<s>
A	O
permutation	B-General_Concept
test	I-General_Concept
(	O
also	O
called	O
re-randomization	B-General_Concept
test	I-General_Concept
)	O
is	O
an	O
exact	B-General_Concept
statistical	O
hypothesis	O
test	O
making	O
use	O
of	O
the	O
proof	O
by	O
contradiction	O
.	O
</s>
<s>
A	O
permutation	B-General_Concept
test	I-General_Concept
involves	O
two	O
or	O
more	O
samples	O
.	O
</s>
<s>
The	O
null	B-General_Concept
hypothesis	I-General_Concept
is	O
that	O
all	O
samples	O
come	O
from	O
the	O
same	O
distribution	O
.	O
</s>
<s>
Under	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
,	O
the	O
distribution	O
of	O
the	O
test	B-General_Concept
statistic	I-General_Concept
is	O
obtained	O
by	O
calculating	O
all	O
possible	O
values	O
of	O
the	O
test	B-General_Concept
statistic	I-General_Concept
under	O
possible	O
rearrangements	O
of	O
the	O
observed	O
data	O
.	O
</s>
<s>
Permutation	B-General_Concept
tests	I-General_Concept
are	O
,	O
therefore	O
,	O
a	O
form	O
of	O
resampling	B-General_Concept
.	O
</s>
<s>
Permutation	B-General_Concept
tests	I-General_Concept
can	O
be	O
understood	O
as	O
surrogate	B-General_Concept
data	I-General_Concept
testing	I-General_Concept
where	O
the	O
surrogate	O
data	O
under	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
are	O
obtained	O
through	O
permutations	O
of	O
the	O
original	O
data	O
.	O
</s>
<s>
If	O
the	O
labels	O
are	O
exchangeable	O
under	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
,	O
then	O
the	O
resulting	O
tests	O
yield	O
exact	B-General_Concept
significance	O
levels	O
;	O
see	O
also	O
exchangeability	O
.	O
</s>
<s>
Permutation	B-General_Concept
tests	I-General_Concept
should	O
not	O
be	O
confused	O
with	O
randomized	B-General_Concept
tests	I-General_Concept
.	O
</s>
<s>
To	O
illustrate	O
the	O
basic	O
idea	O
of	O
a	O
permutation	B-General_Concept
test	I-General_Concept
,	O
suppose	O
we	O
collect	O
random	O
variables	O
and	O
for	O
each	O
individual	O
from	O
two	O
groups	O
and	O
whose	O
sample	O
means	O
are	O
and	O
,	O
and	O
that	O
we	O
want	O
to	O
know	O
whether	O
and	O
come	O
from	O
the	O
same	O
distribution	O
.	O
</s>
<s>
The	O
permutation	B-General_Concept
test	I-General_Concept
is	O
designed	O
to	O
determine	O
whether	O
the	O
observed	O
difference	O
between	O
the	O
sample	O
means	O
is	O
large	O
enough	O
to	O
reject	O
,	O
at	O
some	O
significance	O
level	O
,	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
H	O
that	O
the	O
data	O
drawn	O
from	O
is	O
from	O
the	O
same	O
distribution	O
as	O
the	O
data	O
drawn	O
from	O
.	O
</s>
<s>
First	O
,	O
the	O
difference	O
in	O
means	O
between	O
the	O
two	O
samples	O
is	O
calculated	O
:	O
this	O
is	O
the	O
observed	O
value	O
of	O
the	O
test	B-General_Concept
statistic	I-General_Concept
,	O
.	O
</s>
<s>
The	O
set	O
of	O
these	O
calculated	O
differences	O
is	O
the	O
exact	B-General_Concept
distribution	O
of	O
possible	O
differences	O
(	O
for	O
this	O
sample	O
)	O
under	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
