<s>
In	O
statistics	O
,	O
one	O
purpose	O
for	O
the	O
analysis	B-General_Concept
of	I-General_Concept
variance	I-General_Concept
(	O
ANOVA	B-General_Concept
)	O
is	O
to	O
analyze	O
differences	O
in	O
means	O
between	O
groups	O
.	O
</s>
<s>
ANOVA	B-General_Concept
on	I-General_Concept
ranks	I-General_Concept
is	O
a	O
statistic	O
designed	O
for	O
situations	O
when	O
the	O
normality	O
assumption	O
has	O
been	O
violated	O
.	O
</s>
<s>
Consider	O
randomly	O
selected	O
subjects	O
that	O
are	O
subsequently	O
randomly	O
assigned	O
to	O
groups	O
A	O
,	O
B	O
,	O
and	O
C	O
.	O
Under	O
the	O
truth	O
of	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
,	O
the	O
variability	O
(	O
or	O
sum	O
of	O
squares	O
)	O
of	O
scores	O
on	O
some	O
dependent	O
variable	O
will	O
be	O
the	O
same	O
within	O
each	O
group	O
.	O
</s>
<s>
Under	O
the	O
truth	O
of	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
,	O
the	O
sampling	O
distribution	O
of	O
the	O
F	O
ratio	O
depends	O
on	O
the	O
degrees	O
of	O
freedom	O
for	O
the	O
numerator	O
and	O
the	O
denominator	O
.	O
</s>
<s>
If	O
the	O
resulting	O
F	O
ratio	O
raises	O
the	O
value	O
to	O
such	O
an	O
extent	O
that	O
it	O
exceeds	O
the	O
threshold	O
of	O
what	O
constitutes	O
a	O
rare	O
event	O
(	O
called	O
the	O
Alpha	O
level	O
)	O
,	O
the	O
Anova	B-General_Concept
F	O
test	O
is	O
said	O
to	O
reject	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
of	O
equal	O
means	O
between	O
the	O
three	O
groups	O
,	O
in	O
favor	O
of	O
the	O
alternative	O
hypothesis	O
that	O
at	O
least	O
one	O
of	O
the	O
groups	O
has	O
a	O
larger	O
mean	O
(	O
which	O
in	O
this	O
example	O
,	O
is	O
group	O
A	O
)	O
.	O
</s>
<s>
It	O
may	O
result	O
in	O
a	O
known	O
statistic	O
(	O
e.g.	O
,	O
in	O
the	O
two	O
independent	O
samples	O
layout	O
ranking	O
results	O
in	O
the	O
Wilcoxon	B-General_Concept
rank-sum	I-General_Concept
/	O
Mann	B-General_Concept
–	I-General_Concept
Whitney	I-General_Concept
U	I-General_Concept
test	I-General_Concept
)	O
,	O
and	O
provides	O
the	O
desired	O
robustness	O
and	O
increased	O
statistical	B-General_Concept
power	I-General_Concept
that	O
is	O
sought	O
.	O
</s>
<s>
For	O
example	O
,	O
Monte	B-Algorithm
Carlo	I-Algorithm
studies	I-Algorithm
have	O
shown	O
that	O
the	O
rank	O
transformation	O
in	O
the	O
two	O
independent	O
samples	O
t-test	B-General_Concept
layout	O
can	O
be	O
successfully	O
extended	O
to	O
the	O
one-way	O
independent	O
samples	O
ANOVA	B-General_Concept
,	O
as	O
well	O
as	O
the	O
two	O
independent	O
samples	O
multivariate	O
Hotelling	O
's	O
T2	O
layouts	O
Commercial	O
statistical	B-Algorithm
software	I-Algorithm
packages	O
(	O
e.g.	O
,	O
SAS	O
)	O
followed	O
with	O
recommendations	O
to	O
data	O
analysts	O
to	O
run	O
their	O
data	O
sets	O
through	O
a	O
ranking	O
procedure	O
(	O
e.g.	O
,	O
PROC	O
RANK	O
)	O
prior	O
to	O
conducting	O
standard	O
analyses	O
using	O
parametric	O
procedures	O
.	O
</s>
<s>
ANOVA	B-General_Concept
on	I-General_Concept
ranks	I-General_Concept
means	O
that	O
a	O
standard	O
analysis	B-General_Concept
of	I-General_Concept
variance	I-General_Concept
is	O
calculated	O
on	O
the	O
rank-transformed	O
data	O
.	O
</s>
<s>
Conducting	O
factorial	B-General_Concept
ANOVA	I-General_Concept
on	O
the	O
ranks	O
of	O
original	O
scores	O
has	O
also	O
been	O
suggested	O
.	O
</s>
<s>
However	O
,	O
Monte	B-Algorithm
Carlo	I-Algorithm
studies	I-Algorithm
,	O
and	O
subsequent	O
asymptotic	O
studies	O
found	O
that	O
the	O
rank	O
transformation	O
is	O
inappropriate	O
for	O
testing	O
interaction	O
effects	O
in	O
a	O
4x3	O
and	O
a	O
2x2x2	O
factorial	O
design	O
.	O
</s>
<s>
However	O
,	O
two	O
specific	O
types	O
of	O
secondary	O
transformations	O
,	O
the	O
random	O
normal	O
scores	O
and	O
expected	O
normal	O
scores	O
transformation	O
,	O
have	O
been	O
shown	O
to	O
greatly	O
inflate	O
Type	O
I	O
errors	O
and	O
severely	O
reduce	O
statistical	B-General_Concept
power	I-General_Concept
.	O
</s>
<s>
The	O
ANOVA	B-General_Concept
on	I-General_Concept
ranks	I-General_Concept
has	O
never	O
been	O
recommended	O
when	O
the	O
underlying	O
assumption	O
of	O
homogeneous	O
variances	O
has	O
been	O
violated	O
,	O
either	O
by	O
itself	O
,	O
or	O
in	O
conjunction	O
with	O
a	O
violation	O
of	O
the	O
assumption	O
of	O
population	O
normality	O
.	O
</s>
<s>
According	O
to	O
Hettmansperger	O
and	O
McKean	O
,	O
"	O
Sawilowsky	O
(	O
1990	O
)	O
provides	O
an	O
excellent	O
review	O
of	O
nonparametric	O
approaches	O
to	O
testing	O
for	O
interaction	O
"	O
in	O
ANOVA	B-General_Concept
.	O
</s>
