<s>
In	O
statistics	O
,	O
the	O
Bonferroni	B-General_Concept
correction	I-General_Concept
is	O
a	O
method	O
to	O
counteract	O
the	O
multiple	B-General_Concept
comparisons	I-General_Concept
problem	I-General_Concept
.	O
</s>
<s>
Statistical	O
hypothesis	O
testing	O
is	O
based	O
on	O
rejecting	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
if	O
the	O
likelihood	O
of	O
the	O
observed	O
data	O
under	O
the	O
null	B-General_Concept
hypotheses	I-General_Concept
is	O
low	O
.	O
</s>
<s>
If	O
multiple	O
hypotheses	O
are	O
tested	O
,	O
the	O
probability	O
of	O
observing	O
a	O
rare	O
event	O
increases	O
,	O
and	O
therefore	O
,	O
the	O
likelihood	O
of	O
incorrectly	O
rejecting	O
a	O
null	B-General_Concept
hypothesis	I-General_Concept
(	O
i.e.	O
,	O
making	O
a	O
Type	O
I	O
error	O
)	O
increases	O
.	O
</s>
<s>
The	O
Bonferroni	B-General_Concept
correction	I-General_Concept
compensates	O
for	O
that	O
increase	O
by	O
testing	O
each	O
individual	O
hypothesis	O
at	O
a	O
significance	O
level	O
of	O
,	O
where	O
is	O
the	O
desired	O
overall	O
alpha	O
level	O
and	O
is	O
the	O
number	O
of	O
hypotheses	O
.	O
</s>
<s>
For	O
example	O
,	O
if	O
a	O
trial	O
is	O
testing	O
hypotheses	O
with	O
a	O
desired	O
,	O
then	O
the	O
Bonferroni	B-General_Concept
correction	I-General_Concept
would	O
test	O
each	O
individual	O
hypothesis	O
at	O
.	O
</s>
<s>
Let	O
be	O
a	O
family	O
of	O
hypotheses	O
and	O
their	O
corresponding	O
p-values	B-General_Concept
.	O
</s>
<s>
Let	O
be	O
the	O
total	O
number	O
of	O
null	B-General_Concept
hypotheses	I-General_Concept
,	O
and	O
let	O
be	O
the	O
number	O
of	O
true	O
null	B-General_Concept
hypotheses	I-General_Concept
(	O
which	O
is	O
presumably	O
unknown	O
to	O
the	O
researcher	O
)	O
.	O
</s>
<s>
The	O
Bonferroni	B-General_Concept
correction	I-General_Concept
rejects	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
for	O
each	O
,	O
thereby	O
controlling	O
the	O
FWER	O
at	O
.	O
</s>
<s>
This	O
control	O
does	O
not	O
require	O
any	O
assumptions	O
about	O
dependence	O
among	O
the	O
p-values	B-General_Concept
or	O
about	O
how	O
many	O
of	O
the	O
null	B-General_Concept
hypotheses	I-General_Concept
are	O
true	O
.	O
</s>
<s>
When	O
searching	O
for	O
a	O
signal	O
in	O
a	O
continuous	O
parameter	O
space	O
there	O
can	O
also	O
be	O
a	O
problem	B-General_Concept
of	I-General_Concept
multiple	I-General_Concept
comparisons	I-General_Concept
,	O
or	O
look-elsewhere	O
effect	O
.	O
</s>
<s>
In	O
such	O
cases	O
,	O
one	O
can	O
apply	O
a	O
continuous	O
generalization	O
of	O
the	O
Bonferroni	B-General_Concept
correction	I-General_Concept
by	O
employing	O
Bayesian	O
logic	O
to	O
relate	O
the	O
effective	O
number	O
of	O
trials	O
,	O
,	O
to	O
the	O
prior-to-posterior	O
volume	O
ratio	O
.	O
</s>
<s>
For	O
example	O
,	O
the	O
Holm	B-General_Concept
–	I-General_Concept
Bonferroni	I-General_Concept
method	I-General_Concept
and	O
the	O
Šidák	O
correction	O
are	O
universally	O
more	O
powerful	O
procedures	O
than	O
the	O
Bonferroni	B-General_Concept
correction	I-General_Concept
,	O
meaning	O
that	O
they	O
are	O
always	O
at	O
least	O
as	O
powerful	O
.	O
</s>
<s>
With	O
respect	O
to	O
FWER	O
control	O
,	O
the	O
Bonferroni	B-General_Concept
correction	I-General_Concept
can	O
be	O
conservative	O
if	O
there	O
are	O
a	O
large	O
number	O
of	O
tests	O
and/or	O
the	O
test	O
statistics	O
are	O
positively	O
correlated	O
.	O
</s>
<s>
The	O
correction	O
comes	O
at	O
the	O
cost	O
of	O
increasing	O
the	O
probability	O
of	O
producing	O
false	O
negatives	O
,	O
i.e.	O
,	O
reducing	O
statistical	B-General_Concept
power	I-General_Concept
.	O
</s>
<s>
Such	O
criticisms	O
apply	O
to	O
FWER	O
control	O
in	O
general	O
,	O
and	O
are	O
not	O
specific	O
to	O
the	O
Bonferroni	B-General_Concept
correction	I-General_Concept
.	O
</s>
