<s>
In	O
statistics	O
,	O
Spearman	B-General_Concept
's	I-General_Concept
rank	I-General_Concept
correlation	I-General_Concept
coefficient	I-General_Concept
or	O
Spearman	B-General_Concept
's	I-General_Concept
ρ	I-General_Concept
,	O
named	O
after	O
Charles	O
Spearman	O
and	O
often	O
denoted	O
by	O
the	O
Greek	O
letter	O
(	O
rho	O
)	O
or	O
as	O
,	O
is	O
a	O
nonparametric	B-General_Concept
measure	O
of	O
rank	O
correlation	O
(	O
statistical	O
dependence	O
between	O
the	O
rankings	O
of	O
two	O
variables	O
)	O
.	O
</s>
<s>
The	O
Spearman	B-General_Concept
correlation	I-General_Concept
between	O
two	O
variables	O
is	O
equal	O
to	O
the	O
Pearson	O
correlation	O
between	O
the	O
rank	O
values	O
of	O
those	O
two	O
variables	O
;	O
while	O
Pearson	O
's	O
correlation	O
assesses	O
linear	O
relationships	O
,	O
Spearman	B-General_Concept
's	I-General_Concept
correlation	I-General_Concept
assesses	O
monotonic	O
relationships	O
(	O
whether	O
linear	O
or	O
not	O
)	O
.	O
</s>
<s>
If	O
there	O
are	O
no	O
repeated	O
data	O
values	O
,	O
a	O
perfect	O
Spearman	B-General_Concept
correlation	I-General_Concept
of	O
+1	O
or	O
−1	O
occurs	O
when	O
each	O
of	O
the	O
variables	O
is	O
a	O
perfect	O
monotone	O
function	O
of	O
the	O
other	O
.	O
</s>
<s>
Intuitively	O
,	O
the	O
Spearman	B-General_Concept
correlation	I-General_Concept
between	O
two	O
variables	O
will	O
be	O
high	O
when	O
observations	O
have	O
a	O
similar	O
(	O
or	O
identical	O
for	O
a	O
correlation	O
of	O
1	O
)	O
rank	O
(	O
i.e.	O
</s>
<s>
The	O
Spearman	B-General_Concept
correlation	I-General_Concept
coefficient	O
is	O
defined	O
as	O
the	O
Pearson	O
correlation	O
coefficient	O
between	O
the	O
rank	O
variables	O
.	O
</s>
<s>
and	O
are	O
the	O
standard	B-General_Concept
deviations	I-General_Concept
of	O
the	O
rank	O
variables	O
.	O
</s>
<s>
The	O
first	O
equation	O
—	O
normalizing	O
by	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
—	O
may	O
be	O
used	O
even	O
when	O
ranks	O
are	O
normalized	O
to	O
 [ 0 , 1 ] 	O
(	O
"	O
relative	O
ranks	O
"	O
)	O
because	O
it	O
is	O
insensitive	O
both	O
to	O
translation	O
and	O
linear	O
scaling	O
.	O
</s>
<s>
The	O
simplified	O
method	O
should	O
also	O
not	O
be	O
used	O
in	O
cases	O
where	O
the	O
data	O
set	O
is	O
truncated	O
;	O
that	O
is	O
,	O
when	O
the	O
Spearman	B-General_Concept
's	I-General_Concept
correlation	I-General_Concept
coefficient	O
is	O
desired	O
for	O
the	O
top	O
X	O
records	O
(	O
whether	O
by	O
pre-change	O
rank	O
or	O
post-change	O
rank	O
,	O
or	O
both	O
)	O
,	O
the	O
user	O
should	O
use	O
the	O
Pearson	O
correlation	O
coefficient	O
formula	O
given	O
above	O
.	O
</s>
<s>
The	O
most	O
common	O
of	O
these	O
is	O
the	O
Pearson	O
product-moment	O
correlation	O
coefficient	O
,	O
which	O
is	O
a	O
similar	O
correlation	O
method	O
to	O
Spearman	B-General_Concept
's	I-General_Concept
rank	I-General_Concept
,	O
that	O
measures	O
the	O
“	O
linear	O
