<s>
In	O
statistics	O
,	O
the	O
false	B-General_Concept
discovery	I-General_Concept
rate	I-General_Concept
(	O
FDR	O
)	O
is	O
a	O
method	O
of	O
conceptualizing	O
the	O
rate	O
of	O
type	O
I	O
errors	O
in	O
null	B-General_Concept
hypothesis	I-General_Concept
testing	O
when	O
conducting	O
multiple	B-General_Concept
comparisons	I-General_Concept
.	O
</s>
<s>
FDR-controlling	O
procedures	O
are	O
designed	O
to	O
control	O
the	O
FDR	O
,	O
which	O
is	O
the	O
expected	O
proportion	O
of	O
"	O
discoveries	O
"	O
(	O
rejected	O
null	B-General_Concept
hypotheses	I-General_Concept
)	O
that	O
are	O
false	O
(	O
incorrect	O
rejections	O
of	O
the	O
null	O
)	O
.	O
</s>
<s>
FDR-controlling	O
procedures	O
provide	O
less	O
stringent	O
control	O
of	O
Type	O
I	O
errors	O
compared	O
to	O
family-wise	O
error	O
rate	O
(	O
FWER	O
)	O
controlling	O
procedures	O
(	O
such	O
as	O
the	O
Bonferroni	B-General_Concept
correction	I-General_Concept
)	O
,	O
which	O
control	O
the	O
probability	O
of	O
at	O
least	O
one	O
Type	O
I	O
error	O
.	O
</s>
<s>
Thus	O
,	O
FDR-controlling	O
procedures	O
have	O
greater	O
power	B-General_Concept
,	O
at	O
the	O
cost	O
of	O
increased	O
numbers	O
of	O
Type	O
I	O
errors	O
.	O
</s>
<s>
This	O
,	O
coupled	O
with	O
the	O
growth	O
in	O
computing	O
power	B-General_Concept
,	O
made	O
it	O
possible	O
to	O
seamlessly	O
perform	O
a	O
very	O
high	O
number	O
of	O
statistical	O
tests	O
on	O
a	O
given	O
data	O
set	O
.	O
</s>
<s>
In	O
these	O
datasets	O
,	O
too	O
few	O
of	O
the	O
measured	O
variables	O
showed	O
statistical	O
significance	O
after	O
classic	O
correction	O
for	O
multiple	O
tests	O
with	O
standard	O
multiple	B-General_Concept
comparison	I-General_Concept
procedures	I-General_Concept
.	O
</s>
<s>
In	O
1979	O
,	O
Holm	O
proposed	O
the	O
Holm	B-General_Concept
procedure	I-General_Concept
,	O
a	O
stepwise	O
algorithm	O
for	O
controlling	O
the	O
FWER	O
that	O
is	O
at	O
least	O
as	O
powerful	O
as	O
the	O
well-known	O
Bonferroni	B-General_Concept
adjustment	I-General_Concept
.	O
</s>
<s>
This	O
stepwise	O
algorithm	O
sorts	O
the	O
p-values	B-General_Concept
and	O
sequentially	O
rejects	O
the	O
hypotheses	O
starting	O
from	O
the	O
smallest	O
p-values	B-General_Concept
.	O
</s>
<s>
Benjamini	O
(	O
2010	O
)	O
said	O
that	O
the	O
false	B-General_Concept
discovery	I-General_Concept
rate	I-General_Concept
,	O
and	O
the	O
paper	O
Benjamini	O
and	O
Hochberg	O
(	O
1995	O
)	O
,	O
had	O
its	O
origins	O
in	O
two	O
papers	O
concerned	O
with	O
multiple	B-General_Concept
testing	I-General_Concept
:	O
</s>
<s>
The	O
first	O
paper	O
is	O
by	O
Schweder	O
and	O
Spjotvoll	O
(	O
1982	O
)	O
who	O
suggested	O
plotting	O
the	O
ranked	O
p-values	B-General_Concept
and	O
assessing	O
the	O
number	O
of	O
true	O
null	B-General_Concept
