<s>
In	O
statistics	O
,	O
the	O
Mann	B-General_Concept
–	I-General_Concept
Whitney	I-General_Concept
U	I-General_Concept
test	I-General_Concept
(	O
also	O
called	O
the	O
Mann	B-General_Concept
–	I-General_Concept
Whitney	I-General_Concept
–	O
Wilcoxon	O
(	O
MWW/MWU	O
)	O
,	O
Wilcoxon	B-General_Concept
rank-sum	I-General_Concept
test	I-General_Concept
,	O
or	O
Wilcoxon	O
–	O
Mann	B-General_Concept
–	I-General_Concept
Whitney	I-General_Concept
test	I-General_Concept
)	O
is	O
a	O
nonparametric	B-General_Concept
test	I-General_Concept
of	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
that	O
,	O
for	O
randomly	O
selected	O
values	O
X	O
and	O
Y	O
from	O
two	O
populations	O
,	O
the	O
probability	O
of	O
X	O
being	O
greater	O
than	O
Y	O
is	O
equal	O
to	O
the	O
probability	O
of	O
Y	O
being	O
greater	O
thanX	O
.	O
</s>
<s>
Nonparametric	B-General_Concept
tests	I-General_Concept
used	O
on	O
two	O
dependent	O
samples	O
are	O
the	O
sign	B-General_Concept
test	I-General_Concept
and	O
the	O
Wilcoxon	B-General_Concept
signed-rank	I-General_Concept
test	I-General_Concept
.	O
</s>
<s>
Although	O
Mann	O
and	O
Whitney	O
developed	O
the	O
Mann	B-General_Concept
–	I-General_Concept
Whitney	I-General_Concept
U	I-General_Concept
test	I-General_Concept
under	O
the	O
assumption	O
of	O
continuous	O
responses	O
with	O
the	O
alternative	B-General_Concept
hypothesis	I-General_Concept
being	O
that	O
one	O
distribution	O
is	O
stochastically	O
greater	O
than	O
the	O
other	O
,	O
there	O
are	O
many	O
other	O
ways	O
to	O
formulate	O
the	O
null	B-General_Concept
and	O
alternative	B-General_Concept
hypotheses	I-General_Concept
such	O
that	O
the	O
Mann	B-General_Concept
–	I-General_Concept
Whitney	I-General_Concept
U	I-General_Concept
test	I-General_Concept
will	O
give	O
a	O
valid	O
test	O
.	O
</s>
<s>
Under	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
H0	O
,	O
the	O
distributions	O
of	O
both	O
populations	O
are	O
identical	O
.	O
</s>
<s>
The	O
alternative	B-General_Concept
hypothesis	I-General_Concept
H1	O
is	O
that	O
the	O
distributions	O
are	O
not	O
identical	O
.	O
</s>
<s>
Under	O
more	O
strict	O
assumptions	O
than	O
the	O
general	O
formulation	O
above	O
,	O
e.g.	O
,	O
if	O
the	O
responses	O
are	O
assumed	O
to	O
be	O
continuous	O
and	O
the	O
alternative	O
is	O
restricted	O
to	O
a	O
shift	O
in	O
location	O
,	O
i.e.	O
,	O
,	O
we	O
can	O
interpret	O
a	O
significant	O
Mann	B-General_Concept
–	I-General_Concept
Whitney	I-General_Concept
U	I-General_Concept
test	I-General_Concept
as	O
showing	O
a	O
difference	O
in	O
medians	O
.	O
</s>
<s>
Under	O
this	O
location	O
shift	O
assumption	O
,	O
we	O
can	O
also	O
interpret	O
the	O
Mann	B-General_Concept
–	I-General_Concept
Whitney	I-General_Concept
U	I-General_Concept
test	I-General_Concept
as	O
assessing	O
whether	O
the	O
Hodges	B-General_Concept
–	I-General_Concept
Lehmann	I-General_Concept
estimate	I-General_Concept
of	O
the	O
difference	O
in	O
