<s>
The	O
sign	B-General_Concept
test	I-General_Concept
is	O
a	O
statistical	O
method	O
to	O
test	O
for	O
consistent	O
differences	O
between	O
pairs	O
of	O
observations	O
,	O
such	O
as	O
the	O
weight	O
of	O
subjects	O
before	O
and	O
after	O
treatment	O
.	O
</s>
<s>
Given	O
pairs	O
of	O
observations	O
(	O
such	O
as	O
weight	O
pre	O
-	O
and	O
post-treatment	O
)	O
for	O
each	O
subject	O
,	O
the	O
sign	B-General_Concept
test	I-General_Concept
determines	O
if	O
one	O
member	O
of	O
the	O
pair	O
(	O
such	O
as	O
pre-treatment	O
)	O
tends	O
to	O
be	O
greater	O
than	O
(	O
or	O
less	O
than	O
)	O
the	O
other	O
member	O
of	O
the	O
pair	O
(	O
such	O
as	O
post-treatment	O
)	O
.	O
</s>
<s>
For	O
comparisons	O
of	O
paired	O
observations	O
(	O
x	O
,	O
y	O
)	O
,	O
the	O
sign	B-General_Concept
test	I-General_Concept
is	O
most	O
useful	O
if	O
comparisons	O
can	O
only	O
be	O
expressed	O
as	O
x	O
>	O
y	O
,	O
x	O
=	O
y	O
,	O
or	O
x	O
<	O
y	O
.	O
</s>
<s>
or	O
the	O
Wilcoxon	B-General_Concept
signed-rank	I-General_Concept
test	I-General_Concept
will	O
usually	O
have	O
greater	O
power	O
than	O
the	O
sign	B-General_Concept
test	I-General_Concept
to	O
detect	O
consistent	O
differences	O
.	O
</s>
<s>
If	O
X	O
and	O
Y	O
are	O
quantitative	O
variables	O
,	O
the	O
sign	B-General_Concept
test	I-General_Concept
can	O
be	O
used	O
to	O
test	O
the	O
hypothesis	O
that	O
the	O
difference	O
between	O
the	O
X	O
and	O
Y	O
has	O
zero	O
median	O
,	O
assuming	O
continuous	O
distributions	O
of	O
the	O
two	O
random	O
variables	O
X	O
and	O
Y	O
,	O
in	O
the	O
situation	O
when	O
we	O
can	O
draw	O
paired	B-General_Concept
samples	I-General_Concept
from	O
X	O
and	O
Y	O
.	O
</s>
<s>
The	O
sign	B-General_Concept
test	I-General_Concept
can	O
also	O
test	O
if	O
the	O
median	O
of	O
a	O
collection	O
of	O
numbers	O
is	O
significantly	O
greater	O
than	O
or	O
less	O
than	O
a	O
specified	O
value	O
.	O
</s>
<s>
For	O
example	O
,	O
given	O
a	O
list	O
of	O
student	O
grades	O
in	O
a	O
class	O
,	O
the	O
sign	B-General_Concept
test	I-General_Concept
can	O
determine	O
if	O
the	O
median	O
grade	O
is	O
significantly	O
different	O
from	O
,	O
say	O
,	O
75	O
out	O
of	O
100	O
.	O
</s>
<s>
The	O
sign	B-General_Concept
test	I-General_Concept
is	O
a	O
non-parametric	B-General_Concept
test	I-General_Concept
which	O
makes	O
very	O
few	O
assumptions	O
about	O
the	O
nature	O
of	O
the	O
distributions	O
under	O
test	O
–	O
this	O
means	O
that	O
it	O
has	O
very	O
general	O
applicability	O
but	O
may	O
lack	O
the	O
statistical	B-General_Concept
power	I-General_Concept
of	O
the	O
alternative	O
tests	O
.	O
</s>
<s>
The	O
two	O
conditions	O
for	O
the	O
paired-sample	O
sign	B-General_Concept
test	I-General_Concept
are	O
that	O
a	O
sample	O
must	O
be	O
randomly	O
selected	O
from	O
each	O
population	O
,	O
and	O
the	O
samples	O
must	O
be	O
dependent	O
,	O
or	O
paired	O
.	O
</s>
<s>
Since	O
the	O
test	O
is	O