hypotheses	I-General_Concept
(	O
)	O
via	O
an	O
eye-fitted	O
line	O
starting	O
from	O
the	O
largest	O
p-values	B-General_Concept
.	O
</s>
<s>
The	O
p-values	B-General_Concept
that	O
deviate	O
from	O
this	O
straight	O
line	O
then	O
should	O
correspond	O
to	O
the	O
false	O
null	B-General_Concept
hypotheses	I-General_Concept
.	O
</s>
<s>
In	O
1986	O
,	O
R	B-Language
.	O
J	O
.	O
Simes	O
offered	O
the	O
same	O
procedure	O
as	O
the	O
"	O
Simes	O
procedure	O
"	O
,	O
in	O
order	O
to	O
control	O
the	O
FWER	O
in	O
the	O
weak	O
sense	O
(	O
under	O
the	O
intersection	O
null	B-General_Concept
hypothesis	I-General_Concept
)	O
when	O
the	O
statistics	O
are	O
independent	O
.	O
</s>
<s>
Based	O
on	O
definitions	O
below	O
we	O
can	O
define	O
as	O
the	O
proportion	O
of	O
false	O
discoveries	O
among	O
the	O
discoveries	O
(	O
rejections	O
of	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
)	O
:	O
</s>
<s>
The	O
false	B-General_Concept
discovery	I-General_Concept
rate	I-General_Concept
(	O
FDR	O
)	O
is	O
then	O
simply	O
:	O
</s>
<s>
To	O
avoid	O
division	B-Algorithm
by	I-Algorithm
zero	I-Algorithm
,	O
is	O
defined	O
to	O
be	O
0	O
when	O
.	O
</s>
<s>
The	O
settings	O
for	O
many	O
procedures	O
is	O
such	O
that	O
we	O
have	O
null	B-General_Concept
hypotheses	I-General_Concept
tested	O
and	O
their	O
corresponding	O
p-values	B-General_Concept
.	O
</s>
<s>
We	O
list	O
these	O
p-values	B-General_Concept
in	O
ascending	O
order	O
and	O
denote	O
them	O
by	O
.	O
</s>
<s>
A	O
procedure	O
that	O
goes	O
from	O
a	O
small	O
p-value	B-General_Concept
to	O
a	O
large	O
one	O
will	O
be	O
called	O
a	O
step-up	O
procedure	O
.	O
</s>
<s>
The	O
Benjamini	O
–	O
Yekutieli	O
procedure	O
controls	O
the	O
false	B-General_Concept
discovery	I-General_Concept
rate	I-General_Concept
under	O
arbitrary	O
dependence	O
assumptions	O
.	O
</s>
<s>
In	O
the	O
Storey-Tibshirani	O
procedure	O
,	O
q-values	B-General_Concept
are	O
used	O
for	O
controlling	O
the	O
FDR	O
.	O
</s>
<s>
Using	O
a	O
multiplicity	O
procedure	O
that	O
controls	O
the	O
FDR	O
criterion	O
is	O
adaptive	O
and	O
scalable	B-Architecture
.	O
</s>
<s>
The	O
FDR	O
criterion	O
adapts	O
so	O
that	O
the	O
same	O
number	O
of	O
false	O
discoveries	O
(	O
V	O
)	O
will	O
have	O
different	O
implications	O
,	O
depending	O
on	O
the	O
total	O
number	O
of	O
discoveries	O
(	O
R	B-Language
)	O
.	O
</s>
<s>
If	O
we	O
make	O
4	O
discoveries	O
(	O
R	B-Language
)	O
,	O
having	O
2	O
of	O
them	O
be	O
false	O
discoveries	O
(	O
V	O
)	O
is	O
often	O
very	O
costly	O
.	O
</s>
<s>
If	O
we	O
make	O
50	O
discoveries	O
(	O
R	B-Language
)	O
,	O
having	O
2	O
of	O
them	O
be	O
false	O
discoveries	O
(	O
V	O
)	O
is	O
often	O
not	O
very	O
costly	O
.	O
</s>
<s>
The	O
FDR	O
criterion	O
is	O
scalable	B-Architecture
