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hypothesis at some
of the permutation sets must equal or exceed the
observed F-value. So, for example, to reject the null hypothesis at
α
value, less than
α
α
5
0.05, less than 5% of
the F-ratios obtained from the permuted data can equal or exceed the observed one.
APPENDIX: AN OVERVIEW OF RANDOMIZATION AND MONTE
CARLO METHODS
In this Appendix, we aim to present the principles underlying randomization methods
in a coherent fashion. More complete discussions of the topics presented in this chapter
can be found in the texts by
Efron and Tibshirani (1993)
and
Manly (1997)
, and the work
of Anderson and colleagues (
Anderson, 2001a,b, 2006; Anderson and Robinson 2001;
Anderson and Ter Braak, 2003
). We discuss four classes of methods, including the boot-
strap, jackknife, and permutation tests, and Monte Carlo simulations. To illustrate these
methods, we focus on a few univariate statistical tests. The extension to multivariate statis-
tics is not difficult but, by examining applications to univariate statistics, it may be easier
to acquire an intuitive understanding of how these methods work. The basic ideas
appeared in the work of R.A. Fisher in the 1930s, but the ideas and techniques were nei-
ther developed extensively nor used widely until recently. Perhaps the best summary is
contained in the title of
Efron's (1979)
paper, Computers and the theory of statistics, thinking
the unthinkable. The approach he outlined was indeed unthinkable prior to the advent of
computers, and could not be used widely until computers became fast and inexpensive
enough to be generally available to researchers, which accounts for the long time lag
between the development of the ideas and their widespread application. These methods
are computationally intensive because they replace the complex analytic mathematical
methods of classical statistics by an extensive use of randomization and repeated calcula-
tions. The enormous number of calculations required by these methods makes them
unthinkable without inexpensive (and fast) computers.
Resampling Statistics
Classical statistics relies on algebraic derivations of formulae based on a limited number
of well-studied distributions, particularly the normal (Gaussian), F-, gamma, chi-square,
uniform, and Poisson distributions. To see how resampling-based methods can provide an
alternative, we will work through one simple example.
Suppose X is a set of 31 observations of a length:
X
5
f
2
;
2
;
3
;
4
;
2
;
5
;
3
;
2
;
6
;
2
;
3
;
4
;
6
;
2
;
1
;
4
;
3
;
7
;
2
;
3
;
4
;
4
;
5
;
8
;
5
;
2
;
1
;
3
;
4
;
4
;
3
g
(8A.1)
In this case, N
5
31. We can compute the mean (denoted
,
X
.
for “the expectation of X”)
by:
X
N
X
i
N
,
X
.5
(8A.2)
i
1
5
