<s>
In	O
statistics	O
,	O
an	O
exact	O
(	O
significance	O
)	O
test	O
is	O
a	O
test	O
such	O
that	O
if	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
is	O
true	O
,	O
then	O
all	O
assumptions	O
made	O
during	O
the	O
derivation	O
of	O
the	O
distribution	O
of	O
the	O
test	B-General_Concept
statistic	I-General_Concept
are	O
met	O
.	O
</s>
<s>
Using	O
an	O
exact	B-General_Concept
test	I-General_Concept
provides	O
a	O
significance	O
test	O
that	O
maintains	O
the	O
type	O
I	O
error	O
rate	O
of	O
the	O
test	O
(	O
)	O
at	O
the	O
desired	O
significance	B-General_Concept
level	I-General_Concept
of	O
the	O
test	O
.	O
</s>
<s>
For	O
example	O
,	O
an	O
exact	B-General_Concept
test	I-General_Concept
at	O
a	O
significance	B-General_Concept
level	I-General_Concept
of	O
,	O
when	O
repeated	O
over	O
many	O
samples	O
where	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
is	O
true	O
,	O
will	O
reject	O
at	O
most	O
of	O
the	O
time	O
.	O
</s>
<s>
Exact	B-General_Concept
tests	I-General_Concept
that	O
are	O
based	O
on	O
discrete	O
test	B-General_Concept
statistics	I-General_Concept
may	O
be	O
conservative	O
,	O
indicating	O
that	O
the	O
actual	O
rejection	O
rate	O
lies	O
below	O
the	O
nominal	O
significance	B-General_Concept
level	I-General_Concept
.	O
</s>
<s>
As	O
an	O
example	O
,	O
this	O
is	O
the	O
case	O
for	O
Fisher	B-General_Concept
's	I-General_Concept
exact	I-General_Concept
test	I-General_Concept
and	O
its	O
more	O
powerful	O
alternative	O
,	O
Boschloo	B-General_Concept
's	I-General_Concept
test	I-General_Concept
.	O
</s>
<s>
If	O
the	O
test	B-General_Concept
statistic	I-General_Concept
is	O
continuous	O
,	O
it	O
will	O
reach	O
the	O
significance	B-General_Concept
level	I-General_Concept
exactly	O
.	O
</s>
<s>
Parametric	B-General_Concept
tests	I-General_Concept
,	O
such	O
as	O
those	O
used	O
in	O
exact	B-General_Concept
statistics	I-General_Concept
,	O
are	O
exact	B-General_Concept
tests	I-General_Concept
when	O
the	O
parametric	O
assumptions	O
are	O
fully	O
met	O
,	O
but	O
in	O
practice	O
,	O
the	O
use	O
of	O
the	O
term	O
exact	O
(	O
significance	O
)	O
test	O
is	O
reserved	O
for	O
non-parametric	O
tests	O
,	O
i.e.	O
,	O
tests	O
that	O
do	O
not	O
rest	O
on	O
parametric	O
assumptions	O
.	O
</s>
<s>
Hence	O
,	O
when	O
a	O
result	O
of	O
statistical	O
analysis	O
is	O
termed	O
an	O
“	O
exact	B-General_Concept
test	I-General_Concept
”	O
or	O
specifies	O
an	O
“	O
exact	O
p-value	B-General_Concept
”	O
,	O
this	O
implies	O
that	O
the	O
test	O
is	O
defined	O
without	O
parametric	O
assumptions	O
and	O
is	O
evaluated	O
without	O
making	O
use	O
of	O
approximate	O
algorithms	O
.	O
</s>
<s>
In	O
principle	O
,	O
however	O
,	O
this	O
could	O
also	O
signify	O
that	O
a	O
parametric	B-General_Concept
test	I-General_Concept
has	O
been	O
employed	O
in	O
a	O
situation	O
where	O
all	O
parametric	O
assumptions	O
are	O
fully	O
met	O
,	O
but	O
it	O
is	O
in	O
most	O
cases	O
impossible	O
to	O
prove	O
this	O
completely	O
in	O
a	O
real-world	O
situation	O
.	O
</s>
<s>
Exceptions	O
in	O
which	O
it	O
is	O
certain	O
that	O
parametric	B-General_Concept
tests	I-General_Concept
are	O
exact	O
include	O
tests	O
based	O
on	O
the	O
binomial	O
or	O
Poisson	O
distributions	O
.	O
</s>
<s>
The	O
term	O
permutation	B-General_Concept
test	I-General_Concept
is	O
sometimes	O
used	O
as	O
a	O
synonym	O
for	O
exact	B-General_Concept
test	I-General_Concept
,	O
but	O
it	O
should	O
be	O
kept	O
in	O