”	O
relationships	O
between	O
the	O
raw	O
numbers	O
rather	O
than	O
between	O
their	O
ranks	O
.	O
</s>
<s>
An	O
alternative	O
name	O
for	O
the	O
Spearman	B-General_Concept
rank	I-General_Concept
correlation	I-General_Concept
is	O
the	O
“	O
grade	O
correlation	O
”	O
;	O
in	O
this	O
,	O
the	O
“	O
rank	O
”	O
of	O
an	O
observation	O
is	O
replaced	O
by	O
the	O
“	O
grade	O
”	O
.	O
</s>
<s>
The	O
sign	O
of	O
the	O
Spearman	B-General_Concept
correlation	I-General_Concept
indicates	O
the	O
direction	O
of	O
association	O
between	O
X	O
(	O
the	O
independent	O
variable	O
)	O
and	O
Y	O
(	O
the	O
dependent	O
variable	O
)	O
.	O
</s>
<s>
If	O
Y	O
tends	O
to	O
increase	O
when	O
X	O
increases	O
,	O
the	O
Spearman	B-General_Concept
correlation	I-General_Concept
coefficient	O
is	O
positive	O
.	O
</s>
<s>
If	O
Y	O
tends	O
to	O
decrease	O
when	O
X	O
increases	O
,	O
the	O
Spearman	B-General_Concept
correlation	I-General_Concept
coefficient	O
is	O
negative	O
.	O
</s>
<s>
A	O
Spearman	B-General_Concept
correlation	I-General_Concept
of	O
zero	O
indicates	O
that	O
there	O
is	O
no	O
tendency	O
for	O
Y	O
to	O
either	O
increase	O
or	O
decrease	O
when	O
X	O
increases	O
.	O
</s>
<s>
The	O
Spearman	B-General_Concept
correlation	I-General_Concept
increases	O
in	O
magnitude	O
as	O
X	O
and	O
Y	O
become	O
closer	O
to	O
being	O
perfectly	O
monotone	O
functions	O
of	O
each	O
other	O
.	O
</s>
<s>
When	O
X	O
and	O
Y	O
are	O
perfectly	O
monotonically	O
related	O
,	O
the	O
Spearman	B-General_Concept
correlation	I-General_Concept
coefficient	O
becomes	O
1	O
.	O
</s>
<s>
The	O
Spearman	B-General_Concept
correlation	I-General_Concept
coefficient	O
is	O
often	O
described	O
as	O
being	O
"	O
nonparametric	B-General_Concept
"	O
.	O
</s>
<s>
First	O
,	O
a	O
perfect	O
Spearman	B-General_Concept
correlation	I-General_Concept
results	O
when	O
X	O
and	O
Y	O
are	O
related	O
by	O
any	O
monotonic	O
function	O
.	O
</s>
<s>
The	O
other	O
sense	O
in	O
which	O
the	O
Spearman	B-General_Concept
correlation	I-General_Concept
is	O
nonparametric	B-General_Concept
is	O
that	O
its	O
exact	O
sampling	O
distribution	O
can	O
be	O
obtained	O
without	O
requiring	O
knowledge	O
(	O
i.e.	O
,	O
knowing	O
the	O
parameters	O
)	O
of	O
the	O
joint	O
probability	O
distribution	O
of	O
X	O
and	O
Y	O
.	O
</s>
<s>
In	O
this	O
example	O
,	O
the	O
arbitrary	O
raw	O
data	O
in	O
the	O
table	O
below	O
is	O
used	O
to	O
calculate	O
the	O
correlation	O
between	O
the	O
IQ	O
of	O
a	O
person	O
with	O
the	O
number	O
of	O
hours	O
spent	O
in	O
front	O
of	O
TV	B-Operating_System
per	O
week	O
[	O
fictitious	O
values	O
used ]	O
.	O
</s>
<s>
which	O
evaluates	O
to	O
with	O
a	O
p-value	B-General_Concept
=	O
0.627188	O
(	O
using	O
the	O
t-distribution	O
)	O
.	O
</s>
<s>
That	O
the	O
value	O
is	O
close	O
to	O
zero	O
shows	O
that	O
the	O
correlation	O
between	O
IQ	O
and	O
hours	O
spent	O
watching	O
TV	B-Operating_System
is	O
very	O
low	O
