<s>
In	O
null-hypothesis	B-General_Concept
significance	O
testing	O
,	O
the	O
p-value	B-General_Concept
is	O
the	O
probability	O
of	O
obtaining	O
test	O
results	O
at	O
least	O
as	O
extreme	O
as	O
the	O
result	O
actually	O
observed	O
,	O
under	O
the	O
assumption	O
that	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
is	O
correct	O
.	O
</s>
<s>
A	O
very	O
small	O
p-value	B-General_Concept
means	O
that	O
such	O
an	O
extreme	O
observed	O
outcome	O
would	O
be	O
very	O
unlikely	O
under	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
.	O
</s>
<s>
Reporting	O
p-values	B-General_Concept
of	O
statistical	O
tests	O
is	O
common	O
practice	O
in	O
academic	O
publications	O
of	O
many	O
quantitative	O
fields	O
.	O
</s>
<s>
Since	O
the	O
precise	O
meaning	O
of	O
p-value	B-General_Concept
is	O
hard	O
to	O
grasp	O
,	O
misuse	B-General_Concept
is	I-General_Concept
widespread	I-General_Concept
and	O
has	O
been	O
a	O
major	O
topic	O
in	O
metascience	O
.	O
</s>
<s>
If	O
we	O
state	O
one	O
hypothesis	O
only	O
and	O
the	O
aim	O
of	O
the	O
statistical	O
test	O
is	O
to	O
see	O
whether	O
this	O
hypothesis	O
is	O
tenable	O
,	O
but	O
not	O
to	O
investigate	O
other	O
specific	O
hypotheses	O
,	O
then	O
such	O
a	O
test	O
is	O
called	O
a	O
null	B-General_Concept
hypothesis	I-General_Concept
test	O
.	O
</s>
<s>
As	O
our	O
statistical	O
hypothesis	O
will	O
,	O
by	O
definition	O
,	O
state	O
some	O
property	O
of	O
the	O
distribution	O
,	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
is	O
the	O
default	O
hypothesis	O
under	O
which	O
that	O
property	O
does	O
not	O
exist	O
.	O
</s>
<s>
The	O
null	B-General_Concept
hypothesis	I-General_Concept
is	O
typically	O
that	O
some	O
parameter	O
(	O
such	O
as	O
a	O
correlation	O
or	O
a	O
difference	O
between	O
means	O
)	O
in	O
the	O
populations	O
of	O
interest	O
is	O
zero	O
.	O
</s>
<s>
The	O
p-value	B-General_Concept
is	O
used	O
in	O
the	O
context	O
of	O
null	B-General_Concept
hypothesis	I-General_Concept
testing	O
in	O
order	O
to	O
quantify	O
the	O
statistical	B-General_Concept
significance	I-General_Concept
of	O
a	O
result	O
,	O
the	O
result	O
being	O
the	O
observed	O
value	O
of	O
the	O
chosen	O
statistic	O
.	O
</s>
<s>
The	O
lower	O
the	O
p-value	B-General_Concept
is	O
,	O
the	O
lower	O
the	O
probability	O
of	O
getting	O
that	O
result	O
if	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
were	O
true	O
.	O
</s>
<s>
A	O
result	O
is	O
said	O
to	O
be	O
statistically	B-General_Concept
significant	I-General_Concept
if	O
it	O
allows	O
us	O
to	O
reject	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
.	O
</s>
<s>
All	O
other	O
things	O
being	O
equal	O
,	O
smaller	O
p-values	B-General_Concept
are	O
taken	O
as	O
stronger	O
evidence	O
against	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
.	O
</s>
<s>
Loosely	O
speaking	O
,	O
rejection	O
of	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
implies	O
that	O
there	O
is	O
sufficient	O
evidence	O
against	O
it	O
.	O
</s>
<s>
As	O
a	O
particular	O
example	O
,	O
if	O
a	O
null	B-General_Concept
hypothesis	I-General_Concept
states	O
that	O
a	O
certain	O
summary	O
statistic	O
follows	O
the	O
standard	O
normal	O
distribution	O
N(0,1 )	O
,	O
then	O
the	O
rejection	O
of	O
this	O
null	B-General_Concept
hypothesis	I-General_Concept
could	O
mean	O
that	O
(	O
i	O
)	O
the	O