that	O
group	O
labels	O
are	O
exchangeable	O
(	O
i.e.	O
,	O
are	O
randomly	O
assigned	O
)	O
.	O
</s>
<s>
Many	O
implementations	O
of	O
permutation	B-General_Concept
tests	I-General_Concept
require	O
that	O
the	O
observed	O
data	O
itself	O
be	O
counted	O
as	O
one	O
of	O
the	O
permutations	O
so	O
that	O
the	O
permutation	O
p-value	O
will	O
never	O
be	O
zero	O
.	O
</s>
<s>
Alternatively	O
,	O
if	O
the	O
only	O
purpose	O
of	O
the	O
test	O
is	O
to	O
reject	O
or	O
not	O
reject	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
,	O
one	O
could	O
sort	O
the	O
recorded	O
differences	O
,	O
and	O
then	O
observe	O
if	O
is	O
contained	O
within	O
the	O
middle	O
%	O
of	O
them	O
,	O
for	O
some	O
significance	O
level	O
.	O
</s>
<s>
For	O
paired	O
samples	O
the	O
paired	O
permutation	B-General_Concept
test	I-General_Concept
needs	O
to	O
be	O
applied	O
.	O
</s>
<s>
Permutation	B-General_Concept
tests	I-General_Concept
are	O
a	O
subset	O
of	O
non-parametric	B-General_Concept
statistics	I-General_Concept
.	O
</s>
<s>
In	O
contrast	O
to	O
permutation	B-General_Concept
tests	I-General_Concept
,	O
the	O
distributions	O
underlying	O
many	O
popular	O
"	B-General_Concept
classical	I-General_Concept
"	I-General_Concept
statistical	I-General_Concept
tests	O
,	O
such	O
as	O
the	O
t-test	B-General_Concept
,	O
F-test	B-General_Concept
,	O
z-test	B-General_Concept
,	O
and	O
χ2	B-General_Concept
test	O
,	O
are	O
obtained	O
from	O
theoretical	O
probability	O
distributions	O
.	O
</s>
<s>
Fisher	B-General_Concept
's	I-General_Concept
exact	I-General_Concept
test	I-General_Concept
is	O
an	O
example	O
of	O
a	O
commonly	O
used	O
permutation	B-General_Concept
test	I-General_Concept
for	O
evaluating	O
the	O
association	O
between	O
two	O
dichotomous	O
variables	O
.	O
</s>
<s>
When	O
sample	O
sizes	O
are	O
very	O
large	O
,	O
the	O
Pearson	O
's	O
chi-square	B-General_Concept
test	I-General_Concept
will	O
give	O
accurate	O
results	O
.	O
</s>
<s>
For	O
small	O
samples	O
,	O
the	O
chi-square	B-General_Concept
reference	O
distribution	O
cannot	O
be	O
assumed	O
to	O
give	O
a	O
correct	O
description	O
of	O
the	O
probability	O
distribution	O
of	O
the	O
test	B-General_Concept
statistic	I-General_Concept
,	O
and	O
in	O
this	O
situation	O
the	O
use	O
of	O
Fisher	B-General_Concept
's	I-General_Concept
exact	I-General_Concept
test	I-General_Concept
becomes	O
more	O
appropriate	O
.	O
</s>
<s>
Permutation	B-General_Concept
tests	I-General_Concept
exist	O
in	O
many	O
situations	O
where	O
parametric	O
tests	O
do	O
not	O
(	O
e.g.	O
,	O
when	O
deriving	O
an	O
optimal	O
test	O
when	O
losses	O
are	O
proportional	O
to	O
the	O
size	O
of	O
an	O
error	O
rather	O
than	O
its	O
square	O
)	O
.	O
</s>
<s>
All	O
simple	O
and	O
many	O
relatively	O
complex	O
parametric	O
tests	O
have	O
a	O
corresponding	O
permutation	B-General_Concept
test	I-General_Concept
version	O
that	O
is	O
defined	O
by	O
using	O
the	O
same	O
test	B-General_Concept