central	O
tendency	O
between	O
the	O
two	O
populations	O
differs	O
from	O
zero	O
.	O
</s>
<s>
The	O
Hodges	B-General_Concept
–	I-General_Concept
Lehmann	I-General_Concept
estimate	I-General_Concept
for	O
this	O
two-sample	O
problem	O
is	O
the	O
median	O
of	O
all	O
possible	O
differences	O
between	O
an	O
observation	O
in	O
the	O
first	O
sample	O
and	O
an	O
observation	O
in	O
the	O
second	O
sample	O
.	O
</s>
<s>
Otherwise	O
,	O
if	O
both	O
the	O
dispersions	O
and	O
shapes	O
of	O
the	O
distribution	O
of	O
both	O
samples	O
differ	O
,	O
the	O
Mann	B-General_Concept
–	I-General_Concept
Whitney	I-General_Concept
U	I-General_Concept
test	I-General_Concept
fails	O
a	O
test	O
of	O
medians	O
.	O
</s>
<s>
It	O
is	O
possible	O
to	O
show	O
examples	O
where	O
medians	O
are	O
numerically	O
equal	O
while	O
the	O
test	O
rejects	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
with	O
a	O
small	O
p-value	O
.	O
</s>
<s>
The	O
Mann	B-General_Concept
–	I-General_Concept
Whitney	I-General_Concept
U	I-General_Concept
test	I-General_Concept
/	O
Wilcoxon	B-General_Concept
rank-sum	I-General_Concept
test	I-General_Concept
is	O
not	O
the	O
same	O
as	O
the	O
Wilcoxon	B-General_Concept
signed-rank	I-General_Concept
test	I-General_Concept
,	O
although	O
both	O
are	O
nonparametric	B-General_Concept
and	O
involve	O
summation	O
of	O
ranks	O
.	O
</s>
<s>
The	O
Mann	B-General_Concept
–	I-General_Concept
Whitney	I-General_Concept
U	I-General_Concept
test	I-General_Concept
is	O
applied	O
to	O
independent	O
samples	O
.	O
</s>
<s>
The	O
Wilcoxon	B-General_Concept
signed-rank	I-General_Concept
test	I-General_Concept
is	O
applied	O
to	O
matched	O
or	O
dependent	O
samples	O
.	O
</s>
<s>
The	O
corresponding	O
Mann	B-General_Concept
–	I-General_Concept
Whitney	I-General_Concept
U	I-General_Concept
statistic	O
is	O
defined	O
as	O
:	O
</s>
<s>
The	O
U	O
statistic	O
is	O
equivalent	O
to	O
the	O
area	O
under	O
the	O
receiver	B-Algorithm
operating	I-Algorithm
characteristic	I-Algorithm
curve	I-Algorithm
(	O
AUC	O
)	O
that	O
can	O
be	O
readily	O
calculated	O
.	O
</s>
<s>
Note	O
that	O
this	O
is	O
the	O
same	O
definition	O
as	O
the	O
common	O
language	O
effect	B-Application
size	I-Application
from	O
the	O
section	O
above	O
.	O
</s>
<s>
The	O
test	O
involves	O
the	O
calculation	O
of	O
a	O
statistic	O
,	O
usually	O
called	O
U	O
,	O
whose	O
distribution	O
under	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
is	O
known	O
.	O
</s>
<s>
The	O
Mann	B-General_Concept
–	I-General_Concept
Whitney	I-General_Concept
U	I-General_Concept
test	I-General_Concept
is	O
included	O
in	O
most	O
modern	O
statistical	B-Algorithm
packages	I-Algorithm
.	O
</s>
<s>
For	O
comparing	O
two	O
small	O
sets	O
of	O
observations	O
,	O
a	O
direct	O
method	O
is	O
quick	O
,	O
and	O
gives	O
insight	O
into	O
the	O
meaning	O
of	O
the	O
U	O
statistic	O
,	O
which	O
corresponds	O
to	O
the	O
number	O
of	O
wins	O
out	O
of	O
all	O
pairwise	O
contests	O