nonparametric	B-General_Concept
,	O
the	O
samples	O
need	O
not	O
come	O
from	O
normally	O
distributed	O
populations	O
.	O
</s>
<s>
Let	O
p	O
=	O
Pr( X	O
>	O
Y	O
)	O
,	O
and	O
then	O
test	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
H0	O
:	O
p	O
=	O
0.50	O
.	O
</s>
<s>
In	O
other	O
words	O
,	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
states	O
that	O
given	O
a	O
random	O
pair	O
of	O
measurements	O
(	O
xi	O
,	O
yi	O
)	O
,	O
then	O
xi	O
and	O
yi	O
are	O
equally	O
likely	O
to	O
be	O
larger	O
than	O
the	O
other	O
.	O
</s>
<s>
To	O
test	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
,	O
independent	O
pairs	O
of	O
sample	O
data	O
are	O
collected	O
from	O
the	O
populations	O
{( 	O
x1	O
,	O
y1	O
)	O
,	O
(	O
x2	O
,	O
y2	O
)	O
,	O
.	O
</s>
<s>
Since	O
the	O
test	O
statistic	O
is	O
expected	O
to	O
follow	O
a	O
binomial	O
distribution	O
,	O
the	O
standard	O
binomial	B-General_Concept
test	I-General_Concept
is	O
used	O
to	O
calculate	O
significance	B-General_Concept
.	O
</s>
<s>
The	O
left-tail	O
value	O
is	O
computed	O
by	O
Pr( W	O
≤	O
w	O
)	O
,	O
which	O
is	O
the	O
p-value	B-General_Concept
for	O
the	O
alternative	O
H1	O
:	O
p	O
<	O
0.50	O
.	O
</s>
<s>
The	O
right-tail	O
value	O
is	O
computed	O
by	O
Pr( W	O
≥	O
w	O
)	O
,	O
which	O
is	O
the	O
p-value	B-General_Concept
for	O
the	O
alternative	O
H1	O
:	O
p	O
>	O
0.50	O
.	O
</s>
<s>
For	O
a	O
two-sided	O
alternative	O
H1	O
the	O
p-value	B-General_Concept
is	O
twice	O
the	O
smaller	O
tail-value	O
.	O
</s>
<s>
Zar	O
gives	O
the	O
following	O
example	O
of	O
the	O
sign	B-General_Concept
test	I-General_Concept
for	O
matched	O
pairs	O
.	O
</s>
<s>
The	O
null	B-General_Concept
hypothesis	I-General_Concept
is	O
that	O
there	O
is	O
no	O
difference	O
between	O
the	O
hind	O
leg	O
and	O
foreleg	O
length	O
in	O
deer	O
.	O
</s>
<s>
If	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
is	O
true	O
,	O
that	O
there	O
is	O
no	O
difference	O
in	O
hind	O
leg	O
and	O
foreleg	O
lengths	O
,	O
then	O
the	O
expected	O
number	O
of	O
positive	O
differences	O
is	O
5	O
out	O
of	O
10	O
.	O
</s>
<s>
The	O
probabilities	O
can	O
be	O
calculated	O
using	O
the	O
binomial	B-General_Concept
test	I-General_Concept
,	O
with	O
the	O
probability	O
of	O
heads	O
=	O
probability	O
of	O
tails	O
=	O
0.5	O
.	O
</s>
<s>
The	O
null	B-General_Concept
hypothesis	I-General_Concept
is	O
not	O
rejected	O
at	O
a	O
significance	B-General_Concept
level	I-General_Concept
of	O
p	O
=	O
0.05	O
.	O
</s>
<s>
With	O
a	O
larger	O
sample	O
size	O
,	O
the	O
evidence	O
might	O
be	O
sufficient	O
to	O
reject	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
.	O
</s>
<s>
Because	O
the	O
observations	O
can	O
be	O
expressed	O
as	O
numeric	O
quantities	O
(	O
actual	O
leg	O
length	O
)	O
,	O
the	O
paired	B-General_Concept
t-test	I-General_Concept
or	O
Wilcoxon	B-General_Concept
signed	I-General_Concept
rank	I-General_Concept
test	I-General_Concept