statistic	I-General_Concept
as	O
the	O
parametric	O
test	O
,	O
but	O
obtains	O
the	O
p-value	O
from	O
the	O
sample-specific	O
permutation	O
distribution	O
of	O
that	O
statistic	O
,	O
rather	O
than	O
from	O
the	O
theoretical	O
distribution	O
derived	O
from	O
the	O
parametric	O
assumption	O
.	O
</s>
<s>
For	O
example	O
,	O
it	O
is	O
possible	O
in	O
this	O
manner	O
to	O
construct	O
a	O
permutation	O
t-test	B-General_Concept
,	O
a	O
permutation	O
χ2	B-General_Concept
test	O
of	O
association	O
,	O
a	O
permutation	O
version	O
of	O
Aly	O
's	O
test	O
for	O
comparing	O
variances	O
and	O
so	O
on	O
.	O
</s>
<s>
Permutation	B-General_Concept
tests	I-General_Concept
exist	O
for	O
any	O
test	B-General_Concept
statistic	I-General_Concept
,	O
regardless	O
of	O
whether	O
or	O
not	O
its	O
distribution	O
is	O
known	O
.	O
</s>
<s>
Permutation	B-General_Concept
tests	I-General_Concept
can	O
be	O
used	O
for	O
analyzing	O
unbalanced	O
designs	O
and	O
for	O
combining	O
dependent	O
tests	O
on	O
mixtures	O
of	O
categorical	O
,	O
ordinal	O
,	O
and	O
metric	O
data	O
(	O
Pesarin	O
,	O
2001	O
)	O
.	O
</s>
<s>
Permutation	B-General_Concept
tests	I-General_Concept
may	O
be	O
ideal	O
for	O
analyzing	O
quantitized	O
data	O
that	O
do	O
not	O
satisfy	O
statistical	O
assumptions	O
underlying	O
traditional	O
parametric	O
tests	O
(	O
e.g.	O
,	O
t-tests	B-General_Concept
,	O
ANOVA	O
)	O
,	O
see	O
PERMANOVA	B-General_Concept
.	O
</s>
<s>
Since	O
the	O
1980s	O
,	O
the	O
confluence	O
of	O
relatively	O
inexpensive	O
fast	O
computers	O
and	O
the	O
development	O
of	O
new	O
sophisticated	O
path	O
algorithms	O
applicable	O
in	O
special	O
situations	O
made	O
the	O
application	O
of	O
permutation	B-General_Concept
test	I-General_Concept
methods	O
practical	O
for	O
a	O
wide	O
range	O
of	O
problems	O
.	O
</s>
<s>
It	O
also	O
initiated	O
the	O
addition	O
of	O
exact-test	O
options	O
in	O
the	O
main	O
statistical	O
software	O
packages	O
and	O
the	O
appearance	O
of	O
specialized	O
software	O
for	O
performing	O
a	O
wide	O
range	O
of	O
uni	O
-	O
and	O
multi-variable	O
exact	B-General_Concept
tests	I-General_Concept
and	O
computing	O
test-based	O
"	O
exact	B-General_Concept
"	O
confidence	O
intervals	O
.	O
</s>
<s>
An	O
important	O
assumption	O
behind	O
a	O
permutation	B-General_Concept
test	I-General_Concept
is	O
that	O
the	O
observations	O
are	O
exchangeable	O
under	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
.	O
</s>
<s>
An	O
important	O
consequence	O
of	O
this	O
assumption	O
is	O
that	O
tests	O
of	O
difference	O
in	O
location	O
(	O
like	O
a	O
permutation	O
t-test	B-General_Concept
)	O
require	O
equal	O
variance	O
under	O
the	O
normality	O
assumption	O
.	O
</s>
<s>
In	O
this	O
respect	O
,	O
the	O
permutation	O
t-test	B-General_Concept
shares	O
the	O
same	O
weakness	O
as	O
the	O
classical	O
Student	B-General_Concept
's	I-General_Concept
t-test	I-General_Concept
(	O
the	O
Behrens	B-Algorithm
–	I-Algorithm
Fisher	I-Algorithm
problem	I-Algorithm