,	O
although	O
the	O
negative	O
value	O
suggests	O
that	O
the	O
longer	O
the	O
time	O
spent	O
watching	O
television	O
the	O
lower	O
the	O
IQ	O
.	O
</s>
<s>
Confidence	O
intervals	O
for	O
Spearman	B-General_Concept
's	I-General_Concept
ρ	I-General_Concept
can	O
be	O
easily	O
obtained	O
using	O
the	O
Jackknife	O
Euclidean	O
likelihood	O
approach	O
in	O
de	O
Carvalho	O
and	O
Marques	O
(	O
2012	O
)	O
.	O
</s>
<s>
This	O
approach	O
is	O
implemented	O
in	O
the	O
R	B-Language
package	O
.	O
</s>
<s>
One	O
approach	O
to	O
test	O
whether	O
an	O
observed	O
value	O
of	O
ρ	O
is	O
significantly	O
different	O
from	O
zero	O
(	O
r	B-Language
will	O
always	O
maintain	O
)	O
is	O
to	O
calculate	O
the	O
probability	O
that	O
it	O
would	O
be	O
greater	O
than	O
or	O
equal	O
to	O
the	O
observed	O
r	B-Language
,	O
given	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
,	O
by	O
using	O
a	O
permutation	O
test	O
.	O
</s>
<s>
Another	O
approach	O
parallels	O
the	O
use	O
of	O
the	O
Fisher	B-Algorithm
transformation	I-Algorithm
in	O
the	O
case	O
of	O
the	O
Pearson	O
product-moment	O
correlation	O
coefficient	O
.	O
</s>
<s>
That	O
is	O
,	O
confidence	O
intervals	O
and	O
hypothesis	O
tests	O
relating	O
to	O
the	O
population	O
value	O
ρ	O
can	O
be	O
carried	O
out	O
using	O
the	O
Fisher	B-Algorithm
transformation	I-Algorithm
:	O
</s>
<s>
is	O
a	O
z-score	O
for	O
r	B-Language
,	O
which	O
approximately	O
follows	O
a	O
standard	O
normal	O
distribution	O
under	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
of	O
statistical	O
independence	O
(	O
)	O
.	O
</s>
<s>
which	O
is	O
distributed	O
approximately	O
as	O
Student	O
's	O
t-distribution	O
with	O
degrees	O
of	O
freedom	O
under	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
.	O
</s>
<s>
A	O
generalization	O
of	O
the	O
Spearman	B-General_Concept
coefficient	I-General_Concept
is	O
useful	O
in	O
the	O
situation	O
where	O
there	O
are	O
three	O
or	O
more	O
conditions	O
,	O
a	O
number	O
of	O
subjects	O
are	O
all	O
observed	O
in	O
each	O
of	O
them	O
,	O
and	O
it	O
is	O
predicted	O
that	O
the	O
observations	O
will	O
have	O
a	O
particular	O
order	O
.	O
</s>
<s>
Page	O
and	O
is	O
usually	O
referred	O
to	O
as	O
Page	B-General_Concept
's	I-General_Concept
trend	I-General_Concept
test	I-General_Concept
for	O
ordered	O
alternatives	O
.	O
</s>
<s>
Classic	O
correspondence	B-Algorithm
analysis	I-Algorithm
is	O
a	O
statistical	O
method	O
that	O
gives	O
a	O
score	O
to	O
every	O
value	O
of	O
two	O
nominal	O
variables	O
.	O
</s>
<s>
There	O
exists	O
an	O
equivalent	O
of	O
this	O
method	O
,	O
called	O
grade	O
correspondence	B-Algorithm
analysis	I-Algorithm
,	O
which	O
maximizes	O
Spearman	B-General_Concept
's	I-General_Concept
ρ	I-General_Concept
or	O
Kendall	B-General_Concept
's	I-General_Concept
τ	I-General_Concept
.	O
</s>
<s>
There	O
are	O
two	O
existing	O
approaches	O
to	O
approximating	O
the	O
Spearman	B-General_Concept
