<s>
The	O
harmonic	B-General_Concept
mean	I-General_Concept
p-value	I-General_Concept
(	O
HMP	O
)	O
is	O
a	O
statistical	O
technique	O
for	O
addressing	O
the	O
multiple	B-General_Concept
comparisons	I-General_Concept
problem	I-General_Concept
that	O
controls	O
the	O
strong-sense	O
family-wise	O
error	O
rate	O
(	O
this	O
claim	O
has	O
been	O
disputed	O
)	O
.	O
</s>
<s>
It	O
improves	O
on	O
the	O
power	B-General_Concept
of	O
Bonferroni	B-General_Concept
correction	I-General_Concept
by	O
performing	O
combined	O
tests	O
,	O
i.e.	O
</s>
<s>
by	O
testing	O
whether	O
groups	O
of	O
p-values	B-General_Concept
are	O
statistically	O
significant	O
,	O
like	O
Fisher	B-General_Concept
's	I-General_Concept
method	I-General_Concept
.	O
</s>
<s>
However	O
,	O
it	O
avoids	O
the	O
restrictive	O
assumption	O
that	O
the	O
p-values	B-General_Concept
are	O
independent	O
,	O
unlike	O
Fisher	B-General_Concept
's	I-General_Concept
method	I-General_Concept
.	O
</s>
<s>
Consequently	O
,	O
it	O
controls	O
the	O
false	O
positive	O
rate	O
when	O
tests	O
are	O
dependent	O
,	O
at	O
the	O
expense	O
of	O
less	O
power	B-General_Concept
(	O
i.e.	O
</s>
<s>
Besides	O
providing	O
an	O
alternative	O
to	O
approaches	O
such	O
as	O
Bonferroni	B-General_Concept
correction	I-General_Concept
that	O
controls	O
the	O
stringent	O
family-wise	O
error	O
rate	O
,	O
it	O
also	O
provides	O
an	O
alternative	O
to	O
the	O
widely-used	O
Benjamini-Hochberg	O
procedure	O
(	O
BH	O
)	O
for	O
controlling	O
the	O
less-stringent	O
false	B-General_Concept
discovery	I-General_Concept
rate	I-General_Concept
.	O
</s>
<s>
This	O
is	O
because	O
the	O
power	B-General_Concept
of	O
the	O
HMP	O
to	O
detect	O
significant	O
groups	O
of	O
hypotheses	O
is	O
greater	O
than	O
the	O
power	B-General_Concept
of	O
BH	O
to	O
detect	O
significant	O
individual	O
hypotheses	O
.	O
</s>
<s>
There	O
are	O
two	O
versions	O
of	O
the	O
technique	O
:	O
(	O
i	O
)	O
direct	O
interpretation	O
of	O
the	O
HMP	O
as	O
an	O
approximate	O
p-value	B-General_Concept
and	O
(	O
ii	O
)	O
a	O
procedure	O
for	O
transforming	O
the	O
HMP	O
into	O
an	O
asymptotically	O
exact	O
p-value	B-General_Concept
.	O
</s>
<s>
The	O
approach	O
provides	O
a	O
multilevel	O
test	O
procedure	O
in	O
which	O
the	O
smallest	O
groups	O
of	O
p-values	B-General_Concept
that	O
are	O
statistically	O
significant	O
may	O
be	O
sought	O
.	O
</s>
<s>
The	O
weighted	O
harmonic	O
mean	O
of	O
p-values	B-General_Concept
is	O
defined	O
as	O
where	O
are	O
weights	O
that	O
must	O
sum	O
to	O
one	O
,	O
i.e.	O
</s>
<s>
In	O
general	O
,	O
interpreting	O
the	O
HMP	O
directly	O
as	O
a	O
p-value	B-General_Concept
is	O
anti-conservative	O
,	O
meaning	O
that	O
the	O
false	O
positive	O
rate	O
is	O
higher	O
than	O
expected	O
.	O
</s>
<s>
Rather	O
than	O
applying	O
these	O
bounds	O
,	O
asymptotically	O
exact	O
p-values	B-General_Concept
can	O
be	O