(	O
see	O
the	O
tortoise	O
and	O
hare	B-Application
example	O
under	O
Examples	O
below	O
)	O
.	O
</s>
<s>
Suppose	O
that	O
Aesop	O
is	O
dissatisfied	O
with	O
his	O
classic	O
experiment	O
in	O
which	O
one	O
tortoise	O
was	O
found	O
to	O
beat	O
one	O
hare	B-Application
in	O
a	O
race	O
,	O
and	O
decides	O
to	O
carry	O
out	O
a	O
significance	O
test	O
to	O
discover	O
whether	O
the	O
results	O
could	O
be	O
extended	O
to	O
tortoises	O
and	O
hares	B-Application
in	O
general	O
.	O
</s>
<s>
He	O
collects	O
a	O
sample	O
of	O
6	O
tortoises	O
and	O
6	O
hares	B-Application
,	O
and	O
makes	O
them	O
all	O
run	O
his	O
race	O
at	O
once	O
.	O
</s>
<s>
The	O
order	O
in	O
which	O
they	O
reach	O
the	O
finishing	O
post	O
(	O
their	O
rank	O
order	O
,	O
from	O
first	O
to	O
last	O
crossing	O
the	O
finish	O
line	O
)	O
is	O
as	O
follows	O
,	O
writing	O
T	O
for	O
a	O
tortoise	O
and	O
H	O
for	O
a	O
hare	B-Application
:	O
</s>
<s>
Using	O
the	O
direct	O
method	O
,	O
we	O
take	O
each	O
tortoise	O
in	O
turn	O
,	O
and	O
count	O
the	O
number	O
of	O
hares	B-Application
it	O
beats	O
,	O
getting	O
6	O
,	O
1	O
,	O
1	O
,	O
1	O
,	O
1	O
,	O
1	O
,	O
which	O
means	O
that	O
.	O
</s>
<s>
Alternatively	O
,	O
we	O
could	O
take	O
each	O
hare	B-Application
in	O
turn	O
,	O
and	O
count	O
the	O
number	O
of	O
tortoises	O
it	O
beats	O
.	O
</s>
<s>
The	O
sum	O
of	O
the	O
ranks	O
achieved	O
by	O
the	O
hares	B-Application
is	O
,	O
leading	O
to	O
.	O
</s>
<s>
In	O
reporting	O
the	O
results	O
of	O
a	O
Mann	B-General_Concept
–	I-General_Concept
Whitney	I-General_Concept
U	I-General_Concept
test	I-General_Concept
,	O
it	O
is	O
important	O
to	O
state	O
:	O
</s>
<s>
The	O
value	O
of	O
U	O
(	O
perhaps	O
with	O
some	O
measure	O
of	O
effect	B-Application
size	I-Application
,	O
such	O
as	O
common	O
language	O
effect	B-Application
size	I-Application
or	O
rank-biserial	O
correlation	O
)	O
.	O
</s>
<s>
"	O
Median	O
latencies	O
in	O
groups	O
E	O
and	O
C	O
were	O
153	O
and	O
247	O
ms	O
;	O
the	O
distributions	O
in	O
the	O
two	O
groups	O
differed	O
significantly	O
(	O
Mann	B-General_Concept
–	I-General_Concept
Whitney	I-General_Concept
,	O
,	O
two-tailed	O
)	O
.	O
"	O
</s>
<s>
"	O
Outcomes	O
of	O
the	O
two	O
treatments	O
were	O
compared	O
using	O
the	O
Wilcoxon	O
–	O
Mann	B-General_Concept
–	I-General_Concept
Whitney	I-General_Concept
two-sample	O
rank-sum	B-General_Concept
test	I-General_Concept
.	O
</s>
<s>
The	O
treatment	O
effect	O
(	O
difference	O
between	O
treatments	O
)	O
was	O
quantified	O
using	O
the	O
Hodges	B-General_Concept
–	I-General_Concept
Lehmann	I-General_Concept
(	O
HL	O
)	O
estimator	O
,	O
which	O
is	O
consistent	O
with	O
the	O
Wilcoxon	B-General_Concept
test	I-General_Concept
.	O
</s>
<s>
A	O
non-parametric	B-General_Concept
0.95	O
confidence	O
interval	O
for	O
HLΔ	O
accompanies	O
these	O
estimates	O
as	O
does	O
ρ	O
,	O
an	O
estimate	O
of	O
the	O
probability	O
that	O
a	O
randomly	O
chosen	O
subject	O
from	O
population	O