's	I-General_Concept
rank	I-General_Concept
correlation	I-General_Concept
coefficient	I-General_Concept
from	O
streaming	O
data	O
.	O
</s>
<s>
For	O
non-stationary	O
streaming	O
data	O
,	O
where	O
the	O
Spearman	B-General_Concept
's	I-General_Concept
rank	I-General_Concept
correlation	I-General_Concept
coefficient	I-General_Concept
may	O
change	O
over	O
time	O
,	O
the	O
same	O
procedure	O
can	O
be	O
applied	O
,	O
but	O
to	O
a	O
moving	O
window	O
of	O
observations	O
.	O
</s>
<s>
The	O
second	O
approach	O
to	O
approximating	O
the	O
Spearman	B-General_Concept
's	I-General_Concept
rank	I-General_Concept
correlation	I-General_Concept
coefficient	I-General_Concept
from	O
streaming	O
data	O
involves	O
the	O
use	O
of	O
Hermite	O
series	O
based	O
estimators	O
.	O
</s>
<s>
Spearman	B-General_Concept
's	I-General_Concept
rank	I-General_Concept
correlation	I-General_Concept
coefficient	I-General_Concept
estimator	O
,	O
to	O
give	O
a	O
sequential	O
Spearman	B-General_Concept
's	I-General_Concept
correlation	I-General_Concept
estimator	O
.	O
</s>
<s>
Instead	O
,	O
the	O
Hermite	O
series	O
based	O
estimator	O
uses	O
an	O
exponential	O
weighting	O
scheme	O
to	O
track	O
time-varying	O
Spearman	B-General_Concept
's	I-General_Concept
rank	I-General_Concept
correlation	I-General_Concept
from	O
streaming	O
data	O
,	O
</s>
<s>
A	O
software	O
implementation	O
of	O
these	O
Hermite	O
series	O
based	O
algorithms	O
exists	O
and	O
is	O
discussed	O
in	O
Software	B-General_Concept
implementations	I-General_Concept
.	O
</s>
<s>
R	B-Language
's	O
statistics	O
base-package	O
implements	O
the	O
test	O
in	O
its	O
"	O
stats	O
"	O
package	O
(	O
also	O
cor( x	O
,	O
y	O
,	O
method	O
=	O
"	O
spearman	O
"	O
)	O
will	O
work	O
.	O
</s>
<s>
The	O
package	O
computes	O
fast	O
batch	O
estimates	O
of	O
the	O
Spearman	B-General_Concept
correlation	I-General_Concept
along	O
with	O
sequential	O
estimates	O
(	O
i.e.	O
</s>
<s>
Stata	B-Algorithm
implementation	O
:	O
spearman	O
varlist	O
calculates	O
all	O
pairwise	O
correlation	O
coefficients	O
for	O
all	O
variables	O
in	O
varlist	O
.	O
</s>
<s>
MATLAB	B-Language
implementation	O
:	O
 [ r , p ] 	O
=	O
corr(x,y,'Type','Spearman' )	O
where	O
r	B-Language
is	O
the	O
Spearman	B-General_Concept
's	I-General_Concept
rank	I-General_Concept
correlation	I-General_Concept
coefficient	I-General_Concept
,	O
p	O
is	O
the	O
p-value	B-General_Concept
,	O
and	O
x	O
and	O
y	O
are	O
vectors	O
.	O
</s>
<s>
Python	B-Language
has	O
many	O
different	O
implementation	O
of	O
the	O
spearman	B-General_Concept
correlation	I-General_Concept
statistic	O
:	O
it	O
can	O
be	O
computed	O
with	O
the	O
function	O
of	O
the	O
scipy.stats	O
module	O
,	O
as	O
well	O
as	O
with	O
the	O
DataFrame.corr(method='spearman' )	O
method	O
from	O
the	O
library	O
,	O
and	O
the	O
corr( x	O
,	O
y	O
,	O
method='spearman'	O
)	O
function	O
from	O
the	O
statistical	O
package	O
.	O
</s>