mean	O
of	O
is	O
not	O
0	O
,	O
or	O
(	O
ii	O
)	O
the	O
variance	O
of	O
is	O
not	O
1	O
,	O
or	O
(	O
iii	O
)	O
is	O
not	O
normally	O
distributed	O
.	O
</s>
<s>
Different	O
tests	O
of	O
the	O
same	O
null	B-General_Concept
hypothesis	I-General_Concept
would	O
be	O
more	O
or	O
less	O
sensitive	O
to	O
different	O
alternatives	O
.	O
</s>
<s>
However	O
,	O
even	O
if	O
we	O
do	O
manage	O
to	O
reject	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
for	O
all	O
3	O
alternatives	O
,	O
and	O
even	O
if	O
we	O
know	O
the	O
distribution	O
is	O
normal	O
and	O
variance	O
is	O
1	O
,	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
test	O
does	O
not	O
tell	O
us	O
which	O
non-zero	O
values	O
of	O
the	O
mean	O
are	O
now	O
most	O
plausible	O
.	O
</s>
<s>
Then	O
the	O
p-value	B-General_Concept
is	O
what	O
the	O
prior	O
probability	O
would	O
be	O
of	O
observing	O
a	O
test-statistic	O
value	O
at	O
least	O
as	O
"	O
extreme	O
"	O
as	O
if	O
null	B-General_Concept
hypothesis	I-General_Concept
were	O
true	O
.	O
</s>
<s>
for	O
a	O
two-sided	B-General_Concept
test	I-General_Concept
.	O
</s>
<s>
In	O
a	O
significance	O
test	O
,	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
is	O
rejected	O
if	O
the	O
p-value	B-General_Concept
is	O
less	O
than	O
or	O
equal	O
to	O
a	O
predefined	O
threshold	O
value	O
,	O
which	O
is	O
referred	O
to	O
as	O
the	O
alpha	O
level	O
or	O
significance	B-General_Concept
level	I-General_Concept
.	O
</s>
<s>
In	O
2018	O
,	O
a	O
group	O
of	O
statisticians	O
led	O
by	O
Daniel	O
Benjamin	O
proposed	O
the	O
adaptation	O
of	O
the	O
0.005	O
value	O
as	O
standard	O
value	O
for	O
statistical	B-General_Concept
significance	I-General_Concept
worldwide	O
.	O
</s>
<s>
The	O
p-value	B-General_Concept
is	O
a	O
function	O
of	O
the	O
chosen	O
test	B-General_Concept
statistic	I-General_Concept
and	O
is	O
therefore	O
a	O
random	O
variable	O
.	O
</s>
<s>
If	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
fixes	O
the	O
probability	O
distribution	O
of	O
precisely	O
,	O
and	O
if	O
that	O
distribution	O
is	O
continuous	O
,	O
then	O
when	O
the	O
null-hypothesis	B-General_Concept
is	O
true	O
,	O
the	O
p-value	B-General_Concept
is	O
uniformly	O
distributed	O
between	O
0	O
and	O
1	O
.	O
</s>
<s>
Thus	O
,	O
the	O
p-value	B-General_Concept
is	O
not	O
fixed	O
.	O
</s>
<s>
If	O
the	O
same	O
test	O
is	O
repeated	O
independently	O
with	O
fresh	O
data	O
,	O
one	O
will	O
typically	O
obtain	O
a	O
different	O
p-value	B-General_Concept
in	O
each	O
iteration	O
.	O
</s>
<s>
If	O
the	O
null-hypothesis	B-General_Concept
is	O
composite	O
,	O
or	O
the	O
distribution	O
of	O
the	O
statistic	O
is	O
discrete	O
,	O
the	O
probability	O
of	O
obtaining	O
a	O
p-value	B-General_Concept
less	O
than	O
or	O
equal	O
to	O
any	O
number	O
between	O
0	O
and	O
1	O
is	O
less	O
than	O
or	O
equal	O
to	O
that	O
number	O
,	O
if	O
the	O
null-hypothesis	B-General_Concept
is	O
true	O
.	O
</s>
<s>
It	O
remains	O
the	O
case	O
that	O
very	O
small	O
values	O
are	O
relatively	O
unlikely	O
if	O
the	O
null-hypothesis	B-General_Concept
is	O
true	O
,	O
and	O
that	O
a	O
significance	O
test	O
at	O
level	O
is	O
obtained	O
by	O
rejecting	O
the	O
null-hypothesis	B-General_Concept
if	O
the	O
significance	B-General_Concept
level	I-General_Concept
is	O
less	O
than	O
or	O
equal	O
to	O
.	O
</s>
<s>
Different	O