mind	O
that	O
all	O
permutation	B-General_Concept
tests	I-General_Concept
are	O
exact	B-General_Concept
tests	I-General_Concept
,	O
but	O
not	O
all	O
exact	B-General_Concept
tests	I-General_Concept
are	O
permutation	B-General_Concept
tests	I-General_Concept
.	O
</s>
<s>
Pr(y )	O
is	O
the	O
probability	O
under	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
of	O
a	O
potentially	O
observed	O
outcome	O
y	O
,	O
</s>
<s>
T(y )	O
is	O
the	O
value	O
of	O
the	O
test	B-General_Concept
statistic	I-General_Concept
for	O
an	O
outcome	O
y	O
,	O
with	O
larger	O
values	O
of	O
T	O
representing	O
cases	O
which	O
notionally	O
represent	O
greater	O
departures	O
from	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
,	O
</s>
<s>
and	O
where	O
the	O
sum	O
ranges	O
over	O
all	O
outcomes	O
y	O
(	O
including	O
the	O
observed	O
one	O
)	O
that	O
have	O
the	O
same	O
value	O
of	O
the	O
test	B-General_Concept
statistic	I-General_Concept
obtained	O
for	O
the	O
observed	O
sample	O
x	O
,	O
or	O
a	O
larger	O
one	O
.	O
</s>
<s>
A	O
simple	O
example	O
of	O
this	O
concept	O
involves	O
the	O
observation	O
that	O
Pearson	B-General_Concept
's	I-General_Concept
chi-squared	I-General_Concept
test	I-General_Concept
is	O
an	O
approximate	O
test	O
.	O
</s>
<s>
Suppose	O
Pearson	B-General_Concept
's	I-General_Concept
chi-squared	I-General_Concept
test	I-General_Concept
is	O
used	O
to	O
ascertain	O
whether	O
a	O
six-sided	O
die	O
is	O
"	O
fair	O
"	O
,	O
indicating	O
that	O
it	O
renders	O
each	O
of	O
the	O
six	O
possible	O
outcomes	O
equally	O
often	O
.	O
</s>
<s>
If	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
of	O
"	O
fairness	O
"	O
is	O
true	O
,	O
then	O
the	O
probability	O
distribution	O
of	O
the	O
test	B-General_Concept
statistic	I-General_Concept
can	O
be	O
made	O
as	O
close	O
as	O
desired	O
to	O
the	O
chi-squared	O
distribution	O
with	O
5	O
degrees	O
of	O
freedom	O
by	O
making	O
the	O
sample	O
size	O
n	O
sufficiently	O
large	O
.	O
</s>
<s>
Finding	O
the	O
exact	O
probability	O
that	O
this	O
test	B-General_Concept
statistic	I-General_Concept
exceeds	O
a	O
certain	O
value	O
would	O
then	O
require	O
a	O
combinatorial	O
enumeration	O
of	O
all	O
outcomes	O
of	O
the	O
experiment	O
that	O
gives	O
rise	O
to	O
such	O
a	O
large	O
value	O
of	O
the	O
test	B-General_Concept
statistic	I-General_Concept
.	O
</s>
<s>
It	O
is	O
then	O
questionable	O
whether	O
the	O
same	O
test	B-General_Concept
statistic	I-General_Concept
ought	O
to	O
be	O
used	O
.	O
</s>
<s>
A	O
likelihood-ratio	B-General_Concept
test	I-General_Concept
might	O
be	O
preferred	O
,	O
and	O
the	O
test	B-General_Concept
statistic	I-General_Concept
might	O
not	O
be	O
a	O
monotone	O
function	O
of	O
the	O
one	O
above	O
.	O
</s>
<s>
Fisher	B-General_Concept
's	I-General_Concept
exact	I-General_Concept
test	I-General_Concept
,	O
based	O
on	O
the	O
work	O
of	O
Ronald	O
Fisher	O
and	O
E	O
.	O
J	O
.	O
G	O
.	O
Pitman	O
in	O
the	O
1930s	O
,	O
is	O
exact	O
because	O
the	O
sampling	O
distribution	O
(	O
conditional	O
on	O
the	O
marginals	O
)	O
is	O
known	O
exactly	O
.	O
</s>
<s>
This	O
should	O
be	O
compared	O
with	O
Pearson	B-General_Concept
's	I-General_Concept
chi-squared	I-General_Concept
test	I-General_Concept
,	O
which	O
(	O
although	O
it	O
tests	O
the	O
same	O
null	O
)	O
is	O
not	O
exact	O
because	O
the	O
distribution	O
of	O
the	O
test	B-General_Concept
statistic	I-General_Concept
is	O
only	O
asymptotically	O
correct	O
.	O
</s>