in	O
that	O
the	O
same	O
proportion	O
of	O
false	O
discoveries	O
out	O
of	O
the	O
total	O
number	O
of	O
discoveries	O
(	O
Q	O
)	O
,	O
remains	O
sensible	O
for	O
different	O
number	O
of	O
total	O
discoveries	O
(	O
R	B-Language
)	O
.	O
</s>
<s>
If	O
we	O
make	O
100	O
discoveries	O
(	O
R	B-Language
)	O
,	O
having	O
5	O
of	O
them	O
be	O
false	O
discoveries	O
(	O
)	O
may	O
not	O
be	O
very	O
costly	O
.	O
</s>
<s>
Similarly	O
,	O
if	O
we	O
make	O
1000	O
discoveries	O
(	O
R	B-Language
)	O
,	O
having	O
50	O
of	O
them	O
be	O
false	O
discoveries	O
(	O
as	O
before	O
,	O
)	O
may	O
still	O
not	O
be	O
very	O
costly	O
.	O
</s>
<s>
Controlling	O
the	O
FDR	O
using	O
the	O
linear	O
step-up	O
BH	O
procedure	O
,	O
at	O
level	O
q	O
,	O
has	O
several	O
properties	O
related	O
to	O
the	O
dependency	O
structure	O
between	O
the	O
test	O
statistics	O
of	O
the	O
null	B-General_Concept
hypotheses	I-General_Concept
that	O
are	O
being	O
corrected	O
for	O
.	O
</s>
<s>
If	O
all	O
of	O
the	O
null	B-General_Concept
hypotheses	I-General_Concept
are	O
true	O
(	O
)	O
,	O
then	O
controlling	O
the	O
FDR	O
at	O
level	O
guarantees	O
control	O
over	O
the	O
FWER	O
(	O
this	O
is	O
also	O
called	O
"	O
weak	O
control	O
of	O
the	O
FWER	O
"	O
)	O
:	O
,	O
simply	O
because	O
the	O
event	O
of	O
rejecting	O
at	O
least	O
one	O
true	O
null	B-General_Concept
hypothesis	I-General_Concept
is	O
exactly	O
the	O
event	O
,	O
and	O
the	O
event	O
is	O
exactly	O
the	O
event	O
(	O
when	O
,	O
by	O
definition	O
)	O
.	O
</s>
<s>
In	O
that	O
case	O
there	O
will	O
be	O
room	O
for	O
improving	O
detection	O
power	B-General_Concept
.	O
</s>
<s>
JD	O
Storey	O
promoted	O
the	O
use	O
of	O
the	O
pFDR	O
(	O
a	O
close	O
relative	O
of	O
the	O
FDR	O
)	O
,	O
and	O
the	O
q-value	B-General_Concept
,	O
which	O
can	O
be	O
viewed	O
as	O
the	O
proportion	O
of	O
false	O
discoveries	O
that	O
we	O
expect	O
in	O
an	O
ordered	O
table	O
of	O
results	O
,	O
up	O
to	O
the	O
current	O
line	O
.	O
</s>
<s>
Storey	O
also	O
promoted	O
the	O
idea	O
(	O
also	O
mentioned	O
by	O
BH	O
)	O
that	O
the	O
actual	O
number	O
of	O
null	B-General_Concept
hypotheses	I-General_Concept
,	O
,	O
can	O
be	O
estimated	O
from	O
the	O
shape	O
of	O
the	O
probability	O
distribution	O
curve	O
.	O
</s>
<s>
For	O
example	O
,	O
in	O
a	O
set	O
of	O
data	O
where	O
all	O
null	B-General_Concept
hypotheses	I-General_Concept
are	O
true	O
,	O
50%	O
of	O
results	O
will	O
yield	O
probabilities	O
between	O
0.5	O
and	O
1.0	O
(	O
and	O
the	O
other	O
50%	O
will	O
yield	O
probabilities	O
between	O
0.0	O
and	O
0.5	O
)	O
.	O
</s>
<s>
The	O
false	B-General_Concept
coverage	I-General_Concept
rate	I-General_Concept
(	O
FCR	O
)	O
is	O
,	O
in	O
a	O
sense	O
,	O
the	O
FDR	O
analog	O
to	O
the	O
confidence	O
interval	O
.	O
</s>