B	O
has	O
a	O
higher	O
weight	O
than	O
a	O
randomly	O
chosen	O
subject	O
from	O
populationA	O
.	O
</s>
<s>
The	O
computer	O
statistical	B-Algorithm
packages	I-Algorithm
will	O
use	O
the	O
correctly	O
adjusted	O
formula	O
as	O
a	O
matter	O
of	O
routine	O
.	O
</s>
<s>
It	O
is	O
a	O
widely	O
recommended	O
practice	O
for	O
scientists	O
to	O
report	O
an	O
effect	B-Application
size	I-Application
for	O
an	O
inferential	O
test	O
.	O
</s>
<s>
One	O
method	O
of	O
reporting	O
the	O
effect	B-Application
size	I-Application
for	O
the	O
Mann	B-General_Concept
–	I-General_Concept
Whitney	I-General_Concept
U	I-General_Concept
test	I-General_Concept
is	O
with	O
f	O
,	O
the	O
common	O
language	O
effect	B-Application
size	I-Application
.	O
</s>
<s>
As	O
a	O
sample	O
statistic	O
,	O
the	O
common	O
language	O
effect	B-Application
size	I-Application
is	O
computed	O
by	O
forming	O
all	O
possible	O
pairs	O
between	O
the	O
two	O
groups	O
,	O
then	O
finding	O
the	O
proportion	O
of	O
pairs	O
that	O
support	O
a	O
direction	O
(	O
say	O
,	O
that	O
items	O
from	O
group	O
1	O
are	O
larger	O
than	O
items	O
from	O
group	O
2	O
)	O
.	O
</s>
<s>
To	O
illustrate	O
,	O
in	O
a	O
study	O
with	O
a	O
sample	O
of	O
ten	O
hares	B-Application
and	O
ten	O
tortoises	O
,	O
the	O
total	O
number	O
of	O
ordered	O
pairs	O
is	O
ten	O
times	O
ten	O
or	O
100	O
pairs	O
of	O
hares	B-Application
and	O
tortoises	O
.	O
</s>
<s>
Suppose	O
the	O
results	O
show	O
that	O
the	O
hare	B-Application
ran	O
faster	O
than	O
the	O
tortoise	O
in	O
90	O
of	O
the	O
100	O
sample	O
pairs	O
;	O
in	O
that	O
case	O
,	O
the	O
sample	O
common	O
language	O
effect	B-Application
size	I-Application
is	O
90%	O
.	O
</s>
<s>
This	O
sample	O
value	O
is	O
an	O
unbiased	O
estimator	O
of	O
the	O
population	O
value	O
,	O
so	O
the	O
sample	O
suggests	O
that	O
the	O
best	O
estimate	O
of	O
the	O
common	O
language	O
effect	B-Application
size	I-Application
in	O
the	O
population	O
is	O
90%	O
.	O
</s>
<s>
The	O
relationship	O
between	O
f	O
and	O
the	O
Mann	B-General_Concept
–	I-General_Concept
Whitney	I-General_Concept
U	I-General_Concept
(	O
specifically	O
)	O
is	O
as	O
follows	O
:	O
</s>
<s>
This	O
is	O
the	O
same	O
as	O
the	O
area	O
under	O
the	O
curve	O
(	O
AUC	O
)	O
for	O
the	O
ROC	B-Algorithm
curve	I-Algorithm
.	O
</s>
<s>
ρ	O
is	O
thus	O
a	O
non-parametric	B-General_Concept
measure	O
of	O
the	O
overlap	O
between	O
two	O
distributions	O
;	O
it	O
can	O
take	O
values	O
between	O
0	O
and	O
1	O
,	O
and	O
it	O
is	O
an	O
estimate	O
of	O
,	O
where	O
X	O
and	O
Y	O
are	O
randomly	O
chosen	O
observations	O
from	O
the	O
two	O
distributions	O
.	O
</s>
<s>
The	O
usefulness	O
of	O
the	O
ρ	O
statistic	O
can	O
be	O
seen	O
in	O
the	O
case	O
of	O
the	O
odd	O
example	O
used	O
above	O
,	O
where	O
two	O
distributions	O
that	O
were	O
significantly	O
different	O
on	O
a	O
Mann	B-General_Concept
–	I-General_Concept
Whitney	I-General_Concept
U	I-General_Concept
test	I-General_Concept
