<s>
In	O
statistics	O
,	O
Fisher	B-General_Concept
's	I-General_Concept
method	I-General_Concept
,	O
also	O
known	O
as	O
Fisher	B-General_Concept
's	I-General_Concept
combined	I-General_Concept
probability	I-General_Concept
test	I-General_Concept
,	O
is	O
a	O
technique	O
for	O
data	B-General_Concept
fusion	I-General_Concept
or	O
"	O
meta-analysis	O
"	O
(	O
analysis	O
of	O
analyses	O
)	O
.	O
</s>
<s>
In	O
its	O
basic	O
form	O
,	O
it	O
is	O
used	O
to	O
combine	O
the	O
results	O
from	O
several	O
independence	B-General_Concept
tests	I-General_Concept
bearing	O
upon	O
the	O
same	O
overall	O
hypothesis	O
(	O
H0	O
)	O
.	O
</s>
<s>
where	O
pi	O
is	O
the	O
p-value	B-General_Concept
for	O
the	O
ith	O
hypothesis	O
test	O
.	O
</s>
<s>
When	O
the	O
p-values	B-General_Concept
tend	O
to	O
be	O
small	O
,	O
the	O
test	B-General_Concept
statistic	I-General_Concept
X2	O
will	O
be	O
large	O
,	O
which	O
suggests	O
that	O
the	O
null	O
hypotheses	O
are	O
not	O
true	O
for	O
every	O
test	O
.	O
</s>
<s>
When	O
all	O
the	O
null	O
hypotheses	O
are	O
true	O
,	O
and	O
the	O
pi	O
(	O
or	O
their	O
corresponding	O
test	B-General_Concept
statistics	I-General_Concept
)	O
are	O
independent	O
,	O
X2	O
has	O
a	O
chi-squared	O
distribution	O
with	O
2k	O
degrees	O
of	O
freedom	O
,	O
where	O
k	O
is	O
the	O
number	O
of	O
tests	O
being	O
combined	O
.	O
</s>
<s>
This	O
fact	O
can	O
be	O
used	O
to	O
determine	O
the	O
p-value	B-General_Concept
for	O
X2	O
.	O
</s>
<s>
The	O
distribution	O
of	O
X2	O
is	O
a	O
chi-squared	O
distribution	O
for	O
the	O
following	O
reason	O
;	O
under	O
the	O
null	O
hypothesis	O
for	O
test	O
i	O
,	O
the	O
p-value	B-General_Concept
pi	O
follows	O
a	O
uniform	O
distribution	O
on	O
the	O
interval	O
 [ 0 , 1 ] 	O
.	O
</s>
<s>
Dependence	O
among	O
statistical	O
tests	O
is	O
generally	O
positive	O
,	O
which	O
means	O
that	O
the	O
p-value	B-General_Concept
of	O
X2	O
is	O
too	O
small	O
(	O
anti-conservative	O
)	O
if	O
the	O
dependency	O
is	O
not	O
taken	O
into	O
account	O
.	O
</s>
<s>
Thus	O
,	O
if	O
Fisher	B-General_Concept
's	I-General_Concept
method	I-General_Concept
for	O
independent	O
tests	O
is	O
applied	O
in	O
a	O
dependent	O
setting	O
,	O
and	O
the	O
p-value	B-General_Concept
is	O
not	O
small	O
enough	O
to	O
reject	O
the	O
null	O
hypothesis	O
,	O
then	O
that	O
conclusion	O
will	O
continue	O
to	O
hold	O
even	O
if	O
the	O
dependence	O
is	O
not	O
properly	O
accounted	O
for	O
.	O
</s>
<s>
However	O
,	O
if	O
positive	O
dependence	O
is	O
not	O
accounted	O
for	O
,	O
and	O
the	O
meta-analysis	O
p-value	B-General_Concept
is	O
found	O
to	O
be	O
small	O
,	O
the	O
evidence	O
against	O
the	O
null	O
hypothesis	O
is	O
generally	O
overstated	O
.	O
</s>
<s>
The	O
mean	O
false	O
discovery	O
rate	O
,	O
,	O
reduced	O
for	O
k	O
independent	O
or	O
positively	O
correlated	O
tests	O
,	O
may	O
suffice	O
to	O
control	O
alpha	O
for	O
useful	O
comparison	O
to	O
an	O
over-small	O
p-value	B-General_Concept
from	O
Fisher'sX2	O
.	O
</s>
<s>
Different	O
approaches	O
may	O
be	O
used	O
depending	O
on	O
whether	O
or	O
not	O
the	O
covariance	O
between	O
the	O
different	O
p-values	B-General_Concept
is	O
known	O
.	O
</s>
<s>
Brown	O
's	O
method	O
can	O
be	O
used	O
to	O
