<s>
The	O
Wilcoxon	B-General_Concept
signed-rank	I-General_Concept
test	I-General_Concept
is	O
a	O
non-parametric	B-General_Concept
statistical	O
hypothesis	O
test	O
used	O
either	O
to	O
test	O
the	O
location	O
of	O
a	O
population	O
based	O
on	O
a	O
sample	O
of	O
data	O
,	O
or	O
to	O
compare	O
the	O
locations	O
of	O
two	O
populations	O
using	O
two	O
matched	O
samples	O
.	O
</s>
<s>
The	O
one-sample	O
version	O
serves	O
a	O
purpose	O
similar	O
to	O
that	O
of	O
the	O
one-sample	O
Student	B-General_Concept
's	I-General_Concept
t-test	I-General_Concept
.	O
</s>
<s>
For	O
two	O
matched	O
samples	O
,	O
it	O
is	O
a	O
paired	B-General_Concept
difference	I-General_Concept
test	I-General_Concept
like	O
the	O
paired	O
Student	B-General_Concept
's	I-General_Concept
t-test	I-General_Concept
(	O
also	O
known	O
as	O
the	O
"	O
t-test	B-General_Concept
for	O
matched	O
pairs	O
"	O
or	O
"	O
t-test	B-General_Concept
for	O
dependent	O
samples	O
"	O
)	O
.	O
</s>
<s>
The	O
Wilcoxon	B-General_Concept
test	I-General_Concept
can	O
be	O
a	O
good	O
alternative	O
to	O
the	O
t-test	B-General_Concept
when	O
population	O
means	O
are	O
not	O
of	O
interest	O
;	O
for	O
example	O
,	O
when	O
one	O
wishes	O
to	O
test	O
whether	O
a	O
population	O
's	O
median	O
is	O
nonzero	O
,	O
or	O
whether	O
there	O
is	O
a	O
better	O
than	O
50%	O
chance	O
that	O
a	O
sample	O
from	O
one	O
population	O
is	O
greater	O
than	O
a	O
sample	O
from	O
another	O
population	O
.	O
</s>
<s>
The	O
test	O
is	O
named	O
for	O
Frank	O
Wilcoxon	O
(	O
1892	O
–	O
1965	O
)	O
who	O
,	O
in	O
a	O
single	O
paper	O
,	O
proposed	O
both	O
it	O
and	O
the	O
rank-sum	B-General_Concept
test	I-General_Concept
for	O
two	O
independent	O
samples	O
.	O
</s>
<s>
The	O
test	O
was	O
popularized	O
by	O
Sidney	O
Siegel	O
(	O
1956	O
)	O
in	O
his	O
influential	O
textbook	O
on	O
non-parametric	B-General_Concept
statistics	I-General_Concept
.	O
</s>
<s>
Siegel	O
used	O
the	O
symbol	O
T	O
for	O
the	O
test	B-General_Concept
statistic	I-General_Concept
,	O
and	O
consequently	O
,	O
the	O
test	O
is	O
sometimes	O
referred	O
to	O
as	O
the	O
Wilcoxon	O
T-test	B-General_Concept
.	O
</s>
<s>
There	O
are	O
two	O
variants	O
of	O
the	O
signed-rank	B-General_Concept
test	I-General_Concept
.	O
</s>
<s>
However	O
,	O
most	O
practical	O
applications	O
of	O
the	O
signed-rank	B-General_Concept
test	I-General_Concept
arise	O
from	O
paired	O
data	O
.	O
</s>
<s>
The	O
test	B-General_Concept
statistic	I-General_Concept
is	O
the	O
signed-rank	O
sum	O
:	O
</s>
<s>
The	O
signed-rank	O
sum	O
is	O
closely	O
related	O
to	O
two	O
other	O
test	B-General_Concept
statistics	I-General_Concept
.	O
</s>
<s>
Because	O
,	O
,	O
and	O
carry	O
the	O
same	O
information	O
,	O
any	O
of	O
them	O
may	O
be	O
used	O
as	O
the	O
test	B-General_Concept
statistic	I-General_Concept
.	O
</s>
<s>
The	O
one-sample	O
Wilcoxon	B-General_Concept
signed-rank	I-General_Concept
test	I-General_Concept
can	O
be	O
used	O
to	O
test	O
whether	O
data	O
comes	O
from	O
a	O
symmetric	O
population	O
with	O
a	O
specified	O
median	O
.	O
</s>
<s>
The	O
alternative	O
hypothesis	O
being	O
tested	O
depends	O
on	O
whether	O
the	O
test	B-General_Concept
statistic	I-General_Concept
is	O
used	O
to	O
compute	O
a	O
one-sided	O
or	O
two-sided	O
p-value	O
(	O
and	O
if	O
one-sided	O
,	O
which	O
side	O
)	O
.	O
</s>
<s>
It	O
can	O
also	O
be	O
derived	O
from	O
the	O
description	O
of	O
and	O
in	O
terms	O
of	O
Walsh	O
averages	O
,	O
since	O
that	O
description	O
shows	O
that	O
the	O
Wilcoxon	B-General_Concept
test	I-General_Concept
is	O
the	O
same	O
as	O
the	O
sign	B-General_Concept
test	I-General_Concept
applied	O
to	O
the	O
set	O
of	O
Walsh	O
averages	O
.	O
</s>
<s>
Because	O
the	O
one-sample	O
test	O
can	O
be	O
used	O
as	O
a	O
one-sided	O
test	O
for	O
stochastic	O
dominance	O
,	O
the	O
paired	O
difference	O
Wilcoxon	B-General_Concept