nonetheless	O
had	O
nearly	O
identical	O
medians	O
:	O
the	O
ρ	O
value	O
in	O
this	O
case	O
is	O
approximately	O
0.723	O
in	O
favour	O
of	O
the	O
hares	B-Application
,	O
correctly	O
reflecting	O
the	O
fact	O
that	O
even	O
though	O
the	O
median	O
tortoise	O
beat	O
the	O
median	O
hare	B-Application
,	O
the	O
hares	B-Application
collectively	O
did	O
better	O
than	O
the	O
tortoises	O
collectively	O
.	O
</s>
<s>
A	O
method	O
of	O
reporting	O
the	O
effect	B-Application
size	I-Application
for	O
the	O
Mann	B-General_Concept
–	I-General_Concept
Whitney	I-General_Concept
U	I-General_Concept
test	I-General_Concept
is	O
with	O
a	O
measure	O
of	O
rank	O
correlation	O
known	O
as	O
the	O
rank-biserial	O
correlation	O
.	O
</s>
<s>
There	O
is	O
a	O
simple	O
difference	O
formula	O
to	O
compute	O
the	O
rank-biserial	O
correlation	O
from	O
the	O
common	O
language	O
effect	B-Application
size	I-Application
:	O
the	O
correlation	O
is	O
the	O
difference	O
between	O
the	O
proportion	O
of	O
pairs	O
favorable	O
to	O
the	O
hypothesis	O
(	O
f	O
)	O
minus	O
its	O
complement	O
(	O
i.e.	O
</s>
<s>
This	O
simple	O
difference	O
formula	O
is	O
just	O
the	O
difference	O
of	O
the	O
common	O
language	O
effect	B-Application
size	I-Application
of	O
each	O
group	O
,	O
and	O
is	O
as	O
follows	O
:	O
</s>
<s>
For	O
example	O
,	O
consider	O
the	O
example	O
where	O
hares	B-Application
run	O
faster	O
than	O
tortoises	O
in	O
90	O
of	O
100	O
pairs	O
.	O
</s>
<s>
The	O
common	O
language	O
effect	B-Application
size	I-Application
is	O
90%	O
,	O
so	O
the	O
rank-biserial	O
correlation	O
is	O
90%	O
minus	O
10%	O
,	O
and	O
the	O
rank-biserial	O
.	O
</s>
<s>
An	O
alternative	O
formula	O
for	O
the	O
rank-biserial	O
can	O
be	O
used	O
to	O
calculate	O
it	O
from	O
the	O
Mann	B-General_Concept
–	I-General_Concept
Whitney	I-General_Concept
U	I-General_Concept
(	O
either	O
or	O
)	O
and	O
the	O
sample	O
sizes	O
of	O
each	O
group	O
:	O
</s>
<s>
Using	O
the	O
example	O
above	O
with	O
90	O
pairs	O
that	O
favor	O
the	O
hares	B-Application
and	O
10	O
pairs	O
that	O
favor	O
the	O
tortoise	O
,	O
U2	O
is	O
the	O
smaller	O
of	O
the	O
two	O
,	O
so	O
.	O
</s>
<s>
The	O
Mann	B-General_Concept
–	I-General_Concept
Whitney	I-General_Concept
U	I-General_Concept
test	I-General_Concept
tests	O
a	O
null	B-General_Concept
hypothesis	I-General_Concept
of	O
that	O
the	O
probability	O
distribution	O
of	O
a	O
randomly	O
drawn	O
observation	O
from	O
one	O
group	O
is	O
the	O
same	O
as	O
the	O
probability	O
distribution	O
of	O
a	O
randomly	O
drawn	O
observation	O
from	O
the	O
other	O
group	O
against	O
an	O
alternative	O
that	O
those	O
distributions	O
are	O
not	O
equal	O
(	O
see	O
Mann	B-General_Concept
–	I-General_Concept
Whitney	I-General_Concept
U	I-General_Concept
test	I-General_Concept
#Assumptions	O
and	O
formal	O
statement	O
of	O
hypotheses	O
)	O
.	O
</s>
<s>
In	O
contrast	O
,	O
a	O
t-test	B-General_Concept
tests	O
a	O
null	B-General_Concept