p-values	B-General_Concept
based	O
on	O
independent	O
sets	O
of	O
data	O
can	O
be	O
combined	O
,	O
for	O
instance	O
using	O
Fisher	B-General_Concept
's	I-General_Concept
combined	I-General_Concept
probability	I-General_Concept
test	I-General_Concept
.	O
</s>
<s>
When	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
is	O
true	O
,	O
if	O
it	O
takes	O
the	O
form	O
,	O
and	O
the	O
underlying	O
random	O
variable	O
is	O
continuous	O
,	O
then	O
the	O
probability	O
distribution	O
of	O
the	O
p-value	B-General_Concept
is	O
uniform	O
on	O
the	O
interval	O
 [ 0 , 1 ] 	O
.	O
</s>
<s>
The	O
distribution	O
of	O
p-values	B-General_Concept
for	O
a	O
group	O
of	O
studies	O
is	O
sometimes	O
called	O
a	O
p-curve	O
.	O
</s>
<s>
A	O
p-curve	O
can	O
be	O
used	O
to	O
assess	O
the	O
reliability	O
of	O
scientific	O
literature	O
,	O
such	O
as	O
by	O
detecting	O
publication	O
bias	O
or	O
p-hacking	B-Algorithm
.	O
</s>
<s>
For	O
example	O
,	O
when	O
testing	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
that	O
a	O
distribution	O
is	O
normal	O
with	O
a	O
mean	O
less	O
than	O
or	O
equal	O
to	O
zero	O
against	O
the	O
alternative	O
that	O
the	O
mean	O
is	O
greater	O
than	O
zero	O
(	O
variance	O
known	O
)	O
,	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
does	O
not	O
specify	O
the	O
probability	O
distribution	O
of	O
the	O
appropriate	O
test	B-General_Concept
statistic	I-General_Concept
.	O
</s>
<s>
In	O
the	O
just	O
mentioned	O
example	O
that	O
would	O
be	O
the	O
Z-statistic	O
belonging	O
to	O
the	O
one-sided	O
one-sample	O
Z-test	B-General_Concept
.	O
</s>
<s>
For	O
each	O
possible	O
value	O
of	O
the	O
theoretical	O
mean	O
,	O
the	O
Z-test	B-General_Concept
statistic	O
has	O
a	O
different	O
probability	O
distribution	O
.	O
</s>
<s>
In	O
these	O
circumstances	O
(	O
the	O
case	O
of	O
a	O
so-called	O
composite	O
null	B-General_Concept
hypothesis	I-General_Concept
)	O
the	O
p-value	B-General_Concept
is	O
defined	O
by	O
taking	O
the	O
least	O
favourable	O
null-hypothesis	B-General_Concept
case	O
,	O
which	O
is	O
typically	O
on	O
the	O
border	O
between	O
null	O
and	O
alternative	O
.	O
</s>
<s>
This	O
definition	O
ensures	O
the	O
complementarity	O
of	O
p-values	B-General_Concept
and	O
alpha-levels	O
.	O
</s>
<s>
If	O
we	O
set	O
the	O
significance	B-General_Concept
level	I-General_Concept
alpha	O
to	O
0.05	O
,	O
and	O
only	O
reject	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
if	O
the	O
p-value	B-General_Concept
is	O
less	O
than	O
or	O
equal	O
to	O
0.05	O
,	O
then	O
our	O
hypothesis	O
test	O
will	O
indeed	O
have	O
significance	B-General_Concept
level	I-General_Concept
(	O
maximal	O
type	O
1	O
error	O
rate	O
)	O
0.05	O
.	O
</s>
<s>
The	O
p-value	B-General_Concept
is	O
widely	O
used	O
in	O
statistical	O
hypothesis	O
testing	O
,	O
specifically	O
in	O
null	B-General_Concept
hypothesis	I-General_Concept
significance	O
testing	O
.	O
</s>
<s>
In	O
this	O
method	O
,	O
before	O
conducting	O
the	O
study	O
,	O
one	O
first	O
chooses	O
a	O
model	O
(	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
)	O
and	O
the	O
alpha	O
level	O
α	O
(	O
most	O
commonly	O
.05	O
)	O
.	O
</s>
<s>
After	O
analyzing	O
the	O
data	O
,	O
if	O
the	O
p-value	B-General_Concept
is	O
less	O
than	O
α	O
,	O
that	O
is	O
taken	O
to	O
mean	O
that	O
the	O
observed	O
data	O
is	O
sufficiently	O
inconsistent	O