test	I-General_Concept
can	O
be	O
used	O
to	O
compare	O
the	O
following	O
hypotheses	O
:	O
</s>
<s>
Because	O
of	O
this	O
,	O
the	O
test	B-General_Concept
statistic	I-General_Concept
needs	O
to	O
be	O
modified	O
.	O
</s>
<s>
Then	O
the	O
standard	O
signed-rank	B-General_Concept
test	I-General_Concept
could	O
be	O
applied	O
to	O
the	O
resulting	O
data	O
,	O
as	O
long	O
as	O
there	O
were	O
no	O
ties	O
.	O
</s>
<s>
However	O
,	O
it	O
excludes	O
them	O
from	O
the	O
test	B-General_Concept
statistic	I-General_Concept
,	O
or	O
equivalently	O
it	O
defines	O
.	O
</s>
<s>
The	O
signed-rank	O
zero	O
procedure	O
has	O
the	O
disadvantage	O
that	O
,	O
when	O
zeros	O
occur	O
,	O
the	O
null	O
distribution	O
of	O
the	O
test	B-General_Concept
statistic	I-General_Concept
changes	O
,	O
so	O
tables	O
of	O
p-values	O
can	O
no	O
longer	O
be	O
used	O
.	O
</s>
<s>
When	O
the	O
data	O
does	O
not	O
have	O
ties	O
,	O
the	O
ranks	O
are	O
used	O
to	O
calculate	O
the	O
test	B-General_Concept
statistic	I-General_Concept
.	O
</s>
<s>
Once	O
the	O
ranks	O
are	O
assigned	O
,	O
the	O
test	B-General_Concept
statistic	I-General_Concept
is	O
computed	O
in	O
the	O
same	O
way	O
as	O
usual	O
.	O
</s>
<s>
Observations	O
with	O
smaller	O
absolute	O
values	O
are	O
always	O
given	O
smaller	O
ranks	O
,	O
just	O
as	O
in	O
the	O
standard	O
rank-sum	B-General_Concept
test	I-General_Concept
.	O
</s>
<s>
In	O
the	O
average	O
statistic	O
method	O
,	O
the	O
test	B-General_Concept
statistic	I-General_Concept
is	O
computed	O
for	O
every	O
possible	O
way	O
of	O
breaking	O
ties	O
,	O
and	O
the	O
final	O
statistic	O
is	O
the	O
mean	O
of	O
the	O
tie-broken	O
statistics	O
.	O
</s>
<s>
Wilcoxon	O
originally	O
defined	O
the	O
Wilcoxon	B-General_Concept
rank-sum	I-General_Concept
statistic	O
to	O
be	O
.	O
</s>
<s>
To	O
compute	O
an	O
effect	B-Application
size	I-Application
for	O
the	O
signed-rank	B-General_Concept
test	I-General_Concept
,	O
one	O
can	O
use	O
the	O
rank-biserial	O
correlation	O
.	O
</s>
<s>
If	O
the	O
test	B-General_Concept
statistic	I-General_Concept
T	O
is	O
reported	O
,	O
the	O
rank	O
correlation	O
r	B-Language
is	O
equal	O
to	O
the	O
test	B-General_Concept
statistic	I-General_Concept
T	O
divided	O
by	O
the	O
total	O
rank	O
sum	O
S	O
,	O
orr	O
=	O
T/S	O
.	O
</s>
<s>
Using	O
the	O
above	O
example	O
,	O
the	O
test	B-General_Concept
statistic	I-General_Concept
is	O
T	O
=	O
9	O
.	O
</s>
<s>
Hence	O
,	O
the	O
rank	O
correlation	O
is	O
9/45	O
,	O
so	O
r	B-Language
=	O
0.20	O
.	O
</s>
<s>
If	O
the	O
test	B-General_Concept
statistic	I-General_Concept
T	O
is	O
reported	O
,	O
an	O
equivalent	O
way	O
to	O
compute	O
the	O
rank	O
correlation	O
is	O
with	O
the	O
difference	O
in	O
proportion	O
between	O
the	O
two	O
rank	O
sums	O
,	O
which	O
is	O
the	O
Kerby	O
(	O
2014	O
)	O
simple	O
difference	O
formula	O
.	O
</s>
<s>
Finally	O
,	O
the	O
rank	O
correlation	O
is	O
the	O
difference	O
between	O
the	O
two	O
proportions	O
(	O
.60	O
minus	O
.40	O
)	O
,	O
hence	O
r	B-Language
=	O
.20	O
.	O
</s>
<s>
R	B-Language
includes	O
an	O
implementation	O
of	O
the	O
test	O
as	O
wilcox.test(x,y, paired=TRUE )	O
,	O
where	O
x	O
and	O
y	O
are	O
vectors	O
of	O
equal	O
length	O
.	O
</s>
<s>
includes	O
implementation	O
of	O
the	O
Wilcoxon	B-General_Concept
signed-rank	I-General_Concept
test	I-General_Concept
in	O
C++	O
,	O
C#	O
,	O
Delphi	O
,	O
Visual	O
Basic	O
,	O
etc	O
.	O
</s>
<s>
GNU	B-Language
Octave	I-Language
implements	O
various	O
one-tailed	O
and	O
two-tailed	O
versions	O
of	O
the	O
test	O
in	O
the	O
wilcoxon_test	O
function	O
.	O
</s>
<s>
implements	O
this	O
test	O
using	O
"	O
Wilcoxon	B-General_Concept
rank	I-General_Concept
sum	I-General_Concept
test	I-General_Concept
"	O
as	O
 [ p , h ] 	O
=	O
signrank(x,y )	O
also	O
returns	O
a	O
logical	O
value	O
indicating	O
the	O
test	O
decision	O
.	O
</s>