produced	O
by	O
transforming	O
the	O
HMP	O
.	O
</s>
<s>
Generalized	O
central	O
limit	O
theorem	O
shows	O
that	O
an	O
asymptotically	O
exact	O
p-value	B-General_Concept
,	O
,	O
can	O
be	O
computed	O
from	O
the	O
HMP	O
,	O
,	O
using	O
the	O
formula	O
Subject	O
to	O
the	O
assumptions	O
of	O
generalized	O
central	O
limit	O
theorem	O
,	O
this	O
transformed	O
p-value	B-General_Concept
becomes	O
exact	O
as	O
the	O
number	O
of	O
tests	O
,	O
,	O
becomes	O
large	O
.	O
</s>
<s>
If	O
the	O
HMP	O
is	O
significant	O
at	O
some	O
level	O
for	O
a	O
group	O
of	O
p-values	B-General_Concept
,	O
one	O
may	O
search	O
all	O
subsets	O
of	O
the	O
p-values	B-General_Concept
for	O
the	O
smallest	O
significant	O
group	O
,	O
while	O
maintaining	O
the	O
strong-sense	O
family-wise	O
error	O
rate	O
.	O
</s>
<s>
Formally	O
,	O
this	O
constitutes	O
a	O
closed-testing	B-General_Concept
procedure	I-General_Concept
.	O
</s>
<s>
Reject	O
the	O
null	O
hypothesis	O
that	O
none	O
of	O
the	O
p-values	B-General_Concept
in	O
subset	O
are	O
significant	O
if	O
,	O
where	O
.	O
</s>
<s>
An	O
asymptotically	O
exact	O
version	O
of	O
the	O
above	O
replaces	O
in	O
step	O
2	O
with	O
where	O
gives	O
the	O
number	O
of	O
p-values	B-General_Concept
,	O
not	O
just	O
those	O
in	O
subset	O
.	O
</s>
<s>
Since	O
direct	O
interpretation	O
of	O
the	O
HMP	O
is	O
faster	O
,	O
a	O
two-pass	O
procedure	O
may	O
be	O
used	O
to	O
identify	O
subsets	O
of	O
p-values	B-General_Concept
that	O
are	O
likely	O
to	O
be	O
significant	O
using	O
direct	O
interpretation	O
,	O
subject	O
to	O
confirmation	O
using	O
the	O
asymptotically	O
exact	O
formula	O
.	O
</s>
<s>
Robust	O
to	O
positive	O
dependency	O
between	O
the	O
p-values	B-General_Concept
.	O
</s>
<s>
Most	O
influenced	O
by	O
the	O
smallest	O
p-values	B-General_Concept
.	O
</s>
<s>
Conversely	O
,	O
when	O
the	O
multilevel	O
test	O
deems	O
a	O
subset	O
of	O
p-values	B-General_Concept
to	O
be	O
significant	O
,	O
the	O
HMP	O
for	O
all	O
the	O
p-values	B-General_Concept
combined	O
is	O
likely	O
to	O
be	O
significant	O
;	O
this	O
is	O
certain	O
when	O
the	O
HMP	O
is	O
interpreted	O
directly	O
.	O
</s>
<s>
When	O
the	O
goal	O
is	O
to	O
assess	O
the	O
significance	O
of	O
individual	O
p-values	B-General_Concept
,	O
so	O
that	O
combined	O
tests	O
concerning	O
groups	O
of	O
p-values	B-General_Concept
are	O
of	O
no	O
interest	O
,	O
the	O
HMP	O
is	O
equivalent	O
to	O
the	O
Bonferroni	B-General_Concept
procedure	O
but	O
subject	O
to	O
the	O
more	O
stringent	O
significance	O
threshold	O
(	O
Table	O
1	O
)	O
.	O
</s>
<s>
The	O
HMP	O
assumes	O
the	O
individual	O
p-values	B-General_Concept
have	O
(	O
not	O
necessarily	O
independent	O
)	O
standard	O
uniform	O
distributions	O
when	O
their	O
null	O
hypotheses	O
are	O
true	O
.	O
</s>
<s>
Large	O
numbers	O
of	O
underpowered	O
tests	O
can	O
therefore	O
harm	O
the	O
power	B-General_Concept