with	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
for	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
to	O
be	O
rejected	O
.	O
</s>
<s>
However	O
,	O
that	O
does	O
not	O
prove	O
that	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
is	O
false	O
.	O
</s>
<s>
The	O
p-value	B-General_Concept
does	O
not	O
,	O
in	O
itself	O
,	O
establish	O
probabilities	O
of	O
hypotheses	O
.	O
</s>
<s>
Rather	O
,	O
it	O
is	O
a	O
tool	O
for	O
deciding	O
whether	O
to	O
reject	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
.	O
</s>
<s>
According	O
to	O
the	O
ASA	O
,	O
there	O
is	O
widespread	O
agreement	O
that	O
p-values	B-General_Concept
are	O
often	O
misused	O
and	O
misinterpreted	O
.	O
</s>
<s>
One	O
practice	O
that	O
has	O
been	O
particularly	O
criticized	O
is	O
accepting	O
the	O
alternative	O
hypothesis	O
for	O
any	O
p-value	B-General_Concept
nominally	O
less	O
than	O
.05	O
without	O
other	O
supporting	O
evidence	O
.	O
</s>
<s>
Although	O
p-values	B-General_Concept
are	O
helpful	O
in	O
assessing	O
how	O
incompatible	O
the	O
data	O
are	O
with	O
a	O
specified	O
statistical	O
model	O
,	O
contextual	O
factors	O
must	O
also	O
be	O
considered	O
,	O
such	O
as	O
"	O
the	O
design	O
of	O
a	O
study	O
,	O
the	O
quality	O
of	O
the	O
measurements	O
,	O
the	O
external	O
evidence	O
for	O
the	O
phenomenon	O
under	O
study	O
,	O
and	O
the	O
validity	O
of	O
assumptions	O
that	O
underlie	O
the	O
data	O
analysis	O
"	O
.	O
</s>
<s>
Another	O
concern	O
is	O
that	O
the	O
p-value	B-General_Concept
is	O
often	O
misunderstood	O
as	O
being	O
the	O
probability	O
that	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
is	O
true	O
.	O
</s>
<s>
Some	O
statisticians	O
have	O
proposed	O
abandoning	O
p-values	B-General_Concept
and	O
focusing	O
more	O
on	O
other	O
inferential	O
statistics	O
,	O
such	O
as	O
confidence	O
intervals	O
,	O
likelihood	O
ratios	O
,	O
or	O
Bayes	B-General_Concept
factors	I-General_Concept
,	O
but	O
there	O
is	O
heated	O
debate	O
on	O
the	O
feasibility	O
of	O
these	O
alternatives	O
.	O
</s>
<s>
Others	O
have	O
suggested	O
to	O
remove	O
fixed	O
significance	O
thresholds	O
and	O
to	O
interpret	O
p-values	B-General_Concept
as	O
continuous	O
indices	O
of	O
the	O
strength	O
of	O
evidence	O
against	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
.	O
</s>
<s>
Yet	O
others	O
suggested	O
to	O
report	O
alongside	O
p-values	B-General_Concept
the	O
prior	O
probability	O
of	O
a	O
real	O
effect	O
that	O
would	O
be	O
required	O
to	O
obtain	O
a	O
false	O
positive	O
risk	O
(	O
i.e.	O
</s>
<s>
Usually	O
,	O
is	O
a	O
test	B-General_Concept
statistic	I-General_Concept
.	O
</s>
<s>
A	O
test	B-General_Concept
statistic	I-General_Concept
is	O
the	O
output	O
of	O
a	O
scalar	O
function	O
of	O
all	O
the	O
observations	O
.	O
</s>
<s>
As	O
such	O
,	O
the	O
test	B-General_Concept
statistic	I-General_Concept
follows	O
a	O
distribution	O
determined	O
by	O
the	O
function	O
used	O
to	O
define	O
that	O
test	B-General_Concept
statistic	I-General_Concept
and	O
the	O
distribution	O
of	O
the	O
input	O
observational	O
data	O
.	O
</s>
<s>
For	O
the	O
important	O
case	O
in	O
which	O
the	O
data	O
are	O
hypothesized	O
to	O
be	O
a	O
random	O
sample	O
from	O
a	O
normal	O
distribution	O
,	O
depending	O
on	O
the	O
nature	O
of	O
the	O
test	B-General_Concept
statistic	I-General_Concept