)	O
.	O
</s>
<s>
Good	O
(	O
2005	O
)	O
explains	O
the	O
difference	O
between	O
permutation	B-General_Concept
tests	I-General_Concept
and	O
bootstrap	O
tests	O
the	O
following	O
way	O
:	O
"	O
Permutations	O
test	O
hypotheses	O
concerning	O
distributions	O
;	O
bootstraps	O
test	O
hypotheses	O
concerning	O
parameters	O
.	O
</s>
<s>
Bootstrap	O
tests	O
are	O
not	O
exact	B-General_Concept
.	O
</s>
<s>
In	O
some	O
cases	O
,	O
a	O
permutation	B-General_Concept
test	I-General_Concept
based	O
on	O
a	O
properly	O
studentized	O
statistic	O
can	O
be	O
asymptotically	O
exact	B-General_Concept
even	O
when	O
the	O
exchangeability	O
assumption	O
is	O
violated	O
.	O
</s>
<s>
Bootstrap-based	O
tests	O
can	O
test	O
with	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
and	O
,	O
therefore	O
,	O
are	O
suited	O
for	O
performing	O
equivalence	O
testing	O
.	O
</s>
<s>
An	O
asymptotically	O
equivalent	O
permutation	B-General_Concept
test	I-General_Concept
can	O
be	O
created	O
when	O
there	O
are	O
too	O
many	O
possible	O
orderings	O
of	O
the	O
data	O
to	O
allow	O
complete	O
enumeration	O
in	O
a	O
convenient	O
manner	O
.	O
</s>
<s>
This	O
is	O
done	O
by	O
generating	O
the	O
reference	O
distribution	O
by	O
Monte	B-Algorithm
Carlo	I-Algorithm
sampling	I-Algorithm
,	O
which	O
takes	O
a	O
small	O
(	O
relative	O
to	O
the	O
total	O
number	O
of	O
permutations	O
)	O
random	O
sample	O
of	O
the	O
possible	O
replicates	O
.	O
</s>
<s>
The	O
realization	O
that	O
this	O
could	O
be	O
applied	O
to	O
any	O
permutation	B-General_Concept
test	I-General_Concept
on	O
any	O
dataset	O
was	O
an	O
important	O
breakthrough	O
in	O
the	O
area	O
of	O
applied	O
statistics	O
.	O
</s>
<s>
This	O
type	O
of	O
permutation	B-General_Concept
test	I-General_Concept
is	O
known	O
under	O
various	O
names	O
:	O
approximate	O
permutation	B-General_Concept
test	I-General_Concept
,	O
Monte	O
Carlo	O
permutation	B-General_Concept
tests	I-General_Concept
or	O
random	O
permutation	B-General_Concept
tests	I-General_Concept
.	O
</s>
<s>
On	O
the	O
other	O
hand	O
,	O
the	O
purpose	O
of	O
estimating	O
the	O
p-value	O
is	O
most	O
often	O
to	O
decide	O
whether	O
,	O
where	O
is	O
the	O
threshold	O
at	O
which	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
will	O
be	O
rejected	O
(	O
typically	O
)	O
.	O
</s>
<s>
it	O
is	O
completely	O
unclear	O
whether	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
should	O
be	O
rejected	O
at	O
a	O
level	O
.	O
</s>
<s>
Edgington	O
,	O
E	O
.	O
S.	O
,	O
&	O
Onghena	O
,	O
P	O
.	O
(	O
2007	O
)	O
Randomization	B-General_Concept
tests	I-General_Concept
,	O
4th	O
ed	O
.	O
</s>
<s>
(	O
1999	O
)	O
Data	O
Analysis	O
by	O
Resampling	B-General_Concept
,	O
Duxbury	O
Press	O
.	O
</s>
<s>
Multivariate	O
Permutation	B-General_Concept
Tests	I-General_Concept
:	O
With	O
Applications	O
in	O
Biostatistics	O
,	O
John	O
Wiley	O
&	O
Sons	O
.	O
</s>
<s>
(	O
2012	O
)	O
Practitioner	O
's	O
Guide	O
to	O
Resampling	B-General_Concept
Methods	O
.	O
</s>