will	O
usually	O
have	O
greater	O
power	O
than	O
the	O
sign	B-General_Concept
test	I-General_Concept
to	O
detect	O
consistent	O
differences	O
.	O
</s>
<s>
For	O
this	O
example	O
,	O
the	O
paired	B-General_Concept
t-test	I-General_Concept
for	O
differences	O
indicates	O
that	O
there	O
is	O
a	O
significant	B-General_Concept
difference	I-General_Concept
between	O
hind	O
leg	O
length	O
and	O
foreleg	O
length	O
(	O
p	O
=	O
0.007	O
)	O
.	O
</s>
<s>
If	O
the	O
observed	O
result	O
was	O
9	O
positive	O
differences	O
in	O
10	O
comparisons	O
,	O
the	O
sign	B-General_Concept
test	I-General_Concept
would	O
be	O
significant	O
.	O
</s>
<s>
Conover	O
gives	O
the	O
following	O
example	O
using	O
a	O
one-sided	O
sign	B-General_Concept
test	I-General_Concept
for	O
matched	O
pairs	O
.	O
</s>
<s>
The	O
null	B-General_Concept
hypothesis	I-General_Concept
is	O
that	O
consumers	O
do	O
not	O
prefer	O
product	O
B	O
over	O
product	O
A	O
.	O
</s>
<s>
What	O
is	O
the	O
probability	O
of	O
a	O
result	O
as	O
extreme	O
as	O
8	O
positives	O
in	O
favor	O
of	O
B	O
in	O
9	O
pairs	O
,	O
if	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
is	O
true	O
,	O
that	O
consumers	O
have	O
no	O
preference	O
for	O
B	O
over	O
A	O
?	O
</s>
<s>
The	O
null	B-General_Concept
hypothesis	I-General_Concept
is	O
rejected	O
,	O
and	O
the	O
manufacturer	O
concludes	O
that	O
consumers	O
prefer	O
product	O
B	O
over	O
product	O
A	O
.	O
</s>
<s>
Sprent	O
gives	O
the	O
following	O
example	O
of	O
a	O
sign	B-General_Concept
test	I-General_Concept
for	O
a	O
median	O
.	O
</s>
<s>
The	O
null	B-General_Concept
hypothesis	I-General_Concept
is	O
that	O
median	O
survival	O
is	O
200	O
weeks	O
.	O
</s>
<s>
If	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
is	O
true	O
,	O
that	O
the	O
median	O
survival	O
is	O
200	O
weeks	O
,	O
then	O
,	O
in	O
a	O
random	O
sample	O
approximately	O
half	O
the	O
subjects	O
should	O
survive	O
less	O
than	O
200	O
weeks	O
,	O
and	O
half	O
should	O
survive	O
more	O
than	O
200	O
weeks	O
.	O
</s>
<s>
The	O
expected	O
number	O
of	O
plus	O
signs	O
is	O
5	O
if	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
is	O
true	O
.	O
</s>
<s>
The	O
null	B-General_Concept
hypothesis	I-General_Concept
is	O
not	O
rejected	O
.	O
</s>
<s>
The	O
sign	B-General_Concept
test	I-General_Concept
is	O
a	O
special	O
case	O
of	O
the	O
binomial	B-General_Concept
test	I-General_Concept
where	O
the	O
probability	O
of	O
success	O
under	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
is	O
p	O
=	O
0.5	O
.	O
</s>
<s>
Thus	O
,	O
the	O
sign	B-General_Concept
test	I-General_Concept
can	O
be	O
performed	O
using	O
the	O
binomial	B-General_Concept
test	I-General_Concept
,	O
which	O
is	O
provided	O
in	O
most	O
statistical	O
software	O
programs	O
.	O
</s>
<s>
On-line	O
calculators	O
for	O
the	O
sign	B-General_Concept
test	I-General_Concept
can	O
be	O
founded	O
by	O
searching	O
for	O