and	O
the	O
hypotheses	O
of	O
interest	O
about	O
its	O
distribution	O
,	O
different	O
null	B-General_Concept
hypothesis	I-General_Concept
tests	O
have	O
been	O
developed	O
.	O
</s>
<s>
Some	O
such	O
tests	O
are	O
the	O
z-test	B-General_Concept
for	O
hypotheses	O
concerning	O
the	O
mean	O
of	O
a	O
normal	O
distribution	O
with	O
known	O
variance	O
,	O
the	O
t-test	B-General_Concept
based	O
on	O
Student	O
's	O
t-distribution	O
of	O
a	O
suitable	O
statistic	O
for	O
hypotheses	O
concerning	O
the	O
mean	O
of	O
a	O
normal	O
distribution	O
when	O
the	O
variance	O
is	O
unknown	O
,	O
the	O
F-test	B-General_Concept
based	O
on	O
the	O
F-distribution	B-General_Concept
of	O
yet	O
another	O
statistic	O
for	O
hypotheses	O
concerning	O
the	O
variance	O
.	O
</s>
<s>
For	O
data	O
of	O
other	O
nature	O
,	O
for	O
instance	O
categorical	O
(	O
discrete	O
)	O
data	O
,	O
test	B-General_Concept
statistics	I-General_Concept
might	O
be	O
constructed	O
whose	O
null	B-General_Concept
hypothesis	I-General_Concept
distribution	O
is	O
based	O
on	O
normal	O
approximations	O
to	O
appropriate	O
statistics	O
obtained	O
by	O
invoking	O
the	O
central	O
limit	O
theorem	O
for	O
large	O
samples	O
,	O
as	O
in	O
the	O
case	O
of	O
Pearson	B-General_Concept
's	I-General_Concept
chi-squared	I-General_Concept
test	I-General_Concept
.	O
</s>
<s>
Thus	O
computing	O
a	O
p-value	B-General_Concept
requires	O
a	O
null	B-General_Concept
hypothesis	I-General_Concept
,	O
a	O
test	B-General_Concept
statistic	I-General_Concept
(	O
together	O
with	O
deciding	O
whether	O
the	O
researcher	O
is	O
performing	O
a	O
one-tailed	B-General_Concept
test	I-General_Concept
or	O
a	O
two-tailed	B-General_Concept
test	I-General_Concept
)	O
,	O
and	O
data	O
.	O
</s>
<s>
Even	O
though	O
computing	O
the	O
test	B-General_Concept
statistic	I-General_Concept
on	O
given	O
data	O
may	O
be	O
easy	O
,	O
computing	O
the	O
sampling	O
distribution	O
under	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
,	O
and	O
then	O
computing	O
its	O
cumulative	O
distribution	O
function	O
(	O
CDF	O
)	O
is	O
often	O
a	O
difficult	O
problem	O
.	O
</s>
<s>
Today	O
,	O
this	O
computation	O
is	O
done	O
using	O
statistical	O
software	O
,	O
often	O
via	O
numeric	O
methods	O
(	O
rather	O
than	O
exact	O
formulae	O
)	O
,	O
but	O
,	O
in	O
the	O
early	O
and	O
mid	O
20th	O
century	O
,	O
this	O
was	O
instead	O
done	O
via	O
tables	O
of	O
values	O
,	O
and	O
one	O
interpolated	O
or	O
extrapolated	O
p-values	B-General_Concept
from	O
these	O
discrete	O
values	O
.	O
</s>
<s>
Rather	O
than	O
using	O
a	O
table	O
of	O
p-values	B-General_Concept
,	O
Fisher	O
instead	O
inverted	O
the	O
CDF	O
,	O
publishing	O
a	O
list	O
of	O
values	O
of	O
the	O
test	B-General_Concept
statistic	I-General_Concept
for	O
given	O
fixed	O
p-values	B-General_Concept
;	O
this	O
corresponds	O
to	O
computing	O
the	O
Quantile	O
function	O
(	O
inverse	O
CDF	O
)	O
.	O
</s>
<s>
The	O
null	B-General_Concept
hypothesis	I-General_Concept
is	O
that	O
the	O
coin	O
is	O
fair	O
,	O
and	O
coin	O
tosses	O
are	O
independent	O
of	O
one	O
another	O
.	O
</s>
<s>
If	O
a	O
right-tailed	O
test	O
is	O
considered	O
,	O
which	O
would	O
be	O
the	O
case	O
if	O
one	O
is	O
actually	O
interested	O
in	O
the	O
possibility	O
that	O
the	O
coin	O
is	O
biased	O
towards	O
falling	O
heads	O
,	O
then	O
the	O