hypothesis	I-General_Concept
of	O
equal	O
means	O
in	O
two	O
groups	O
against	O
an	O
alternative	O
of	O
unequal	O
means	O
.	O
</s>
<s>
Hence	O
,	O
except	O
in	O
special	O
cases	O
,	O
the	O
Mann	B-General_Concept
–	I-General_Concept
Whitney	I-General_Concept
U	I-General_Concept
test	I-General_Concept
and	O
the	O
t-test	B-General_Concept
do	O
not	O
test	O
the	O
same	O
hypotheses	O
and	O
should	O
be	O
compared	O
with	O
this	O
in	O
mind	O
.	O
</s>
<s>
Ordinal	O
data	O
The	O
Mann	B-General_Concept
–	I-General_Concept
Whitney	I-General_Concept
U	I-General_Concept
test	I-General_Concept
is	O
preferable	O
to	O
the	O
t-test	B-General_Concept
when	O
the	O
data	O
are	O
ordinal	O
but	O
not	O
interval	O
scaled	O
,	O
in	O
which	O
case	O
the	O
spacing	O
between	O
adjacent	O
values	O
of	O
the	O
scale	O
cannot	O
be	O
assumed	O
to	O
be	O
constant	O
.	O
</s>
<s>
RobustnessAs	O
it	O
compares	O
the	O
sums	O
of	O
ranks	O
,	O
the	O
Mann	B-General_Concept
–	I-General_Concept
Whitney	I-General_Concept
U	I-General_Concept
test	I-General_Concept
is	O
less	O
likely	O
than	O
the	O
t-test	B-General_Concept
to	O
spuriously	O
indicate	O
significance	O
because	O
of	O
the	O
presence	O
of	O
outliers	O
.	O
</s>
<s>
However	O
,	O
the	O
Mann	B-General_Concept
–	I-General_Concept
Whitney	I-General_Concept
U	I-General_Concept
test	I-General_Concept
may	O
have	O
worse	O
type	O
I	O
error	O
control	O
when	O
data	O
are	O
both	O
heteroscedastic	O
and	O
non-normal	O
.	O
</s>
<s>
EfficiencyWhen	O
normality	O
holds	O
,	O
the	O
Mann	B-General_Concept
–	I-General_Concept
Whitney	I-General_Concept
U	I-General_Concept
test	I-General_Concept
has	O
an	O
(	O
asymptotic	O
)	O
efficiency	O
of	O
3/	O
or	O
about0.95	O
when	O
compared	O
to	O
the	O
t-test	B-General_Concept
.	O
</s>
<s>
For	O
distributions	O
sufficiently	O
far	O
from	O
normal	O
and	O
for	O
sufficiently	O
large	O
sample	O
sizes	O
,	O
the	O
Mann	B-General_Concept
–	I-General_Concept
Whitney	I-General_Concept
U	I-General_Concept
test	I-General_Concept
is	O
considerably	O
more	O
efficient	O
than	O
the	O
t	O
.	O
This	O
comparison	O
in	O
efficiency	O
,	O
however	O
,	O
should	O
be	O
interpreted	O
with	O
caution	O
,	O
as	O
Mann	B-General_Concept
–	I-General_Concept
Whitney	I-General_Concept
and	O
the	O
t-test	B-General_Concept
do	O
not	O
test	O
the	O
same	O
quantities	O
.	O
</s>
<s>
If	O
,	O
for	O
example	O
,	O
a	O
difference	O
of	O
group	O
means	O
is	O
of	O
primary	O
interest	O
,	O
Mann	B-General_Concept
–	I-General_Concept
Whitney	I-General_Concept
is	O
not	O
an	O
appropriate	O
test	O
.	O
</s>
<s>
The	O
Mann	B-General_Concept
–	I-General_Concept
Whitney	I-General_Concept
U	I-General_Concept
test	I-General_Concept
will	O
give	O
very	O
similar	O
results	O
to	O
performing	O
an	O
ordinary	O
parametric	O
two-sample	B-General_Concept
t-test	I-General_Concept
on	O
the	O
rankings	O
of	O
the	O
data	O
.	O
</s>
<s>
The	O
Mann	B-General_Concept
–	I-General_Concept
Whitney	I-General_Concept