"	O
sign	B-General_Concept
test	I-General_Concept
calculator	O
"	O
.	O
</s>
<s>
Many	O
websites	O
offer	O
the	O
binomial	B-General_Concept
test	I-General_Concept
,	O
but	O
generally	O
offer	O
only	O
a	O
two-sided	O
version	O
.	O
</s>
<s>
In	O
R	B-Language
,	O
the	O
binomial	B-General_Concept
test	I-General_Concept
can	O
be	O
performed	O
using	O
the	O
function	O
binom.test( )	O
.	O
</s>
<s>
The	O
sign	B-General_Concept
test	I-General_Concept
example	O
from	O
Zar	O
compared	O
the	O
length	O
of	O
hind	O
legs	O
and	O
forelegs	O
of	O
deer	O
.	O
</s>
<s>
The	O
hypothesized	O
probability	O
of	O
success	O
(	O
defined	O
as	O
hind	O
leg	O
longer	O
than	O
foreleg	O
)	O
is	O
p	O
=	O
0.5	O
under	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
that	O
hind	O
legs	O
and	O
forelegs	O
do	O
not	O
differ	O
in	O
length	O
.	O
</s>
<s>
The	O
R	B-Language
command	O
gives	O
p	O
=	O
0.1094	O
,	O
as	O
in	O
the	O
example	O
.	O
</s>
<s>
The	O
sign	B-General_Concept
test	I-General_Concept
example	O
in	O
Conover	O
examined	O
consumer	O
preference	O
for	O
product	O
A	O
vs.	O
product	O
B	O
.	O
</s>
<s>
The	O
null	B-General_Concept
hypothesis	I-General_Concept
was	O
that	O
consumers	O
do	O
not	O
prefer	O
product	O
B	O
over	O
product	O
A	O
.	O
</s>
<s>
The	O
R	B-Language
command	O
gives	O
p	O
=	O
0.01953	O
,	O
as	O
in	O
the	O
example	O
.	O
</s>
<s>
describe	O
John	O
Arbuthnot	O
's	O
use	O
of	O
the	O
sign	B-General_Concept
test	I-General_Concept
in	O
1710	O
.	O
</s>
<s>
If	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
of	O
equal	O
number	O
of	O
births	O
is	O
true	O
,	O
the	O
probability	O
of	O
the	O
observed	O
outcome	O
is	O
1/282	O
,	O
leading	O
Arbuthnot	O
to	O
conclude	O
that	O
the	O
probability	O
of	O
male	O
and	O
female	O
births	O
were	O
not	O
exactly	O
equal	O
.	O
</s>
<s>
For	O
his	O
publications	O
in	O
1692	O
and	O
1710	O
,	O
Arbuthnot	O
is	O
credited	O
with	O
"	O
…	O
the	O
first	O
use	O
of	O
significance	B-General_Concept
tests	O
…	O
"	O
</s>
<s>
,	O
the	O
first	O
example	O
of	O
reasoning	O
about	O
statistical	B-General_Concept
significance	I-General_Concept
and	O
moral	O
certainty	O
,	O
and	O
"	O
…	O
perhaps	O
the	O
first	O
published	O
report	O
of	O
a	O
nonparametric	B-General_Concept
test	I-General_Concept
…	O
"	O
.	O
</s>
<s>
Hence	O
we	O
here	O
have	O
a	O
test	O
of	O
significance	B-General_Concept
rejecting	O
the	O
hypothesis	O
p	O
=	O
0.5	O
followed	O
by	O
an	O
estimation	O
of	O
p	O
and	O
a	O
discussion	O
of	O
the	O
goodness	O
of	O
fit	O
…	O
"	O
</s>
<s>
The	O
sign	B-General_Concept
test	I-General_Concept
requires	O
only	O
that	O
the	O
observations	O
in	O
a	O
pair	O
be	O
ordered	O
,	O
for	O
example	O
x	O
>	O
y	O
.	O
</s>
<s>
If	O
the	O
observations	O
can	O
be	O
ranked	O
,	O
and	O
each	O
observation	O
in	O
a	O
pair	O
is	O
a	O
random	O
sample	O
from	O
a	O
symmetric	O
distribution	O
,	O
then	O
the	O
Wilcoxon	B-General_Concept