p-value	B-General_Concept
of	O
this	O
result	O
is	O
the	O
chance	O
of	O
a	O
fair	O
coin	O
landing	O
on	O
heads	O
at	O
least	O
14	O
times	O
out	O
of	O
20	O
flips	O
.	O
</s>
<s>
This	O
probability	O
is	O
the	O
p-value	B-General_Concept
,	O
considering	O
only	O
extreme	O
results	O
that	O
favor	O
heads	O
.	O
</s>
<s>
This	O
is	O
called	O
a	O
one-tailed	B-General_Concept
test	I-General_Concept
.	O
</s>
<s>
The	O
two-tailed	O
p-value	B-General_Concept
,	O
which	O
considers	O
deviations	O
favoring	O
either	O
heads	O
or	O
tails	O
,	O
may	O
instead	O
be	O
calculated	O
.	O
</s>
<s>
As	O
the	O
binomial	O
distribution	O
is	O
symmetrical	O
for	O
a	O
fair	O
coin	O
,	O
the	O
two-sided	O
p-value	B-General_Concept
is	O
simply	O
twice	O
the	O
above	O
calculated	O
single-sided	O
p-value	B-General_Concept
:	O
the	O
two-sided	O
p-value	B-General_Concept
is	O
0.115	O
.	O
</s>
<s>
Two-tailed	O
p-value	B-General_Concept
of	O
observation	O
O	O
given	O
H0	O
=	O
2	O
×	O
min( Pr( no	O
.	O
</s>
<s>
Here	O
,	O
the	O
calculated	O
p-value	B-General_Concept
exceeds	O
.05	O
,	O
meaning	O
that	O
the	O
data	O
falls	O
within	O
the	O
range	O
of	O
what	O
would	O
happen	O
95%	O
of	O
the	O
time	O
,	O
if	O
the	O
coin	O
were	O
fair	O
.	O
</s>
<s>
Hence	O
,	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
is	O
not	O
rejected	O
at	O
the	O
.05	O
level	O
.	O
</s>
<s>
However	O
,	O
had	O
one	O
more	O
head	O
been	O
obtained	O
,	O
the	O
resulting	O
p-value	B-General_Concept
(	O
two-tailed	O
)	O
would	O
have	O
been0.0414	O
(	O
4.14	O
%	O
)	O
,	O
in	O
which	O
case	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
would	O
be	O
rejected	O
at	O
the	O
.05	O
level	O
.	O
</s>
<s>
We	O
now	O
calculate	O
the	O
p-value	B-General_Concept
of	O
the	O
"	O
3	O
heads	O
3	O
tails	O
"	O
outcome	O
.	O
</s>
<s>
If	O
we	O
use	O
the	O
test	B-General_Concept
statistic	I-General_Concept
,	O
then	O
under	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
is	O
exactly	O
1	O
for	O
two-sided	O
p-value	B-General_Concept
,	O
and	O
exactly	O
for	O
one-sided	O
left-tail	O
p-value	B-General_Concept
,	O
and	O
same	O
for	O
one-sided	O
right-tail	O
p-value	B-General_Concept
.	O
</s>
<s>
If	O
we	O
consider	O
every	O
outcome	O
that	O
has	O
equal	O
or	O
lower	O
probability	O
than	O
"	O
3	O
heads	O
3	O
tails	O
"	O
as	O
"	O
at	O
least	O
as	O
extreme	O
"	O
,	O
then	O
the	O
p-value	B-General_Concept
is	O
exactly	O
.	O
</s>
<s>
However	O
,	O
suppose	O
we	O
have	O
planned	O
to	O
simply	O
flip	O
the	O
coin	O
6	O
times	O
no	O
matter	O
what	O
happens	O
,	O
then	O
the	O
second	O
definition	O
of	O
p-value	B-General_Concept
would	O
mean	O
that	O
the	O
p-value	B-General_Concept
of	O
"	O
3	O
heads	O
3	O
tails	O
"	O
is	O
exactly	O
1	O
.	O
</s>
<s>
Thus	O
,	O
the	O
"	O
at	O
least	O
as	O
extreme	O
"	O
definition	O
of	O
p-value	B-General_Concept
is	O
deeply	O
contextual	O
,	O
and	O
depends	O
on	O
what	O
the	O
experimenter	O
planned	O
to	O
do	O
even	O
in	O
situations	O
that	O
did	O
not	O
occur	O
.	O
</s>
<s>
P-value	B-General_Concept
computations	O
date	O
back	O
to	O
the	O
1700s	O
,	O
where	O
they	O
were	O
computed	O
for	O
the	O
human	O
sex	O
ratio	O
at	O
birth	O
,	O
and	O
used	O
to	O
compute	O
statistical	B-General_Concept
significance	I-General_Concept
compared	O
to	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
of	O
equal	O
probability	O
