null
hypothesis. Because both samples appear to have non-normal distributions, it seems reason-
able to attribute the difference between results to violating the assumption of normality.
and
Permutation Tests
Permutation tests pre-date bootstrap tests, having been introduced by R.A. Fisher in the
1930s as a basis for supporting the ideas of the Student's t-test rather than as a tool for
computation. With the advent of computers, permutation methods could be used prof-
itably for statistical inference. Permutation tests operate in much the same manner as
bootstrap tests, but differ in that they resample groups without replacement. This makes
permutation tests suitable for hypothesis testing, but not for the estimation of confidence
intervals (
Efron and Tibshirani, 1993; Good, 1994; Manly, 1997
).
Again, we can look at a simple, abstract example of how a permutation set is formed to
get a sense of how the approach works, and how it differs from the bootstrap. Consider
two data sets
C
and
D
:
C
5
f
C
1
;
C
2
;
C
3
;
C
4
;
C
5
g
(8A.21)
D
5
f
D
1
;
D
2
;
D
3
;
D
4
g
(8A.22)