combine	O
dependent	O
p-values	B-General_Concept
whose	O
underlying	O
test	B-General_Concept
statistics	I-General_Concept
have	O
a	O
multivariate	O
normal	O
distribution	O
with	O
a	O
known	O
covariance	O
matrix	O
.	O
</s>
<s>
Kost	O
's	O
method	O
extends	O
Brown	O
's	O
to	O
allow	O
one	O
to	O
combine	O
p-values	B-General_Concept
when	O
the	O
covariance	O
matrix	O
is	O
known	O
only	O
up	O
to	O
a	O
scalar	O
multiplicative	O
factor	O
.	O
</s>
<s>
The	O
harmonic	B-General_Concept
mean	I-General_Concept
p-value	I-General_Concept
offers	O
an	O
alternative	O
to	O
Fisher	B-General_Concept
's	I-General_Concept
method	I-General_Concept
for	O
combining	O
p-values	B-General_Concept
when	O
the	O
dependency	O
structure	O
is	O
unknown	O
but	O
the	O
tests	O
cannot	O
be	O
assumed	O
to	O
be	O
independent	O
.	O
</s>
<s>
Fisher	B-General_Concept
's	I-General_Concept
method	I-General_Concept
is	O
typically	O
applied	O
to	O
a	O
collection	O
of	O
independent	O
test	B-General_Concept
statistics	I-General_Concept
,	O
usually	O
from	O
separate	O
studies	O
having	O
the	O
same	O
null	O
hypothesis	O
.	O
</s>
<s>
A	O
common	O
reason	O
for	O
the	O
latter	O
form	O
of	O
heterogeneity	O
is	O
that	O
effect	B-Application
sizes	I-Application
may	O
differ	O
among	O
populations	O
.	O
</s>
<s>
Any	O
discrepancies	O
among	O
the	O
results	O
from	O
separate	O
studies	O
or	O
experiments	O
must	O
be	O
due	O
to	O
chance	O
,	O
possibly	O
driven	O
by	O
differences	O
in	O
power	B-General_Concept
.	O
</s>
<s>
A	O
closely	O
related	O
approach	O
to	O
Fisher	B-General_Concept
's	I-General_Concept
method	I-General_Concept
is	O
Stouffer	O
's	O
Z	O
,	O
based	O
on	O
Z-scores	O
rather	O
than	O
p-values	B-General_Concept
,	O
allowing	O
incorporation	O
of	O
study	O
weights	O
.	O
</s>
<s>
This	O
Z-score	O
is	O
appropriate	O
for	O
one-sided	O
right-tailed	O
p-values	B-General_Concept
;	O
minor	O
modifications	O
can	O
be	O
made	O
if	O
two-sided	O
or	O
left-tailed	O
p-values	B-General_Concept
are	O
being	O
analysed	O
.	O
</s>
<s>
Specifically	O
,	O
if	O
two-sided	O
p-values	B-General_Concept
are	O
being	O
analyzed	O
,	O
the	O
two-sided	O
p-value	B-General_Concept
(	O
pi/2	O
)	O
is	O
used	O
,	O
or	O
1-pi	O
if	O
left-tailed	O
p-values	B-General_Concept
are	O
used	O
.	O
</s>
<s>
Since	O
Fisher	B-General_Concept
's	I-General_Concept
method	I-General_Concept
is	O
based	O
on	O
the	O
average	O
of	O
log(pi )	O
values	O
,	O
and	O
the	O
Z-score	O
method	O
is	O
based	O
on	O
the	O
average	O
of	O
the	O
Zi	O
values	O
,	O
the	O
relationship	O
between	O
these	O
two	O
approaches	O
follows	O
from	O
the	O
relationship	O
between	O
z	O
and	O
log(p )	O
=	O
log(1Φ(z )	O
)	O
.	O
</s>
<s>
As	O
a	O
result	O
,	O
the	O
power	B-General_Concept
of	O
the	O
Z-score	O
method	O
is	O
nearly	O
identical	O
to	O
the	O
power	B-General_Concept
of	O
Fisher	B-General_Concept
's	I-General_Concept
method	I-General_Concept
.	O
</s>
<s>
While	O
weighted	O
versions	O
of	O
Fisher	O
's	O
statistic	O
can	O
be	O
derived	O
,	O
the	O
null	O
distribution	O
becomes	O
a	O
weighted	O
sum	O
of	O
independent	O
chi-squared	B-General_Concept
statistics	I-General_Concept
,	O
which	O
is	O
less	O
convenient	O
to	O
work	O
with	O
.	O
</s>