of	O
male	O
and	O
female	O
births	O
.	O
</s>
<s>
Considering	O
more	O
male	O
or	O
more	O
female	O
births	O
as	O
equally	O
likely	O
,	O
the	O
probability	O
of	O
the	O
observed	O
outcome	O
is	O
1/282	O
,	O
or	O
about	O
1	O
in	O
4,836,000,000,000,000,000,000,000	O
;	O
in	O
modern	O
terms	O
,	O
the	O
p-value	B-General_Concept
.	O
</s>
<s>
In	O
modern	O
terms	O
,	O
he	O
rejected	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
of	O
equally	O
likely	O
male	O
and	O
female	O
births	O
at	O
the	O
p	O
=	O
1/282	O
significance	B-General_Concept
level	I-General_Concept
.	O
</s>
<s>
This	O
and	O
other	O
work	O
by	O
Arbuthnot	O
is	O
credited	O
as	O
"	O
…	O
the	O
first	O
use	O
of	O
significance	O
tests	O
…	O
"	O
the	O
first	O
example	O
of	O
reasoning	O
about	O
statistical	B-General_Concept
significance	I-General_Concept
,	O
and	O
"	O
…	O
perhaps	O
the	O
first	O
published	O
report	O
of	O
a	O
nonparametric	B-General_Concept
test	I-General_Concept
…	O
"	O
,	O
specifically	O
the	O
sign	B-General_Concept
test	I-General_Concept
;	O
see	O
details	O
at	O
.	O
</s>
<s>
The	O
p-value	B-General_Concept
was	O
first	O
formally	O
introduced	O
by	O
Karl	O
Pearson	O
,	O
in	O
his	O
Pearson	B-General_Concept
's	I-General_Concept
chi-squared	I-General_Concept
test	I-General_Concept
,	O
using	O
the	O
chi-squared	O
distribution	O
and	O
notated	O
as	O
capital	O
P	O
.	O
The	O
p-values	B-General_Concept
for	O
the	O
chi-squared	O
distribution	O
(	O
for	O
various	O
values	O
of	O
χ2	O
and	O
degrees	O
of	O
freedom	O
)	O
,	O
now	O
notated	O
as	O
P	O
,	O
were	O
calculated	O
in	O
,	O
collected	O
in	O
.	O
</s>
<s>
The	O
use	O
of	O
the	O
p-value	B-General_Concept
in	O
statistics	O
was	O
popularized	O
by	O
Ronald	O
Fisher	O
,	O
and	O
it	O
plays	O
a	O
central	O
role	O
in	O
his	O
approach	O
to	O
the	O
subject	O
.	O
</s>
<s>
In	O
his	O
influential	O
book	O
Statistical	O
Methods	O
for	O
Research	O
Workers	O
(	O
1925	O
)	O
,	O
Fisher	O
proposed	O
the	O
level	O
p	O
=	O
0.05	O
,	O
or	O
a	O
1	O
in	O
20	O
chance	O
of	O
being	O
exceeded	O
by	O
chance	O
,	O
as	O
a	O
limit	O
for	O
statistical	B-General_Concept
significance	I-General_Concept
,	O
and	O
applied	O
this	O
to	O
a	O
normal	O
distribution	O
(	O
as	O
a	O
two-tailed	B-General_Concept
test	I-General_Concept
)	O
,	O
thus	O
yielding	O
the	O
rule	O
of	O
two	O
standard	O
deviations	O
(	O
on	O
a	O
normal	O
distribution	O
)	O
for	O
statistical	B-General_Concept
significance	I-General_Concept
(	O
see	O
68	O
–	O
95	O
–	O
99.7	O
rule	O
)	O
.	O
</s>
<s>
He	O
then	O
computed	O
a	O
table	O
of	O
values	O
,	O
similar	O
to	O
Elderton	O
but	O
,	O
importantly	O
,	O
reversed	O
the	O
roles	O
of	O
χ2	O
and	O
p	O
.	O
That	O
is	O
,	O
rather	O
than	O
computing	O
p	O
for	O
different	O
values	O
of	O
χ2	O
(	O
and	O
degrees	O
of	O
freedom	O
n	O
)	O
,	O
he	O
computed	O
values	O
of	O
χ2	O
that	O
yield	O
specified	O
p-values	B-General_Concept
,	O
specifically	O
0.99	O
,	O
0.98	O
,	O
0.95	O
,	O
0	O
,	O
90	O
,	O
0.80	O
,	O
0.70	O
,	O
0.50	O
,	O
0.30	O
,	O
0.20	O
,	O
0.10	O
,	O
0.05	O
,	O
0.02	O
,	O
and	O
0.01	O
.	O
</s>
<s>
That	O
allowed	O
computed	O
values	O
of	O
χ2	O
to	O
be	O
compared	O
against	O
cutoffs	O
and	O
encouraged	O
the	O
use	O
of	O
p-values	B-General_Concept
(	O
especially	O
0.05	O
,	O
0.02	O
,	O
and	O
0.01	O
)	O
as	O
cutoffs	O
,	O
instead	O
of	O
computing	O