U	I-General_Concept
test	I-General_Concept
is	O
not	O
valid	O
for	O
testing	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
against	O
the	O
alternative	B-General_Concept
hypothesis	I-General_Concept
)	O
,	O
without	O
assuming	O
that	O
the	O
distributions	O
are	O
the	O
same	O
under	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
(	O
i.e.	O
,	O
assuming	O
)	O
.	O
</s>
<s>
Specifically	O
,	O
under	O
the	O
more	O
general	O
null	B-General_Concept
hypothesis	I-General_Concept
,	O
the	O
Mann	B-General_Concept
–	I-General_Concept
Whitney	I-General_Concept
U	I-General_Concept
test	I-General_Concept
can	O
have	O
inflated	O
type	O
I	O
error	O
rates	O
even	O
in	O
large	O
samples	O
(	O
especially	O
if	O
the	O
variances	O
of	O
two	O
populations	O
are	O
unequal	O
and	O
the	O
sample	O
sizes	O
are	O
different	O
)	O
,	O
a	O
problem	O
the	O
better	O
alternatives	O
solve	O
.	O
</s>
<s>
As	O
a	O
result	O
,	O
it	O
has	O
been	O
suggested	O
to	O
use	O
one	O
of	O
the	O
alternatives	O
(	O
specifically	O
the	O
Brunner	O
–	O
Munzel	O
test	O
)	O
if	O
it	O
cannot	O
be	O
assumed	O
that	O
the	O
distributions	O
are	O
equal	O
under	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
.	O
</s>
<s>
If	O
one	O
desires	O
a	O
simple	O
shift	O
interpretation	O
,	O
the	O
Mann	B-General_Concept
–	I-General_Concept
Whitney	I-General_Concept
U	I-General_Concept
test	I-General_Concept
should	O
not	O
be	O
used	O
when	O
the	O
distributions	O
of	O
the	O
two	O
samples	O
are	O
very	O
different	O
,	O
as	O
it	O
can	O
give	O
erroneous	O
interpretation	O
of	O
significant	O
results	O
.	O
</s>
<s>
In	O
that	O
situation	O
,	O
the	O
unequal	B-General_Concept
variances	I-General_Concept
version	O
of	O
the	O
t-test	B-General_Concept
may	O
give	O
more	O
reliable	O
results	O
.	O
</s>
<s>
Similarly	O
,	O
some	O
authors	O
(	O
e.g.	O
,	O
Conover	O
)	O
suggest	O
transforming	O
the	O
data	O
to	O
ranks	O
(	O
if	O
they	O
are	O
not	O
already	O
ranks	O
)	O
and	O
then	O
performing	O
the	O
t-test	B-General_Concept
on	O
the	O
transformed	O
data	O
,	O
the	O
version	O
of	O
the	O
t-test	B-General_Concept
used	O
depending	O
on	O
whether	O
or	O
not	O
the	O
population	O
variances	O
are	O
suspected	O
to	O
be	O
different	O
.	O
</s>
<s>
The	O
Brown	B-General_Concept
–	I-General_Concept
Forsythe	I-General_Concept
test	I-General_Concept
has	O
been	O
suggested	O
as	O
an	O
appropriate	O
non-parametric	B-General_Concept
equivalent	O
to	O
the	O
F-test	B-General_Concept
for	O
equal	O
variances	O
.	O
</s>
<s>
A	O
more	O
powerful	O
test	O
is	O
the	O
Brunner	O
–	O
Munzel	O
test	O
,	O
outperforming	O
the	O
Mann	B-General_Concept
–	I-General_Concept
Whitney	I-General_Concept
U	I-General_Concept
test	I-General_Concept
in	O
case	O
of	O
violated	O
assumption	O
of	O
exchangeability	O
.	O
</s>
<s>
The	O
Mann	B-General_Concept
–	I-General_Concept
Whitney	I-General_Concept
U	I-General_Concept
test	I-General_Concept
is	O
a	O
special	O
case	O
of	O
the	O