of	O
the	O
HMP	O
.	O
</s>
<s>
While	O
the	O
choice	O
of	O
weights	O
is	O
unimportant	O
for	O
the	O
validity	O
of	O
the	O
HMP	O
under	O
the	O
null	O
hypothesis	O
,	O
the	O
weights	O
influence	O
the	O
power	B-General_Concept
of	O
the	O
procedure	O
.	O
</s>
<s>
The	O
HMP	O
was	O
conceived	O
by	O
analogy	O
to	O
Bayesian	O
model	O
averaging	O
and	O
can	O
be	O
interpreted	O
as	O
inversely	O
proportional	O
to	O
a	O
model-averaged	O
Bayes	B-General_Concept
factor	I-General_Concept
when	O
combining	O
p-values	B-General_Concept
from	O
likelihood	B-General_Concept
ratio	I-General_Concept
tests	I-General_Concept
.	O
</s>
<s>
Good	O
reported	O
an	O
empirical	O
relationship	O
between	O
the	O
Bayes	B-General_Concept
factor	I-General_Concept
and	O
the	O
p-value	B-General_Concept
from	O
a	O
likelihood	B-General_Concept
ratio	I-General_Concept
test	I-General_Concept
.	O
</s>
<s>
For	O
a	O
null	O
hypothesis	O
nested	O
in	O
a	O
more	O
general	O
alternative	O
hypothesis	O
he	O
observed	O
that	O
often	O
,	O
where	O
denotes	O
the	O
Bayes	B-General_Concept
factor	I-General_Concept
in	O
favour	O
of	O
versus	O
Extrapolating	O
,	O
he	O
proposed	O
a	O
rule	O
of	O
thumb	O
in	O
which	O
the	O
HMP	O
is	O
taken	O
to	O
be	O
inversely	O
proportional	O
to	O
the	O
model-averaged	O
Bayes	B-General_Concept
factor	I-General_Concept
for	O
a	O
collection	O
of	O
tests	O
with	O
common	O
null	O
hypothesis:For	O
Good	O
,	O
his	O
rule-of-thumb	O
supported	O
an	O
interchangeability	O
between	O
Bayesian	O
and	O
classical	B-General_Concept
approaches	O
to	O
hypothesis	O
testing	O
.	O
</s>
<s>
For	O
likelihood	B-General_Concept
ratio	I-General_Concept
tests	I-General_Concept
with	O
exactly	O
two	O
degrees	O
of	O
freedom	O
,	O
Wilks	B-General_Concept
 '	I-General_Concept
theorem	I-General_Concept
implies	O
that	O
,	O
where	O
is	O
the	O
maximized	O
likelihood	O
ratio	O
in	O
favour	O
of	O
alternative	O
hypothesis	O
and	O
therefore	O
,	O
where	O
is	O
the	O
weighted	O
mean	O
maximized	O
likelihood	O
ratio	O
,	O
using	O
weights	O
Since	O
is	O
an	O
upper	O
bound	O
on	O
the	O
Bayes	B-General_Concept
factor	I-General_Concept
,	O
,	O
then	O
is	O
an	O
upper	O
bound	O
on	O
the	O
model-averaged	O
Bayes	O
factor:While	O
the	O
equivalence	O
holds	O
only	O
for	O
two	O
degrees	O
of	O
freedom	O
,	O
the	O
relationship	O
between	O
and	O
and	O
therefore	O
behaves	O
similarly	O
for	O
other	O
degrees	O
of	O
freedom	O
.	O
</s>
<s>
Under	O
the	O
assumption	O
that	O
the	O
distributions	O
of	O
the	O
p-values	B-General_Concept
under	O
the	O
alternative	O
hypotheses	O
follow	O
Beta	O
distributions	O
with	O
parameters	O
and	O
that	O
the	O
weights	O
the	O
HMP	O
provides	O
a	O
tighter	O
upper	O
bound	O
on	O
the	O
model-averaged	O
Bayes	O
factor:a	O
result	O
that	O
again	O
reproduces	O
the	O
inverse	O
proportionality	O
of	O
Good	O
's	O
empirical	O
relationship	O
.	O
</s>