and	O
reporting	O
p-values	B-General_Concept
themselves	O
.	O
</s>
<s>
As	O
an	O
illustration	O
of	O
the	O
application	O
of	O
p-values	B-General_Concept
to	O
the	O
design	O
and	O
interpretation	O
of	O
experiments	O
,	O
in	O
his	O
following	O
book	O
The	O
Design	O
of	O
Experiments	O
(	O
1935	O
)	O
,	O
Fisher	O
presented	O
the	O
lady	O
tasting	O
tea	O
experiment	O
,	O
which	O
is	O
the	O
archetypal	O
example	O
of	O
the	O
p-value	B-General_Concept
.	O
</s>
<s>
In	O
that	O
case	O
,	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
was	O
that	O
she	O
had	O
no	O
special	O
ability	O
,	O
the	O
test	O
was	O
Fisher	B-General_Concept
's	I-General_Concept
exact	I-General_Concept
test	I-General_Concept
,	O
and	O
the	O
p-value	B-General_Concept
was	O
so	O
Fisher	O
was	O
willing	O
to	O
reject	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
(	O
consider	O
the	O
outcome	O
highly	O
unlikely	O
to	O
be	O
due	O
to	O
chance	O
)	O
if	O
all	O
were	O
classified	O
correctly	O
.	O
</s>
<s>
He	O
also	O
applies	O
this	O
threshold	O
to	O
the	O
design	O
of	O
experiments	O
,	O
noting	O
that	O
had	O
only	O
6	O
cups	O
been	O
presented	O
(	O
3	O
of	O
each	O
)	O
,	O
a	O
perfect	O
classification	O
would	O
have	O
only	O
yielded	O
a	O
p-value	B-General_Concept
of	O
which	O
would	O
not	O
have	O
met	O
this	O
level	O
of	O
significance	O
.	O
</s>
<s>
Fisher	O
also	O
underlined	O
the	O
interpretation	O
of	O
p	O
,	O
as	O
the	O
long-run	O
proportion	O
of	O
values	O
at	O
least	O
as	O
extreme	O
as	O
the	O
data	O
,	O
assuming	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
is	O
true	O
.	O
</s>
<s>
In	O
later	O
editions	O
,	O
Fisher	O
explicitly	O
contrasted	O
the	O
use	O
of	O
the	O
p-value	B-General_Concept
for	O
statistical	O
inference	O
in	O
science	O
with	O
the	O
Neyman	O
–	O
Pearson	O
method	O
,	O
which	O
he	O
terms	O
"	O
Acceptance	O
Procedures	O
"	O
.	O
</s>
<s>
Fisher	O
emphasizes	O
that	O
while	O
fixed	O
levels	O
such	O
as	O
5%	O
,	O
2%	O
,	O
and	O
1%	O
are	O
convenient	O
,	O
the	O
exact	O
p-value	B-General_Concept
can	O
be	O
used	O
,	O
and	O
the	O
strength	O
of	O
evidence	O
can	O
and	O
will	O
be	O
revised	O
with	O
further	O
experimentation	O
.	O
</s>
<s>
The	O
E-value	B-General_Concept
corresponds	O
to	O
the	O
expected	O
number	O
of	O
times	O
in	O
multiple	B-General_Concept
testing	I-General_Concept
that	O
one	O
expects	O
to	O
obtain	O
a	O
test	B-General_Concept
statistic	I-General_Concept
at	O
least	O
as	O
extreme	O
as	O
the	O
one	O
that	O
was	O
actually	O
observed	O
if	O
one	O
assumes	O
that	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
is	O
true	O
.	O
</s>
<s>
The	O
E-value	B-General_Concept
is	O
the	O
product	O
of	O
the	O
number	O
of	O
tests	O
and	O
the	O
p-value	B-General_Concept
.	O
</s>
<s>
The	O
q-value	B-General_Concept
is	O
the	O
analog	O
of	O
the	O
p-value	B-General_Concept
with	O
respect	O
to	O
the	O
positive	O
false	O
discovery	O
rate	O
.	O
</s>
<s>
It	O
is	O
used	O
in	O
multiple	B-General_Concept
hypothesis	I-General_Concept
testing	I-General_Concept
to	O
maintain	O
statistical	O
power	O
while	O
minimizing	O
the	O
false	B-General_Concept
positive	I-General_Concept
rate	I-General_Concept
.	O
</s>
<s>
The	O
Probability	O
of	O
Direction	O
(	O
pd	O
)	O
is	O
the	O
Bayesian	O
numerical	O
equivalent	O
of	O
the	O
p-value	B-General_Concept
.	O
</s>
