<s>
In	O
statistical	O
hypothesis	O
testing	O
,	O
specifically	O
multiple	B-General_Concept
hypothesis	I-General_Concept
testing	I-General_Concept
,	O
the	O
q-value	B-General_Concept
in	O
the	O
Storey-Tibshirani	O
procedure	O
provides	O
a	O
means	O
to	O
control	O
the	O
positive	O
false	B-General_Concept
discovery	I-General_Concept
rate	I-General_Concept
(	O
pFDR	O
)	O
.	O
</s>
<s>
Just	O
as	O
the	O
p-value	B-General_Concept
gives	O
the	O
expected	O
false	B-General_Concept
positive	I-General_Concept
rate	I-General_Concept
obtained	O
by	O
rejecting	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
for	O
any	O
result	O
with	O
an	O
equal	O
or	O
smaller	O
p-value	B-General_Concept
,	O
the	O
q-value	B-General_Concept
gives	O
the	O
expected	O
pFDR	O
obtained	O
by	O
rejecting	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
for	O
any	O
result	O
with	O
an	O
equal	O
or	O
smaller	O
q-value	B-General_Concept
.	O
</s>
<s>
In	O
statistics	O
,	O
testing	O
multiple	O
hypotheses	O
simultaneously	O
using	O
methods	O
appropriate	O
for	O
testing	O
single	O
hypotheses	O
tends	O
to	O
yield	O
many	O
false	O
positives	O
:	O
the	O
so-called	O
multiple	B-General_Concept
comparisons	I-General_Concept
problem	I-General_Concept
.	O
</s>
<s>
For	O
example	O
,	O
assume	O
that	O
one	O
were	O
to	O
test	O
1,000	O
null	B-General_Concept
hypotheses	I-General_Concept
,	O
all	O
of	O
which	O
are	O
true	O
,	O
and	O
(	O
as	O
is	O
conventional	O
in	O
single	O
hypothesis	O
testing	O
)	O
to	O
reject	O
null	B-General_Concept
hypotheses	I-General_Concept
with	O
a	O
significance	B-General_Concept
level	I-General_Concept
of	O
0.05	O
;	O
due	O
to	O
random	O
chance	O
,	O
one	O
would	O
expect	O
5%	O
of	O
the	O
results	O
to	O
appear	O
significant	O
(	O
P	B-General_Concept
<	O
0.05	O
)	O
,	O
yielding	O
50	O
false	O
positives	O
(	O
rejections	O
of	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
)	O
.	O
</s>
<s>
Since	O
the	O
1950s	O
,	O
statisticians	O
had	O
been	O
developing	O
methods	O
for	O
multiple	B-General_Concept
comparisons	I-General_Concept
that	O
reduced	O
the	O
number	O
of	O
false	O
positives	O
,	O
such	O
as	O
controlling	O
the	O
family-wise	O
error	O
rate	O
(	O
FWER	O
)	O
using	O
the	O
Bonferroni	B-General_Concept
correction	I-General_Concept
,	O
but	O
these	O
methods	O
also	O
increased	O
the	O
number	O
of	O
false	O
negatives	O
(	O
i.e.	O
</s>
<s>
reduced	O
the	O
statistical	B-General_Concept
power	I-General_Concept
)	O
.	O
</s>
<s>
In	O
1995	O
,	O
Yoav	O
Benjamini	O
and	O
Yosef	O
Hochberg	O
proposed	O
controlling	O
the	O
false	B-General_Concept
discovery	I-General_Concept
rate	I-General_Concept
(	O
FDR	O
)	O
as	O
a	O
more	O
statistically	O
powerful	O
alternative	O
to	O
controlling	O
the	O
FWER	O
in	O
multiple	B-General_Concept
hypothesis	I-General_Concept
testing	I-General_Concept
.	O
</s>
<s>
The	O
pFDR	O
and	O
the	O
q-value	B-General_Concept
were	O
introduced	O
by	O
John	O
D	O
.	O
Storey	O
in	O
2002	O
in	O
order	O
to	O
improve	O
upon	O
a	O
limitation	O
of	O
the	O
FDR	O
,	O
namely	O
that	O
the	O
FDR	O
is	O
not	O
defined	O
when	O
there	O
are	O
no	O
positive	O
results	O
.	O
</s>
<s>
Let	O
there	O
be	O
a	O
null	B-General_Concept
hypothesis	I-General_Concept
and	O
an	O
alternative	B-General_Concept
hypothesis	I-General_Concept
.	O
</s>
<s>
Perform	O
hypothesis	O
tests	O
;	O
let	O
the	O
test	B-General_Concept
statistics	I-General_Concept
be	O
i.i.d.	O
</s>
<s>
That	O
is	O
,	O
if	O
is	O
true	O
for	O
test	O
(	O
)	O
,	O
then	O
follows	O
the	O
null	B-General_Concept