proportional	O
odds	O
model	O
,	O
allowing	O
for	O
covariate-adjustment	O
.	O
</s>
<s>
See	O
also	O
Kolmogorov	B-General_Concept
–	I-General_Concept
Smirnov	I-General_Concept
test	I-General_Concept
.	O
</s>
<s>
The	O
Mann	B-General_Concept
–	I-General_Concept
Whitney	I-General_Concept
U	I-General_Concept
test	I-General_Concept
is	O
related	O
to	O
a	O
number	O
of	O
other	O
non-parametric	B-General_Concept
statistical	O
procedures	O
.	O
</s>
<s>
For	O
example	O
,	O
it	O
is	O
equivalent	O
to	O
Kendall	B-General_Concept
's	I-General_Concept
tau	I-General_Concept
correlation	O
coefficient	O
if	O
one	O
of	O
the	O
variables	O
is	O
binary	O
(	O
that	O
is	O
,	O
it	O
can	O
only	O
take	O
two	O
values	O
)	O
.	O
</s>
<s>
In	O
many	O
software	O
packages	O
,	O
the	O
Mann	B-General_Concept
–	I-General_Concept
Whitney	I-General_Concept
U	I-General_Concept
test	I-General_Concept
(	O
of	O
the	O
hypothesis	O
of	O
equal	O
distributions	O
against	O
appropriate	O
alternatives	O
)	O
has	O
been	O
poorly	O
documented	O
.	O
</s>
<s>
MATLAB	B-Language
has	O
in	O
its	O
Statistics	O
Toolbox	O
.	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
</s>
<s>
The	O
R	B-Language
package	O
will	O
calculate	O
the	O
z	O
statistic	O
for	O
a	O
Wilcoxon	O
two-sample	O
,	O
paired	O
,	O
or	O
one-sample	O
test	O
.	O
</s>
<s>
SAS	B-Language
implements	O
the	O
test	O
in	O
its	O
PROC	O
NPAR1WAY	O
procedure	O
.	O
</s>
<s>
Julia	B-Application
(	O
programming	O
language	O
)	O
has	O
implementations	O
of	O
this	O
test	O
through	O
several	O
packages	O
.	O
</s>
<s>
StatsDirect	B-Algorithm
(	O
StatsDirect	B-Algorithm
Ltd	O
,	O
Manchester	O
,	O
UK	O
)	O
implements	O
.	O
</s>
<s>
Stata	B-Algorithm
(	O
Stata	B-Algorithm
Corporation	O
,	O
College	O
Station	O
,	O
TX	O
)	O
implements	O
the	O
test	O
in	O
its	O
command	O
.	O
</s>
<s>
PSPP	B-Language
implements	O
the	O
test	O
in	O
its	O
function	O
.	O
</s>
<s>
KNIME	B-Language
implements	O
the	O
test	O
in	O
its	O
node	O
.	O
</s>
<s>
In	O
a	O
single	O
paper	O
in	O
1945	O
,	O
Frank	O
Wilcoxon	O
proposed	O
both	O
the	O
one-sample	O
signed	O
rank	O
and	O
the	O
two-sample	O
rank	B-General_Concept
sum	I-General_Concept
test	I-General_Concept
,	O
in	O
a	O
test	O
of	O
significance	O
with	O
a	O
point	O
null-hypothesis	B-General_Concept
against	O
its	O
complementary	O
alternative	O
(	O
that	O
is	O
,	O
equal	O
versus	O
not	O
equal	O
)	O
.	O
</s>
<s>
This	O
article	O
discussed	O
alternative	B-General_Concept
hypotheses	I-General_Concept
,	O
including	O
a	O
stochastic	O
ordering	O
(	O
where	O
the	O
cumulative	O
distribution	O
functions	O
satisfied	O
the	O
pointwise	O
inequality	O
)	O
.	O
</s>
<s>
This	O
paper	O
also	O
computed	O
the	O
first	O
four	O
moments	O
and	O
established	O
the	O
limiting	O
normality	O
of	O
the	O
statistic	O
under	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
,	O
so	O
establishing	O
that	O
it	O
is	O
asymptotically	O
distribution-free	B-General_Concept
.	O
</s>
