<s>
In	O
statistics	O
,	O
the	O
multiple	B-General_Concept
comparisons	I-General_Concept
,	O
multiplicity	O
or	O
multiple	B-General_Concept
testing	I-General_Concept
problem	I-General_Concept
occurs	O
when	O
one	O
considers	O
a	O
set	O
of	O
statistical	O
inferences	O
simultaneously	O
or	O
infers	O
a	O
subset	O
of	O
parameters	O
selected	O
based	O
on	O
the	O
observed	O
values	O
.	O
</s>
<s>
The	O
problem	B-General_Concept
of	I-General_Concept
multiple	I-General_Concept
comparisons	I-General_Concept
received	O
increased	O
attention	O
in	O
the	O
1950s	O
with	O
the	O
work	O
of	O
statisticians	O
such	O
as	O
Tukey	O
and	O
Scheffé	O
.	O
</s>
<s>
In	O
1996	O
,	O
the	O
first	O
international	O
conference	O
on	O
multiple	B-General_Concept
comparison	I-General_Concept
procedures	O
took	O
place	O
in	O
Tel	O
Aviv	O
.	O
</s>
<s>
Multiple	B-General_Concept
comparisons	I-General_Concept
arise	O
when	O
a	O
statistical	O
analysis	O
involves	O
multiple	O
simultaneous	O
statistical	O
tests	O
,	O
each	O
of	O
which	O
has	O
a	O
potential	O
to	O
produce	O
a	O
"	O
discovery.	O
"	O
</s>
<s>
Failure	O
to	O
compensate	O
for	O
multiple	B-General_Concept
comparisons	I-General_Concept
can	O
have	O
important	O
real-world	O
consequences	O
,	O
as	O
illustrated	O
by	O
the	O
following	O
examples	O
:	O
</s>
<s>
Our	O
confidence	O
that	O
a	O
result	O
will	O
generalize	O
to	O
independent	O
data	O
should	O
generally	O
be	O
weaker	O
if	O
it	O
is	O
observed	O
as	O
part	O
of	O
an	O
analysis	O
that	O
involves	O
multiple	B-General_Concept
comparisons	I-General_Concept
,	O
rather	O
than	O
an	O
analysis	O
that	O
involves	O
only	O
a	O
single	O
comparison	O
.	O
</s>
<s>
The	O
multiple	B-General_Concept
comparisons	I-General_Concept
problem	I-General_Concept
also	O
applies	O
to	O
confidence	O
intervals	O
.	O
</s>
<s>
Techniques	O
have	O
been	O
developed	O
to	O
prevent	O
the	O
inflation	O
of	O
false	B-General_Concept
positive	I-General_Concept
rates	I-General_Concept
and	O
non-coverage	O
rates	O
that	O
occur	O
with	O
multiple	O
statistical	O
tests	O
.	O
</s>
<s>
The	O
most	O
conservative	O
method	O
,	O
which	O
is	O
free	O
of	O
dependence	O
and	O
distributional	O
assumptions	O
,	O
is	O
the	O
Bonferroni	B-General_Concept
correction	I-General_Concept
.	O
</s>
<s>
Another	O
procedure	O
is	O
the	O
Holm	B-General_Concept
–	I-General_Concept
Bonferroni	I-General_Concept
method	I-General_Concept
,	O
which	O
uniformly	O
delivers	O
more	O
power	O
than	O
the	O
simple	O
Bonferroni	B-General_Concept
correction	I-General_Concept
,	O
by	O
testing	O
only	O
the	O
lowest	O
p-value	O
(	O
)	O
against	O
the	O
strictest	O
criterion	O
,	O
and	O
the	O
higher	O
p-values	O
(	O
)	O
against	O
progressively	O
less	O
strict	O
criteria	O
.	O
</s>
<s>
Continuous	O
generalizations	O
of	O
the	O
Bonferroni	B-General_Concept
and	O
Šidák	O
correction	O
are	O
presented	O
in	O
.	O
</s>
<s>
Multiple	B-General_Concept
testing	I-General_Concept
correction	I-General_Concept
refers	O
to	O
making	O
statistical	O
tests	O
more	O
stringent	O
in	O
order	O
to	O
counteract	O
the	O
problem	O
of	O
multiple	B-General_Concept
testing	I-General_Concept
.	O
</s>
<s>
The	O
best	O
known	O
such	O
adjustment	O
is	O
the	O
Bonferroni	B-General_Concept
correction	I-General_Concept
,	O
but	O
other	O
methods	O
have	O
been	O
developed	O
.	O
</s>
<s>