distribution	I-General_Concept
;	O
while	O
if	O
is	O
true	O
(	O
)	O
,	O
then	O
follows	O
the	O
alternative	O
distribution	O
.	O
</s>
<s>
Denote	O
the	O
critical	O
region	O
(	O
the	O
values	O
of	O
for	O
which	O
is	O
rejected	O
)	O
at	O
significance	B-General_Concept
level	I-General_Concept
by	O
.	O
</s>
<s>
Let	O
an	O
experiment	O
yield	O
a	O
value	O
for	O
the	O
test	B-General_Concept
statistic	I-General_Concept
.	O
</s>
<s>
That	O
is	O
,	O
the	O
q-value	B-General_Concept
is	O
the	O
infimum	O
of	O
the	O
pFDR	O
if	O
is	O
rejected	O
for	O
test	B-General_Concept
statistics	I-General_Concept
with	O
values	O
.	O
</s>
<s>
which	O
is	O
the	O
infimum	O
of	O
the	O
probability	O
that	O
is	O
true	O
given	O
that	O
is	O
rejected	O
(	O
the	O
false	B-General_Concept
discovery	I-General_Concept
rate	I-General_Concept
)	O
.	O
</s>
<s>
the	O
infimum	O
of	O
the	O
probability	O
that	O
is	O
rejected	O
given	O
that	O
is	O
true	O
(	O
the	O
false	B-General_Concept
positive	I-General_Concept
rate	I-General_Concept
)	O
.	O
</s>
<s>
Comparing	O
the	O
definitions	O
of	O
the	O
p	B-General_Concept
-	O
and	O
q-values	B-General_Concept
,	O
it	O
can	O
be	O
seen	O
that	O
the	O
q-value	B-General_Concept
is	O
the	O
minimum	O
posterior	O
probability	O
that	O
is	O
true	O
.	O
</s>
<s>
The	O
q-value	B-General_Concept
can	O
be	O
interpreted	O
as	O
the	O
false	B-General_Concept
discovery	I-General_Concept
rate	I-General_Concept
(	O
FDR	O
)	O
:	O
the	O
proportion	O
of	O
false	O
positives	O
among	O
all	O
positive	O
results	O
.	O
</s>
<s>
Given	O
a	O
set	O
of	O
test	B-General_Concept
statistics	I-General_Concept
and	O
their	O
associated	O
q-values	B-General_Concept
,	O
rejecting	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
for	O
all	O
tests	O
whose	O
q-value	B-General_Concept
is	O
less	O
than	O
or	O
equal	O
to	O
some	O
threshold	O
ensures	O
that	O
the	O
expected	O
value	O
of	O
the	O
false	B-General_Concept
discovery	I-General_Concept
rate	I-General_Concept
is	O
.	O
</s>
<s>
detecting	O
differential	O
expression	O
in	O
a	O
gene	O
that	O
is	O
not	O
differentially	O
expressed	O
)	O
but	O
imposes	O
a	O
strict	O
threshold	O
for	O
the	O
p-value	B-General_Concept
that	O
results	O
in	O
many	O
false	O
negatives	O
(	O
many	O
differentially	O
expressed	O
genes	O
are	O
overlooked	O
)	O
.	O
</s>
<s>
However	O
,	O
controlling	O
the	O
pFDR	O
by	O
selecting	O
genes	O
with	O
significant	O
q-values	B-General_Concept
lowers	O
the	O
number	O
of	O
false	O
negatives	O
(	O
increases	O
the	O
statistical	B-General_Concept
power	I-General_Concept
)	O
while	O
ensuring	O
that	O
the	O
expected	O
value	O
of	O
the	O
proportion	O
of	O
false	O
positives	O
among	O
all	O
positive	O
results	O
is	O
low	O
(	O
e.g.	O
</s>
<s>
If	O
we	O
consider	O
every	O
gene	O
with	O
a	O
p-value	B-General_Concept
of	O
less	O
than	O
0.05	O
to	O
be	O
differentially	O
expressed	O
,	O
we	O
expect	O
that	O
450	O
(	O
5%	O
)	O
of	O
the	O
9,000	O
genes	O
that	O
are	O
not	O
differentially	O
expressed	O
will	O
appear	O
to	O
be	O
differentially	O
expressed	O
(	O
450	O
false	O
positives	O
)	O
.	O
</s>
<s>
If	O
we	O
control	O
the	O
pFDR	O
to	O
0.05	O
by	O
considering	O
all	O
genes	O
with	O
a	O
q-value	B-General_Concept
of	O
less	O
than	O
0.05	O
to	O
be	O
differentially	O
expressed	O
,	O
then	O
we	O
expect	O
5%	O
of	O
the	O
positive	O
results	O
to	O
be	O
false	O
positives	O
(	O
e.g.	O
</s>
<s>
The	O
package	O
in	O
R	B-Language
estimates	O
q-values	B-General_Concept
from	O
a	O
list	O
of	O
p-values	B-General_Concept
.	O
</s>