signed-rank	I-General_Concept
test	I-General_Concept
is	O
appropriate	O
.	O
</s>
<s>
The	O
Wilcoxon	B-General_Concept
test	I-General_Concept
will	O
generally	O
have	O
greater	O
power	O
to	O
detect	O
differences	O
than	O
the	O
sign	B-General_Concept
test	I-General_Concept
.	O
</s>
<s>
The	O
asymptotic	O
relative	O
efficiency	O
of	O
the	O
sign	B-General_Concept
test	I-General_Concept
to	O
the	O
Wilcoxon	B-General_Concept
signed	I-General_Concept
rank	I-General_Concept
test	I-General_Concept
,	O
under	O
these	O
circumstances	O
,	O
is	O
0.67	O
.	O
</s>
<s>
If	O
the	O
paired	O
observations	O
are	O
numeric	O
quantities	O
(	O
such	O
as	O
the	O
actual	O
length	O
of	O
the	O
hind	O
leg	O
and	O
foreleg	O
in	O
the	O
Zar	O
example	O
)	O
,	O
and	O
the	O
differences	O
between	O
paired	O
observations	O
are	O
random	O
samples	O
from	O
a	O
single	O
normal	O
distribution	O
,	O
then	O
the	O
paired	B-General_Concept
t-test	I-General_Concept
is	O
appropriate	O
.	O
</s>
<s>
The	O
paired	B-General_Concept
t-test	I-General_Concept
will	O
generally	O
have	O
greater	O
power	O
to	O
detect	O
differences	O
than	O
the	O
sign	B-General_Concept
test	I-General_Concept
.	O
</s>
<s>
The	O
asymptotic	O
relative	O
efficiency	O
of	O
the	O
sign	B-General_Concept
test	I-General_Concept
to	O
the	O
paired	B-General_Concept
t-test	I-General_Concept
,	O
under	O
these	O
circumstances	O
,	O
is	O
0.637	O
.	O
</s>
<s>
However	O
,	O
if	O
the	O
distribution	O
of	O
the	O
differences	O
between	O
pairs	O
is	O
not	O
normal	O
,	O
but	O
instead	O
is	O
heavy-tailed	O
(	O
platykurtic	B-Error_Name
distribution	I-Error_Name
)	O
,	O
the	O
sign	B-General_Concept
test	I-General_Concept
can	O
have	O
more	O
power	O
than	O
the	O
paired	B-General_Concept
t-test	I-General_Concept
,	O
with	O
asymptotic	O
relative	O
efficiency	O
of	O
2.0	O
relative	O
to	O
the	O
paired	B-General_Concept
t-test	I-General_Concept
and	O
1.3	O
relative	O
to	O
the	O
Wilcoxon	B-General_Concept
signed	I-General_Concept
rank	I-General_Concept
test	I-General_Concept
.	O
</s>
<s>
In	O
these	O
cases	O
,	O
the	O
same	O
procedure	O
as	O
the	O
sign	B-General_Concept
test	I-General_Concept
is	O
used	O
,	O
but	O
is	O
known	O
as	O
McNemar	B-General_Concept
's	I-General_Concept
test	I-General_Concept
.	O
</s>
<s>
If	O
the	O
individual	O
observations	O
can	O
be	O
ordered	O
in	O
the	O
same	O
way	O
as	O
for	O
the	O
sign	B-General_Concept
test	I-General_Concept
,	O
for	O
example	O
B	O
>	O
C	O
>	O
A	O
,	O
then	O
the	O
Friedman	B-General_Concept
test	I-General_Concept
may	O
be	O
used	O
.	O
</s>
<s>
Bian	O
,	O
McAleer	O
and	O
Wong	O
proposed	O
in	O
2011	O
a	O
non-parametric	B-General_Concept
test	I-General_Concept
for	O
paired	O
data	O
when	O
there	O
are	O
many	O
ties	O
.	O
</s>
<s>
They	O
showed	O
that	O
their	O
trinomial	O
test	O
is	O
superior	O
to	O
the	O
sign	B-General_Concept
test	I-General_Concept
in	O
presence	O
of	O
ties	O
.	O
</s>