Such	O
methods	O
are	O
typically	O
designed	O
to	O
control	O
the	O
family-wise	O
error	O
rate	O
or	O
the	O
false	B-General_Concept
discovery	I-General_Concept
rate	I-General_Concept
.	O
</s>
<s>
Traditional	O
methods	O
for	O
multiple	B-General_Concept
comparisons	I-General_Concept
adjustments	O
focus	O
on	O
correcting	O
for	O
modest	O
numbers	O
of	O
comparisons	O
,	O
often	O
in	O
an	O
analysis	B-General_Concept
of	I-General_Concept
variance	I-General_Concept
.	O
</s>
<s>
A	O
different	O
set	O
of	O
techniques	O
have	O
been	O
developed	O
for	O
"	O
large-scale	O
multiple	B-General_Concept
testing	I-General_Concept
"	O
,	O
in	O
which	O
thousands	O
or	O
even	O
greater	O
numbers	O
of	O
tests	O
are	O
performed	O
.	O
</s>
<s>
Such	O
non-replication	O
can	O
have	O
many	O
causes	O
,	O
but	O
it	O
is	O
widely	O
considered	O
that	O
failure	O
to	O
fully	O
account	O
for	O
the	O
consequences	O
of	O
making	O
multiple	B-General_Concept
comparisons	I-General_Concept
is	O
one	O
of	O
the	O
causes	O
.	O
</s>
<s>
It	O
has	O
been	O
argued	O
that	O
advances	O
in	O
measurement	O
and	O
information	B-General_Concept
technology	I-General_Concept
have	O
made	O
it	O
far	O
easier	O
to	O
generate	O
large	O
datasets	O
for	O
exploratory	B-General_Concept
analysis	I-General_Concept
,	O
often	O
leading	O
to	O
the	O
testing	O
of	O
large	O
numbers	O
of	O
hypotheses	O
with	O
no	O
prior	O
basis	O
for	O
expecting	O
many	O
of	O
the	O
hypotheses	O
to	O
be	O
true	O
.	O
</s>
<s>
In	O
this	O
situation	O
,	O
very	O
high	O
false	B-General_Concept
positive	I-General_Concept
rates	I-General_Concept
are	O
expected	O
unless	O
multiple	B-General_Concept
comparisons	I-General_Concept
adjustments	O
are	O
made	O
.	O
</s>
<s>
Alternatively	O
,	O
if	O
a	O
study	O
is	O
viewed	O
as	O
exploratory	O
,	O
or	O
if	O
significant	O
results	O
can	O
be	O
easily	O
re-tested	O
in	O
an	O
independent	O
study	O
,	O
control	O
of	O
the	O
false	B-General_Concept
discovery	I-General_Concept
rate	I-General_Concept
(	O
FDR	O
)	O
is	O
often	O
preferred	O
.	O
</s>
<s>
The	O
practice	O
of	O
trying	O
many	O
unadjusted	O
comparisons	O
in	O
the	O
hope	O
of	O
finding	O
a	O
significant	O
one	O
is	O
a	O
known	O
problem	O
,	O
whether	O
applied	O
unintentionally	O
or	O
deliberately	O
,	O
is	O
sometimes	O
called	O
"	O
p-hacking.	O
"	O
</s>
<s>
A	O
drawback	O
of	O
this	O
approach	O
is	O
that	O
it	O
overstates	O
the	O
evidence	O
that	O
some	O
of	O
the	O
alternative	O
hypotheses	O
are	O
true	O
when	O
the	O
test	B-General_Concept
statistics	I-General_Concept
are	O
positively	O
correlated	O
,	O
which	O
commonly	O
occurs	O
in	O
practice	O
.	O
</s>
<s>
On	O
the	O
other	O
hand	O
,	O
the	O
approach	O
remains	O
valid	O
even	O
in	O
the	O
presence	O
of	O
correlation	O
among	O
the	O
test	B-General_Concept
statistics	I-General_Concept
,	O
as	O
long	O
as	O
the	O
Poisson	O
distribution	O
can	O
be	O
shown	O
to	O
provide	O
a	O
good	O
approximation	O
for	O
the	O
number	O
of	O
significant	O
results	O
.	O
</s>
<s>
Another	O
common	O
approach	O
that	O
can	O
be	O
used	O
in	O
situations	O
where	O
the	O
test	B-General_Concept
statistics	I-General_Concept
can	O
be	O
standardized	O
to	O
Z-scores	O
is	O
to	O
make	O
a	O
normal	B-Application
quantile	I-Application
plot	I-Application
of	O
the	O
test	B-General_Concept
statistics	I-General_Concept
.	O
</s>
